HP-JEPA: Hierarchical Partitioning for Multi-Resolution Graph Joint-Embedding Predictive Learning

# Hierarchical Partitioning for Multi-Resolution Graph Joint-Embedding Predictive Learning
Source: [https://arxiv.org/html/2608.00491](https://arxiv.org/html/2608.00491)
,Jingxiang QuDepartment of Computer ScienceStony Brook UniversityStony BrookNYUSA,Wenhan GaoDepartment of Applied Mathematics and StatisticsStony Brook UniversityStony BrookNYUSA,Jiaxing ZhangTikTokBellevueWAUSA,Linsey PangPayPalSan JoseCAUSA,Ravid Shwartz\-ZivCenter for Data ScienceNew York UniversityNew YorkNYUSA,Yann LeCunCenter for Data ScienceNew York UniversityNew YorkNYUSAandYuefan DengDepartment of Applied Mathematics and StatisticsStony Brook UniversityStony BrookNYUSA

###### Abstract\.

Graph self\-supervised learning aims to learn transferable representations from large\-scale unlabeled graph data\. Joint\-embedding predictive architectures \(JEPAs\) avoid explicit negative\-pair construction and raw\-input reconstruction by predicting masked targets directly in latent space\. However, existing graph JEPAs typically rely on a single predefined graph partition, biasing the learned representations toward one structural granularity and limiting their ability to capture complementary patterns at different graph scales\. To address this limitation, we propose HP\-JEPA, a hierarchical partitioning framework for multi\-resolution graph joint\-embedding prediction\. HP\-JEPA organizes each graph into an ordered bank of coarse\-to\-fine partition resolutions and performs context–target latent prediction separately at each resolution using an online encoder, an exponential\-moving\-average target encoder, and a latent predictor\. The resulting resolution\-specific graph representations are subsequently integrated through concatenation or task\-specific resolution weighting, allowing downstream models to combine complementary local, regional, and global structural information\. Experiments on seven graph classification benchmarks and one graph regression benchmark show that HP\-JEPA outperforms the fixed\-resolution Graph\-JEPA baseline on 6 of 8 tasks, improving upon Graph\-JEPA on most evaluated benchmarks\. Size\-stratified analyses further show that HP\-JEPA achieves higher accuracy than Graph\-JEPA in most evaluated graph\-size quartiles on three representative datasets\. These results highlight the effectiveness of hierarchical multi\-resolution partitioning for transferable graph representation learning\.

Graph representation learning, self\-supervised learning, joint\-embedding predictive architecture, graph neural networks

## 1\.Introduction

![Refer to caption](https://arxiv.org/html/2608.00491v1/x1.png)Figure 1\.Motivation and overview of HP\-JEPA\.Real\-world graphs contain complementary structural information at multiple resolutions, including fine\-resolution local motifs, intermediate\-resolution regional context, and coarse\-resolution global topology\. A single fixed resolution can be overly fine, fragmenting global context, or overly coarse, obscuring local structure\. HP\-JEPA constructs graph\-region tokens at multiple resolutions and integrates their resolution\-specific representations\. Task\-specific resolution weighting allows downstream models to emphasize the most informative resolutions\.Graph representation learning aims to encode graph\-structured data into transferable representations that support downstream classification and regression tasks\(Hamilton et al\.,[2017b](https://arxiv.org/html/2608.00491#bib.bib6); Wu et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib22); Xu et al\.,[2019](https://arxiv.org/html/2608.00491#bib.bib24)\)\. Because task\-specific annotations are often limited or expensive to obtain, self\-supervised learning has become an important paradigm for learning from large collections of unlabeled graphs\. Existing graph self\-supervised methods primarily construct their training signals through contrastive learning\(Sun et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib16); You et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib25); Hassani and Khasahmadi,[2020](https://arxiv.org/html/2608.00491#bib.bib7); Suresh et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib17)\), generative architectures\(Hou et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib8); Tan et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib18)\), self\-predictive\(Thakoor et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib19); Xie et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib23)\)or joint\-embedding predictive architectures \(JEPAs\)\(Assran et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib2); Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)\. Contrastive methods depend heavily on graph augmentations and, in many cases, negative\-pair construction\(Veličković et al\.,[2019](https://arxiv.org/html/2608.00491#bib.bib20); Qiu et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib14); Zhang et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib26); Huang et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib9)\), while generative architectures optimize the reconstruction of raw node attributes or graph structures\. Self\-predictive methods try to predict the unobserved components of the graph, and supervise this process on original data space\. JEPAs instead predict held\-out targets directly in representation space, avoiding both explicit negative pairs and raw\-input reconstruction\(Bardes et al\.,[2024](https://arxiv.org/html/2608.00491#bib.bib3); Lei et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib11)\)\. This formulation offers a promising route toward learning high\-level structural and semantic dependencies in graphs\.

Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)extends joint\-embedding prediction to graph\-level representation learning by partitioning each graph into a collection of regions or patches\. During pretraining, visible context regions are encoded by an online encoder, while held\-out target regions are represented by an exponential\-moving\-average target encoder\. A predictor then estimates the latent target representations from the visible context and target\-specific structural queries\. This formulation successfully transfers JEPA\-style learning to graph\-structured data without reconstructing raw node or edge information\. However, the graph is tokenized using a single predefined partition resolution throughout pretraining\. Consequently, the predictive objective is restricted to dependencies expressed at one structural granularity, and the resulting representation may become specialized to the particular resolution induced by that partition\.

The fixed\-resolution assumption is restrictive because graph semantics are inherently multi\-granular\(Wen et al\.,[2024](https://arxiv.org/html/2608.00491#bib.bib21); Zhao et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib27); Perozzi et al\.,[2017](https://arxiv.org/html/2608.00491#bib.bib13)\)\. Fine\-grained partitions preserve localized motifs and short\-range interactions, but may fragment regional organization and long\-range structural dependencies\. Conversely, coarse partitions capture broader graph topology and regional context, but can blur discriminative local patterns\. The most informative granularity may also vary across graphs of different sizes and across downstream tasks: a resolution suitable for identifying local biochemical motifs, for example, may not be optimal for characterizing global interaction patterns in social graphs\. Therefore, no single partition resolution can be expected to preserve all task\-relevant information or generalize uniformly across heterogeneous graph resolutions\. These observations motivate graph representation learning over an ordered hierarchy of partition resolutions rather than a single fixed partition, as shown in Fig\.[1](https://arxiv.org/html/2608.00491#S1.F1)\.

