Generative Diffusion Models of Stochastic Graph Signals

# Generative Diffusion Models of Stochastic Graph Signals
Source: [https://arxiv.org/html/2607.06833](https://arxiv.org/html/2607.06833)
Yiğit Berkay Uslu, Samar Hadou, Sergio Rozada, Shirin Saeedi Bidokhti Alejandro RibeiroPreliminary results of this work were presented in part at the ICASSP 2026 conference\[[38](https://arxiv.org/html/2607.06833#bib.bib44)\]and appear in the preprint\[[37](https://arxiv.org/html/2607.06833#bib.bib45)\]\. Y\. Berkay Uslu, Samar Hadou, Shirin S\. Bidokhti and Alejandro Ribeiro are with the Dept\. of Electrical and Systems Eng\., University of Pennsylvania, Philadelphia, PA 19104 USA\. Sergio Rozada is with the Department of Signal Processing and Communications, King Juan Carlos University, Madrid, Spain\. The implementation code is available at[https://github\.com/yigit\-uslu/graph\-signal\-diffusion](https://github.com/yigit-uslu/graph-signal-diffusion-modeling)\.

###### Abstract

Sampling stochastic signals supported on a graph underlies many graph machine learning tasks, including recommender systems, forecasting in financial markets, and wireless network optimization\. In these settings, the target signals are realizations of unknown conditional distributions\. However, prevailing approaches rely mostly on intricate, application\-tailored designs that often regress to a conditional mean instead of sampling from the conditional law\. This paper unifies such problems as conditional graph signal generative modeling and tackles them with a single denoising diffusion framework\. We learn a reverse diffusion process, parametrized by graph neural networks \(GNNs\), that draws graph signals conditioned directly on the graph topology and on node\-feature side information\. The reverse process is realized by a novel architecture, the U\-Graph Neural Network \(U\-GNN\), which generalizes the image\-convolutional U\-Net to graph\-structured signals\. The U\-GNN performs multi\-resolution encoder–decoder processing in which pooling and unpooling reduce to a learned node selection, expressed by nested selection matrices, and a zero\-padded lifting of coarse signals back to the full node set\. The graph convolutions are carried out on the original graph, with a stride that sets their hop reach, so the U\-GNN bypasses explicit graph coarsening at every resolution\. We demonstrate our method on two generative tasks: stock price forecasting and optimal wireless resource allocation, with extensive numerical results in both domains\.

## IIntroduction

Signals defined over irregular, graph\-structured domains are pervasive, spanning application areas such as recommender systems\[[41](https://arxiv.org/html/2607.06833#bib.bib29),[24](https://arxiv.org/html/2607.06833#bib.bib30),[23](https://arxiv.org/html/2607.06833#bib.bib31)\], wireless communications\[[7](https://arxiv.org/html/2607.06833#bib.bib1),[31](https://arxiv.org/html/2607.06833#bib.bib2),[46](https://arxiv.org/html/2607.06833#bib.bib3),[42](https://arxiv.org/html/2607.06833#bib.bib4),[4](https://arxiv.org/html/2607.06833#bib.bib5),[39](https://arxiv.org/html/2607.06833#bib.bib7)\], and forecasting in financial markets\[[3](https://arxiv.org/html/2607.06833#bib.bib21),[30](https://arxiv.org/html/2607.06833#bib.bib22),[35](https://arxiv.org/html/2607.06833#bib.bib23)\]\. In many such settings, these signals are realizations from unknown probability distributions rather than deterministic quantities\[[25](https://arxiv.org/html/2607.06833#bib.bib28)\]\. This paper proposes a generative modeling framework that learns to sample from such graph signal distributions\.Generative modeling has proved effective for synthesizing graph topologies themselves\[[16](https://arxiv.org/html/2607.06833#bib.bib56),[40](https://arxiv.org/html/2607.06833#bib.bib57)\], with molecular generation serving as a canonical example\[[14](https://arxiv.org/html/2607.06833#bib.bib58)\]\. We instead take the graph structure as given and learn to generate node\-defined signals\.

A wide array of generative paradigms exists for learning to sample from unknown distributions\[[18](https://arxiv.org/html/2607.06833#bib.bib53),[10](https://arxiv.org/html/2607.06833#bib.bib54),[26](https://arxiv.org/html/2607.06833#bib.bib55)\], such as variational autoencoders, generative adversarial networks, and normalizing flows\. Among them, denoising diffusion models and their score\- and flow\-based variants stand out for their sample quality, training stability, and principled conditional inference in high dimensions\[[13](https://arxiv.org/html/2607.06833#bib.bib47),[32](https://arxiv.org/html/2607.06833#bib.bib49),[33](https://arxiv.org/html/2607.06833#bib.bib48),[22](https://arxiv.org/html/2607.06833#bib.bib51)\]\. While these models differ in what the parametric model approximates and in the sampling dynamics, they share a common structure\. A fixed forward process gradually corrupts the data into a tractable reference prior, and a learned reverse process denoises samples from that prior back toward the data distribution\. Graph signals have recently become a target for generative, particularly diffusion, modeling in domains such as the forecasting of traffic and financial series\[[38](https://arxiv.org/html/2607.06833#bib.bib44),[43](https://arxiv.org/html/2607.06833#bib.bib26),[5](https://arxiv.org/html/2607.06833#bib.bib24),[2](https://arxiv.org/html/2607.06833#bib.bib25),[28](https://arxiv.org/html/2607.06833#bib.bib43)\], and wireless optimization\[[6](https://arxiv.org/html/2607.06833#bib.bib12),[21](https://arxiv.org/html/2607.06833#bib.bib13),[36](https://arxiv.org/html/2607.06833#bib.bib11),[11](https://arxiv.org/html/2607.06833#bib.bib16),[12](https://arxiv.org/html/2607.06833#bib.bib15),[37](https://arxiv.org/html/2607.06833#bib.bib45),[20](https://arxiv.org/html/2607.06833#bib.bib18)\]\. However, these efforts have mostly been application\-specific, with task\-tailored denoiser architectures\. Their formulations also often remain deterministic, seeking solution distributions whose mass concentrates near an optimal deterministic solution rather than drawing diversified samples from the conditional law\.

In this paper, we propose a general methodological framework for graph signal generation\. At its core, this requires designing a diffusion model whose denoising network encodes inductive biases tailored to the data domain\. For images, the standard denoising backbone is the U\-Net with a symmetric encoder–decoder structure and multi\-resolution hierarchy\[[27](https://arxiv.org/html/2607.06833#bib.bib59),[15](https://arxiv.org/html/2607.06833#bib.bib64)\]\. The U\-Net is built on convolutional neural networks \(CNNs\), whose layers apply translation\-equivariant filters on the regular pixel grid\. The natural counterpart for graph\-structured data is the graph neural network \(GNN\), whose layers apply graph\-shift\-equivariant filters, thereby extending convolutional processing to irregular domains\[[8](https://arxiv.org/html/2607.06833#bib.bib42)\]\. Replacing CNNs with GNN layers does not, on its own, suffice to adapt the U\-Net design to the denoising of graph signals\. The main obstacle is the pooling stage\. In the image setting, the pixel grid provides a regular pooling lattice that persists after each down\-sampling step, thus, feature extraction and spatial reduction compose seamlessly\. Graph data lacks such a canonical lattice\. A common workaround coarsens the graph at each resolution through a learned pooling operator, as in differentiable cluster\-assignment pooling and top\-kknode\-scoring schemes\[[44](https://arxiv.org/html/2607.06833#bib.bib60),[9](https://arxiv.org/html/2607.06833#bib.bib61)\]\. These methods are mainly designed for graph\-level tasks, such as classification\. Under arbitrary coarsening, however, the pooled signal loses a clear interpretation as a convolutional signal on the original graph\.

We take a different route in this paper\. We formalize down\- and up\-sampling, equivalently graph or node \(un\)pooling, as a*learned node \(de\)selection*operation, expressed through nested selection matrices\[[8](https://arxiv.org/html/2607.06833#bib.bib42)\]\. These matrices are trained end\-to\-end with the denoising network, so each resolution retains the nodes most informative for the denoising task at hand\. The selection operator and its transpose play two complementary roles\. First, the nested hierarchy of selections instantiates the multi\-resolution encoder–decoder cascade\. Each encoder selection picks a smaller active node subset that defines the next coarser resolution, and the matching transpose in the decoder lifts coarse signals back to the finer one\. Second, at a fixed resolution, the same operator pair keeps GNN filtering interpretable as a convolution on the original graph\. The transpose zero\-pads a node\-reduced signal to the full vertex set, the GNN layer filters there, and the selection returns the result to the reduced domain\. Unlike the coarsening schemes above, filtering therefore stays convolutional on the original graph at every resolution\. We further introduce a stride parameter that sets the hop distance of each graph shift\. This mirrors how dilated convolutions in CNNs replace fixed pooling rules, folding down\-sampling and receptive\-field growth into the convolution itself\[[34](https://arxiv.org/html/2607.06833#bib.bib76),[45](https://arxiv.org/html/2607.06833#bib.bib77)\]\. On the active nodes, a larger stride enlarges the aggregation neighborhoods and prevents the graph filters from drawing mostly from zero\-padded positions under heavy down\-sampling\. This construction avoids explicit graph coarsening and lets convolutional filtering and pooling compose across arbitrary resolutions\.

Building on this mechanism, we propose the*U\-Graph Neural Network \(U\-GNN\)*architecture as a graph\-domain adaptation of the classical U\-Net\. It comprises a U\-shaped encoder–decoder pipeline with skip connections, built from GNN layers and the pooling and unpooling operators above\. We then cast graph signal generation as a conditional denoising diffusion process and parametrize its reverse process with a U\-GNN\. The generations are thereby conditioned directly on the graph topology and on node\-feature side information\. Although we focus on diffusion denoising, the U\-GNN is a general\-purpose architecture for learning over graph signals\. It inherits the permutation equivariance, stability to perturbations, and cross\-topology transferability of GNNs\[[29](https://arxiv.org/html/2607.06833#bib.bib70),[31](https://arxiv.org/html/2607.06833#bib.bib2)\]\.

We demonstrate the framework and the U\-GNN architecture on two generative tasks\. The first is stock price forecasting over a correlation graph of financial indicators\. Given recent market history, we generate future price trajectories as graph signals, capturing the market uncertainty and rare tail events that point forecasts and statistical baselines often miss\. The second is resource allocation in wireless networks under ergodic quality\-of\-service \(QoS\) requirements, where optimal policies are inherently probabilistic and are realized through time\-sharing\. Treating solution samples from an expert primal–dual algorithm as graph signals on channel \(interference\) graphs, we learn to sample from the expert allocation distribution given channel and user\-state information\.

A brief summary of our contributions is as follows\.

1. \(C1\)\.We develop a graph signal generative modeling approach that combines denoising diffusion models with GNNs to sample stochastic graph signals\.
2. \(C2\)\.We parametrize the reverse diffusion process with a novel U\-GNN architecture, whose building blocks are GNN layers with stride and zero\-padded pooling, that generalizes image\-convolutional U\-Nets to graph\-structured signals\.
3. \(C3\)\.We apply our approach to stock price forecasting and optimal wireless resource allocation, with extensive numerical results in both domains\.

The rest of the paper is organized as follows\. Section[II](https://arxiv.org/html/2607.06833#S2)presents the conditional graph signal generative diffusion framework\. Section[III](https://arxiv.org/html/2607.06833#S3)introduces GNNs with built\-in stride and pooling\. Section[IV](https://arxiv.org/html/2607.06833#S4)assembles the U\-GNN architecture\. Section[V](https://arxiv.org/html/2607.06833#S5)validates the method on wireless resource allocation and financial forecasting\. Section[VI](https://arxiv.org/html/2607.06833#S6)concludes the paper\.

## IIGraph Signal Generative Diffusion Models

Consider a weighted graph𝒢=\(𝒱,ℰ,𝒲\)\{\\mathcal\{G\}\}=\(\{\\mathcal\{V\}\},\{\\mathcal\{E\}\},\{\\mathcal\{W\}\}\)onNNnodes, with node set𝒱=\{1,…,N\}\{\\mathcal\{V\}\}=\\\{1,\\ldots,N\\\}, edge setℰ⊆𝒱×𝒱\{\\mathcal\{E\}\}\\subseteq\{\\mathcal\{V\}\}\\times\{\\mathcal\{V\}\}, and weight map𝒲:ℰ→ℝ\{\\mathcal\{W\}\}:\{\\mathcal\{E\}\}\\to\{\\mathbb\{R\}\}\. A graph signal assigns features to the nodes and is collected in a matrix𝐗=\[𝐱1,…,𝐱F\]∈ℝN×F\{\\mathbf\{X\}\}=\[\{\\mathbf\{x\}\}^\{1\},\\ldots,\{\\mathbf\{x\}\}^\{F\}\]\\in\{\\mathbb\{R\}\}^\{N\\times F\}, whoseffth column𝐱f∈ℝN\{\\mathbf\{x\}\}^\{f\}\\in\{\\mathbb\{R\}\}^\{N\}gathers theffth feature across all nodes\. We endow𝒢\{\\mathcal\{G\}\}with a graph shift operator \(GSO\)𝐒∈ℝN×N\{\\mathbf\{S\}\}\\in\{\\mathbb\{R\}\}^\{N\\times N\}, a sparse matrix with\[𝐒\]m,n=0\[\{\\mathbf\{S\}\}\]\_\{m,n\}=0wheneverm≠nm\\neq nand\(m,n\)∉ℰ\(m,n\)\\notin\{\\mathcal\{E\}\}\. We note that this GSO definition allows the diagonal entries to be chosen freely\. The GSO encodes the graph topology, and successive products \(shifts\)𝐒𝐗\{\\mathbf\{S\}\}\{\\mathbf\{X\}\}propagate signal values along the edges\. Common GSO choices are the \(weighted\) adjacency matrix𝐀\{\\mathbf\{A\}\}and the graph Laplacian𝐋=diag​\(𝐀𝟏N\)−𝐀\{\\mathbf\{L\}\}=\\mathrm\{diag\}\(\{\\mathbf\{A\}\}\{\\mathbf\{1\}\}\_\{N\}\)\-\{\\mathbf\{A\}\}\.

We address the conditional generation of graph signals\. The target is a graph signal𝐱0∈ℝN\{\\mathbf\{x\}\}\_\{0\}\\in\{\\mathbb\{R\}\}^\{N\}on a known graph𝒢\{\\mathcal\{G\}\}\. The conditioning collects the topology, through the GSO𝐒\{\\mathbf\{S\}\}, and auxiliary node states𝐮∈ℝN×U\{\\mathbf\{u\}\}\\in\{\\mathbb\{R\}\}^\{N\\times U\}\(side information\) into the variable𝝃=\(𝐒,𝐮\)\\boldsymbol\{\\xi\}=\(\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)\. Letq𝝃q\_\{\\boldsymbol\{\\xi\}\}denote the distribution of the conditioning andq𝐱\|𝝃\(⋅\|𝝃\)q\_\{\{\\mathbf\{x\}\}\|\\boldsymbol\{\\xi\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)the law of the target given𝝃\\boldsymbol\{\\xi\}\. These define the joint data distributionqdata=q𝐱\|𝝃​q𝝃q\_\{\\mathrm\{data\}\}=q\_\{\{\\mathbf\{x\}\}\|\\boldsymbol\{\\xi\}\}\\,q\_\{\\boldsymbol\{\\xi\}\}, from which we draw pairs\(𝐱0,𝝃\)\(\{\\mathbf\{x\}\}\_\{0\},\\boldsymbol\{\\xi\}\)\. We seek a parametric generatorp𝜽⋆\(⋅\|𝝃\)=p\(⋅\|𝝃;𝜽∗\)p\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)=p\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\};\\boldsymbol\{\\theta\}^\{\*\}\)whose conditional law matchesq𝐱\|𝝃\(⋅\|𝝃\)q\_\{\{\\mathbf\{x\}\}\|\\boldsymbol\{\\xi\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)forq𝝃q\_\{\\boldsymbol\{\\xi\}\}\-almost every𝝃\\boldsymbol\{\\xi\}, i\.e\.,

𝜽⋆∈argmin𝜽𝔼𝝃∼q𝝃\[DKL\(q𝐱\|𝝃\(⋅\|𝝃\)∥p𝜽\(⋅\|𝝃\)\)\]\.\\displaystyle\\boldsymbol\{\\theta\}^\{\\star\}\\in\\operatornamewithlimits\{argmin\}\_\{\\boldsymbol\{\\theta\}\}\\mathbb\{E\}\_\{\\boldsymbol\{\\xi\}\\sim q\_\{\\boldsymbol\{\\xi\}\}\}\\big\[\\mathrm\{D\_\{KL\}\}\\big\(q\_\{\{\\mathbf\{x\}\}\|\\boldsymbol\{\\xi\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)\\;\\big\\\|\\;p\_\{\\boldsymbol\{\\theta\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)\\big\)\\big\]\.\(1\)The conditional laws are unknown in closed form\. We assume samples\(𝐱0,𝝃\)∼qdata\(\{\\mathbf\{x\}\}\_\{0\},\\boldsymbol\{\\xi\}\)\\sim q\_\{\\mathrm\{data\}\}are available\. Therefore, we realize the optimal generatorp𝜽⋆p\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}as a conditional denoising diffusion model, trained on a dataset\{\(𝐱0\(i\),𝝃\(i\)\)\}i=1\|𝒟\|∼qdata\\\{\(\{\\mathbf\{x\}\}^\{\(i\)\}\_\{0\},\\boldsymbol\{\\xi\}^\{\(i\)\}\)\\\}\_\{i=1\}^\{\|\{\\mathcal\{D\}\}\|\}\\sim q\_\{\\mathrm\{data\}\}\.

### II\-ADenoising Diffusion Models for Graph Signals

![Refer to caption](https://arxiv.org/html/2607.06833v1/x1.png)Figure 1:A denoising diffusion model of graph signals\.A forward noising processq​\(𝐱k\|𝐱k−1\)q\(\{\\mathbf\{x\}\}\_\{k\}\\,\|\\,\{\\mathbf\{x\}\}\_\{k\-1\}\)gradually removes structure from the original graph signals by adding white noise, with𝐱K≈𝒩​\(𝟎,𝐈\)\{\\mathbf\{x\}\}\_\{K\}\\approx\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)\. A denoising diffusion processp𝜽​\(𝐱k−1\|𝐱k;𝐒,𝐮\)p\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\-1\}\|\{\\mathbf\{x\}\}\_\{k\};\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)is trained to reverse the forward process and generate novel graph signal samples𝐱0\{\\mathbf\{x\}\}\_\{0\}distributed approximately by the conditional distributionsqdata​\(𝐱0\|𝐒,𝐮\)q\_\{\\mathrm\{data\}\}\(\{\\mathbf\{x\}\}\_\{0\}\\,\|\\,\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)for a given GSO𝐒\{\\mathbf\{S\}\}and observation values represented by another graph signal𝐮\{\\mathbf\{u\}\}\.A denoising diffusion model pairs a fixed forward processq​\(𝐱1:K\|𝐱0\)q\(\{\\mathbf\{x\}\}\_\{1:K\}\\,\|\\,\{\\mathbf\{x\}\}\_\{0\}\), which noises clean targets toward a tractable reference prior, with a learned reverse processp𝜽​\(𝐱k−1\|𝐱k\)p\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\-1\}\\,\|\\,\{\\mathbf\{x\}\}\_\{k\}\), which denoises prior samples back toward the data\. Conditional generation requires no additional machinery\. We condition the reverse process on𝝃\\boldsymbol\{\\xi\}asp𝜽​\(𝐱k−1\|𝐱k;𝝃\)p\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\-1\}\\,\|\\,\{\\mathbf\{x\}\}\_\{k\};\\,\\boldsymbol\{\\xi\}\), leaving the forward process and the training objective unchanged\. Fig\.[1](https://arxiv.org/html/2607.06833#S2.F1)illustrates this construction, which we detail next\.

Forward noising process\.The forward process is a Markov chain that adds noise to the target𝐱0\{\\mathbf\{x\}\}\_\{0\}overKKsteps\. Given\(𝐱0,𝝃\)∼q0=qdata\(\{\\mathbf\{x\}\}\_\{0\},\\boldsymbol\{\\xi\}\)\\sim q\_\{0\}=q\_\{\\mathrm\{data\}\}with𝝃\\boldsymbol\{\\xi\}held fixed, we corrupt𝐱0\{\\mathbf\{x\}\}\_\{0\}with white noise on an increasing schedule\{βk\}k=1K\\\{\\beta\_\{k\}\\\}\_\{k=1\}^\{K\}\(e\.g\., linear\), producing iterates\{𝐱k\}k=1K\\\{\{\\mathbf\{x\}\}\_\{k\}\\\}\_\{k=1\}^\{K\}through the Gaussian transitions,

q​\(𝐱k\|𝐱k−1\)=𝒩​\(𝐱k;1−βk​𝐱k−1,βk​𝐈\)\.\\displaystyle q\(\{\\mathbf\{x\}\}\_\{k\}\\,\|\\,\{\\mathbf\{x\}\}\_\{k\-1\}\)=\{\\mathcal\{N\}\}\(\{\\mathbf\{x\}\}\_\{k\};\\sqrt\{1\-\\beta\_\{k\}\}\\,\{\\mathbf\{x\}\}\_\{k\-1\},\\beta\_\{k\}\{\\mathbf\{I\}\}\)\.\(2\)Since the drift is linear and the noise increments are independent, the conditionals remain Gaussian at each stepkk, i\.e\.,

qk​\(𝐱k\|𝐱0\)=𝒩​\(𝐱k;α¯k​𝐱0,\(1−α¯k\)​𝐈\),\\displaystyle q\_\{k\}\(\{\\mathbf\{x\}\}\_\{k\}\\,\|\\,\{\\mathbf\{x\}\}\_\{0\}\)=\\mathcal\{N\}\\big\(\{\\mathbf\{x\}\}\_\{k\};\\,\\sqrt\{\\bar\{\\alpha\}\_\{k\}\}\\,\{\\mathbf\{x\}\}\_\{0\},\\,\(1\-\\bar\{\\alpha\}\_\{k\}\)\\,\{\\mathbf\{I\}\}\\big\),\(3\)withαk≔1−βk\\alpha\_\{k\}\\coloneqq 1\-\\beta\_\{k\}andα¯k≔∏m=1kαm\\bar\{\\alpha\}\_\{k\}\\coloneqq\\prod\_\{m=1\}^\{k\}\\alpha\_\{m\}fork=1,…,Kk=1,\\ldots,K\. A standard reparametrization of \([3](https://arxiv.org/html/2607.06833#S2.E3)\) expresses𝐱k\{\\mathbf\{x\}\}\_\{k\}as a deterministic map of\(𝐱0,k\)\(\{\\mathbf\{x\}\}\_\{0\},k\)plus independent standard noise\[[13](https://arxiv.org/html/2607.06833#bib.bib47)\],

𝐱k​\(𝐱0,k,ϵ\)=α¯k​𝐱0\+1−α¯k​ϵ,ϵ∼𝒩​\(𝟎,𝐈\)⟂𝐱0,\\displaystyle\\hskip\-2\.5pt\{\\mathbf\{x\}\}\_\{k\}\(\{\\mathbf\{x\}\}\_\{0\},k,\\boldsymbol\{\\epsilon\}\)\\\!=\\\!\\sqrt\{\\bar\{\\alpha\}\_\{k\}\}\\,\{\\mathbf\{x\}\}\_\{0\}\\\!\+\\\!\\sqrt\{1\-\\bar\{\\alpha\}\_\{k\}\}\\,\\boldsymbol\{\\epsilon\},\\quad\\boldsymbol\{\\epsilon\}\\sim\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)\\\!\\perp\\\!\{\\mathbf\{x\}\}\_\{0\},\(4\)and it permits drawing any iterate𝐱k∼qk\(⋅\|𝐱0,𝝃\)\{\\mathbf\{x\}\}\_\{k\}\\sim q\_\{k\}\(\\cdot\\,\|\\,\{\\mathbf\{x\}\}\_\{0\},\\boldsymbol\{\\xi\}\)directly, without simulating the chain in \([2](https://arxiv.org/html/2607.06833#S2.E2)\)\. For a suitable schedule and a large enoughKK, the process drives𝐱0∼q0\(⋅\|𝝃\)\{\\mathbf\{x\}\}\_\{0\}\\sim q\_\{0\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)toward a Gaussian prior, with𝐱K∼qK≈𝒩​\(𝟎,𝐈\)\{\\mathbf\{x\}\}\_\{K\}\\sim q\_\{K\}\\approx\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)for every𝝃\\boldsymbol\{\\xi\}\.

Backward \(reverse\) denoising process\.Denoising diffusion fixespK\(⋅\|𝝃\)=qK≈𝒩\(𝟎,𝐈\)p\_\{K\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)=q\_\{K\}\\approx\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)and learns a reverse Markov chain whose kernelsp𝜽​\(𝐱k−1\|𝐱k;𝝃\)p\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\-1\}\\,\|\\,\{\\mathbf\{x\}\}\_\{k\};\\boldsymbol\{\\xi\}\)approximate the time reversal of the forward chain\. Consistent with the continuous\-time limit, we model each reverse step as Gaussian transitions,

p𝜽​\(𝐱k−1\|𝐱k;𝝃\)=𝒩​\(𝐱k−1;𝝁𝜽​\(𝐱k,k;𝝃\),𝚺𝜽​\(𝐱k,k;𝝃\)\)\.\\displaystyle\\hskip\-2\.5ptp\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\-1\}\\\!\{\\,\\big\|\\,\}\\\!\{\\mathbf\{x\}\}\_\{k\};\\boldsymbol\{\\xi\}\)\\\!=\\\!\\mathcal\{N\}\\\!\\left\(\{\\mathbf\{x\}\}\_\{k\-1\};\\boldsymbol\{\\mu\}\_\{\\boldsymbol\{\\theta\}\}\\big\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\\big\),\\boldsymbol\{\\Sigma\}\_\{\\boldsymbol\{\\theta\}\}\\big\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\\big\)\\right\)\.\(5\)As is standard, we fix the covariance𝚺𝜽=σk2​𝐈\\boldsymbol\{\\Sigma\}\_\{\\boldsymbol\{\\theta\}\}\\\!=\\\!\\sigma\_\{k\}^\{2\}\{\\mathbf\{I\}\}for a predefinedσk\\sigma\_\{k\}tied to the scheduleβk\\beta\_\{k\}and learn only the mean𝝁𝜽\\boldsymbol\{\\mu\}\_\{\\boldsymbol\{\\theta\}\}\.

Training maximizes the evidence lower bound \(ELBO\) on the log\-likelihoodlogp𝜽\(⋅\|𝝃\)\\log p\_\{\\boldsymbol\{\\theta\}\}\(\\cdot\{\\,\\big\|\\,\}\\boldsymbol\{\\xi\}\)\. Under the fixed covariance and the reparametrization \([4](https://arxiv.org/html/2607.06833#S2.E4)\), this is equivalent to a noise\-prediction objective\. Omitting its per\-step ELBO weights matches, and often improves upon, the weighted objective in practice\[[13](https://arxiv.org/html/2607.06833#bib.bib47)\]\. We therefore minimize the following objective,

𝜽⋆∈argmin𝜽ℒ​\(𝜽\)≔𝔼​\[‖ϵ−ϵ𝜽​\(𝐱k​\(𝐱0,k,ϵ\),k;𝝃\)‖2\],\\displaystyle\\boldsymbol\{\\theta\}^\{\\star\}\\in\\operatornamewithlimits\{argmin\}\_\{\\boldsymbol\{\\theta\}\}\{\\mathcal\{L\}\}\(\\boldsymbol\{\\theta\}\)\\coloneqq\\mathbb\{E\}\\Big\[\\big\\\|\\boldsymbol\{\\epsilon\}\-\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}\\big\(\{\\mathbf\{x\}\}\_\{k\}\(\{\\mathbf\{x\}\}\_\{0\},k,\\boldsymbol\{\\epsilon\}\),k;\\boldsymbol\{\\xi\}\\big\)\\big\\\|^\{2\}\\Big\],\(6\)where the expectation is over\(𝐱0,𝝃\)∼qdata\(\{\\mathbf\{x\}\}\_\{0\},\\boldsymbol\{\\xi\}\)\\sim q\_\{\\mathrm\{data\}\}and diffusion stepsk∼unif​\{1,…,K\}k\\sim\\mathrm\{unif\}\\\{1,\\ldots,K\\\}\.

By the reparametrization \([4](https://arxiv.org/html/2607.06833#S2.E4)\), the trained noise predictorϵ𝜽⋆​\(𝐱k,k;𝝃\)\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\)yields both a*graph signal denoiser*and the mean of the reverse kernel in \([5](https://arxiv.org/html/2607.06833#S2.E5)\),

𝐱^0​\(𝐱k,k;𝝃\)\\displaystyle\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\)=𝐱k−1−α¯k​ϵ𝜽⋆α¯k,\\displaystyle=\\frac\{\{\\mathbf\{x\}\}\_\{k\}\-\\sqrt\{1\-\\bar\{\\alpha\}\_\{k\}\}\\,\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\}\{\\sqrt\{\\bar\{\\alpha\}\_\{k\}\}\},\(7\)𝝁𝜽⋆​\(𝐱k,k;𝝃\)\\displaystyle\\boldsymbol\{\\mu\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\)=1αk​\(𝐱k−βk1−α¯k​ϵ𝜽⋆\)\.\\displaystyle=\\frac\{1\}\{\\sqrt\{\\alpha\_\{k\}\}\}\\left\(\{\\mathbf\{x\}\}\_\{k\}\-\\frac\{\\beta\_\{k\}\}\{\\sqrt\{1\-\\bar\{\\alpha\}\_\{k\}\}\}\\,\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\\right\)\.\(8\)Thusϵ𝜽⋆\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}and the fixed𝚺𝜽=σk2​𝐈\\boldsymbol\{\\Sigma\}\_\{\\boldsymbol\{\\theta\}\}\\\!=\\\!\\sigma\_\{k\}^\{2\}\{\\mathbf\{I\}\}specify \([5](https://arxiv.org/html/2607.06833#S2.E5)\) completely\. For sampling, we construct a family of generally non\-Markovian reverse processes\[[32](https://arxiv.org/html/2607.06833#bib.bib49)\]that preserve the forward marginals \([3](https://arxiv.org/html/2607.06833#S2.E3)\),

𝐱k−1=α¯k−1​𝐱^0\+1−α¯k−1−σk2​ϵ𝜽⋆\+σk​𝐰,\\displaystyle\{\\mathbf\{x\}\}\_\{k\-1\}=\\sqrt\{\\bar\{\\alpha\}\_\{k\-1\}\}\\,\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}\+\\sqrt\{1\-\\bar\{\\alpha\}\_\{k\-1\}\-\\sigma\_\{k\}^\{2\}\}\\,\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\+\\sigma\_\{k\}\{\\mathbf\{w\}\},\(9\)with𝐰∼𝒩​\(𝟎,𝐈\)\{\\mathbf\{w\}\}\\sim\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)\. A single factorη∈\[0,1\]\\eta\\in\[0,1\]tunes the noise scaleσk\\sigma\_\{k\}\[cf\. \([29](https://arxiv.org/html/2607.06833#A1.E29)\)\]\. Each step draws𝐱k−1\{\\mathbf\{x\}\}\_\{k\-1\}exactly from the reverse kernel \([5](https://arxiv.org/html/2607.06833#S2.E5)\) atη=1\\eta=1, and smaller values yield near\-deterministic and usually faster sampling\.

At inference, we initialize𝐱K∼𝒩​\(𝟎,𝐈\)\{\\mathbf\{x\}\}\_\{K\}\\sim\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)and iterate \([9](https://arxiv.org/html/2607.06833#S2.E9)\) fork=K,…,1k=K,\\ldots,1to draw samples𝐱0∼p𝜽⋆\(⋅\|𝝃\)\{\\mathbf\{x\}\}\_\{0\}\\sim p\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)\. These samples are distributed approximately according to the conditional data distributionq𝐱\|𝝃\(⋅\|𝝃\)q\_\{\{\\mathbf\{x\}\}\|\\boldsymbol\{\\xi\}\}\(\\cdot\\,\|\\,\\boldsymbol\{\\xi\}\)for the given conditioning𝝃=\(𝐒,𝐮\)\\boldsymbol\{\\xi\}=\(\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\), thereby realizing the conditional generative model sought in \([1](https://arxiv.org/html/2607.06833#S2.E1)\)\. To reduce the number of model evaluations, we run an accelerated variant of \([9](https://arxiv.org/html/2607.06833#S2.E9)\) \(see Appendix[A](https://arxiv.org/html/2607.06833#A1)\)\.

U\-Nets have been instrumental in image generative models\. Accordingly, we particularizeϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}to a U\-Graph Neural Network \(U\-GNN\) that tailors U\-Nets for graph signal denoising\.

## IIIPooling & Stride in Graph Neural Networks

We first review graph filters, convolutions, and a standard GNN layer\. We then propose a GNN module with built\-in pooling through selection matrices and stride, which Section[IV](https://arxiv.org/html/2607.06833#S4)arranges into the U\-GNN encoder–decoder pipeline\.

Graph filters and convolutions\.Graph filters are the building blocks of GNNs and process input graph signals through successive graph shifts\. A single shift𝐒𝐱\{\\mathbf\{S\}\}\{\\mathbf\{x\}\}mixes each node’s value with those of its one\-hop neighbors, andkkrepeated shifts yield𝐒k​𝐱\{\\mathbf\{S\}\}^\{k\}\{\\mathbf\{x\}\}, which aggregates at each node the signals within itskk\-hop neighborhood\. A linear shift\-invariant graph filter of orderKKis the matrix polynomial𝐡​\(𝐒\)≔∑k=0Khk​𝐒k\{\\mathbf\{h\}\}\(\{\\mathbf\{S\}\}\)\\coloneqq\\sum\_\{k=0\}^\{K\}h\_\{k\}\{\\mathbf\{S\}\}^\{k\}with taps \(coefficients\)𝐡=\[h0,…,hK\]\{\\mathbf\{h\}\}=\[h\_\{0\},\\ldots,h\_\{K\}\]\. It processes an input signal𝐱\{\\mathbf\{x\}\}by graph convolution,

𝐲=𝐡∗𝐒𝐱=𝐡​\(𝐒\)​𝐱=∑k=0Khk​𝐒k​𝐱,\\displaystyle\{\\mathbf\{y\}\}=\{\\mathbf\{h\}\}\*\_\{\{\\mathbf\{S\}\}\}\{\\mathbf\{x\}\}=\{\\mathbf\{h\}\}\(\{\\mathbf\{S\}\}\)\\,\{\\mathbf\{x\}\}=\\sum\_\{k=0\}^\{K\}h\_\{k\}\{\\mathbf\{S\}\}^\{k\}\{\\mathbf\{x\}\},\(10\)and produces an output graph signal𝐲∈ℝN\{\\mathbf\{y\}\}\\in\{\\mathbb\{R\}\}^\{N\}as a weighted sum of the shifted signals𝐒k​𝐱\{\\mathbf\{S\}\}^\{k\}\{\\mathbf\{x\}\}\.

Graph neural networks\.A GNN stacks graph convolutional layers, each composing a graph filter with a pointwise nonlinearity\[[8](https://arxiv.org/html/2607.06833#bib.bib42)\]\. Theℓ\\ellth layer of anLL\-layer GNN reads

𝐗ℓ=σℓ​𝐇ℓ​\(𝐒\)​𝐗ℓ−1=σℓ​\(∑k=0K𝐒k​𝐗ℓ−1​𝚯ℓ,k\),\\displaystyle\{\\mathbf\{X\}\}\_\{\\ell\}=\\sigma\_\{\\ell\}\\,\{\\mathbf\{H\}\}\_\{\\ell\}\(\{\\mathbf\{S\}\}\)\\,\{\\mathbf\{X\}\}\_\{\\ell\-1\}=\\sigma\_\{\\ell\}\\\!\\left\(\\sum\_\{k=0\}^\{K\}\{\\mathbf\{S\}\}^\{k\}\\,\{\\mathbf\{X\}\}\_\{\\ell\-1\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\right\),\(11\)forℓ=1,…,L\\ell=1,\\ldots,L\. Here, each scalar tap of \([10](https://arxiv.org/html/2607.06833#S3.E10)\) becomes a MIMO filter bank𝚯ℓ,k\\boldsymbol\{\\Theta\}\_\{\\ell,k\}that maps theFℓ−1F\_\{\\ell\-1\}input features in𝐗ℓ−1\{\\mathbf\{X\}\}\_\{\\ell\-1\}toFℓF\_\{\\ell\}output features withinKK\-hop neighborhoods, with learnable coefficientsΘℓ=\{𝚯ℓ,k\}k=0K\\Theta\_\{\\ell\}=\\\{\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\\}\_\{k=0\}^\{K\}and a pointwise nonlinearityσℓ\\sigma\_\{\\ell\}, e\.g\., a ReLU\. Stacking these layers defines theΘ\\Theta\-parametrized GNN𝚽​\(⋅,𝐒;Θ\)=𝚽L​\(⋅,𝐒;ΘL\)∘⋯∘𝚽1​\(⋅,𝐒;Θ1\)\\boldsymbol\{\\Phi\}\(\\cdot,\{\\mathbf\{S\}\};\\Theta\)=\\boldsymbol\{\\Phi\}\_\{L\}\(\\cdot,\{\\mathbf\{S\}\};\\Theta\_\{L\}\)\\circ\\cdots\\circ\\boldsymbol\{\\Phi\}\_\{1\}\(\\cdot,\{\\mathbf\{S\}\};\\Theta\_\{1\}\), with trainable parametersΘ=\{Θℓ\}ℓ=1L\\Theta=\\\{\\Theta\_\{\\ell\}\\\}\_\{\\ell=1\}^\{L\}\. It maps an input𝐗0≔𝐗∈ℝN×F0\{\\mathbf\{X\}\}\_\{0\}\\coloneqq\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}to an output𝐗L≕𝐘∈ℝN×FL\{\\mathbf\{X\}\}\_\{L\}\\eqqcolon\{\\mathbf\{Y\}\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{L\}\},

𝐘=𝚽​\(𝐗,𝐒;Θ\)\.\\displaystyle\{\\mathbf\{Y\}\}=\\boldsymbol\{\\Phi\}\(\{\\mathbf\{X\}\},\{\\mathbf\{S\}\};\\Theta\)\.\(12\)
The GNN in \([12](https://arxiv.org/html/2607.06833#S3.E12)\) is general\-purpose but not tailored to diffusion\-based generation\. Since graph convolutions already extract features on arbitrary graphs, adapting the U\-Net backbone additionally requires only node \(un\)pooling, which we introduce next together with a stride on the graph shift\.