To address this limitation, we proposeHP\-JEPA, a hierarchical partitioning framework for multi\-resolution graph joint\-embedding prediction\. HP\-JEPA represents each graph using an ordered bank of coarse\-to\-fine partition resolutions\. Here,*hierarchical*refers to the ordering of structural granularities: partitions at different resolutions are constructed from the full graph\. At each active resolution, graph regions are encoded as patch\-content tokens together with structural queries that describe their positions and contexts in the original graph\. HP\-JEPA then performs resolution\-wise context–target prediction using a shared online encoder, an exponential\-moving\-average target encoder, and a latent predictor\. By applying the predictive objective separately across resolutions, HP\-JEPA exposes the representation learner to structural dependencies ranging from fine local patterns to coarse graph\-level organization, while retaining the JEPA advantages of label\-free latent prediction without negative pairs or raw graph reconstruction\.

After self\-supervised pretraining, HP\-JEPA pools the valid region\-token representations within each resolution to obtain a set of resolution\-specific graph embeddings\. These embeddings can be concatenated to preserve information from all configured resolutions or combined through a smoothed, task\-specific weighting mechanism that learns the relative importance of each resolution while keeping the pretrained encoder frozen\. This design separates label\-free multi\-resolution representation learning from lightweight task\-specific resolution selection\. The PROTEINS case study further shows that HP\-JEPA produces a visually more organized embedding with clearer separation across graph\-size quartiles and class labels\. Together, these results indicate that learning and integrating representations across partition resolutions provides a more flexible alternative to fixed\-resolution graph joint\-embedding prediction\.

Our main contributions are summarized as follows:

- •We identify the fixed\-resolution bias in existing graph joint\-embedding predictive architectures and formulate multi\-resolution graph tokenization as a key requirement for learning representations across heterogeneous graph granularities\.
- •We propose HP\-JEPA, which combines an ordered hierarchy of graph partition resolutions, resolution\-wise latent context–target prediction, and multi\-resolution readout with optional task\-specific resolution weighting\.
- •We evaluate HP\-JEPA through downstream graph classification and regression, a size\-stratified representation case study, and sensitivity analyses over graph\-size quartiles and resolution\-bank configurations\. HP\-JEPA improves over Graph\-JEPA on six of eight tasks and maintains its advantage in most evaluated graph\-size quartiles and resolution\-bank configurations\.

## 2\.Related Work

Table 1\.Design\-level comparison of graph representation learning methods\. The check marks \(✓\) and cross marks \(✗\) indicate that the property is explicitly satisfied/unsatisfied by design, and a triangle \(△\\triangle\) indicates a partial match\.Notes\.\(1\) “No Neg\.” means the objective does not require explicit negative pairs\. \(2\) “No Raw Recon\.” means the method does not reconstruct raw node attributes, edges, or graph structure\. \(3\) “Latent Pred\.” means the target is a representation\-space target\. \(4\) “Region Tokens” means graph regions, subgraphs, or patches are explicit learning units\. \(5\) “Multi\-Resolution Tokens” means that graph\-region tokens are explicitly constructed at multiple partition resolutions\. \(6\) For F\-GIN, check marks under “No Neg\.” and “No Raw Recon\.” only indicate that these mechanisms are absent; F\-GIN is still supervised\. AD\-GCL\-FIX and AD\-GCL\-OPT are merged because they share the same method design\.

Graph neural networks provide the standard encoder backbone for graph representation learning by aggregating information over graph neighborhoods\. Representative message\-passing models include GCN, GraphSAGE, MPNN, and GIN\(Kipf and Welling,[2017](https://arxiv.org/html/2608.00491#bib.bib10); Hamilton et al\.,[2017a](https://arxiv.org/html/2608.00491#bib.bib5); Gilmer et al\.,[2017](https://arxiv.org/html/2608.00491#bib.bib4); Xu et al\.,[2019](https://arxiv.org/html/2608.00491#bib.bib24)\)\. Based on these encoders, contrastive and mutual\-information\-based graph self\-supervised methods learn representations by comparing graph views or local–global summaries\. InfoGraph maximizes mutual information between graph\-level and substructure representations\(Sun et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib16)\), GraphCL contrasts augmented graph views\(You et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib25)\), MVGRL contrasts first\-order and diffusion views\(Hassani and Khasahmadi,[2020](https://arxiv.org/html/2608.00491#bib.bib7)\), and AD\-GCL learns adversarial graph augmentations\(Suresh et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib17)\)\. These methods improve graph\-level representation learning, but their supervision is mainly defined by view agreement or augmentation design rather than latent prediction over graph\-region structures\.HP\-JEPAaddresses this gap by learning from context–target prediction in graph\-region latent space instead of relying on a fixed contrastive view construction\.

Masked graph autoencoding provides another line of graph self\-supervised learning\. GraphMAE masks node attributes and reconstructs masked features\(Hou et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib8)\), while S2GAE masks graph structure and reconstructs missing edges\(Tan et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib18)\)\. These methods avoid explicit negative pairs, but their objectives are still tied to reconstructing raw graph observations\. Non\-contrastive methods move closer to representation\-space learning: BGRL predicts target representations from another augmented view without negative samples\(Thakoor et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib19)\), and LaGraph formulates self\-supervised learning as latent graph prediction\(Xie et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib23)\)\. However, these methods do not explicitly organize graph\-region targets across a hierarchy of structural granularities\.HP\-JEPAaddresses this limitation by predicting latent targets without reconstructing node attributes or edges, while using graph regions at multiple partition resolutions as the learning units\.

Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)is the closest prior method to our work because it applies JEPA\-style context–target prediction to graph patches\. It partitions a graph into subgraphs and predicts latent target\-patch representations from context\-patch representations, avoiding raw reconstruction\. However, Graph\-JEPA exposes only a single patch\-token resolution to the predictive objective\. This fixed\-resolution design can miss dependencies that appear across local, mesoscopic, and coarse graph structures\.HP\-JEPAextends Graph\-JEPA by constructing graph tokens over an ordered bank of partition resolutions, performing latent prediction independently within each resolution, and integrating the resulting resolution\-specific representations during downstream readout\. Here, the coarse\-to\-fine resolution hierarchy refers to the ordering of structural granularities rather than nested or parent–child partitions\. The difference betweenHP\-JEPAand existing methods is summarized in Table[1](https://arxiv.org/html/2608.00491#S2.T1)\.