Pooling\.Consider a signal𝐗∈ℝN×F\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{N\\times F\}onNNnodes\. A sampling \(selection\) matrix𝐃∈\{0,1\}N′×N\{\\mathbf\{D\}\}\\in\\\{0,1\\\}^\{N^\{\\prime\}\\times N\}withN′<NN^\{\\prime\}<Nretains the rows of𝐈N\{\\mathbf\{I\}\}\_\{N\}indexed by a sampling setΩ⊆𝒱\\Omega\\subseteq\{\\mathcal\{V\}\}with\|Ω\|=N′\|\\Omega\|=N^\{\\prime\}\.111We present𝐃\{\\mathbf\{D\}\}as binary for exposition\. We relax it to a soft selection mask during training and binarize it to pickN′N^\{\\prime\}nodes at inference \(Appendix[C](https://arxiv.org/html/2607.06833#A3)\)\.We learnΩ\\Omega, and hence𝐃\{\\mathbf\{D\}\}, end\-to\-end, so it is input\-dependent rather than fixed \(Section[IV\-B](https://arxiv.org/html/2607.06833#S4.SS2)\)\. We callρ≥1\\rho\\geq 1withN′=⌊N/ρ⌋N^\{\\prime\}=\\lfloor N/\\rho\\rfloorthe*down\-sampling \(pooling\) factor*that sets the resolution reduction\. By construction,𝐃\{\\mathbf\{D\}\}has orthonormal rows,𝐃𝐃⊤=𝐈N′\{\\mathbf\{D\}\}\{\\mathbf\{D\}\}^\{\\top\}=\{\\mathbf\{I\}\}\_\{N^\{\\prime\}\}, and𝐃⊤​𝐃\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{D\}\}is a diagonal mask with ones onΩ\\Omega\. The forward map𝐙=𝐃𝐗∈ℝN′×F\{\\mathbf\{Z\}\}=\{\\mathbf\{D\}\}\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\}down\-samples𝐗\{\\mathbf\{X\}\}\. Its adjoint𝐃⊤​𝐙\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{Z\}\}up\-samples back toNNnodes by*zero\-padding*, i\.e\., it restores the active entries onΩ\\Omegaand inserts zeros on the inactive nodes𝒱∖Ω\{\\mathcal\{V\}\}\\setminus\\Omega\.

A*graph convolutional layer with pooling*processes a down\-sampled signal𝐙ℓ−1∈ℝN′×Fℓ−1\{\\mathbf\{Z\}\}\_\{\\ell\-1\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{\\ell\-1\}\}into𝐙ℓ∈ℝN′×Fℓ\{\\mathbf\{Z\}\}\_\{\\ell\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{\\ell\}\},

𝐙ℓ=𝐃​σℓ​\(∑k=0K𝐒k​𝐃⊤​𝐙ℓ−1​𝚯ℓ,k\)\.\\displaystyle\{\\mathbf\{Z\}\}\_\{\\ell\}=\{\\mathbf\{D\}\}\\,\\sigma\_\{\\ell\}\\\!\\left\(\\sum\_\{k=0\}^\{K\}\{\\mathbf\{S\}\}^\{k\}\\,\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{Z\}\}\_\{\\ell\-1\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\right\)\.\(13\)We call \([13](https://arxiv.org/html/2607.06833#S3.E13)\) a*lift–filter–reduce*layer\. We lift𝐙ℓ−1\{\\mathbf\{Z\}\}\_\{\\ell\-1\}to the full vertex set by zero\-padding through𝐃⊤\{\\mathbf\{D\}\}^\{\\top\}, filter on𝐒\{\\mathbf\{S\}\}, and reduce the result back to the active set through𝐃\{\\mathbf\{D\}\}\. Because the inactive nodes are zero, each active node aggregates only the active values in its neighborhood on𝐒\{\\mathbf\{S\}\}, which can be few under heavy pooling and motivates the stride introduced next\.

Stride\.Modern CNNs, and U\-Net variants in particular, often replace fixed pooling rules such as max\- or sum\-pooling with strided or dilated convolutions for learned down\-sampling and receptive\-field growth, e\.g\.,\[[15](https://arxiv.org/html/2607.06833#bib.bib64),[45](https://arxiv.org/html/2607.06833#bib.bib77)\]\. A dilated convolution spaces its taps by a fixed step, so each tap reads from a point several positions away rather than from an immediate neighbor\. On a graph, the natural spacing is the hop count\. Thus, we introduce a stride \(dilation\) parameterγ∈ℤ\+\\gamma\\in\\mathbb\{Z\}^\{\+\}and replace the unit shift𝐒\{\\mathbf\{S\}\}with theγ\\gamma\-hop shift𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}\. The order\-KKfilter then aggregates from the dilated hops\{0,γ,2​γ,…,K​γ\}\\\{0,\\gamma,2\\gamma,\\ldots,K\\gamma\\\}rather than the consecutive hops\{0,1,…,K\}\\\{0,1,\\ldots,K\\\}\. This yields the*strided graph convolutional layer with pooling*,

𝐙ℓ=𝚽ℓ​\(𝐙ℓ−1,𝐒γ;Θℓ,𝐃\)=𝐃​σℓ​\(∑k=0K\(𝐒γ\)k​𝐃⊤​𝐙ℓ−1​𝚯ℓ,k\),\\displaystyle\{\\mathbf\{Z\}\}\_\{\\ell\}=\\boldsymbol\{\\Phi\}\_\{\\ell\}\(\{\\mathbf\{Z\}\}\_\{\\ell\-1\},\{\\mathbf\{S\}\}^\{\\gamma\};\\Theta\_\{\\ell\},\{\\mathbf\{D\}\}\)=\{\\mathbf\{D\}\}\\,\\sigma\_\{\\ell\}\\\!\\left\(\\sum\_\{k=0\}^\{K\}\(\{\\mathbf\{S\}\}^\{\\gamma\}\)^\{k\}\\,\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{Z\}\}\_\{\\ell\-1\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\right\),\(14\)which recovers the plain layer \([11](https://arxiv.org/html/2607.06833#S3.E11)\) when𝐃=𝐈N\{\\mathbf\{D\}\}=\{\\mathbf\{I\}\}\_\{N\}\(ρ=1\\rho=1\) andγ=1\\gamma=1\. The pooling factorρ\\rhoand strideγ\\gammaserve complementary roles\. Through𝐃\{\\mathbf\{D\}\}, the factorρ\\rhosets how many nodes survive and hence the sparsity of the zero\-padded signal\. Through𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}, the strideγ\\gammasets how far each tap reaches \[cf\. Remark[1](https://arxiv.org/html/2607.06833#Thmremark1)\]\.

The strided GNN layer \([14](https://arxiv.org/html/2607.06833#S3.E14)\) keeps the lift–filter–reduce form of \([13](https://arxiv.org/html/2607.06833#S3.E13)\), filtering on𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}instead of𝐒\{\\mathbf\{S\}\}\. The computations remain on the original vertex set, and𝐒\{\\mathbf\{S\}\}is reused at every resolution without explicitly forming𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}in our implementation\. Instead, we filter throughγ​K\\gamma Ksuccessive unit shifts and tap the signal everyγ\\gammahops\. Thus, each layer reduces to sparse matrix–vector products, which we realize with the sparse routines of thePyGlibrary\. We refer to Appendix[B](https://arxiv.org/html/2607.06833#A2)for details\.

Reduced GSOs\.The strided layer \([14](https://arxiv.org/html/2607.06833#S3.E14)\) also admits an equivalent form that operates directly on theN′N^\{\\prime\}\-dimensional signals\. Pushing𝐃\{\\mathbf\{D\}\}and𝐃⊤\{\\mathbf\{D\}\}^\{\\top\}through the sum defines thekk\-hop*reduced graph shift operators \(GSOs\)*,

𝐒\(k\)≔𝐃​\(𝐒γ\)k​𝐃⊤∈ℝN′×N′,\\displaystyle\{\\mathbf\{S\}\}^\{\(k\)\}\\coloneqq\{\\mathbf\{D\}\}\\,\(\{\\mathbf\{S\}\}^\{\\gamma\}\)^\{k\}\\,\{\\mathbf\{D\}\}^\{\\top\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times N^\{\\prime\}\},\(15\)whose one\-hop instance𝐒\(1\)\{\\mathbf\{S\}\}^\{\(1\)\}is the support of a coarse subgraph onΩ\\Omega\. The family\{𝐒\(k\)\}k=0K\\\{\{\\mathbf\{S\}\}^\{\(k\)\}\\\}\_\{k=0\}^\{K\}then processes the low\-dimensional signals directly,

𝐙ℓ=σℓ​\(∑k=0K𝐒\(k\)​𝐙ℓ−1​𝚯ℓ,k\)\.\\displaystyle\{\\mathbf\{Z\}\}\_\{\\ell\}=\\sigma\_\{\\ell\}\\\!\\left\(\\sum\_\{k=0\}^\{K\}\{\\mathbf\{S\}\}^\{\(k\)\}\\,\{\\mathbf\{Z\}\}\_\{\\ell\-1\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\right\)\.\(16\)The two forms are equal because row selection commutes with the pointwise nonlinearity,𝐃​σℓ​\(𝐌\)=σℓ​\(𝐃𝐌\)\{\\mathbf\{D\}\}\\,\\sigma\_\{\\ell\}\(\{\\mathbf\{M\}\}\)=\\sigma\_\{\\ell\}\(\{\\mathbf\{D\}\}\{\\mathbf\{M\}\}\), and they differ only in the domain of computation\. The reduced form avoids computation on the inactive nodes but is ill\-suited to our setting\. The reduced GSOs do not compose, i\.e\.,𝐒\(m\+n\)≠𝐒\(m\)​𝐒\(n\)\{\\mathbf\{S\}\}^\{\(m\+n\)\}\\neq\{\\mathbf\{S\}\}^\{\(m\)\}\{\\mathbf\{S\}\}^\{\(n\)\}, since𝐃⊤​𝐃≠𝐈N\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{D\}\}\\neq\{\\mathbf\{I\}\}\_\{N\}in general\. Therefore, each𝐒\(k\)\{\\mathbf\{S\}\}^\{\(k\)\}must be formed separately rather than by powering𝐒\(1\)\{\\mathbf\{S\}\}^\{\(1\)\}\. Moreover,𝐃\{\\mathbf\{D\}\}is input\-dependent, so the family\{𝐒\(k\)\}\\\{\{\\mathbf\{S\}\}^\{\(k\)\}\\\}is rebuilt on every forward pass, negating any gains from pre\-computing\. We therefore adopt the lift–filter–reduce form\.

GNNs with pooling and stride\.StackingLLstrided graph convolutional layers with pooling \[cf\. \([14](https://arxiv.org/html/2607.06833#S3.E14)\)\] defines the\(Θ,𝐃\)\(\\Theta,\{\\mathbf\{D\}\}\)\-parametrized GNN module𝚽​\(⋅,𝐒;Θ,𝐃\)=𝚽L​\(⋅,𝐒γ;ΘL,𝐃\)∘⋯∘𝚽1​\(⋅,𝐒γ;Θ1,𝐃\)\\boldsymbol\{\\Phi\}\(\\cdot,\{\\mathbf\{S\}\};\\Theta,\{\\mathbf\{D\}\}\)=\\boldsymbol\{\\Phi\}\_\{L\}\(\\cdot,\{\\mathbf\{S\}\}^\{\\gamma\};\\Theta\_\{L\},\{\\mathbf\{D\}\}\)\\circ\\cdots\\circ\\boldsymbol\{\\Phi\}\_\{1\}\(\\cdot,\{\\mathbf\{S\}\}^\{\\gamma\};\\Theta\_\{1\},\{\\mathbf\{D\}\}\)\. It maps an input𝐙0≔𝐗∈ℝN′×F0\{\\mathbf\{Z\}\}\_\{0\}\\coloneqq\{\\mathbf\{X\}\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{0\}\}to an output𝐙L≕𝐘∈ℝN′×FL\{\\mathbf\{Z\}\}\_\{L\}\\eqqcolon\{\\mathbf\{Y\}\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{L\}\},

𝐘=𝚽​\(𝐗,𝐒;Θ,𝐃\),\\displaystyle\{\\mathbf\{Y\}\}=\\boldsymbol\{\\Phi\}\(\{\\mathbf\{X\}\},\{\\mathbf\{S\}\};\\Theta,\{\\mathbf\{D\}\}\),\(17\)with parametersΘ=\{Θℓ\}ℓ=1L\\Theta=\\\{\\Theta\_\{\\ell\}\\\}\_\{\\ell=1\}^\{L\}and𝐃∈\{0,1\}N′×N\{\\mathbf\{D\}\}\\in\\\{0,1\\\}^\{N^\{\\prime\}\\times N\}\. This GNN module is the main processing unit of each block in the U\-GNN architecture, which is introduced next\.

## IVU\-Graph Neural Networks

![Refer to caption](https://arxiv.org/html/2607.06833v1/x2.png)Figure 2:U\-GNN denoiser architecture\.A U\-GNN ofB=4B\{=\}4depths maps the noisy graph signal𝐱k\{\\mathbf\{x\}\}\_\{k\}to a noise estimateϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}\. At each depth, a projection𝚷b\\boldsymbol\{\\Pi\}\_\{b\}\(purple\) fuses the global embeddings\[𝐔0;𝐊0\]\[\{\\mathbf\{U\}\}\_\{0\};\{\\mathbf\{K\}\}\_\{0\}\]of node states𝐮\{\\mathbf\{u\}\}and diffusion stepkkinto the signal path before a GNN module𝚽bE/D\\boldsymbol\{\\Phi\}\_\{b\}^\{\\mathrm\{E\}/\\mathrm\{D\}\}\(green\), and decoder blocks additionally merge the encoder skips\. Learned selectors𝚿b\\boldsymbol\{\\Psi\}\_\{b\}\(orange\) produce selection matrices𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}that down\-sample the signal between depths and, transposed, up\-sample it in the decoder\. The shift operator𝐒\{\\mathbf\{S\}\}and the global embeddings\[𝐔0;𝐊0\]\[\{\\mathbf\{U\}\}\_\{0\};\{\\mathbf\{K\}\}\_\{0\}\]are shared across all blocks\. Side panels trace one fixed input graph \(signal\) sample through all four depths, with the encoding \(down\-sampling\) path on the left and decoding \(up\-sampling\) on the right\. On the left, a clean𝐱0\{\\mathbf\{x\}\}\_\{0\}\(top\-left\) is corrupted into the input𝐱k\{\\mathbf\{x\}\}\_\{k\}at depth11via the forward diffusion process \[cf\. \([3](https://arxiv.org/html/2607.06833#S2.E3)\) and \([4](https://arxiv.org/html/2607.06833#S2.E4)\)\] and then down\-sampled across depths by the selectors𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}\(↓\\downarrow\)\. Node colors show a low\-pass view of𝐱k\{\\mathbf\{x\}\}\_\{k\}whose residual noise fades with depth, and hollow markers denote inactive nodes\. On the right, the transposes𝐂b\+1⊤\{\\mathbf\{C\}\}\_\{b\+1\}^\{\\top\}up\-sample \(↑\\uparrow\) the estimate to full resolution, yieldingϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}at depth11, from which the clean𝐱^0\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}\(top\-right\) is estimated via \([7](https://arxiv.org/html/2607.06833#S2.E7)\)\.The GNN in \([17](https://arxiv.org/html/2607.06833#S3.E17)\) processes graph signals at a single node resolution fixed by the sampling matrix𝐃\{\\mathbf\{D\}\}\. To build the multi\-resolution encoder–decoder hierarchy of a U\-Net on top of this module, we first define the resolutions through depth\-dependent selection matrices\{𝐂b\}b=1B\\\{\{\\mathbf\{C\}\}\_\{b\}\\\}\_\{b=1\}^\{B\}, where𝐂b∈\{0,1\}Nb×Nb−1\{\\mathbf\{C\}\}\_\{b\}\\in\\\{0,1\\\}^\{N\_\{b\}\\times N\_\{b\-1\}\}selectsNbN\_\{b\}nodes from theNb−1N\_\{b\-1\}nodes active at the preceding depth, withN0=NN\_\{0\}=NandN=N1≥N2≥⋯≥NBN=N\_\{1\}\\geq N\_\{2\}\\geq\\cdots\\geq N\_\{B\}\. We form the composite \(nested\) sampling matrices,

𝐃b≔∏j=1b𝐂j=𝐂b​⋯​𝐂1∈\{0,1\}Nb×N,\\displaystyle\{\\mathbf\{D\}\}\_\{b\}\\coloneqq\\prod\_\{j=1\}^\{b\}\{\\mathbf\{C\}\}\_\{j\}=\{\\mathbf\{C\}\}\_\{b\}\\cdots\{\\mathbf\{C\}\}\_\{1\}\\in\\\{0,1\\\}^\{N\_\{b\}\\times N\},\(18\)which relate the depth\-bbnodes to the fullNN\-node domain and induce a nested hierarchy of sampling setsΩB⊆⋯⊆Ω1⊆𝒱\\Omega\_\{B\}\\subseteq\\cdots\\subseteq\\Omega\_\{1\}\\subseteq\{\\mathcal\{V\}\}\. We set𝐃1=𝐂1=𝐈N\{\\mathbf\{D\}\}\_\{1\}=\{\\mathbf\{C\}\}\_\{1\}=\{\\mathbf\{I\}\}\_\{N\}by convention, so the first depth operates at full resolution\. Here,𝐂b\{\\mathbf\{C\}\}\_\{b\}and𝐃b\{\\mathbf\{D\}\}\_\{b\}carry the*per\-level*and*cumulative*down\-sampling factorsρb≔Nb−1/Nb\\rho\_\{b\}\\coloneqq N\_\{b\-1\}/N\_\{b\}\(ρ1=1\\rho\_\{1\}=1\) andρ¯b≔∏i=1bρi=N/Nb\\bar\{\\rho\}\_\{b\}\\coloneqq\\prod\_\{i=1\}^\{b\}\\rho\_\{i\}=N/N\_\{b\}, respectively\.

We call the resulting architecture a*U\-Graph Neural Network \(U\-GNN\)*\. It spansBBresolution levelsN=N1≥⋯≥NBN=N\_\{1\}\\geq\\cdots\\geq N\_\{B\}, withB−1B\-1encoder–decoder block pairs and a bottleneck block at the coarsest resolutionNBN\_\{B\}\.

We present the U\-GNN in two stages\. We first fix a pure encoder–decoder cascade whose blocks are single encoding, decoding, and bottleneck GNN modules \[cf\. \([17](https://arxiv.org/html/2607.06833#S3.E17)\)\], written𝚽bE\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}\_\{b\},𝚽bD\\boldsymbol\{\\Phi\}^\{\{\\text\{D\}\}\}\_\{b\}, or𝚽BB\\boldsymbol\{\\Phi\}^\{\{\\text\{B\}\}\}\_\{B\}with weightsΘbE\\Theta^\{\{\\text\{E\}\}\}\_\{b\},ΘbD\\Theta^\{\{\\text\{D\}\}\}\_\{b\}, andΘBB\\Theta^\{\{\\text\{B\}\}\}\_\{B\}, respectively, along with the signal flow among them in \([19](https://arxiv.org/html/2607.06833#S4.E19)\)–\([21](https://arxiv.org/html/2607.06833#S4.E21)\)\. These equations include the skip connections, take the selection matrices𝐂b\{\\mathbf\{C\}\}\_\{b\}as given, and omit the node\-state and diffusion\-step embeddings\. We then complete the blocks with projection \(fusion\) layers and, on the encoder side, node\-selection heads \(Sections[IV\-A](https://arxiv.org/html/2607.06833#S4.SS1)and[IV\-B](https://arxiv.org/html/2607.06833#S4.SS2)\)\. These only enlarge the block interiors and parameter sets, leaving the composition here unchanged\. The U\-GNN overview in Fig\.[2](https://arxiv.org/html/2607.06833#S4.F2)and the complementary block interface details in Fig\.[7](https://arxiv.org/html/2607.06833#A3.F7)\(deferred to Appendix[E](https://arxiv.org/html/2607.06833#A5)due to limited space\) may be consulted throughout this section\.