## 3\.Preliminaries

This section establishes the notation and background used by HP\-JEPA\. We first formulate graph representation learning for an attributed graphG=\(V,E,𝐗\)G=\(V,E,\\mathbf\{X\}\)and then review a single\-resolution graph JEPA formulation\. In this formulation, the graph is decomposed into a collection of regions, one context region and a set of target regions are sampled, and the target latent states are predicted using an online token encoderfθf\_\{\\theta\}, an exponential\-moving\-average \(EMA\) target token encoderfθ¯f\_\{\\bar\{\\theta\}\}, and a predictorqψq\_\{\\psi\}\. Section[4](https://arxiv.org/html/2608.00491#S4)extends this formulation toL=L𝒟L=L\_\{\\mathcal\{D\}\}independently constructed graph resolutions, performs prediction separately within each resolution, maps each target\-token state to a two\-coordinate latent target, and combines the resulting resolution\-specific graph representations\{hG\(ℓ\)\}ℓ=1L\\\{h\_\{G\}^\{\(\\ell\)\}\\\}\_\{\\ell=1\}^\{L\}\.

### 3\.1\.Graph Representation Learning

LetG=\(V,E,𝐗\)G=\(V,E,\\mathbf\{X\}\)denote an attributed graph, whereVVandEEare the node and edge sets,𝐗∈ℝn×dx\\mathbf\{X\}\\in\\mathbb\{R\}^\{n\\times d\_\{x\}\}is the node\-feature matrix, andn=\|V\|n=\|V\|\. Edge attributes, when available, are treated as part of the graph input but omitted from the notation for simplicity\. We consider a graph dataset𝒟=\{Gi\}i=1N𝒟\\mathcal\{D\}=\\\{G\_\{i\}\\\}\_\{i=1\}^\{N\_\{\\mathcal\{D\}\}\}sampled from an underlying graph distributionp𝒢p\_\{\\mathcal\{G\}\}\.

Graph representation learning aims to map each graphGGto a fixed\-dimensional representationhG∈ℝdh\_\{G\}\\in\\mathbb\{R\}^\{d\}that summarizes its structural and semantic information\. A typical graph encoderℰϑ\\mathcal\{E\}\_\{\\vartheta\}first produces node representations\{rv∈ℝd:v∈V\}\\\{r\_\{v\}\\in\\mathbb\{R\}^\{d\}:v\\in V\\\}and then applies a permutation\-invariant readout according to\{rv\}v∈V:=ℰϑ​\(G\)\\\{r\_\{v\}\\\}\_\{v\\in V\}:=\\mathcal\{E\}\_\{\\vartheta\}\(G\)andhG:=Readout⁡\(\{rv:v∈V\}\)∈ℝdh\_\{G\}:=\\operatorname\{Readout\}\(\\\{r\_\{v\}:v\\in V\\\}\)\\in\\mathbb\{R\}^\{d\}\. The readout may be instantiated using mean pooling, sum pooling, or another permutation\-invariant aggregation operator, ensuring thathGh\_\{G\}is unaffected by node ordering and has a fixed dimensionality across graphs of different sizes\.

For a downstream graph\-level task with labelyGy\_\{G\}, a task\-specific predictorπϕ\\pi\_\{\\phi\}producesy^G:=πϕ​\(hG\)\\widehat\{y\}\_\{G\}:=\\pi\_\{\\phi\}\(h\_\{G\}\)\. Given a labeled downstream dataset𝒟task=\{\(Gi,yi\)\}i=1Ntask\\mathcal\{D\}\_\{\\mathrm\{task\}\}=\\\{\(G\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{\\mathrm\{task\}\}\}, the predictor is trained by minimizingℒtask:=Ntask−1​∑i=1Ntaskℓtask​\(πϕ​\(hGi\),yi\)\\mathcal\{L\}\_\{\\mathrm\{task\}\}:=N\_\{\\mathrm\{task\}\}^\{\-1\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{task\}\}\}\\ell\_\{\\mathrm\{task\}\}\(\\pi\_\{\\phi\}\(h\_\{G\_\{i\}\}\),y\_\{i\}\), whereℓtask\\ell\_\{\\mathrm\{task\}\}denotes a classification or regression loss\. In self\-supervised graph representation learning, the representation encoder is pretrained without downstream labels, after which the learned graph representations are evaluated using a lightweight task\-specific predictor with the pretrained encoder either frozen or fine\-tuned\.

![Refer to caption](https://arxiv.org/html/2608.00491v1/x2.png)Figure 2\.Overview of HP\-JEPA\.\(a\)The input graph is partitioned independently atLLconfigured resolutions\. At resolutionℓ\\ell, each core regionPi\(ℓ\)P\_\{i\}^\{\(\\ell\)\}defines a graph\-patch supportP~i\(ℓ\)\\widetilde\{P\}\_\{i\}^\{\(\\ell\)\}, while padded and inactive token slots are masked and ignored\.\(b\)At each eligible resolution, the sampled context indexcℓ​\(G\)c\_\{\\ell\}\(G\)and target\-index set𝒯ℓ​\(G\)\\mathcal\{T\}\_\{\\ell\}\(G\)determine the context and target\-content inputs processed by the online token encoderfθf\_\{\\theta\}and EMA target token encoderfθ¯f\_\{\\bar\{\\theta\}\}, respectively\. For eacht∈𝒯ℓ​\(G\)t\\in\\mathcal\{T\}\_\{\\ell\}\(G\), the predictorqψq\_\{\\psi\}estimatesy^t\(ℓ\)\\widehat\{y\}\_\{t\}^\{\(\\ell\)\}from the context statescℓ\(ℓ\)s\_\{c\_\{\\ell\}\}^\{\(\\ell\)\}and target structural querypt\(ℓ\)p\_\{t\}^\{\(\\ell\)\}, supervised by the two\-coordinate targetyt,tgt\(ℓ\)y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}using a coordinate\-wise SmoothL1L\_\{1\}loss\.\(c\)During downstream readout, the frozen EMA target token encoder processes each active resolution independently, and its valid token representationsrG,i\(ℓ\)r\_\{G,i\}^\{\(\\ell\)\}are mean\-pooled to obtainhG\(ℓ\)h\_\{G\}^\{\(\\ell\)\}\. The resolution\-specific representations are subsequently combined through concatenationhGconcath\_\{G\}^\{\\mathrm\{concat\}\}or uniformly smoothed task\-specific resolution weightinghGtaskh\_\{G\}^\{\\mathrm\{task\}\}for graph classification or regression\.HP\-JEPA independently constructs graph regions at multiple resolutions, performs two\-coordinate latent prediction separately within each eligible resolution, mean\-pools valid token representations to obtain resolution\-specific graph representations, and combines those representations through concatenation or task\-specific resolution weighting\.
### 3\.2\.JEPA for Graphs

Joint\-embedding predictive architectures learn representations by predicting a target in latent space from a partially observed context rather than reconstructing the target in the original input space\. Given a context inputxctxx\_\{\\mathrm\{ctx\}\}and a target inputxtgtx\_\{\\mathrm\{tgt\}\}, the online token encoder producessctx:=fθ​\(xctx\)∈ℝds\_\{\\mathrm\{ctx\}\}:=f\_\{\\theta\}\(x\_\{\\mathrm\{ctx\}\}\)\\in\\mathbb\{R\}^\{d\}, while the EMA target token encoder producesutgt:=fθ¯​\(sg⁡\(xtgt\)\)∈ℝdu\_\{\\mathrm\{tgt\}\}:=f\_\{\\bar\{\\theta\}\}\(\\operatorname\{sg\}\(x\_\{\\mathrm\{tgt\}\}\)\)\\in\\mathbb\{R\}^\{d\}, wheresg⁡\(⋅\)\\operatorname\{sg\}\(\\cdot\)denotes stop\-gradient\. A structural queryptgt∈ℝdp\_\{\\mathrm\{tgt\}\}\\in\\mathbb\{R\}^\{d\}identifies the target whose representation should be predicted, and the predictor producesu^tgt:=qψ​\(sctx\+ptgt\)\\widehat\{u\}\_\{\\mathrm\{tgt\}\}:=q\_\{\\psi\}\(s\_\{\\mathrm\{ctx\}\}\+p\_\{\\mathrm\{tgt\}\}\)\. A generic JEPA objective is then written asℒJEPA:=ℓpred​\(u^tgt,sg⁡\(utgt\)\)\\mathcal\{L\}\_\{\\mathrm\{JEPA\}\}:=\\ell\_\{\\mathrm\{pred\}\}\(\\widehat\{u\}\_\{\\mathrm\{tgt\}\},\\operatorname\{sg\}\(u\_\{\\mathrm\{tgt\}\}\)\), whereℓpred\\ell\_\{\\mathrm\{pred\}\}measures discrepancy in latent space\. By defining supervision in the representation space, JEPA avoids explicit reconstruction of low\-level input details and does not require negative samples\.

To specialize this formulation to graph regions using the notation adopted in Section[4](https://arxiv.org/html/2608.00491#S4), consider a single predefined graph partition𝒫​\(G\):=\{Pi\}i=1k​\(G\)\\mathcal\{P\}\(G\):=\\\{P\_\{i\}\\\}\_\{i=1\}^\{k\(G\)\}, where everyPiP\_\{i\}is nonempty,⋃i=1k​\(G\)Pi=V\\bigcup\_\{i=1\}^\{k\(G\)\}P\_\{i\}=V, andPi∩Pj=∅P\_\{i\}\\cap P\_\{j\}=\\varnothingfori≠ji\\neq j\. Here,k​\(G\)k\(G\)denotes the number of valid regions in the partition\. For each core regionPiP\_\{i\}, letP~i⊇Pi\\widetilde\{P\}\_\{i\}\\supseteq P\_\{i\}denote the support of its associated graph patch, and letρi\\rho\_\{i\}denote a structural descriptor characterizing the position and structural context of that patch in the original graph\.

A patch\-token encodergηg\_\{\\eta\}maps the graph patch to a content tokenzi:=gη​\(G​\[P~i\]\)∈ℝdz\_\{i\}:=g\_\{\\eta\}\(G\[\\widetilde\{P\}\_\{i\}\]\)\\in\\mathbb\{R\}^\{d\}, while a structural\-query encodereξe\_\{\\xi\}maps its descriptor topi:=eξ​\(ρi\)∈ℝdp\_\{i\}:=e\_\{\\xi\}\(\\rho\_\{i\}\)\\in\\mathbb\{R\}^\{d\}\. The context and downstream readout paths retain both patch content and structural information throughxictx=xifull:=zi\+pix\_\{i\}^\{\\mathrm\{ctx\}\}=x\_\{i\}^\{\\mathrm\{full\}\}:=z\_\{i\}\+p\_\{i\}, whereas the target path receives only the patch\-content inputxitgt:=zix\_\{i\}^\{\\mathrm\{tgt\}\}:=z\_\{i\}\. Consequently, the target token encoder does not directly observe the target structural query supplied to the predictor\.

Letℐ​\(G\):=\{1,…,k​\(G\)\}\\mathcal\{I\}\(G\):=\\\{1,\\ldots,k\(G\)\\\}denote the valid region\-index set\. A context indexc​\(G\)c\(G\)is sampled fromℐ​\(G\)\\mathcal\{I\}\(G\), and a nonempty target\-index set𝒯​\(G\)⊆ℐ​\(G\)∖\{c​\(G\)\}\\mathcal\{T\}\(G\)\\subseteq\\mathcal\{I\}\(G\)\\setminus\\\{c\(G\)\\\}is sampled from the remaining regions; we writeM​\(G\):=\|𝒯​\(G\)\|M\(G\):=\|\\mathcal\{T\}\(G\)\|\. Suppressing the graph argument when it is clear from context, the online token encoder producessc:=fθ​\(xcctx\)s\_\{c\}:=f\_\{\\theta\}\(x\_\{c\}^\{\\mathrm\{ctx\}\}\), while the EMA target token encoder produces\[ut\]t∈𝒯​\(G\):=fθ¯​\(sg⁡\(\[xttgt\]t∈𝒯​\(G\)\)\)\[u\_\{t\}\]\_\{t\\in\\mathcal\{T\}\(G\)\}:=f\_\{\\bar\{\\theta\}\}\(\\operatorname\{sg\}\(\[x\_\{t\}^\{\\mathrm\{tgt\}\}\]\_\{t\\in\\mathcal\{T\}\(G\)\}\)\)\. For each target indext∈𝒯​\(G\)t\\in\\mathcal\{T\}\(G\), the predictor estimatesu^t:=qψ​\(sc\+pt\)\\widehat\{u\}\_\{t\}:=q\_\{\\psi\}\(s\_\{c\}\+p\_\{t\}\), and the single\-resolution graph JEPA objective isℒGraph​\-​JEPA​\(G\):=M​\(G\)−1​∑t∈𝒯​\(G\)ℓpred​\(u^t,sg⁡\(ut\)\)\\mathcal\{L\}\_\{\\mathrm\{Graph\\text\{\-\}JEPA\}\}\(G\):=M\(G\)^\{\-1\}\\sum\_\{t\\in\\mathcal\{T\}\(G\)\}\\ell\_\{\\mathrm\{pred\}\}\(\\widehat\{u\}\_\{t\},\\operatorname\{sg\}\(u\_\{t\}\)\)\.