The encoder blocks form an*encoding path*𝚽E=𝚽B−1E∘⋯∘𝚽1E\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}=\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}\_\{B\-1\}\\circ\\cdots\\circ\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}\_\{1\}, and the decoder blocks a*decoding path*𝚽D=𝚽1D∘⋯∘𝚽B−1D\\boldsymbol\{\\Phi\}^\{\{\\text\{D\}\}\}=\\boldsymbol\{\\Phi\}^\{\{\\text\{D\}\}\}\_\{1\}\\circ\\cdots\\circ\\boldsymbol\{\\Phi\}^\{\{\\text\{D\}\}\}\_\{B\-1\}\. The encoding path extracts graph convolutional features at progressively coarser resolutions, while the decoding path reverses this progression by restoring resolution and reintroducing fine details through skip connections\.

We describe each path in turn, starting from the input features𝐙0\{\\mathbf\{Z\}\}\_\{0\}\. Each encoder block maps its input to an encoded signal𝐙b∈ℝNb×Fb\{\\mathbf\{Z\}\}\_\{b\}\\in\{\\mathbb\{R\}\}^\{N\_\{b\}\\times F\_\{b\}\},

𝐙b=𝚽bE​\(𝐂b​𝐙b−1,𝐒;ΘbE,𝐃b\),\\displaystyle\{\\mathbf\{Z\}\}\_\{b\}=\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}\_\{b\}\\left\(\{\\mathbf\{C\}\}\_\{b\}\{\\mathbf\{Z\}\}\_\{b\-1\},\\,\{\\mathbf\{S\}\};\\,\\Theta^\{\{\\text\{E\}\}\}\_\{b\},\\,\{\\mathbf\{D\}\}\_\{b\}\\right\),\(19\)forb=1,…,B−1b=1,\\ldots,B\-1, where𝐂b\{\\mathbf\{C\}\}\_\{b\}down\-samples the depth\-\(b−1\)\(b\{\-\}1\)feature onto theNbN\_\{b\}nodes processed at depthbb\. At the top level,𝐂1=𝐈N\{\\mathbf\{C\}\}\_\{1\}=\{\\mathbf\{I\}\}\_\{N\}and𝚽1E\\boldsymbol\{\\Phi\}^\{\{\\text\{E\}\}\}\_\{1\}operates at full resolutionN1=NN\_\{1\}=N\.

Each decoder block receives the coarser decoded signal𝐘b\+1∈ℝNb\+1×Fb\+1\{\\mathbf\{Y\}\}\_\{b\+1\}\\in\{\\mathbb\{R\}\}^\{N\_\{b\+1\}\\times F\_\{b\+1\}\}from depthb\+1b\+1, up\-samples it onto theNbN\_\{b\}\-node support through𝐂b\+1⊤\{\\mathbf\{C\}\}^\{\\top\}\_\{b\+1\}, and combines it with the matched encoder feature𝐙b\{\\mathbf\{Z\}\}\_\{b\}through a skip connection,

𝐘b=𝚽bD​\(\[𝐂b\+1⊤​𝐘b\+1;𝐙b\],𝐒;ΘbD,𝐃b\),\\displaystyle\{\\mathbf\{Y\}\}\_\{b\}=\\boldsymbol\{\\Phi\}^\{\{\\text\{D\}\}\}\_\{b\}\\left\(\\big\[\{\\mathbf\{C\}\}^\{\\top\}\_\{b\+1\}\{\\mathbf\{Y\}\}\_\{b\+1\};\\,\{\\mathbf\{Z\}\}\_\{b\}\\big\],\\,\{\\mathbf\{S\}\};\\,\\Theta^\{\{\\text\{D\}\}\}\_\{b\},\\,\{\\mathbf\{D\}\}\_\{b\}\\right\),\(20\)forb=B−1,…,1b=B\-1,\\ldots,1, where\[⋅;⋅\]\[\\,\\cdot\\,;\\,\\cdot\\,\]denotes the skip combination\. The decoded signal𝐘b\{\\mathbf\{Y\}\}\_\{b\}then passes to the next decoder block or to the read\-out layer atb=1b=1\.

The two paths connect at depthBBthrough the bottleneck block, which acts as aBBth encoder block,

𝐘B=𝚽BB​\(𝐂B​𝐙B−1,𝐒;ΘBB,𝐃B\)\.\\displaystyle\{\\mathbf\{Y\}\}\_\{B\}=\\boldsymbol\{\\Phi\}^\{\{\\text\{B\}\}\}\_\{B\}\\left\(\{\\mathbf\{C\}\}\_\{B\}\{\\mathbf\{Z\}\}\_\{B\-1\},\\,\{\\mathbf\{S\}\};\\,\\Theta^\{\{\\text\{B\}\}\}\_\{B\},\\,\{\\mathbf\{D\}\}\_\{B\}\\right\)\.\(21\)The bottleneck output𝐘B\{\\mathbf\{Y\}\}\_\{B\}enters the decoding path \([20](https://arxiv.org/html/2607.06833#S4.E20)\) at depthb=B−1b=B\-1, and the final decoder output𝐘1∈ℝN×F0\{\\mathbf\{Y\}\}\_\{1\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}is mapped by a read\-out layer to the model target, i\.e\., the noise predictionϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}here\. We note that all blocks share the same𝐒\{\\mathbf\{S\}\}, and a depth\-dependent strideγb\\gamma\_\{b\}is implicit in the notation \[cf\. \([14](https://arxiv.org/html/2607.06833#S3.E14)\)\]\. Appendix[B](https://arxiv.org/html/2607.06833#A2)specifies howγb\\gamma\_\{b\}is set and how each block evaluates its strided convolutions on the shared𝐒\{\\mathbf\{S\}\}\.

### IV\-AInput Processing and Block\-Level Conditioning

The U\-GNN realizes the denoiserϵ𝜽​\(𝐱k,k;𝝃\)\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}\(\{\\mathbf\{x\}\}\_\{k\},k;\\boldsymbol\{\\xi\}\)with conditioning𝝃=\(𝐒,𝐮\)\\boldsymbol\{\\xi\}=\(\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)\. It incorporates the conditioning along two routes\. The graph enters as the shared shift operator𝐒\{\\mathbf\{S\}\}of every GNN module \[cf\. Section[III](https://arxiv.org/html/2607.06833#S3)\], while the node states𝐮\{\\mathbf\{u\}\}enter as side information, embedded and fused into each block together with an embedding of the diffusion stepkk\. Note that the stepkkis a model input and not part of the conditioning𝝃\\boldsymbol\{\\xi\}\. We now detail the embedding pipeline\.

A read\-in layer𝚷0\\boldsymbol\{\\Pi\}\_\{0\}maps the noisy signal𝐱k∈ℝN\{\\mathbf\{x\}\}\_\{k\}\\in\{\\mathbb\{R\}\}^\{N\}to the initial node embeddings𝐙0∈ℝN×F0\{\\mathbf\{Z\}\}\_\{0\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}that enter the first encoder block\. The node states𝐮∈ℝN×U\{\\mathbf\{u\}\}\\in\{\\mathbb\{R\}\}^\{N\\times U\}and the stepkkare embedded separately into global representations shared across all blocks\. A node\-wise MLP𝚷0U\\boldsymbol\{\\Pi\}^\{\{\\text\{U\}\}\}\_\{0\}\(a shared per\-node MLP with no cross\-node mixing\) maps𝐮\{\\mathbf\{u\}\}to per\-node embeddings𝐔0=𝚷0U​\(𝐮\)∈ℝN×F0\{\\mathbf\{U\}\}\_\{0\}=\\boldsymbol\{\\Pi\}^\{\{\\text\{U\}\}\}\_\{0\}\(\{\\mathbf\{u\}\}\)\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}\. A diffusion\-step encoder𝚷0k\\boldsymbol\{\\Pi\}^\{\{\\text\{k\}\}\}\_\{0\}, a sinusoidal positional encoding followed by a node\-wise MLP, maps the scalarkkto anF0F\_\{0\}\-dimensional vector broadcast to all nodes as𝐊0=𝟏N​𝚷0k​\(k\)⊤∈ℝN×F0\{\\mathbf\{K\}\}\_\{0\}=\{\\mathbf\{1\}\}\_\{N\}\\,\\boldsymbol\{\\Pi\}^\{\{\\text\{k\}\}\}\_\{0\}\(k\)^\{\\top\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}\.

At each depthbb, a fusion layer𝚷b\\boldsymbol\{\\Pi\}\_\{b\}merges the main signal path with these global embeddings before the GNN module\. The block input𝐕b\{\\mathbf\{V\}\}\_\{b\}and the down\-sampled embeddings𝐃b​\[𝐔0;𝐊0\]\{\\mathbf\{D\}\}\_\{b\}\\,\[\{\\mathbf\{U\}\}\_\{0\};\\,\{\\mathbf\{K\}\}\_\{0\}\]are each projected to a common width, merged along the feature dimension, and mapped toFbF\_\{b\}channels,

𝐏b=𝚷b​\(𝐕b,𝐃b​\[𝐔0;𝐊0\]\)\.\\displaystyle\{\\mathbf\{P\}\}\_\{b\}=\\boldsymbol\{\\Pi\}\_\{b\}\\big\(\{\\mathbf\{V\}\}\_\{b\},\\;\{\\mathbf\{D\}\}\_\{b\}\\,\[\{\\mathbf\{U\}\}\_\{0\};\\,\{\\mathbf\{K\}\}\_\{0\}\]\\big\)\.\(22\)The factor𝐃b\{\\mathbf\{D\}\}\_\{b\}restricts the global embeddings to theNbN\_\{b\}nodes active at depthbb, matching the support of𝐕b\{\\mathbf\{V\}\}\_\{b\}, and the per\-input projection reconciles their differing feature \(channel\) widths\. The block input depends on the path\. Forbbth encoder block,𝐕b=𝐂b​𝐙b−1\{\\mathbf\{V\}\}\_\{b\}=\{\\mathbf\{C\}\}\_\{b\}\\,\{\\mathbf\{Z\}\}\_\{b\-1\}down\-samples the depth\-\(b−1\)\(b\{\-\}1\)feature onto theNbN\_\{b\}active nodes\. Forbbth decoder block, the coarser output𝐘b\+1∈ℝNb\+1×Fb\+1\{\\mathbf\{Y\}\}\_\{b\+1\}\\in\{\\mathbb\{R\}\}^\{N\_\{b\+1\}\\times F\_\{b\+1\}\}is up\-sampled through𝐂b\+1⊤\{\\mathbf\{C\}\}\_\{b\+1\}^\{\\top\}, concatenated with the encoder skip𝐙b\{\\mathbf\{Z\}\}\_\{b\}along the feature dimension, and mapped toFbF\_\{b\}channels by a learned skip\-projection𝚷bskip:ℝFb\+1\+Fb→ℝFb\\boldsymbol\{\\Pi\}^\{\\mathrm\{skip\}\}\_\{b\}:\{\\mathbb\{R\}\}^\{F\_\{b\+1\}\+F\_\{b\}\}\\to\{\\mathbb\{R\}\}^\{F\_\{b\}\},

𝐕b=𝚷bskip​\(\[𝐂b\+1⊤​𝐘b\+1;𝐙b\]\)\.\\displaystyle\{\\mathbf\{V\}\}\_\{b\}=\\boldsymbol\{\\Pi\}^\{\\mathrm\{skip\}\}\_\{b\}\\left\(\\big\[\{\\mathbf\{C\}\}\_\{b\+1\}^\{\\top\}\{\\mathbf\{Y\}\}\_\{b\+1\};\\,\{\\mathbf\{Z\}\}\_\{b\}\\big\]\\right\)\.\(23\)The fusion layer𝚷b\\boldsymbol\{\\Pi\}\_\{b\}is shared between the matched encoder and decoder blocks, so its weights appear in bothΘbE\\Theta^\{\{\\text\{E\}\}\}\_\{b\}andΘbD\\Theta^\{\{\\text\{D\}\}\}\_\{b\}, while the GNN module and skip\-projection remain block\-specific\. The node\-state and diffusion\-step embeddings enter each block by input substitution\. In \([19](https://arxiv.org/html/2607.06833#S4.E19)\) and \([21](https://arxiv.org/html/2607.06833#S4.E21)\), the module input𝐂b​𝐙b−1\{\\mathbf\{C\}\}\_\{b\}\\,\{\\mathbf\{Z\}\}\_\{b\-1\}is replaced by the fused𝐏b\{\\mathbf\{P\}\}\_\{b\}\. In \([20](https://arxiv.org/html/2607.06833#S4.E20)\), the skip combination\[𝐂b\+1⊤​𝐘b\+1;𝐙b\]\[\{\\mathbf\{C\}\}^\{\\top\}\_\{b\+1\}\{\\mathbf\{Y\}\}\_\{b\+1\};\\,\{\\mathbf\{Z\}\}\_\{b\}\]is first mapped by𝚷bskip\\boldsymbol\{\\Pi\}^\{\\mathrm\{skip\}\}\_\{b\}\[cf\. \([23](https://arxiv.org/html/2607.06833#S4.E23)\)\] and then fused into𝐏b\{\\mathbf\{P\}\}\_\{b\}\. The GNN modules and the block composition remain unchanged, and only the module inputs differ\. Figure[2](https://arxiv.org/html/2607.06833#S4.F2)shows this separation, with the fusion drawn as the purple projection𝚷b\\boldsymbol\{\\Pi\}\_\{b\}preceding each green GNN module, and Fig\.[7](https://arxiv.org/html/2607.06833#A3.F7)\(Appendix[E](https://arxiv.org/html/2607.06833#A5)\) details the block interface\. Finally, a read\-out layer𝚷out\\boldsymbol\{\\Pi\}\_\{\\mathrm\{out\}\}, a node\-wise MLP, maps the final decoder output𝐘1∈ℝN×F0\{\\mathbf\{Y\}\}\_\{1\}\\in\{\\mathbb\{R\}\}^\{N\\times F\_\{0\}\}to the predictionϵ𝜽∈ℝN\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}\\in\{\\mathbb\{R\}\}^\{N\}\.

### IV\-BDown\-sampling \(Pooling\) via Learned Node Selection

The selection matrices𝐂b\{\\mathbf\{C\}\}\_\{b\}and their compositions𝐃b\{\\mathbf\{D\}\}\_\{b\}determine the active nodes at each depth\. Rather than fixing them in advance, we learn them end\-to\-end by letting each encoder block score its own GNN output and select the active nodes propagated to the next depth\. The node\-selection head of encoder blockbbderives𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}from the encoded feature𝐙b\{\\mathbf\{Z\}\}\_\{b\}, forb=1,…,B−1b=1,\\ldots,B\-1, producing𝐂2,…,𝐂B\{\\mathbf\{C\}\}\_\{2\},\\ldots,\{\\mathbf\{C\}\}\_\{B\}\. The top level performs no selection since𝐂1=𝐃1=𝐈N\{\\mathbf\{C\}\}\_\{1\}=\{\\mathbf\{D\}\}\_\{1\}=\{\\mathbf\{I\}\}\_\{N\}, and the deepest selection𝐂B\{\\mathbf\{C\}\}\_\{B\}down\-samples into the bottleneck, which itself does not further down\-sample\.

We describe the selection operation using two index systems\.*Global*indices label nodes in𝒱\{\\mathcal\{V\}\}, with the active setΩb⊆𝒱\\Omega\_\{b\}\\subseteq\{\\mathcal\{V\}\},\|Ωb\|=Nb\|\\Omega\_\{b\}\|=N\_\{b\}, nested as in the beginning of Section[IV](https://arxiv.org/html/2607.06833#S4)\.*Local*indices\[Nb\]≔\{1,…,Nb\}\[N\_\{b\}\]\\coloneqq\\\{1,\\ldots,N\_\{b\}\\\}enumerate only the active nodes in a fixed order\. The composite matrix𝐃b∈\{0,1\}Nb×N\{\\mathbf\{D\}\}\_\{b\}\\in\\\{0,1\\\}^\{N\_\{b\}\\times N\}maps a global\-indexed signal to its local form, and𝐃b⊤\{\\mathbf\{D\}\}\_\{b\}^\{\\top\}scatters a local\-indexed signal back to theNN\-node domain\. The selection below acts in the local space\[Nb\]\[N\_\{b\}\]\.

Each encoder block is equipped with a head that scores the active nodes and keeps the highest\-scoring ones\. Concretely, a node\-wise MLP𝚿b\\boldsymbol\{\\Psi\}\_\{b\}, with parameters𝚯bsel\\boldsymbol\{\\Theta\}^\{\\mathrm\{sel\}\}\_\{b\}, takes the block output𝐙b\{\\mathbf\{Z\}\}\_\{b\}, along with the node and step embeddings restricted to the active nodes, and returns a per\-node score vector,

𝐯b=𝚿b​\(𝐙b,𝐃b​\[𝐔0;𝐊0\];𝚯bsel\)∈ℝNb\.\\displaystyle\{\\mathbf\{v\}\}\_\{b\}=\\boldsymbol\{\\Psi\}\_\{b\}\\left\(\{\\mathbf\{Z\}\}\_\{b\},\\,\{\\mathbf\{D\}\}\_\{b\}\\,\[\{\\mathbf\{U\}\}\_\{0\};\\,\{\\mathbf\{K\}\}\_\{0\}\];\\,\\boldsymbol\{\\Theta\}^\{\\mathrm\{sel\}\}\_\{b\}\\right\)\\in\{\\mathbb\{R\}\}^\{N\_\{b\}\}\.\(24\)The dependence on𝐃b​\[𝐔0;𝐊0\]\{\\mathbf\{D\}\}\_\{b\}\\,\[\{\\mathbf\{U\}\}\_\{0\};\\,\{\\mathbf\{K\}\}\_\{0\}\], as in \([22](https://arxiv.org/html/2607.06833#S4.E22)\), allows the selection to adapt to the node states and the diffusion step\. Given a down\-sampling factorρb\+1\>1\\rho\_\{b\+1\}\>1, we keep theNb\+1=⌊Nb/ρb\+1⌋N\_\{b\+1\}=\\lfloor N\_\{b\}/\\rho\_\{b\+1\}\\rfloorhighest\-scoring nodes,

𝒮b=TopK​\(𝐯b,Nb\+1\)⊆\[Nb\]\.\\displaystyle\\mathcal\{S\}\_\{b\}=\\mathrm\{TopK\}\(\{\\mathbf\{v\}\}\_\{b\},\\,N\_\{b\+1\}\)\\subseteq\[N\_\{b\}\]\.\(25\)The selection matrix𝐂b\+1=\[𝐈Nb\]𝒮b,:∈\{0,1\}Nb\+1×Nb\{\\mathbf\{C\}\}\_\{b\+1\}=\\big\[\{\\mathbf\{I\}\}\_\{N\_\{b\}\}\\big\]\_\{\\mathcal\{S\}\_\{b\},\\,:\}\\in\\\{0,1\\\}^\{N\_\{b\+1\}\\times N\_\{b\}\}keeps the rows of𝐈Nb\{\\mathbf\{I\}\}\_\{N\_\{b\}\}indexed by𝒮b\\mathcal\{S\}\_\{b\}, and the composite matrix updates as𝐃b\+1=𝐂b\+1​𝐃b\{\\mathbf\{D\}\}\_\{b\+1\}=\{\\mathbf\{C\}\}\_\{b\+1\}\{\\mathbf\{D\}\}\_\{b\}\[cf\. \([18](https://arxiv.org/html/2607.06833#S4.E18)\)\]\. The retained global setΩb\+1⊆Ωb\\Omega\_\{b\+1\}\\subseteq\\Omega\_\{b\}collects the nonzero columns of𝐃b\+1\{\\mathbf\{D\}\}\_\{b\+1\}, so𝒮b\\mathcal\{S\}\_\{b\}andΩb\+1\\Omega\_\{b\+1\}express the same selection in local and global coordinates, respectively\.