Through this latent prediction objective, the single\-resolution formulation learns statistical dependencies between a visible graph context and held\-out target regions without reconstructing node attributes or edges\. However, it represents the graph using only one predefined partition𝒫​\(G\)\\mathcal\{P\}\(G\)and therefore exposes the model to one structural granularity\. HP\-JEPA generalizes this formulation toLLindependently constructed resolutions\{𝒫\(ℓ\)​\(G\)\}ℓ=1L\\\{\\mathcal\{P\}^\{\(\\ell\)\}\(G\)\\\}\_\{\\ell=1\}^\{L\}with configured capacitiesKℓ=2ℓK\_\{\\ell\}=2^\{\\ell\}and graph\-dependent valid\-region countskℓ​\(G\)k\_\{\\ell\}\(G\)\. It replacesc​\(G\)c\(G\),𝒯​\(G\)\\mathcal\{T\}\(G\), andM​\(G\)M\(G\)with the resolution\-specific quantitiescℓ​\(G\)c\_\{\\ell\}\(G\),𝒯ℓ​\(G\)\\mathcal\{T\}\_\{\\ell\}\(G\), andMℓ​\(G\)M\_\{\\ell\}\(G\), and performs prediction independently within each eligible resolution\.

Unlike the generic full\-vector objective above, HP\-JEPA derives a two\-coordinate target from each EMA target\-token state\. Specifically, forut\(ℓ\)∈ℝdu\_\{t\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}, it definesm​\(ut\(ℓ\)\):=d−1​𝟏d⊤​ut\(ℓ\)m\(u\_\{t\}^\{\(\\ell\)\}\):=d^\{\-1\}\\mathbf\{1\}\_\{d\}^\{\\top\}u\_\{t\}^\{\(\\ell\)\}andyt,tgt\(ℓ\):=Φ​\(ut\(ℓ\)\)=\[cosh⁡\(m​\(ut\(ℓ\)\)\),sinh⁡\(m​\(ut\(ℓ\)\)\)\]⊤∈ℝ2y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}:=\\Phi\(u\_\{t\}^\{\(\\ell\)\}\)=\[\\cosh\(m\(u\_\{t\}^\{\(\\ell\)\}\)\),\\sinh\(m\(u\_\{t\}^\{\(\\ell\)\}\)\)\]^\{\\top\}\\in\\mathbb\{R\}^\{2\}, while the predictor producesy^t\(ℓ\):=qψ​\(scℓ\(ℓ\)\+pt\(ℓ\)\)\\widehat\{y\}\_\{t\}^\{\(\\ell\)\}:=q\_\{\\psi\}\(s\_\{c\_\{\\ell\}\}^\{\(\\ell\)\}\+p\_\{t\}^\{\(\\ell\)\}\)\. After self\-supervised pretraining, valid token representations are pooled separately within each resolution to obtain\{hG\(ℓ\)\}ℓ=1L\\\{h\_\{G\}^\{\(\\ell\)\}\\\}\_\{\\ell=1\}^\{L\}, which are combined through concatenation or task\-specific resolution weighting as described in Section[4](https://arxiv.org/html/2608.00491#S4)\.

## 4\.Method

We consider an attributed graphG=\(V,E,𝐗\)G=\(V,E,\\mathbf\{X\}\), whereVVandEEare the node and edge sets,𝐗∈ℝn×dx\\mathbf\{X\}\\in\\mathbb\{R\}^\{n\\times d\_\{x\}\}is the node\-feature matrix, andn=\|V\|n=\|V\|\. LetL:=L𝒟L:=L\_\{\\mathcal\{D\}\}denote the number of resolutions configured for dataset𝒟\\mathcal\{D\}\. Resolutionℓ∈\{1,…,L\}\\ell\\in\\\{1,\\ldots,L\\\}has region capacityKℓ=2ℓK\_\{\\ell\}=2^\{\\ell\}and, when active forGG, containskℓ​\(G\)k\_\{\\ell\}\(G\)valid graph regions\. HP\-JEPA represents each valid region using a patch\-content token and a structural query, and performs latent prediction independently at each resolution using an online token encoder, an exponential\-moving\-average \(EMA\) target token encoder, and a predictor\. After pretraining, the frozen target branch produces one graph representation per resolution, and these representations are combined through concatenation or task\-specific resolution weighting\. Accordingly, HP\-JEPA consists of three components: i\) multi\-resolution graph tokenization; ii\) resolution\-wise joint\-embedding prediction; and iii\) multi\-resolution readout with task\-specific resolution weighting\.

### 4\.1\.Multi\-Resolution Graph Tokenization

As illustrated in Figure[2](https://arxiv.org/html/2608.00491#S3.F2)\(a\), HP\-JEPA constructs graph tokens atLLconfigured resolutions\. A resolution is active for graphGGwhenKℓ≤2​nK\_\{\\ell\}\\leq 2n; onceKℓ\>2​nK\_\{\\ell\}\>2n, that resolution and all finer resolutions are inactive\. At an active resolutionℓ\\ell, we setkℓ​\(G\):=min⁡\{Kℓ,n\}k\_\{\\ell\}\(G\):=\\min\\\{K\_\{\\ell\},n\\\}and partitionVVinto the nonempty core regions𝒫\(ℓ\)​\(G\):=\{Pi\(ℓ\)\}i=1kℓ​\(G\)\\mathcal\{P\}^\{\(\\ell\)\}\(G\):=\\\{P\_\{i\}^\{\(\\ell\)\}\\\}\_\{i=1\}^\{k\_\{\\ell\}\(G\)\}\. These regions satisfy⋃i=1kℓ​\(G\)Pi\(ℓ\)=V\\bigcup\_\{i=1\}^\{k\_\{\\ell\}\(G\)\}P\_\{i\}^\{\(\\ell\)\}=VandPi\(ℓ\)∩Pj\(ℓ\)=∅P\_\{i\}^\{\(\\ell\)\}\\cap P\_\{j\}^\{\(\\ell\)\}=\\varnothingfori≠ji\\neq j\. For an inactive resolution, we setkℓ​\(G\)=0k\_\{\\ell\}\(G\)=0\. WhenKℓ\>nK\_\{\\ell\}\>nat an active resolution, the remaining configured token slots are padded and excluded by a padding mask\. Each active resolution is constructed independently from the full input graph, and no nesting or parent–child relation is assumed between𝒫\(ℓ\)​\(G\)\\mathcal\{P\}^\{\(\\ell\)\}\(G\)and𝒫\(r\)​\(G\)\\mathcal\{P\}^\{\(r\)\}\(G\)forℓ≠r\\ell\\neq r\. The resolution capacities and partitioning procedure used in our experiments are specified in Appendix[A](https://arxiv.org/html/2608.00491#A1)\.

For each core regionPi\(ℓ\)P\_\{i\}^\{\(\\ell\)\}, letP~i\(ℓ\)⊇Pi\(ℓ\)\\widetilde\{P\}\_\{i\}^\{\(\\ell\)\}\\supseteq P\_\{i\}^\{\(\\ell\)\}denote the node support of its associated graph patch, and letρi\(ℓ\)\\rho\_\{i\}^\{\(\\ell\)\}denote a structural descriptor of the patch in the original graph\. A patch\-token encodergηg\_\{\\eta\}produces the content tokenzi\(ℓ\):=gη​\(G​\[P~i\(ℓ\)\]\)∈ℝdz\_\{i\}^\{\(\\ell\)\}:=g\_\{\\eta\}\(G\[\\widetilde\{P\}\_\{i\}^\{\(\\ell\)\}\]\)\\in\\mathbb\{R\}^\{d\}, while a structural\-query encodereξe\_\{\\xi\}producespi\(ℓ\):=eξ​\(ρi\(ℓ\)\)∈ℝdp\_\{i\}^\{\(\\ell\)\}:=e\_\{\\xi\}\(\\rho\_\{i\}^\{\(\\ell\)\}\)\\in\\mathbb\{R\}^\{d\}\. The content token summarizes the attributes and internal structure of the patch, whereas the structural query identifies its structural context\. Both encoders are shared across all configured resolutions\.

The context and downstream readout paths retain both patch content and structural information throughxictx,\(ℓ\)=xifull,\(ℓ\):=zi\(ℓ\)\+pi\(ℓ\)x\_\{i\}^\{\\mathrm\{ctx\},\(\\ell\)\}=x\_\{i\}^\{\\mathrm\{full\},\(\\ell\)\}:=z\_\{i\}^\{\(\\ell\)\}\+p\_\{i\}^\{\(\\ell\)\}, whereas the target path receives only the patch\-content inputxitgt,\(ℓ\):=zi\(ℓ\)x\_\{i\}^\{\\mathrm\{tgt\},\(\\ell\)\}:=z\_\{i\}^\{\(\\ell\)\}\. Consequently, the EMA target token encoder cannot directly observe the target structural query supplied to the predictor\. At each resolution, the token encoder operates only on the corresponding valid token inputs, and padded slots are excluded by the appropriate mask\. Tokens from different resolutions are not placed in a common sequence during either joint\-embedding pretraining or downstream readout\. Further implementation details of patch construction, structural descriptors, token encoders, and masking are provided in Appendix[A](https://arxiv.org/html/2608.00491#A1)\.

### 4\.2\.Resolution\-Wise Joint\-Embedding Prediction

As illustrated in Figure[2](https://arxiv.org/html/2608.00491#S3.F2)\(b\), HP\-JEPA performs joint\-embedding prediction separately at every eligible resolution\. For graphGGand resolutionℓ\\ellsatisfyingkℓ​\(G\)≥2k\_\{\\ell\}\(G\)\\geq 2, letℐℓ​\(G\):=\{1,…,kℓ​\(G\)\}\\mathcal\{I\}\_\{\\ell\}\(G\):=\\\{1,\\ldots,k\_\{\\ell\}\(G\)\\\}denote the valid region\-index set\. One context indexcℓ​\(G\)c\_\{\\ell\}\(G\)is sampled uniformly fromℐℓ​\(G\)\\mathcal\{I\}\_\{\\ell\}\(G\), and a nonempty target\-index set𝒯ℓ​\(G\)⊆ℐℓ​\(G\)∖\{cℓ​\(G\)\}\\mathcal\{T\}\_\{\\ell\}\(G\)\\subseteq\\mathcal\{I\}\_\{\\ell\}\(G\)\\setminus\\\{c\_\{\\ell\}\(G\)\\\}is sampled without replacement\. We denote the number of selected target tokens byMℓ​\(G\):=\|𝒯ℓ​\(G\)\|M\_\{\\ell\}\(G\):=\|\\mathcal\{T\}\_\{\\ell\}\(G\)\|\. The target\-set size and sampling rule are specified in Appendix[A](https://arxiv.org/html/2608.00491#A1)\. For readability, we subsequently writecℓ=cℓ​\(G\)c\_\{\\ell\}=c\_\{\\ell\}\(G\)and𝒯ℓ=𝒯ℓ​\(G\)\\mathcal\{T\}\_\{\\ell\}=\\mathcal\{T\}\_\{\\ell\}\(G\)when the graph is clear from context\.

The online token encoder produces the context statescℓ\(ℓ\):=fθ​\(xcℓctx,\(ℓ\)\)∈ℝds\_\{c\_\{\\ell\}\}^\{\(\\ell\)\}:=f\_\{\\theta\}\(x\_\{c\_\{\\ell\}\}^\{\\mathrm\{ctx\},\(\\ell\)\}\)\\in\\mathbb\{R\}^\{d\}\. For the target branch, define the target\-input sequenceX𝒯\(ℓ\):=\[xttgt,\(ℓ\)\]t∈𝒯ℓX\_\{\\mathcal\{T\}\}^\{\(\\ell\)\}:=\[x\_\{t\}^\{\\mathrm\{tgt\},\(\\ell\)\}\]\_\{t\\in\\mathcal\{T\}\_\{\\ell\}\}and its encoded sequenceU𝒯\(ℓ\):=fθ¯​\(sg⁡\(X𝒯\(ℓ\)\)\)U\_\{\\mathcal\{T\}\}^\{\(\\ell\)\}:=f\_\{\\bar\{\\theta\}\}\(\\operatorname\{sg\}\(X\_\{\\mathcal\{T\}\}^\{\(\\ell\)\}\)\), wheresg⁡\(⋅\)\\operatorname\{sg\}\(\\cdot\)denotes stop\-gradient\. We useut\(ℓ\)∈ℝdu\_\{t\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}to denote the output inU𝒯\(ℓ\)U\_\{\\mathcal\{T\}\}^\{\(\\ell\)\}associated with target indextt\. Onlyfθ¯f\_\{\\bar\{\\theta\}\}is maintained as an EMA copy offθf\_\{\\theta\}\. The target patch\-content inputs are detached before enteringfθ¯f\_\{\\bar\{\\theta\}\}, whereas the target structural queries are supplied only to the online predictor and never enter the EMA target token encoder\.