The decoder performs no selection\. At each depth, the encoder\-produced𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}is reused by the matched decoder for up\-sampling through𝐂b\+1⊤\{\\mathbf\{C\}\}\_\{b\+1\}^\{\\top\}\[cf\. \([23](https://arxiv.org/html/2607.06833#S4.E23)\)\], while𝐃b\{\\mathbf\{D\}\}\_\{b\}parametrizes the depth\-bbGNN modules \[cf\. \([19](https://arxiv.org/html/2607.06833#S4.E19)\)–\([21](https://arxiv.org/html/2607.06833#S4.E21)\)\]\.

The Top\-K operation in \([25](https://arxiv.org/html/2607.06833#S4.E25)\) is non\-differentiable in𝐯b\{\\mathbf\{v\}\}\_\{b\}\. Therefore, we train the head𝚿b\\boldsymbol\{\\Psi\}\_\{b\}with a straight\-through estimator \(STE\)\[[1](https://arxiv.org/html/2607.06833#bib.bib78)\]that keeps the hard selection𝒮b\\mathcal\{S\}\_\{b\}in the forward pass, while gradients reach the scores𝐯b\{\\mathbf\{v\}\}\_\{b\}through a differentiable sigmoid surrogate of the binary selection mask\. Appendix[C](https://arxiv.org/html/2607.06833#A3)details the surrogate mask, and Appendix[D](https://arxiv.org/html/2607.06833#A4)integrates classical fixed\-pooling rules with the U\-GNN\.

## VNumerical Results

We evaluate the proposed U\-GNN denoiser on stock price forecasting for the S&P 500 index and wireless resource allocation \(WRA\) in a power control setup\. Both use the same depth\-B=4B\{=\}4U\-GNN backboneϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}, diffusion noise schedules, and optimizers\. They differ only in the signal and conditioning interface that each task dictates—a multi\-step return trajectory for forecasting versus a single power allocation for resource allocation\. Table[I](https://arxiv.org/html/2607.06833#S5.T1)summarizes the model, diffusion, and optimization hyperparameters\. We detail the shared and application\-specific configurations in Appendix[E](https://arxiv.org/html/2607.06833#A5)\.

TABLE I:Summary of U\-GNN configuration\.### V\-AStock Price Forecasting

We cast stock price forecasting as a spatio\-temporal generative task\. Given recent market history for a panel of stocks, the goal is to sample from the conditional distribution of their short\-horizon future log\-return trajectories\.

TABLE II:Forecast accuracy and distributional fidelity comparisons on the S&P 500 test split\.Best values per column are shown inbold\. TheU\-GNNoutperforms the GRW baseline in almost every accuracy, calibration and stylized\-fact metrics\.MV\-DA: cumulative\-horizon majority\-vote direction accuracy\.Vol,Mom,Kurt: absolute gap\|sgen−sreal\|\|s\_\{\\text\{gen\}\}\-s\_\{\\text\{real\}\}\|between generated and real volatility\-clustering, momentum, and excess\-kurtosis statistics\. Continuous ranked probability score \(CRPS\), mean 90% prediction interval score \(MIS90\), root mean squared error \(RMSE\), and mean absolute error \(MAE\) are reported on both the return and nominal price scales\.

![Refer to caption](https://arxiv.org/html/2607.06833v1/x3.png)

![Refer to caption](https://arxiv.org/html/2607.06833v1/x4.png)

![Refer to caption](https://arxiv.org/html/2607.06833v1/x5.png)

Figure 3:Example S&P 500 forecasting trajectories from U\-GNN\. Each column is a different test window with three panels for arbitrarily chosen stocks\. In each panel, the solid black line is the observed history whereas the solid blue line is the ground\-truth future, the thin orange lines are individual trajectories sampled from the diffusion ensemble, and the solid orange line is their mean, over the forecast horizon\. The spread of the sampled trajectories reflects the forecast uncertainty\.![Refer to caption](https://arxiv.org/html/2607.06833v1/x6.png)\(a\)
![Refer to caption](https://arxiv.org/html/2607.06833v1/x7.png)\(b\)

![Refer to caption](https://arxiv.org/html/2607.06833v1/x8.png)\(c\)
![Refer to caption](https://arxiv.org/html/2607.06833v1/x9.png)\(d\)

Figure 4:Distributional fidelity diagnostics on the S&P 500 test split\.\(a\)Temporal structure: autocorrelation of returnsrrand squared returnsr2r^\{2\}for real and generated series, probing linear predictability and volatility clustering\.\(b\)Spatial \(cross\-sectional\) structure: the top\-5 eigenvalues of the return covariance matrixC^\\hat\{C\}, capturing how well each model reproduces inter\-stock correlation\.\(c\)–\(d\)Histograms and empirical CDFs of per\-window negative log\-likelihood \(NLL\): values are computed under the GRW model’s analytic likelihood\(c\)and as ELBO estimates under the trained U\-GNN\(d\), each evaluated on the real data and on samples drawn from both models\. Close overlap between the generated and real distributions indicates higher distributional realism; U\-GNN’s samples track the real statistics more closely than the GRW baseline across all four panels\.Setup\.We use daily S&P 500 data obtained with the open\-sourceyfinancelibrary\. After aligning all stocks on a common trading day axis and dropping those with insufficient coverage, we retainN=468N=468stocks across1111GICS sectors over23532353trading days \(2016\-09\-06 to 2026\-02\-13\)\. Let𝐫\(t\)∈ℝN\{\\mathbf\{r\}\}^\{\(t\)\}\\in\{\\mathbb\{R\}\}^\{N\}collect the daily log returns on trading daytt, where\[𝐫\(t\)\]i\[\{\\mathbf\{r\}\}^\{\(t\)\}\]\_\{i\}is the log ratio of stockii’s consecutive closing prices\. For each stock and day, we gatherU=12U=12market features into𝐮\(t\)∈ℝN×U\{\\mathbf\{u\}\}^\{\(t\)\}\\in\{\\mathbb\{R\}\}^\{N\\times U\}, comprising the open, high, low, and standardized closing prices; the log return𝐫\(t\)\{\\mathbf\{r\}\}^\{\(t\)\}and its trailing moving averages; the log\-volume; and two technical indicators, RSI and MACD\.

We build a static, undirected graph𝒢\{\\mathcal\{G\}\}from long\-term company fundamentals \(e\.g\., market capitalization and price\-to\-earnings ratios\)\. Its adjacency𝐖\{\\mathbf\{W\}\}combines the rank correlation between two stocks’ fundamental profiles with a same\-sector bonus\. We threshold and spectrally normalize𝐖\{\\mathbf\{W\}\}to obtain the GSO𝐒\{\\mathbf\{S\}\}, which is shared across all forecast windows\.

We form data samples with a sliding window of length\(Th\+Tp\)\(T\_\{h\}\+T\_\{p\}\)over the panel\. In each window, the firstThT\_\{h\}days are the conditioning history, stacked into𝐮=\[𝐮\(t−Th\+1\),⋯,𝐮\(t\)\]∈ℝN×Th×U\{\\mathbf\{u\}\}=\[\\,\{\\mathbf\{u\}\}^\{\(t\-T\_\{h\}\+1\)\},\\ \\cdots,\\ \{\\mathbf\{u\}\}^\{\(t\)\}\\,\]\\in\{\\mathbb\{R\}\}^\{N\\times T\_\{h\}\\times U\}, and the nextTpT\_\{p\}days are the forecast target, stacked into the clean graph signal𝐱0=\[𝐫\(t\+1\),⋯,𝐫\(t\+Tp\)\]∈ℝN×Tp\{\\mathbf\{x\}\}\_\{0\}=\[\\,\{\\mathbf\{r\}\}^\{\(t\+1\)\},\\ \\cdots,\\ \{\\mathbf\{r\}\}^\{\(t\+T\_\{p\}\)\}\\,\]\\in\{\\mathbb\{R\}\}^\{N\\times T\_\{p\}\}\. We reversibly instance\-normalize \(RevIN\) each target window\[[17](https://arxiv.org/html/2607.06833#bib.bib80)\], using statistics from its conditioning block alone, and we invert this normalization on the DDIM\-generated samples before scoring\. The U\-GNN treats theTpT\_\{p\}horizon steps as a temporal axis\. It applies graph convolutions on𝐒\{\\mathbf\{S\}\}at each step, couples the steps through interleaved temporal\-convolution layers, and fuses a per\-node summary of theThT\_\{h\}history steps into the conditioning context by cross\-attention\.

We setTh=20T\_\{h\}=20andTp=5T\_\{p\}=5, yielding23282328windows in total\. We partition them using an interleaved chronological scheme\. We split the trading\-day axis into1010equal chunks\. Within each chunk, we assign the earliest80%80\\%days to training, the next10%10\\%to validation, and the final10%10\\%to testing, then pool the matching parts across chunks\. This ordering keeps each test block later than its training block, so the model is never fitted on days that follow its test period\. Interleaving also spreads anomalous regimes, such as COVID\-19, across all three splits rather than concentrating them in one, which mitigates the test distribution shift that a single contiguous split would create\.

Results\.We assess the U\-GNN forecasts qualitatively through the sample trajectories of Fig\.[3](https://arxiv.org/html/2607.06833#S5.F3)and quantitatively through the accuracy and distributional fidelity metrics of Table[II](https://arxiv.org/html/2607.06833#S5.T2), both computed on the held\-out test split with100100sampled trajectories per window\. We benchmark U\-GNN against a geometric random walk \(GRW\), in which each stock’s future log\-returns are drawn i\.i\.d\. from a fixed per\-stock Gaussian\. Its drift and volatility are estimated once on the training set, with shrinkage toward the market\-wide statistics, and they are held fixed at inference, independently of the conditioning window\.

U\-GNN outperforms the GRW on nearly every metric, at both the return and price scales\. It reduces price CRPS, RMSE, and MAE by22%22\\%,28%28\\%, and21%21\\%, respectively, and recovers directional information unavailable to a random walk, raising majority\-vote direction accuracy above chance \(0\.530\.53vs\.0\.490\.49\)\. Beyond point and interval accuracy, U\-GNN reproduces stylized facts of returns\. It substantially shrinks the volatility\-clustering gap \(0\.060\.06vs\.0\.100\.10\) and reduces the momentum gap\. The one exception is excess kurtosis, where it overshoots the empirical heavy tails\. This overshoot is the artifact of a desirable property\. The GRW confines nearly all of its mass to a band around the mean, whereas U\-GNN places non\-negligible probability on the rare, far\-from\-mean moves that real markets exhibit\. At the reported checkpoint it allocates somewhat more mass to these extremes than the data warrant, and we read this gap as a fidelity–sharpness trade\-off rather than a fixed limitation\. Checkpoints with a near\-vanishing kurtosis gap are attainable, but only by sacrificing sharpness\. Sharp forecasts emerge early in training and distributional fidelity improves later, with the reported model sitting toward the sharper end \(Fig\.[4\(d\)](https://arxiv.org/html/2607.06833#S5.F4.sf4)\)\.

These dynamics are evident in the individual forecasts of Fig\.[3](https://arxiv.org/html/2607.06833#S5.F3)\. The sampled trajectories spread out from the last observed price, and the spread grows over the forecast days, while their mean stays close to the realized path\. Each sample tends to track the recent behavior of its stock, picking up its trend and volatility rather than flattening into a straight line or spreading evenly in all directions\. A few paths break from the bundle to trace larger excursions, which is the generative model anticipating the occasional market swings noted above\.

Fig\.[4](https://arxiv.org/html/2607.06833#S5.F4)assesses the distributional fidelity of U\-GNN samples against those of GRW along four axes\. Fig\.[4\(a\)](https://arxiv.org/html/2607.06833#S5.F4.sf1)shows that U\-GNN matches the autocorrelation signature of stock returns, which is near\-zero forrrand persistent forr2r^\{2\}, with the latter implying strong volatility clustering\. Fig\.[4\(b\)](https://arxiv.org/html/2607.06833#S5.F4.sf2)pertains to the cross\-sectional structure, where U\-GNN tracks the leading eigenvalues of the return covarianceC^\\hat\{C\}that capture the dominant market directions, whereas the GRW baseline flattens them\. The remaining two plots take a score\-based view\. Each assigns every evaluation window a scalar score, then compares the resulting distribution to the real one through the11\-Wasserstein \(W1W\_\{1\}\) distance between their empirical CDFs\. They differ only in which model supplies the score\.

Fig\.[4\(c\)](https://arxiv.org/html/2607.06833#S5.F4.sf3)uses the GRW score, the analytic NLL per \(stock, step\) in nats, for which an i\.i\.d\. Gaussian is lower\-bounded by its differential entropy12​ln⁡\(2​π​e\)≈1\.42\\tfrac\{1\}\{2\}\\ln\(2\\pi e\)\\approx 1\.42\(dotted line\)\. Scored by the homoskedastic Gaussian that generated them, the GRW samples collapse onto a near\-degenerate spike essentially at this floor, so every window is equally surprising, and the score carries no volatility information\. The real returns, instead, spread out with a rightward skew, as calm windows fall below the floor and heavy\-tailed ones, beyond the reach of the GRW, are pushed into the tail\. U\-GNN’s samples sit just left of the floor \(mean≈1\.25\\approx 1\.25\), which the GRW scorer even rates as more typical than an i\.i\.d\. draw\. Even under the baseline’s own likelihood, U\-GNN lands closer to the real distribution \(W1=0\.14W\_\{1\}\{=\}0\.14\) than GRW’s own samples do \(W1=0\.23W\_\{1\}\{=\}0\.23\)\.

Fig\.[4\(d\)](https://arxiv.org/html/2607.06833#S5.F4.sf4)instead scores each window by a uniformly weighted version of U\-GNN’s training loss in \([6](https://arxiv.org/html/2607.06833#S2.E6)\), estimated by a stratified Monte Carlo average over6464strata of the diffusion step \(one noise draw each\)\. This orders windows by denoising difficulty, and hence by volatility\. U\-GNN’s samples cluster lowest \(mean≈0\.28\\approx 0\.28\), while the real data lies just to their right \(mean≈0\.32\\approx 0\.32\)\. GRW paths form a distinct high\-error mode \(mean≈0\.46\\approx 0\.46\), since memoryless Gaussian increments are hard to denoise for a model that expects temporal and cross\-sectional structures\. Here, U\-GNN nearly coincides with the real data \(W1=0\.04W\_\{1\}\{=\}0\.04\), while GRW sits farther away \(W1=0\.14W\_\{1\}\{=\}0\.14\)\. Neither score is impartial on its own, but read together, they corroborate that U\-GNN better captures the real distribution\. While U\-GNN’s score isolates GRW samples as unrealistic, GRW ranks U\-GNN above the real data\. That said, the temporal nature of stock data may favor attention\-based architectures suited to long\-range dependencies\. We leave the integration of a graph transformer with our U\-GNN backbone to future work\.

### V\-BWireless Resource Allocation

TABLE III:Power control \(WRA\) simulation setup\.We study optimal power control in multi\-user interference networks\. The setup closely follows our preliminary work on diffusion policies for resource allocation\[[36](https://arxiv.org/html/2607.06833#bib.bib11),[37](https://arxiv.org/html/2607.06833#bib.bib45)\]and the related state\-augmented formulations of\[[39](https://arxiv.org/html/2607.06833#bib.bib7)\]\. We summarize it below and collect the simulation parameters in Table[III](https://arxiv.org/html/2607.06833#S5.T3)\.

TABLE IV:Ergodic\-rate performance on the WRA test split \(3232networks,12,80012\{,\}800tx–rx pairs,fmin=0\.6f\_\{\\min\}=0\.6bits/s/Hz\)\.U\-GNNtracks the expert \(PD\) policy on the cell\-edge percentiles and feasibility, where FP and AP collapse\. Per column, the policy performing closest to the reference expert is shown inboldand±\\pmis one standard deviation across networks\.pkk:kk\-th percentile of per\-user ergodic rates \(worst\-served tx–rx pairs\)\.Gap:100×\(fexp−f\)/fexp100\\times\(f^\{\\text\{exp\}\}\-f\)/f^\{\\text\{exp\}\}\(\>0\>0: below expert,<0<0: above\)\.Feasible: Percentage of tx–rx pairs meetingfminf\_\{\\min\}\. FP usesPmaxP\_\{\\max\}on every link; AP uses each network’s mean expert power\.

Setup\.We consider wireless networks ofN=400N=400transmitter–receiver \(tx–rx\) pairs dropped uniformly at random over a square service area\. The pairs share a single channel of bandwidthWWwith noise PSDN0N\_\{0\}and per\-transmitter power budgetPmaxP\_\{\\max\}\(Table[III](https://arxiv.org/html/2607.06833#S5.T3)\)\. The channel gainhi​j,t∈ℝ\+h\_\{ij,t\}\\in\{\\mathbb\{R\}\}\_\{\+\}from transmitteriito receiverjjat stepttcombines a static large\-scale component with a unit\-mean Rayleigh fast\-fading component\. The large\-scale component follows a dual\-slope path\-loss law with log\-normal shadowing and fixes the long\-term average gainhi​jh\_\{ij\}\. We collect the long\-term and instantaneous gains in𝐇,𝐇t∈ℝ\+N×N\{\\mathbf\{H\}\},\{\\mathbf\{H\}\}\_\{t\}\\in\{\\mathbb\{R\}\}^\{N\\times N\}\_\{\+\}, with\[𝐇\]i​j=hi​j\[\{\\mathbf\{H\}\}\]\_\{ij\}=h\_\{ij\}and\[𝐇t\]i​j=hi​j,t\[\{\\mathbf\{H\}\}\_\{t\}\]\_\{ij\}=h\_\{ij,t\}\.

The policy and the instantaneous gains𝐇t\{\\mathbf\{H\}\}\_\{t\}evolve on separate timescales\. The fast\-fading process generatesTTrealizations of𝐇t\{\\mathbf\{H\}\}\_\{t\}, while the policy emits one allocation per slot and holds it constant over a window ofT0T\_\{0\}realizations, givingS=T/T0S=T/T\_\{0\}slots per network \(Table[III](https://arxiv.org/html/2607.06833#S5.T3)\)\. The network configuration𝐇\{\\mathbf\{H\}\}stays fixed over the operation window and is the only channel state available at allocation time\. The policy is conditioned on𝐇\{\\mathbf\{H\}\}via the GSO𝐒\{\\mathbf\{S\}\}and node\-state𝐮\{\\mathbf\{u\}\}\(defined below\), while the instantaneous gains𝐇t\{\\mathbf\{H\}\}\_\{t\}are observed only through the rates they induce\.

We model each configuration as a draw𝐇∼q𝐇\{\\mathbf\{H\}\}\\sim q\_\{\{\\mathbf\{H\}\}\}from a family of network geometries and large\-scale fading\. Sweeping the area side lengthRRover four values yields four user densities, each contributing3232networks for a dataset of128128configurations in total, which we split5:1:25\{:\}1\{:\}2into training, validation, and testing \(Table[III](https://arxiv.org/html/2607.06833#S5.T3)\)\.