For eacht∈𝒯ℓt\\in\\mathcal\{T\}\_\{\\ell\}, the predictor combines the context state with the corresponding target structural query and producesy^t\(ℓ\):=qψ​\(scℓ\(ℓ\)\+pt\(ℓ\)\)∈ℝ2\\widehat\{y\}\_\{t\}^\{\(\\ell\)\}:=q\_\{\\psi\}\(s\_\{c\_\{\\ell\}\}^\{\(\\ell\)\}\+p\_\{t\}^\{\(\\ell\)\}\)\\in\\mathbb\{R\}^\{2\}\. Let𝟏d∈ℝd\\mathbf\{1\}\_\{d\}\\in\\mathbb\{R\}^\{d\}denote the all\-ones vector, definem​\(u\):=d−1​𝟏d⊤​um\(u\):=d^\{\-1\}\\mathbf\{1\}\_\{d\}^\{\\top\}u, and define the coordinate mapΦ​\(u\):=\[cosh⁡\(m​\(u\)\),sinh⁡\(m​\(u\)\)\]⊤\\Phi\(u\):=\[\\cosh\(m\(u\)\),\\sinh\(m\(u\)\)\]^\{\\top\}\. The scalar target statistic is thereforemt\(ℓ\):=m​\(ut\(ℓ\)\)m\_\{t\}^\{\(\\ell\)\}:=m\(u\_\{t\}^\{\(\\ell\)\}\), and the corresponding target coordinate isyt,tgt\(ℓ\):=Φ​\(ut\(ℓ\)\)∈ℝ2y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}:=\\Phi\(u\_\{t\}^\{\(\\ell\)\}\)\\in\\mathbb\{R\}^\{2\}\. WithJL:=diag⁡\(−1,1\)J\_\{\\mathrm\{L\}\}:=\\operatorname\{diag\}\(\-1,1\), the target satisfies\(yt,tgt\(ℓ\)\)⊤​JL​yt,tgt\(ℓ\)=−1\(y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}\)^\{\\top\}J\_\{\\mathrm\{L\}\}y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}=\-1and\(yt,tgt\(ℓ\)\)1\>0\(y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}\)\_\{1\}\>0, and therefore lies on the positive branch of the one\-dimensional Lorentz hyperboloid\. The predictor output is trained to approximate this target in ambient coordinates but is not itself explicitly constrained to lie on the hyperboloid\.

LetℓG,ℓ,t\\ell\_\{G,\\ell,t\}denote the coordinate\-wiseSmoothL1β\\operatorname\{SmoothL1\}\_\{\\beta\}penalty betweeny^t\(ℓ\)\\widehat\{y\}\_\{t\}^\{\(\\ell\)\}andsg⁡\(yt,tgt\(ℓ\)\)\\operatorname\{sg\}\(y\_\{t,\\mathrm\{tgt\}\}^\{\(\\ell\)\}\), averaged over the two output coordinates\. The resolution\-specific graph loss isℒℓ​\(G\):=Mℓ​\(G\)−1​∑t∈𝒯ℓ​\(G\)ℓG,ℓ,t\\mathcal\{L\}\_\{\\ell\}\(G\):=M\_\{\\ell\}\(G\)^\{\-1\}\\sum\_\{t\\in\\mathcal\{T\}\_\{\\ell\}\(G\)\}\\ell\_\{G,\\ell,t\}\. Thus, the objective predicts a two\-coordinate function of the feature mean of each EMA target\-token state rather than reconstructing its completedd\-dimensional representation\. The value ofβ\\betais reported in Appendix[A](https://arxiv.org/html/2608.00491#A1)\.

For a minibatchℬ\\mathcal\{B\}, letℬℓ:=\{G∈ℬ:kℓ​\(G\)≥2\}\\mathcal\{B\}\_\{\\ell\}:=\\\{G\\in\\mathcal\{B\}:k\_\{\\ell\}\(G\)\\geq 2\\\}denote the graphs eligible at resolutionℓ\\ell\. Whenℬℓ≠∅\\mathcal\{B\}\_\{\\ell\}\\neq\\varnothing, define the target\-count normalizerZℓ​\(ℬ\):=∑G∈ℬℓMℓ​\(G\)Z\_\{\\ell\}\(\\mathcal\{B\}\):=\\sum\_\{G\\in\\mathcal\{B\}\_\{\\ell\}\}M\_\{\\ell\}\(G\)and the graph weightwG,ℓ:=Mℓ​\(G\)/Zℓ​\(ℬ\)w\_\{G,\\ell\}:=M\_\{\\ell\}\(G\)/Z\_\{\\ell\}\(\\mathcal\{B\}\)\. The minibatch objective is thenℒℓ​\(ℬ\):=∑G∈ℬℓwG,ℓ​ℒℓ​\(G\)\\mathcal\{L\}\_\{\\ell\}\(\\mathcal\{B\}\):=\\sum\_\{G\\in\\mathcal\{B\}\_\{\\ell\}\}w\_\{G,\\ell\}\\mathcal\{L\}\_\{\\ell\}\(G\)\. This reduction averages uniformly over all selected target tokens in the minibatch, so a graph with more selected targets contributes proportionally more to the update\. Although the context index is excluded from𝒯ℓ​\(G\)\\mathcal\{T\}\_\{\\ell\}\(G\), context and target patches may share nodes when the patch\-construction operator expands core regions into overlapping supports\. The held\-out prediction unit is therefore the target token rather than a strictly node\-disjoint subgraph\.

HP\-JEPA processes the configured resolutions sequentially\. For each minibatch, the model visitsℓ=1,…,L\\ell=1,\\ldots,L, applies one optimizer update usingℒℓ​\(ℬ\)\\mathcal\{L\}\_\{\\ell\}\(\\mathcal\{B\}\)wheneverℬℓ≠∅\\mathcal\{B\}\_\{\\ell\}\\neq\\varnothing, and then updates the EMA target token encoder asθ¯\[ν\+1\]:=μν​θ¯\[ν\]\+\(1−μν\)​θ\[ν\+1\]\\bar\{\\theta\}^\{\[\\nu\+1\]\}:=\\mu\_\{\\nu\}\\bar\{\\theta\}^\{\[\\nu\]\}\+\(1\-\\mu\_\{\\nu\}\)\\theta^\{\[\\nu\+1\]\}, whereν\\nuis the global optimizer\-step index andμν∈\[0,1\)\\mu\_\{\\nu\}\\in\[0,1\)is the corresponding EMA momentum\. Resolution\-specific losses are not summed before a common optimizer update, and no cross\-resolution, adjacent\-resolution, or parent–child consistency term is included in the pretraining objective\. The optimizer and EMA schedules are specified in Appendix[A](https://arxiv.org/html/2608.00491#A1)\.