Given an allocation𝐩\{\\mathbf\{p\}\}and a channel realization𝐇~\\widetilde\{\{\\mathbf\{H\}\}\}, the rate of the receiverjjis

\[𝐟~​\(𝐩,𝐇~\)\]j=log2⁡\(1\+\[𝐩\]j​\[𝐇~\]j​jW​N0\+∑i≠j\[𝐩\]i​\[𝐇~\]i​j\)\.\\big\[\\widetilde\{\{\\mathbf\{f\}\}\}\(\{\\mathbf\{p\}\},\\widetilde\{\{\\mathbf\{H\}\}\}\)\\big\]\_\{j\}=\\log\_\{2\}\\\!\\bigg\(1\+\\frac\{\[\{\\mathbf\{p\}\}\]\_\{j\}\\,\[\\widetilde\{\{\\mathbf\{H\}\}\}\]\_\{jj\}\}\{WN\_\{0\}\+\\sum\_\{i\\neq j\}\[\{\\mathbf\{p\}\}\]\_\{i\}\\,\[\\widetilde\{\{\\mathbf\{H\}\}\}\]\_\{ij\}\}\\bigg\)\.\(26\)We evaluate a policy by its ergodic rate, estimated by averaging the instantaneous rates over the fading realizations,

𝐟​\(𝝅,𝐇\)=𝔼𝐩,𝐇~​\[𝐟~​\(𝐩,𝐇~\)\]≈1T​∑t=1T𝐟~​\(𝐩⌈t/T0⌉,𝐇t\)\.\{\\mathbf\{f\}\}\(\\boldsymbol\{\\pi\},\{\\mathbf\{H\}\}\)=\\mathbb\{E\}\_\{\{\\mathbf\{p\}\},\\widetilde\{\{\\mathbf\{H\}\}\}\}\\big\[\\,\\widetilde\{\{\\mathbf\{f\}\}\}\(\{\\mathbf\{p\}\},\\widetilde\{\{\\mathbf\{H\}\}\}\)\\big\]\\approx\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\widetilde\{\{\\mathbf\{f\}\}\}\\big\(\{\\mathbf\{p\}\}\_\{\\lceil t/T\_\{0\}\\rceil\},\{\\mathbf\{H\}\}\_\{t\}\\big\)\.\(27\)In \([27](https://arxiv.org/html/2607.06833#S5.E27)\), the expectation is over the policy allocations𝐩∼𝝅\(⋅\|𝐇\)\{\\mathbf\{p\}\}\\sim\\boldsymbol\{\\pi\}\(\\cdot\\,\|\\,\{\\mathbf\{H\}\}\)and the fast\-fading channel𝐇~∼q𝐇~\|𝐇\\widetilde\{\{\\mathbf\{H\}\}\}\\sim q\_\{\\widetilde\{\{\\mathbf\{H\}\}\}\|\{\\mathbf\{H\}\}\}\. The allocation𝐩s\{\\mathbf\{p\}\}\_\{s\}is drawn once per slots=⌈t/T0⌉s=\\lceil t/T\_\{0\}\\rceiland held fixed across it, while𝐇t\{\\mathbf\{H\}\}\_\{t\}is the channel realized at steptt\.

Given𝐇\{\\mathbf\{H\}\}, an optimal policy𝝅⋆\(𝐇\)=𝝅⋆\(⋅\|𝐇\)\\boldsymbol\{\\pi\}^\{\\star\}\(\{\\mathbf\{H\}\}\)=\\boldsymbol\{\\pi\}^\{\\star\}\(\\cdot\{\\,\\big\|\\,\}\{\\mathbf\{H\}\}\)maximizes the ergodic sum\-rate subject to a per\-user minimum\-rate \(QoS\) requirementfmin=0\.6f\_\{\\min\}=0\.6bits/s/Hz, i\.e\.,

𝝅⋆​\(𝐇\)∈argmax𝝅\\displaystyle\\boldsymbol\{\\pi\}^\{\\star\}\(\{\\mathbf\{H\}\}\)\\;\\in\\;\\operatornamewithlimits\{argmax\}\_\{\\boldsymbol\{\\pi\}\}\\quad𝟏N⊤​𝐟​\(𝝅,𝐇\)\\displaystyle\\mathbf\{1\}\_\{N\}^\{\\top\}\\,\{\\mathbf\{f\}\}\(\\boldsymbol\{\\pi\},\{\\mathbf\{H\}\}\)\(28a\)subject to𝐟​\(𝝅,𝐇\)≥fmin​1N\.\\displaystyle\{\\mathbf\{f\}\}\(\\boldsymbol\{\\pi\},\{\\mathbf\{H\}\}\)\\geq f\_\{\\min\}\\,\\mathbf\{1\}\_\{N\}\.\(28b\)
The diffusion targets are produced by an*expert primal–dual \(PD\) algorithm*that solves \([28](https://arxiv.org/html/2607.06833#S5.E28)\) via Lagrangian dual \(sub\)gradient ascent\. We train a separate expert per density group, parametrized by a shallow GNN, across all networks\. We then collect200200allocations per training network from its converged primal iterates, giving1600016000conditional samples that the diffusion model \(U\-GNN\) learns to reproduce given\(𝐒,𝐮\)\(\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)\. Running the same expert across all splits, including the validation and test networks held out from U\-GNN, yields the reference allocations against which we benchmark the U\-GNN policy\. Since the expert solves each network directly rather than generalizing, it is a near\-ideal reference that U\-GNN can only approach, so the comparison favors the expert by design\. At inference, U\-GNN thus samples directly from this*distribution*of near\-optimal, feasible allocations rather than re\-solving the optimization per network\. We refer to\[[37](https://arxiv.org/html/2607.06833#bib.bib45)\]for the expert’s algorithmic details\.

We represent𝐇\{\\mathbf\{H\}\}as a directed, weighted channel graph onNNnodes, one per tx–rx pair\. A self\-loop carries the direct\-link gainhi​ih\_\{ii\}, and an edge carries the interference gainhi​jh\_\{ij\},i≠ji\\neq j, which is generally asymmetric, i\.e\.,hi​j≠hj​ih\_\{ij\}\\neq h\_\{ji\}\. We sparsify this graph into the GSO𝐒\{\\mathbf\{S\}\}by retaining, for each receiver, its self\-loop and the incoming edges from them=10m=10strongest interferers, followed by a log\-normalization that compresses the wide dynamic range of the channel gains\. We also derive a two\-feature node\-state vector𝐮\{\\mathbf\{u\}\}that collects each node’s direct\-link strength and its aggregate incoming interference under full\-power transmission\. Both𝐒\{\\mathbf\{S\}\}and𝐮\{\\mathbf\{u\}\}are deterministic functions of𝐇\{\\mathbf\{H\}\}, so they form the conditioning input𝝃=\(𝐒,𝐮\)\\boldsymbol\{\\xi\}=\(\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)of Section[II](https://arxiv.org/html/2607.06833#S2), andq𝐇q\_\{\{\\mathbf\{H\}\}\}induces its distributionq𝝃q\_\{\\boldsymbol\{\\xi\}\}\.

The diffusion variable is the rescaled allocation𝐱0=𝐩/Pmax−12∈\[−12,12\]N\{\\mathbf\{x\}\}\_\{0\}=\{\\mathbf\{p\}\}/P\_\{\\max\}\-\\tfrac\{1\}\{2\}\\in\[\-\\tfrac\{1\}\{2\},\\tfrac\{1\}\{2\}\]^\{N\}, viewed as a scalar graph signal on𝐒\{\\mathbf\{S\}\}\. We realize the U\-GNN policy, which is trained to imitate the optimal policy𝝅⋆\\boldsymbol\{\\pi\}^\{\\star\}, through the conditional diffusion generatorp𝜽⋆\(⋅\|𝐒,𝐮\)p\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\\cdot\\,\|\\,\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)of Section[II](https://arxiv.org/html/2607.06833#S2)\. That is, given𝐇\{\\mathbf\{H\}\}, we sample a batch ofS=100S=100rescaled allocations𝐱0∼p𝜽⋆\(⋅\|𝐒,𝐮\)\{\\mathbf\{x\}\}\_\{0\}\\sim p\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\\cdot\\,\|\\,\{\\mathbf\{S\}\},\{\\mathbf\{u\}\}\)and map them to power allocations𝐩=clip\[0,Pmax\]​\(Pmax​\(𝐱0\+12\)\)\{\\mathbf\{p\}\}=\\mathrm\{clip\}\_\{\[0,P\_\{\\max\}\]\}\\\!\\big\(P\_\{\\max\}\(\{\\mathbf\{x\}\}\_\{0\}\+\\tfrac\{1\}\{2\}\)\\big\)\. We place these allocations in random order as\{𝐩s\}s=1S\\\{\{\\mathbf\{p\}\}\_\{s\}\\\}\_\{s=1\}^\{S\}and execute them sequentially, with slotsstransmitting𝐩s\{\\mathbf\{p\}\}\_\{s\}over itsT0T\_\{0\}\-realization window\.

We compare the U\-GNN policy against two deterministic baselines that forgo time\-sharing\. For every slotssand network𝐇\{\\mathbf\{H\}\}, full power \(FP\) sets𝐩s=Pmax​𝟏N\{\\mathbf\{p\}\}\_\{s\}=P\_\{\\max\}\\mathbf\{1\}\_\{N\}, and average power \(AP\) is the expert’s per\-network mean allocation, i\.e\.,𝐩s​\(𝐇\)≈𝔼𝐩∼𝝅⋆\(⋅\|𝐇\)​\[𝐩\]\{\\mathbf\{p\}\}\_\{s\}\(\{\\mathbf\{H\}\}\)\\approx\\mathbb\{E\}\_\{\{\\mathbf\{p\}\}\\sim\\boldsymbol\{\\pi\}^\{\\star\}\(\\cdot\\,\|\\,\{\\mathbf\{H\}\}\)\}\\,\\big\[\{\\mathbf\{p\}\}\\big\]\.

![Refer to caption](https://arxiv.org/html/2607.06833v1/x10.png)Figure 5:U\-GNN attains ergodic feasibility through time\-sharing\.We zoom in on an example test network, where each node is a tx–rx pair and edges indicate interference strength\. The top row shows the per\-node normalized transmit power𝐩/Pmax\{\\mathbf\{p\}\}/P\_\{\\max\}\(logarithmic color scale\) for three consecutive slots of the sequential allocation sampled by the U\-GNN policy, alongside the mean allocation𝔼​\[𝐩\]\\mathbb\{E\}\[\{\\mathbf\{p\}\}\]\(rightmost\)\. The bottom row shows the per\-receiver rates for each slot \(color bar centered atfminf\_\{\\min\}\) with the per\-slot feasibility annotated above, and the time\-averaged rate delivered by time\-sharing \(rightmost\)\. No single slot serves all users \(5656–86%86\\%feasible\), yet alternating these allocations across slots mitigates interference and achieves 100% feasibility for this network\. In contrast, the mean allocation rates typically stay infeasible for interference\-limited tx–rx pairs\.Results\.We report in Table[IV](https://arxiv.org/html/2607.06833#S5.T4)the ergodic\-rate performance of the U\-GNN policy, expert \(PD\) policy, and baselines\. The U\-GNN diffusion policy closely matches the expert across the full distribution\. Its 5th\- and 10th\-percentiles \(p55, p1010\) rates reach0\.770\.77and0\.920\.92bits/s/Hz, trailing the expert by only5\.3%5\.3\\%and0\.9%0\.9\\%\. The p55rate already exceeds thefmin=0\.6f\_\{\\min\}=0\.6bits/s/Hz floor, and only the bottom∼2%\{\\sim\}2\\%of users fall slightly short\. U\-GNN’s gap to the expert is largest at p11, at17\.8%17\.8\\%\(0\.550\.55vs\.0\.660\.66bits/s/Hz\), whereas FP and AP baselines trail by87\.9%87\.9\\%and71\.8%71\.8\\%at this percentile, making U\-GNN’s degradation minor by comparison\. FP attains the highest mean rate at3\.143\.14bits/s/Hz while collapsing its p11rate to0\.080\.08bits/s/Hz and lowering feasibility to88\.0%88\.0\\%\. U\-GNN instead sustains a mean rate of2\.812\.81bits/s/Hz, on par with the expert \(2\.842\.84\) and AP \(2\.852\.85\)\. AP matches this mean rate but still leaves7\.3%7\.3\\%of users belowfminf\_\{\\min\}, so collapsing the policy to a deterministic mean allocation is not enough\.

U\-GNN’s value rests jointly on its stochastic formulation and generative sampling\. The stochastic formulation treats the policy as a distribution over allocations, sustaining feasibility through randomization where the deterministic AP fails\. The expert shares this formulation but reaches it through thousands of primal–dual iterations per network, whereas U\-GNN bypasses online optimization and attains comparable performance in a single DDIM sampling pass\.

![Refer to caption](https://arxiv.org/html/2607.06833v1/x11.png)Figure 6:Size\-transferability of the U\-GNN policy\.Ergodic55th\-percentile \(p55\) rate vs\. network size for four interference densities \(tx–rx pairs/km2\) atfmin=0\.6f\_\{\\min\}=0\.6bits/s/Hz \(red dashed line\)\. Diamonds mark the native training size \(400400pairs\), and all other sizes are evaluated without retraining\.Fig\.[5](https://arxiv.org/html/2607.06833#S5.F5)illustrates why the optimal policy must be stochastic and how U\-GNN realizes it through*time\-sharing*\. The optimal \(expert\) policy is multi\-modal, activating a different subset of the mutually interfering tx–rx pairs in each slot\. No single allocation serves all users, yet, alternating them across slots renders the time\-averaged rate𝐟​\(𝝅⋆,𝐇\)\{\\mathbf\{f\}\}\(\\boldsymbol\{\\pi\}^\{\\star\},\{\\mathbf\{H\}\}\)feasible for every receiver\. Crucially, this rate averages the per\-slot rates and must not be conflated with the rate of the single mean allocation𝔼​\[𝐩\]\\mathbb\{E\}\[\{\\mathbf\{p\}\}\], which activates every interferer at once and remains infeasible\. This is exactly the failure mode of the AP baseline in Table[IV](https://arxiv.org/html/2607.06833#S5.T4)\.

Fig\.[6](https://arxiv.org/html/2607.06833#S5.F6)examines the size\-transferability of the U\-GNN policy\. The model is trained only on networks of the native size\|𝒱\|=400\|\{\\mathcal\{V\}\}\|=400tx–rx pairs \(diamonds\) with mixed densities\. It is then evaluated without retraining on networks ranging from200200to16001600pairs at similar densities\. The ergodic55th\-percentile rate stays essentially flat and above thefminf\_\{\\min\}floor across this range for every density\. Denser networks \(11\.911\.9vs\.6\.66\.6pairs/km2\) attain lower tail rates due to stronger interference, but their ordering and stability are preserved across all sizes\. This indicates that the permutation\-equivariant GNN backbone is also scalable\. Similar observations hold for the mean and additional tail rates, which are not shown here\.

## VIConclusion & Future Work

We proposed a denoising diffusion framework for generating graph signals\. We cast this as a denoising process and parametrized it with a U\-GNN, a graph\-domain adaptation of the U\-Net\. At its core, a pooling mechanism formalizes down/up\-sampling as learned node selection via nested selection matrices and zero\-padding\. We demonstrated the framework on wireless resource allocation and stock\-price forecasting\. Future directions include designing graph\-aware and latent diffusion processes, and integrating graph transformers into the U\-GNN backbone for spatio\-temporal tasks\.

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## Appendix AAccelerated DDIM Sampling

The base DDIM sampler \([9](https://arxiv.org/html/2607.06833#S2.E9)\) deferred the specification of its noise scaleσk\\sigma\_\{k\}\. We provide it here, where it also governs the accelerated sampler used in our experiments\. A factorη∈\[0,1\]\\eta\\in\[0,1\]controls the per\-step stochasticity through

σk​\(η\)=η​1−α¯k−11−α¯k​1−α¯kα¯k−1,\\displaystyle\\sigma\_\{k\}\(\\eta\)=\\eta\\sqrt\{\\frac\{1\-\\bar\{\\alpha\}\_\{k\-1\}\}\{1\-\\bar\{\\alpha\}\_\{k\}\}\}\\,\\sqrt\{1\-\\frac\{\\bar\{\\alpha\}\_\{k\}\}\{\\bar\{\\alpha\}\_\{k\-1\}\}\},\(29\)which interpolates between the ancestral DDPM update atη=1\\eta=1and the deterministic DDIM map atη=0\\eta=0\.

Because the DDIM updates in \([9](https://arxiv.org/html/2607.06833#S2.E9)\) preserve the marginals of the forward process, we can subsample the denoising steps to accelerate inference without retraining\. Concretely, we trainϵ𝜽⋆\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}on the full grid ofKKsteps but sample along only an increasing subset\{τ1,…,τK′\}⊂\{1,…,K\}\\\{\\tau\_\{1\},\\ldots,\\tau\_\{K^\{\\prime\}\}\\\}\\subset\\\{1,\\ldots,K\\\}withτK′=K\\tau\_\{K^\{\\prime\}\}=KandK′≪KK^\{\\prime\}\\ll K, and we setτ0≔0\\tau\_\{0\}\\coloneqq 0\. We draw𝐱τK′=𝐱K∼𝒩​\(𝟎,𝐈\)\{\\mathbf\{x\}\}\_\{\\tau\_\{K^\{\\prime\}\}\}=\{\\mathbf\{x\}\}\_\{K\}\\sim\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)and iterate the update \([9](https://arxiv.org/html/2607.06833#S2.E9)\) over this sub\-grid,

𝐱τk−1=α¯τk−1​𝐱^0\+1−α¯τk−1−στk2​ϵ𝜽⋆\+στk​𝐰,\\displaystyle\{\\mathbf\{x\}\}\_\{\\tau\_\{k\-1\}\}\\\!=\\\!\\sqrt\{\\bar\{\\alpha\}\_\{\\tau\_\{k\-1\}\}\}\\;\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}\\\!\+\\\!\\sqrt\{1\-\\bar\{\\alpha\}\_\{\\tau\_\{k\-1\}\}\-\\sigma\_\{\\tau\_\{k\}\}^\{2\}\}\\;\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\\\!\+\\\!\\sigma\_\{\\tau\_\{k\}\}\{\\mathbf\{w\}\},\(30\)fork=K′,…,1k=K^\{\\prime\},\\ldots,1, and𝐰∼𝒩​\(𝟎,𝐈\)\{\\mathbf\{w\}\}\\sim\\mathcal\{N\}\(\{\\mathbf\{0\}\},\{\\mathbf\{I\}\}\)\. We note that both𝐱^0=𝐱^0​\(𝐱τk,τk;𝝃\)\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}=\\widehat\{\{\\mathbf\{x\}\}\}\_\{0\}\(\{\\mathbf\{x\}\}\_\{\\tau\_\{k\}\},\\tau\_\{k\};\\boldsymbol\{\\xi\}\)\[cf\. \([7](https://arxiv.org/html/2607.06833#S2.E7)\)\] andϵ𝜽⋆=ϵ𝜽⋆​\(𝐱τk,τk;𝝃\)\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}=\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}^\{\\star\}\}\(\{\\mathbf\{x\}\}\_\{\\tau\_\{k\}\},\\tau\_\{k\};\\boldsymbol\{\\xi\}\)are evaluated at the current stepτk\\tau\_\{k\}\. The noise scaleστk\\sigma\_\{\\tau\_\{k\}\}follows from \([29](https://arxiv.org/html/2607.06833#A1.E29)\) under the substitution\(k,k−1\)↦\(τk,τk−1\)\(k,k\{\-\}1\)\\mapsto\(\\tau\_\{k\},\\tau\_\{k\-1\}\)\.

## Appendix BImplementation of Strided Graph Convolutions

A literal evaluation of the strided graph convolution in \([14](https://arxiv.org/html/2607.06833#S3.E14)\) would form the operator𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}and accumulate its powers\(𝐒γ\)k\(\{\\mathbf\{S\}\}^\{\\gamma\}\)^\{k\},k=0,…,Kk=0,\\ldots,K\. Forγ\>1\\gamma\>1, this densifies𝐒\{\\mathbf\{S\}\}and forfeits its sparsity, much like the reduced GSOs𝐒\(k\)\{\\mathbf\{S\}\}^\{\(k\)\}of \([15](https://arxiv.org/html/2607.06833#S3.E15)\) do\. We instead exploit\(𝐒γ\)k=𝐒γ​k\(\{\\mathbf\{S\}\}^\{\\gamma\}\)^\{k\}=\{\\mathbf\{S\}\}^\{\\gamma k\}and evaluate the layer throughγ​K\\gamma Ksuccessive sparse products with the unit shift𝐒\{\\mathbf\{S\}\}, tapping the signal everyγ\\gammashifts\.

Starting from the lifted signal𝐙~0=𝐃⊤​𝐙ℓ−1\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{0\}=\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{Z\}\}\_\{\\ell\-1\}, we form𝐙~j=𝐒​𝐙~j−1\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\}=\{\\mathbf\{S\}\}\\,\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\-1\}forj=1,…,γ​Kj=1,\\ldots,\\gamma Kand accumulate the tapped terms𝐙~γ​k​𝚯ℓ,k\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{\\gamma k\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,k\},k=0,…,Kk=0,\\ldots,K\. We then apply the pointwise nonlinearityσℓ\\sigma\_\{\\ell\}and the pooling𝐃\{\\mathbf\{D\}\}\. The untapped shifts,j∉\{0,γ,…,γ​K\}j\\notin\\\{0,\\gamma,\\ldots,\\gamma K\\\}, only advance the walk to the next tapped hop\. Algorithm[1](https://arxiv.org/html/2607.06833#alg1)summarizes the procedure\.

Algorithm 1Strided graph convolutions via iterated unit shifts1:Input signal

𝐙ℓ−1∈ℝN′×Fℓ−1\{\\mathbf\{Z\}\}\_\{\\ell\-1\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{\\ell\-1\}\}; shift

𝐒\{\\mathbf\{S\}\}; selection

𝐃\{\\mathbf\{D\}\}; taps

\{𝚯ℓ,k\}k=0K\\\{\\boldsymbol\{\\Theta\}\_\{\\ell,k\}\\\}\_\{k=0\}^\{K\}; stride

γ\\gamma; pointwise nonlinearity

σℓ\\sigma\_\{\\ell\}
2:Output signal

𝐙ℓ∈ℝN′×Fℓ\{\\mathbf\{Z\}\}\_\{\\ell\}\\in\{\\mathbb\{R\}\}^\{N^\{\\prime\}\\times F\_\{\\ell\}\}
3:

𝐙~0←𝐃⊤​𝐙ℓ−1\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{0\}\\leftarrow\{\\mathbf\{D\}\}^\{\\top\}\{\\mathbf\{Z\}\}\_\{\\ell\-1\}⊳\\trianglerightlift to the full vertex set

4:

𝐀←𝐙~0​𝚯ℓ,0\{\\mathbf\{A\}\}\\leftarrow\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{0\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,0\}⊳\\trianglerighttapk=0k=0

5:for

j=1,…,γ​Kj=1,\\ldots,\\gamma Kdo

6:

𝐙~j←𝐒​𝐙~j−1\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\}\\leftarrow\{\\mathbf\{S\}\}\\,\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\-1\}⊳\\trianglerightone sparse shift; a singleN×FN\\times Fbuffer suffices

7:if

γ∣j\\gamma\\mid jthen

8:

𝐀←𝐀\+𝐙~j​𝚯ℓ,j/γ\{\\mathbf\{A\}\}\\leftarrow\{\\mathbf\{A\}\}\+\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\}\\,\\boldsymbol\{\\Theta\}\_\{\\ell,\\,j/\\gamma\}⊳\\trianglerighttap everyγ\\gammashifts

9:endif

10:endfor

11:

𝐙ℓ←𝐃​σℓ​\(𝐀\)\{\\mathbf\{Z\}\}\_\{\\ell\}\\leftarrow\{\\mathbf\{D\}\}\\,\\sigma\_\{\\ell\}\(\{\\mathbf\{A\}\}\)⊳\\trianglerightnonlinearity, then reduce toΩ\\Omega

12:return

𝐙ℓ\{\\mathbf\{Z\}\}\_\{\\ell\}

Notably, we never materialize the reduced GSOs\. The shift𝐒\{\\mathbf\{S\}\}is a sparse weighted aggregation, while the unpooling𝐃⊤\{\\mathbf\{D\}\}^\{\\top\}and pooling𝐃\{\\mathbf\{D\}\}are index scatter and gather operations\. Only the running buffer𝐙~j\\widetilde\{\{\\mathbf\{Z\}\}\}\_\{j\}and one accumulator are kept, each of sizeN×FN\\times F, so each layer runs in roughly𝒪​\(γ​K​\|ℰ\|​F\)\\mathcal\{O\}\(\\gamma K\\,\|\{\\mathcal\{E\}\}\|\\,F\)aggregation time and𝒪​\(N​F\)\\mathcal\{O\}\(NF\)memory, where\|ℰ\|\|\{\\mathcal\{E\}\}\|is the number of edges of𝒢\{\\mathcal\{G\}\}\. Forming the dense𝐒γ\{\\mathbf\{S\}\}^\{\\gamma\}explicitly would instead cost𝒪​\(N2\)\\mathcal\{O\}\(N^\{2\}\)memory\. The filter keepsK\+1K\+1taps for anyγ\\gamma, so the stride adds shifts but no parameters\.

The stride enters only as a multiplier on the shift count and is tied to the pooling ratio\. When the GNN module sits in a depth\-bbblock of the U\-GNN, it instantiates \([14](https://arxiv.org/html/2607.06833#S3.E14)\) with𝐃=𝐃b\{\\mathbf\{D\}\}=\{\\mathbf\{D\}\}\_\{b\}, and all of its layers share the strideγb=min⁡\{⌊ρ¯b⌋,γmax\}\\gamma\_\{b\}=\\min\\\{\\lfloor\\sqrt\{\\bar\{\\rho\}\_\{b\}\}\\,\\rfloor,\\,\\gamma\_\{\\max\}\\\}, whereρ¯b=∏i=1bρi=N/Nb\\bar\{\\rho\}\_\{b\}=\\prod\_\{i=1\}^\{b\}\\rho\_\{i\}=N/N\_\{b\}is the cumulative down\-sampling factor \[cf\. \([18](https://arxiv.org/html/2607.06833#S4.E18)\)\]\. This is the ruleγ=⌊ρ⌋\\gamma=\\lfloor\\sqrt\{\\rho\}\\,\\rfloorof Remark[1](https://arxiv.org/html/2607.06833#Thmremark1)applied at depthbbwithρ=ρ¯b\\rho=\\bar\{\\rho\}\_\{b\}and capped atγmax\\gamma\_\{\\max\}, so the total reachγb​K\\gamma\_\{b\}Kdoes not span the whole graph and we avoid oversmoothing\.

## Appendix CStraight\-Through Estimation and Stochastic Top\-K Selection

To train the score head𝚿b\\boldsymbol\{\\Psi\}\_\{b\}by backpropagation, we apply a straight\-through estimator \(STE\)\[[1](https://arxiv.org/html/2607.06833#bib.bib78)\]to the binary selection mask\. That is, the forward pass selects a hard mask, while gradients reach𝐯b\{\\mathbf\{v\}\}\_\{b\}through a differentiable sigmoid surrogate\. Moreover, we employ a Gumbel\-perturbed Top\-K rule to encourage exploration during training\.

Stochastic selection\.Let𝐡b∈\{0,1\}Nb\{\\mathbf\{h\}\}\_\{b\}\\in\\\{0,1\\\}^\{N\_\{b\}\}be the Top\-K indicator, with\[𝐡b\]n=𝟙​\{n∈𝒮b\}\[\{\\mathbf\{h\}\}\_\{b\}\]\_\{n\}=\\mathbbm\{1\}\\\{n\\in\\mathcal\{S\}\_\{b\}\\\}\. At inference,𝒮b\\mathcal\{S\}\_\{b\}is the deterministic Top\-K of \([25](https://arxiv.org/html/2607.06833#S4.E25)\)\. During training, we draw𝒮b\\mathcal\{S\}\_\{b\}stochastically to encourage exploration of the discrete selection\. Given i\.i\.d\. standard Gumbel variates\[𝐠b\]n=−log⁡\(−log⁡un\)\[\{\\mathbf\{g\}\}\_\{b\}\]\_\{n\}=\-\\log\(\-\\log u\_\{n\}\),un∼Uniform​\(0,1\)u\_\{n\}\\sim\\mathrm\{Uniform\}\(0,1\), and an exploration scaleε∈ℝ\+\\varepsilon\\in\{\\mathbb\{R\}\}\_\{\+\}, we perturb the scores before ranking,

𝒮b=TopK​\(𝐯b\+ε​𝐠b,Nb\+1\)\.\\displaystyle\\mathcal\{S\}\_\{b\}=\\mathrm\{TopK\}\(\{\\mathbf\{v\}\}\_\{b\}\+\\varepsilon\\,\{\\mathbf\{g\}\}\_\{b\},\\,N\_\{b\+1\}\)\.\(31\)Note that Top\-K is invariant to positive scaling, i\.e\.,TopK​\(𝐯b\+ε​𝐠b,Nb\+1\)=TopK​\(𝐯b/ε\+𝐠b,Nb\+1\)\\mathrm\{TopK\}\(\{\\mathbf\{v\}\}\_\{b\}\+\\varepsilon\{\\mathbf\{g\}\}\_\{b\},\\,N\_\{b\+1\}\)=\\mathrm\{TopK\}\(\{\\mathbf\{v\}\}\_\{b\}/\\varepsilon\+\{\\mathbf\{g\}\}\_\{b\},\\,N\_\{b\+1\}\)\. By the Gumbel\-Top\-K trick\[[19](https://arxiv.org/html/2607.06833#bib.bib79)\], \([31](https://arxiv.org/html/2607.06833#A3.E31)\) therefore draws𝒮b\\mathcal\{S\}\_\{b\}without replacement with probabilities proportional toe\[𝐯b\]n/εe^\{\[\{\\mathbf\{v\}\}\_\{b\}\]\_\{n\}/\\varepsilon\}\. The scaleε\\varepsilonacts as a sampling temperature, and asε→0\\varepsilon\\to 0, the sample concentrates on the deterministic Top\-K, which we recover at inference when the perturbation vanishes atε=0\\varepsilon=0\.

Straight\-through gradient\.For the backward pass, we form a surrogate from the*unperturbed*scores\. Given a temperatureτ∈ℝ\+\\tau\\in\{\\mathbb\{R\}\}\_\{\+\}and the logistic sigmoidς​\(⋅\)\\varsigma\(\\cdot\), we set𝐡~b=ς​\(𝐯b/τ\)∈\(0,1\)Nb\\widetilde\{\{\\mathbf\{h\}\}\}\_\{b\}=\\varsigma\(\{\\mathbf\{v\}\}\_\{b\}/\\tau\)\\in\(0,1\)^\{N\_\{b\}\}\. The training mask is

𝐦b=𝐡b\+𝐡~b−sg​\(𝐡~b\),\\displaystyle\{\\mathbf\{m\}\}\_\{b\}=\{\\mathbf\{h\}\}\_\{b\}\+\\widetilde\{\{\\mathbf\{h\}\}\}\_\{b\}\-\\mathrm\{sg\}\(\\widetilde\{\{\\mathbf\{h\}\}\}\_\{b\}\),\(32\)wheresg​\(⋅\)\\mathrm\{sg\}\(\\cdot\)is the stop\-gradient operator\. The soft terms cancel in value, thus𝐦b=𝐡b\{\\mathbf\{m\}\}\_\{b\}=\{\\mathbf\{h\}\}\_\{b\}in the forward pass\. In the backward pass,sg​\(𝐡~b\)\\mathrm\{sg\}\(\\tilde\{\{\\mathbf\{h\}\}\}\_\{b\}\)is constant, so the gradient reaches𝐯b\{\\mathbf\{v\}\}\_\{b\}only through𝐡~b\\tilde\{\{\\mathbf\{h\}\}\}\_\{b\}\. We note that because the surrogate uses the unperturbed scores, the Gumbel noise affects only which nodes enter𝒮b\\mathcal\{S\}\_\{b\}, not the surrogate gradient at any node\.

Gated down\-sampling\.We insert this mask into the encoder down\-sampling\. During training, the depth\-\(b\+1\)\(b\{\+\}1\)input𝐕b\+1=𝐂b\+1​𝐙b\{\\mathbf\{V\}\}\_\{b\+1\}=\{\\mathbf\{C\}\}\_\{b\+1\}\\,\{\\mathbf\{Z\}\}\_\{b\}of \([22](https://arxiv.org/html/2607.06833#S4.E22)\) becomes

𝐕b\+1=𝐂b\+1​\(𝐦b⊙𝐙b\),b=1,…,B−1,\\displaystyle\{\\mathbf\{V\}\}\_\{b\+1\}=\{\\mathbf\{C\}\}\_\{b\+1\}\\big\(\{\\mathbf\{m\}\}\_\{b\}\\odot\{\\mathbf\{Z\}\}\_\{b\}\\big\),\\qquad b=1,\\ldots,B\-1,\(33\)where⊙\\odotscales each row of𝐙b\{\\mathbf\{Z\}\}\_\{b\}by the corresponding entry of𝐦b\{\\mathbf\{m\}\}\_\{b\}\. Since𝐦b=𝐡b\{\\mathbf\{m\}\}\_\{b\}=\{\\mathbf\{h\}\}\_\{b\}in the forward pass, the selected rows pass through unchanged, and𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}discards the rest\. At inference, the STE branch is dropped \(𝐦b=𝐡b\{\\mathbf\{m\}\}\_\{b\}=\{\\mathbf\{h\}\}\_\{b\}\), and \([33](https://arxiv.org/html/2607.06833#A3.E33)\) reverts to the plain down\-sampling via𝐂b\+1​𝐙b\{\\mathbf\{C\}\}\_\{b\+1\}\\,\{\\mathbf\{Z\}\}\_\{b\}\.

Schedules\.We anneal bothτ\\tauandε\\varepsilonlinearly over training\. LetttandTTdenote the current and total number of training epochs, withT=5000T=5000as reported in Table[I](https://arxiv.org/html/2607.06833#S5.T1)\. The schedule uses a warm\-up ofTwT\_\{\\mathrm\{w\}\}epochs followed by an anneal ofTaT\_\{\\mathrm\{a\}\}epochs,

τ​\(t\)\\displaystyle\\tau\(t\)=τ0\+r​\(t\)​\(τmin−τ0\),\\displaystyle=\\tau\_\{0\}\+r\(t\)\\,\(\\tau\_\{\\min\}\-\\tau\_\{0\}\),\(34\)ε​\(t\)\\displaystyle\\varepsilon\(t\)=ε0\+r​\(t\)​\(εmin−ε0\),\\displaystyle=\\varepsilon\_\{0\}\+r\(t\)\\,\(\\varepsilon\_\{\\min\}\-\\varepsilon\_\{0\}\),wherer​\(t\)=min⁡\{max⁡\{\(t−Tw\)/Ta,0\},1\}r\(t\)=\\min\\\{\\max\\\{\(t\-T\_\{\\mathrm\{w\}\}\)/T\_\{\\mathrm\{a\}\},\\,0\\\},\\,1\\\}\. We setTw=0\.02​TT\_\{\\mathrm\{w\}\}=0\.02\\,T,Ta=0\.73​TT\_\{\\mathrm\{a\}\}=0\.73\\,T,\(τ0,τmin\)=\(1,0\.5\)\(\\tau\_\{0\},\\tau\_\{\\min\}\)=\(1,0\.5\), and\(ε0,εmin\)=\(1,0\)\(\\varepsilon\_\{0\},\\varepsilon\_\{\\min\}\)=\(1,0\)\. This way, bothε\\varepsilonandτ\\taureach their floors at epochTw\+Ta=0\.75​TT\_\{\\mathrm\{w\}\}\+T\_\{\\mathrm\{a\}\}=0\.75\\,T, after which selection is \(approximately\) deterministic, and the surrogate sharpness is fixed\. These values worked well in our experiments, but we found the results to be robust, as any choice that explores broadly early and becomes deterministic well before training ends gave comparable performance\.

We remark that the STE in \([32](https://arxiv.org/html/2607.06833#A3.E32)\) supplies a surrogate gradient for the binary mask, and not a differentiable relaxation of the hard Top\-K rule\. In the forward pass,𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}removes the unselected rows, so each training iteration passes diffusion\-loss gradients only to the score entries of the nodes in𝒮b\\mathcal\{S\}\_\{b\}\. The head nonetheless learns a global ranking for two reasons\. First, whileε\>0\\varepsilon\>0, the stochastic selection varies𝒮b\\mathcal\{S\}\_\{b\}across iterations, so the gradient reaches a broader set of nodes over training\. Second, the head shares its parameters across all candidate nodes, so each update reshapes the scoring function at every node in subsequent passes\.

![Refer to caption](https://arxiv.org/html/2607.06833v1/x12.png)Figure 7:Block interface of a U\-GNN encoder–decoder pair\.*Encoder \(left\):*The block input𝐕b=𝐂b​𝐙b−1\{\\mathbf\{V\}\}\_\{b\}\{=\}\{\\mathbf\{C\}\}\_\{b\}\\,\{\\mathbf\{Z\}\}\_\{b\-1\}passes through the fusion layer𝚷b\\boldsymbol\{\\Pi\}\_\{b\}and the GNN module𝚽bE\\boldsymbol\{\\Phi\}^\{\\mathrm\{E\}\}\_\{b\}to produce𝐙b\{\\mathbf\{Z\}\}\_\{b\}\. This feature continues along the skip and also feeds the learned selector𝚿b\\boldsymbol\{\\Psi\}\_\{b\}\. A Top\-K readout of the selector scores then yields the selections𝐂b\+1\{\\mathbf\{C\}\}\_\{b\+1\}\.*Junction \(centre\):*The down\-sampled𝐂b\+1​𝐙b\{\\mathbf\{C\}\}\_\{b\+1\}\{\\mathbf\{Z\}\}\_\{b\}descends into the elided deeper levelsb\+1,…,Bb\{\+\}1,\\dots,B\. The returning𝐘b\+1\{\\mathbf\{Y\}\}\_\{b\+1\}is up\-sampled by𝐂b\+1⊤\{\\mathbf\{C\}\}\_\{b\+1\}^\{\\top\}\.*Decoder \(right\):*The skip\-projection𝚷bskip\\boldsymbol\{\\Pi\}^\{\\mathrm\{skip\}\}\_\{b\}combines the up\-sampled signal with the encoder skip𝐙b\{\\mathbf\{Z\}\}\_\{b\}\. The result passes through𝚷b\\boldsymbol\{\\Pi\}\_\{b\}and the GNN module𝚽bD\\boldsymbol\{\\Phi\}^\{\\mathrm\{D\}\}\_\{b\}, producing𝐘b\{\\mathbf\{Y\}\}\_\{b\}for depthb−1b\{\-\}1\. In every block, the global embeddings\[𝐔0;𝐊0\]\[\{\\mathbf\{U\}\}\_\{0\};\{\\mathbf\{K\}\}\_\{0\}\], which are restricted to the depth\-bbactive nodes as𝐃b​\[𝐔0;𝐊0\]\{\\mathbf\{D\}\}\_\{b\}\[\{\\mathbf\{U\}\}\_\{0\};\{\\mathbf\{K\}\}\_\{0\}\], enter𝚷b\\boldsymbol\{\\Pi\}\_\{b\}and𝚿b\\boldsymbol\{\\Psi\}\_\{b\}\. The composite𝐃b\{\\mathbf\{D\}\}\_\{b\}also parametrizes the GNN modules𝚽bE\\boldsymbol\{\\Phi\}^\{\\mathrm\{E\}\}\_\{b\}and𝚽bD\\boldsymbol\{\\Phi\}^\{\\mathrm\{D\}\}\_\{b\}, which define graph convolutions over the GSO𝐒\{\\mathbf\{S\}\}\.
## Appendix DClassical Sum/Max\-Pooling over Graph Neighborhoods

CNNs with pooling couple neighborhood aggregation and spatial down\-sampling into a single operation that summarizes each pixel window, e\.g\., by its average or maximum, at a reduced resolution\. This coupling exploits the intrinsic geometry of the regular grid\. For a general graph, the GSO𝐒\{\\mathbf\{S\}\}supplies the analogous structural information, which we use to define pooling neighborhoods\.

Our design decouples these two operations\. The strided graph convolution in \([14](https://arxiv.org/html/2607.06833#S3.E14)\) already returns a nonlinear summary of each node’s neighborhood\. In reduced indices, this is theKK\-hop reach of\{𝐒\(k\)\}k=0K\\\{\{\\mathbf\{S\}\}^\{\(k\)\}\\\}\_\{k=0\}^\{K\}, and in the lifted domain, it is theγ​K\\gamma K\-hop reach of𝐒\{\\mathbf\{S\}\}after𝐃b⊤\{\\mathbf\{D\}\}\_\{b\}^\{\\top\}lifts the signal and before𝐃b\{\\mathbf\{D\}\}\_\{b\}reduces it back \[cf\. \([15](https://arxiv.org/html/2607.06833#S3.E15)\)\]\. The learned selectors𝐂b\{\\mathbf\{C\}\}\_\{b\}in turn perform the node\-space down\-sampling \[cf\. \([25](https://arxiv.org/html/2607.06833#S4.E25)\)\]\. An explicit sum\- or max\-pooling over graph neighborhoods is therefore optional rather than essential\. We describe it next for completeness\.

Pooling over the full graph\.For a given𝐒\{\\mathbf\{S\}\}and pooling rangeκ∈ℤ\+\\kappa\\in\\mathbb\{Z\}^\{\+\}, we define the binary*reachability matrix*

\[𝐑​\(𝐒\)\]n,m≔𝟙​\{\[𝐒k\]n,m≠0​for some​0≤k≤κ\}\.\\displaystyle\\big\[\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)\\big\]\_\{n,m\}\\coloneqq\\mathbbm\{1\}\\Big\\\{\\big\[\{\\mathbf\{S\}\}^\{k\}\\big\]\_\{n,m\}\\neq 0\\ \\text\{for some\}\\ 0\\leq k\\leq\\kappa\\Big\\\}\.\(35\)Note that each node reaches itself\. From𝐑=𝐑​\(𝐒\)\{\\mathbf\{R\}\}=\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)we read off, for each noden∈𝒱n\\in\{\\mathcal\{V\}\}, the index set

𝐧​\(n\)≔\{m∈𝒱:\[𝐑\]n,m=1\}\\displaystyle\{\\mathbf\{n\}\}\(n\)\\coloneqq\\left\\\{m\\in\{\\mathcal\{V\}\}\\colon\\big\[\{\\mathbf\{R\}\}\\big\]\_\{n,m\}=1\\right\\\}\(36\)of nodes aggregated atnn\. Letφ:ℝ⋅⁣×F→ℝF\\varphi\\colon\{\\mathbb\{R\}\}^\{\\cdot\\times F\}\\to\{\\mathbb\{R\}\}^\{F\}be a pooling map that is invariant to permutations of its rows \(the pooled nodes\) and acts per feature, e\.g\.,\[φsum​\(𝐗\)\]f=∑i\[𝐗\]i,f\[\\varphi\_\{\\mathrm\{sum\}\}\(\{\\mathbf\{X\}\}\)\]\_\{f\}=\\sum\_\{i\}\[\{\\mathbf\{X\}\}\]\_\{i,f\}for sum\-pooling and\[φmax\(𝐗\)\]f=maxi\[𝐗\]i,f\[\\varphi\_\{\\max\}\(\{\\mathbf\{X\}\}\)\]\_\{f\}=\\max\_\{i\}\[\{\\mathbf\{X\}\}\]\_\{i,f\}for max\-pooling\. In analogy with the graph\-filter notation𝐇​\(𝐒\)​𝐗\{\\mathbf\{H\}\}\(\{\\mathbf\{S\}\}\)\{\\mathbf\{X\}\}, we write𝐑​\(𝐒\)​𝐗\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)\{\\mathbf\{X\}\}for the pooled signal,

\[𝐑​\(𝐒\)​𝐗\]n=φ​\(\[𝐗\]𝐧​\(n\)\),n∈𝒱\.\\displaystyle\\big\[\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)\\,\{\\mathbf\{X\}\}\\big\]\_\{n\}\\;=\\;\\varphi\\big\(\\big\[\{\\mathbf\{X\}\}\\big\]\_\{\{\\mathbf\{n\}\}\(n\)\}\\big\),\\quad n\\in\{\\mathcal\{V\}\}\.\(37\)Note that for sum\-pooling, this operator form coincides with the matrix product𝐑​\(𝐒\)​𝐗\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)\{\\mathbf\{X\}\}\.

Pooling at reduced resolutions\.Pooling at the resolutions of the U\-GNN follows the reduced\-GSO construction\. We define the node\-reduced counterpart of𝐑\{\\mathbf\{R\}\},

𝐑b​\(𝐒\)=𝐃b​𝐑​\(𝐒\)​𝐃b⊤∈\{0,1\}Nb×Nb,\\displaystyle\{\\mathbf\{R\}\}\_\{b\}\(\{\\mathbf\{S\}\}\)=\{\\mathbf\{D\}\}\_\{b\}\\,\{\\mathbf\{R\}\}\(\{\\mathbf\{S\}\}\)\\,\{\\mathbf\{D\}\}\_\{b\}^\{\\top\}\\in\\\{0,1\\\}^\{N\_\{b\}\\times N\_\{b\}\},\(38\)which intersects theκb\\kappa\_\{b\}\-reach of𝐒\{\\mathbf\{S\}\}with the active setΩb\\Omega\_\{b\}\. Like the reduced GSOs in \([15](https://arxiv.org/html/2607.06833#S3.E15)\),𝐑b\{\\mathbf\{R\}\}\_\{b\}is indexed by the local labels\[Nb\]\[N\_\{b\}\]\. Also, the rangeκb\\kappa\_\{b\}is left implicit at depthbb, just as the strideγb\\gamma\_\{b\}is in \([14](https://arxiv.org/html/2607.06833#S3.E14)\)\. We define the set

𝐧b​\(n\)≔\{m∈\[Nb\]:\[𝐑b\]n,m=1\},\\displaystyle\{\\mathbf\{n\}\}\_\{b\}\(n\)\\coloneqq\\left\\\{m\\in\[N\_\{b\}\]\\colon\\big\[\{\\mathbf\{R\}\}\_\{b\}\\big\]\_\{n,m\}=1\\right\\\},\(39\)which collects the active nodes withinκb\\kappa\_\{b\}hops ofnnon𝐒\{\\mathbf\{S\}\}, includingnnitself\. We then pool a down\-sampled signal𝐙∈ℝNb×F\{\\mathbf\{Z\}\}\\in\{\\mathbb\{R\}\}^\{N\_\{b\}\\times F\}as

\[𝐑b​\(𝐒\)​𝐙\]n=φ​\(\[𝐙\]𝐧b​\(n\)\),n∈\[Nb\]\.\\displaystyle\\big\[\{\\mathbf\{R\}\}\_\{b\}\(\{\\mathbf\{S\}\}\)\\,\{\\mathbf\{Z\}\}\\big\]\_\{n\}=\\varphi\\big\(\\big\[\{\\mathbf\{Z\}\}\\big\]\_\{\{\\mathbf\{n\}\}\_\{b\}\(n\)\}\\big\),\\quad n\\in\[N\_\{b\}\]\.\(40\)In \([40](https://arxiv.org/html/2607.06833#A4.E40)\),𝐑b​\(𝐒\)​𝐙\{\\mathbf\{R\}\}\_\{b\}\(\{\\mathbf\{S\}\}\)\{\\mathbf\{Z\}\}admits a lift–pool–reduce form mirroring that of the strided convolutions\. One lifts𝐙\{\\mathbf\{Z\}\}to the full vertex set, pools over theκb\\kappa\_\{b\}\-hop neighborhoods of𝐒\{\\mathbf\{S\}\}, and reduces back toΩb\\Omega\_\{b\}through𝐃b\{\\mathbf\{D\}\}\_\{b\}\. The lift assigns each inactive node the identity element ofφ\\varphi, so that it leaves the pooled output unchanged\. This element is0for sum\-pooling, for which the plain𝐃b⊤\{\\mathbf\{D\}\}\_\{b\}^\{\\top\}suffices, and−∞\-\\infty\(a large negative constant in practice\) for max\-pooling, supplied by a modified lift𝐃~b⊤\\widetilde\{\{\\mathbf\{D\}\}\}\_\{b\}^\{\\top\}\.

Optional placement in the U\-GNN\.At each encoder depthb=1,…,B−1b=1,\\ldots,B\-1, the encoder output𝐙b\{\\mathbf\{Z\}\}\_\{b\}is first pooled as𝐑b​𝐙b\{\\mathbf\{R\}\}\_\{b\}\{\\mathbf\{Z\}\}\_\{b\}and then down\-sampled by𝐂b\+1​𝐑b​𝐙b\{\\mathbf\{C\}\}\_\{b\+1\}\{\\mathbf\{R\}\}\_\{b\}\{\\mathbf\{Z\}\}\_\{b\}, which feeds the block at depthb\+1b\+1\[cf\. \([19](https://arxiv.org/html/2607.06833#S4.E19)\)\]\. The node selectors use the pooled features as well, so we replace𝐙b\{\\mathbf\{Z\}\}\_\{b\}with𝐑b​𝐙b\{\\mathbf\{R\}\}\_\{b\}\{\\mathbf\{Z\}\}\_\{b\}in \([24](https://arxiv.org/html/2607.06833#S4.E24)\)\. Both stages fold into the definition of the encoding GNN module\. Note that the bottleneck at depthBBperforms no node\-space down\-sampling or pooling\.

The case for explicit pooling is weak in our design since the graph convolutional layers in \([14](https://arxiv.org/html/2607.06833#S3.E14)\) and selector heads in \([24](https://arxiv.org/html/2607.06833#S4.E24)\) already provide nonlinear neighborhood aggregation and learned down\-sampling, respectively\. In our experiments, max\-pooling added computational overhead without a noticeable gain\. Hence, we adopt \([14](https://arxiv.org/html/2607.06833#S3.E14)\) as the default GNN operation\.

## Appendix EU\-GNN Denoiser and Diffusion Configuration

This appendix expands Table[I](https://arxiv.org/html/2607.06833#S5.T1)\. In particular, Appendix[E\.1](https://arxiv.org/html/2607.06833#A5.SS1)details the denoiser, diffusion process, and training shared across the two tasks, and Appendix[E\.2](https://arxiv.org/html/2607.06833#A5.SS2)specifies the signal and conditioning interface particular to each\.

### E\.1Shared Denoiser, Diffusion, and Training

Backbone\.The denoiserϵ𝜽\\boldsymbol\{\\epsilon\}\_\{\\boldsymbol\{\\theta\}\}is a U\-GNN spanningB=4B\{=\}4node resolutions\. Each of its33down\-sampling levels reduces the active node set by a factor22, for an23=8×2^\{3\}\{=\}8\\timesreduction overall and four resolutions ofNN,N/2N/2,N/4N/4, andN/8N/8nodes\. Each reduction is realized by a learned node\-selection pooling operator, made differentiable with a straight\-through estimator \(Appendix[C](https://arxiv.org/html/2607.06833#A3)\)\. To counteract the increasing sparsity of the coarsened graphs, the graph convolutions are strided, capped atγmax=2\\gamma\_\{\\max\}\{=\}2\(Appendix[B](https://arxiv.org/html/2607.06833#A2)\)\. The feature \(channel\) width is uniform at6464channels across all levels\.

Each encoding and decoding level applies22GNN layers of orderK=2K\{=\}2over powers of the augmented adjacency𝐈\+𝐀^\{\\mathbf\{I\}\}\+\\hat\{\{\\mathbf\{A\}\}\}, with channel\-wise layer normalization \(LayerNorm\), a rectified linear unit \(ReLU\) nonlinearity, and dropout0\.10\.1; these layers are strided variants of the topology\-adaptive graph convolution \(TAGConv\), following its[PyGimplementation](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.TAGConv.html)\. A22\-layer bottleneck processes the coarsest summary, and the decoder path restores full resolution while concatenating the encoder skip features at each level\. The output head applies LayerNorm, a sigmoid linear unit \(SiLU\) nonlinearity, and a linear map\. Fig\.[7](https://arxiv.org/html/2607.06833#A3.F7)complements Fig\.[2](https://arxiv.org/html/2607.06833#S4.F2)and shows a detailed schematic of one encoder–decoder block pair at depthbb\.

Conditioning and fusion\.The diffusion stepkkis mapped to a128128\-dimensional sinusoidal embedding followed by a multilayer perceptron \(MLP\), and the node conditioning to a128128\-dimensional representation\. Both are projected to the level width6464at every level, where the signal path and the projected step embedding are concatenated and linearly mapped back to6464channels \(128→64128\{\\to\}64\)\. The conditioning is then merged in a task\-specific way: by concatenation into the same projection for WRA, so that it maps192192channels down to6464over the signal, step, and conditioning; and by two\-head cross\-attention for S&P 500—queries from the signal path, keys and values from the conditioning context—which leaves the projection from128128channels down to6464\.

Diffusion\.The forward process applies a linearβ\\betaschedule from10−410^\{\-4\}to2×10−22\\times 10^\{\-2\}over500500steps\. At inference, we sample using5×5\\times\-accelerated denoising diffusion implicit models \(DDIM\) with100100steps and stochasticity parameterη=0\.2\\eta\{=\}0\.2\(Appendix[A](https://arxiv.org/html/2607.06833#A1)\)\. We draw100100samples per conditioning input to form the predictive ensemble\.

Optimization\.We train for50005000epochs with AdamW \(learning rate10−410^\{\-4\},\(β1,β2\)=\(0\.9,0\.98\)\(\\beta\_\{1\},\\beta\_\{2\}\)\{=\}\(0\.9,0\.98\), weight decay10−410^\{\-4\}\), gradient\-norm clipping at1\.01\.0, and automatic mixed precision \(AMP\)\. The learning rate follows a per\-step cosine schedule with a linear warm\-up over the first1%1\\%of training, then a cosine decay to a floor of5%5\\%of the peak over0\.80\.8of training, after which it is held constant\. The node\-selector temperature and exploration noise are annealed over roughly the first75%75\\%of training, moving the selector from soft selection toward nearly hard selection \(Appendix[C](https://arxiv.org/html/2607.06833#A3)\)\. We retain the best checkpoint under a task\-specific composite validation criterion\. Both models were trained on a single NVIDIA RTX 3090\. The S&P 500 model completed training in≈58\.5\{\\approx\}58\.5GPU hours, and the WRA model did so in≈33\{\\approx\}33GPU hours\.

### E\.2Application\-Specific Knobs

Stock\-price forecasting \(S&P 500\)\.The diffused signal is the future log\-return trajectory𝐱0∈ℝN×Tp\{\\mathbf\{x\}\}\_\{0\}\\in\{\\mathbb\{R\}\}^\{N\\times T\_\{p\}\}over theN=468N\{=\}468stocks\. The pooling cascade coarsens the node set as468→234→117→58468\{\\to\}234\{\\to\}117\{\\to\}58across the four levels\. Because returns are unbounded, the predicted clean signal is left unclipped\.

The conditioning comprises1212market features over the pastThT\_\{h\}\-day window\. A per\-node temporal encoder summarizes this window and projects theThT\_\{h\}history steps onto theTpT\_\{p\}horizon steps, forming the per\-level context that the cross\-attention fusion above consumes\. This encoder interleaves dilated 1\-D convolutions with two\-head temporal self\-attention\. A lightweight in\-block temporal mixer further couples theTpT\_\{p\}horizon steps\. The target channel is additionally normalized by reversible instance normalization \(RevIN\)\[[17](https://arxiv.org/html/2607.06833#bib.bib80)\], using statistics from the conditioning window\. RevIN uses a blend weight of0\.70\.7and a scale correction of1\.1071\.107\.

The model has1\.44​M1\.44\\mathrm\{M\}trainable parameters and is trained on batches of6464sliding windows\. We retain the checkpoint that minimizes a composite of the validation noise loss, its train–validation gap, and continuous ranked probability score \(CRPS\)\. This checkpoint is reached near epoch45004500\.

Wireless resource allocation \(WRA\)\.The diffused signal is the centered allocation𝐱0=𝐩/Pmax−12∈\[−12,12\]N\{\\mathbf\{x\}\}\_\{0\}=\{\\mathbf\{p\}\}/P\_\{\\max\}\-\\tfrac\{1\}\{2\}\\in\[\-\\tfrac\{1\}\{2\},\\tfrac\{1\}\{2\}\]^\{N\}over theN=400N\{=\}400transmitter–receiver \(tx–rx\) pairs\. The pooling cascade coarsens the node set as400→200→100→50400\{\\to\}200\{\\to\}100\{\\to\}50\. The powers are recovered as𝐩=clip\[0,Pmax\]​\(Pmax​\(𝐱0\+12\)\)\{\\mathbf\{p\}\}=\\mathrm\{clip\}\_\{\[0,P\_\{\\max\}\]\}\\\!\\big\(P\_\{\\max\}\(\{\\mathbf\{x\}\}\_\{0\}\+\\tfrac\{1\}\{2\}\)\\big\)\. In contrast to the stock model, the allocation is a single snapshot with no time axis, soT=1T\{=\}1\. Thus, the temporal encoder and in\-block mixer are therefore disabled, and RevIN is not used\.

The22per\-node conditioning features are concatenated into the level fusion described above\. These features are the direct\-link gain and the aggregate cross\-link interference\. Because the support is bounded, the predicted clean signal is clipped to the valid power range at each reverse step\.

Target allocations are supplied by a primal–dual expert\. The expert is a shallow GNN with33layers, hidden width6464, andK=2K\{=\}2hops, trained per density group with dual\-ascent step0\.20\.2\. As supervision for the U\-GNN, we collect200200allocations per training network from its converged primal iterates\. This yields1600016000samples in total\. Running the same expert on the held\-out networks yields the reference allocations used for evaluation\. Theoretical and algorithmic details appear in\[[37](https://arxiv.org/html/2607.06833#bib.bib45)\]\.

The model has0\.63​M0\.63\\mathrm\{M\}trainable parameters and is trained on batches of800800network graph and resource allocation signal pairs\. We retain the checkpoint that minimizes a composite of the mean rate\-feasibility gap and the1%1\\%and5%5\\%rate gaps, averaged over the four density groups\. This checkpoint is reached at epoch16001600\.