Appendix[B](https://arxiv.org/html/2608.00491#A2)shows that, with respect to Lorentz geodesic distance,Φ\\Phiis an isometric encoding of the feature\-mean projectionm​\(u\)=d−1​𝟏d⊤​um\(u\)=d^\{\-1\}\\mathbf\{1\}\_\{d\}^\{\\top\}u\. It therefore preserves differences in the supervised feature\-mean component while remaining invariant to components in𝟏d⟂:=\{v∈ℝd:𝟏d⊤​v=0\}\\mathbf\{1\}\_\{d\}^\{\\perp\}:=\\\{v\\in\\mathbb\{R\}^\{d\}:\\mathbf\{1\}\_\{d\}^\{\\top\}v=0\\\}\. The same analysis shows that the population optimum of coordinate\-wise Smooth L1 is a conditional Huber location of the target coordinates given the context state and target structural query, rather than necessarily their conditional mean\.

### 4\.3\.Multi\-Resolution Readout and Task\-Specific Resolution Weighting

As illustrated in Figure[2](https://arxiv.org/html/2608.00491#S3.F2)\(c\), after self\-supervised pretraining, the context–target sampling procedure and predictorqψq\_\{\\psi\}are discarded, whilegηg\_\{\\eta\},eξe\_\{\\xi\}, andfθ¯f\_\{\\bar\{\\theta\}\}are retained and frozen\. For each active resolutionℓ\\ell, define the full\-token sequenceXGfull,\(ℓ\):=\[xifull,\(ℓ\)\]i=1kℓ​\(G\)X\_\{G\}^\{\\mathrm\{full\},\(\\ell\)\}:=\[x\_\{i\}^\{\\mathrm\{full\},\(\\ell\)\}\]\_\{i=1\}^\{k\_\{\\ell\}\(G\)\}and encode it independently asRG\(ℓ\):=fθ¯​\(XGfull,\(ℓ\)\)R\_\{G\}^\{\(\\ell\)\}:=f\_\{\\bar\{\\theta\}\}\(X\_\{G\}^\{\\mathrm\{full\},\(\\ell\)\}\)\. We userG,i\(ℓ\)∈ℝdr\_\{G,i\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}to denote the output inRG\(ℓ\)R\_\{G\}^\{\(\\ell\)\}associated with regionii\. Padded positions are masked and excluded\. Because each resolution is processed separately, no cross\-resolution token interaction is introduced by the frozen encoder; different resolutions are combined only after resolution\-wise pooling\.

For an active resolution, the valid token representations are mean\-pooled ashG\(ℓ\):=kℓ​\(G\)−1​∑i=1kℓ​\(G\)rG,i\(ℓ\)∈ℝdh\_\{G\}^\{\(\\ell\)\}:=k\_\{\\ell\}\(G\)^\{\-1\}\\sum\_\{i=1\}^\{k\_\{\\ell\}\(G\)\}r\_\{G,i\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}\. For an inactive resolution withkℓ​\(G\)=0k\_\{\\ell\}\(G\)=0, we sethG\(ℓ\):=𝟎dh\_\{G\}^\{\(\\ell\)\}:=\\mathbf\{0\}\_\{d\}, where𝟎d∈ℝd\\mathbf\{0\}\_\{d\}\\in\\mathbb\{R\}^\{d\}is the zero vector\. One downstream representation concatenates all configured resolution slots ashGconcat:=hG\(1\)​‖⋯‖​hG\(L\)∈ℝL​dh\_\{G\}^\{\\mathrm\{concat\}\}:=h\_\{G\}^\{\(1\)\}\\\|\\cdots\\\|h\_\{G\}^\{\(L\)\}\\in\\mathbb\{R\}^\{Ld\}, where∥\\\|denotes vector concatenation\. This representation preserves the output associated with every active resolution while maintaining a fixed dimension\.

Alternatively, HP\-JEPA learns a task\-specific mixture of the resolution representations\. Letbtask∈ℝLb^\{\\mathrm\{task\}\}\\in\\mathbb\{R\}^\{L\}be the resolution\-logit vector and define the raw weights byω~task:=softmax⁡\(btask\)\\widetilde\{\\omega\}^\{\\mathrm\{task\}\}:=\\operatorname\{softmax\}\(b^\{\\mathrm\{task\}\}\)\. The uniformly smoothed weight for resolutionℓ\\ellisωℓtask:=\(1−λunif\)​ω~ℓtask\+λunif/L\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}:=\(1\-\\lambda\_\{\\mathrm\{unif\}\}\)\\widetilde\{\\omega\}\_\{\\ell\}^\{\\mathrm\{task\}\}\+\\lambda\_\{\\mathrm\{unif\}\}/L, whereλunif∈\[0,1\]\\lambda\_\{\\mathrm\{unif\}\}\\in\[0,1\]\. These weights satisfyωℓtask\>0\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}\>0and∑ℓ=1Lωℓtask=1\\sum\_\{\\ell=1\}^\{L\}\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}=1\. The resulting task\-specific representation ishGtask:=∑ℓ=1Lωℓtask​hG\(ℓ\)∈ℝdh\_\{G\}^\{\\mathrm\{task\}\}:=\\sum\_\{\\ell=1\}^\{L\}\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}h\_\{G\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}\.

The weight vectorωtask\\omega^\{\\mathrm\{task\}\}is shared by all graphs within the same downstream task and experimental split, so the weighting is task\-specific rather than graph\-specific\. The weights are defined over allLLconfigured resolution slots and are not renormalized over the graph\-specific active subset\. Consequently,λunif=1\\lambda\_\{\\mathrm\{unif\}\}=1gives the uniform representationhGtask:=L−1​∑ℓ=1LhG\(ℓ\)h\_\{G\}^\{\\mathrm\{task\}\}:=L^\{\-1\}\\sum\_\{\\ell=1\}^\{L\}h\_\{G\}^\{\(\\ell\)\}, where inactive slots contribute𝟎d\\mathbf\{0\}\_\{d\}, whereasλunif=0\\lambda\_\{\\mathrm\{unif\}\}=0gives the unsmoothed learned mixture\.

The concatenated representationhGconcath\_\{G\}^\{\\mathrm\{concat\}\}is obtained without downstream labels\. Forλunif<1\\lambda\_\{\\mathrm\{unif\}\}<1, the task\-specific representation uses downstream labels to estimate the relative contribution of each resolution after self\-supervised pretraining, whereasλunif=1\\lambda\_\{\\mathrm\{unif\}\}=1yields label\-independent uniform averaging\. The patch modules and EMA target token encoder remain frozen throughout downstream training\. The procedure used to optimizebtaskb^\{\\mathrm\{task\}\}, select its checkpoint, apply uniform smoothing, and fit the final downstream predictor is specified in Table[5](https://arxiv.org/html/2608.00491#A1.T5)\.

Conditioned on the frozen resolution\-specific representations, Appendix[C](https://arxiv.org/html/2608.00491#A3)shows that the closure of the unsmoothed softmax\-parameterized weighting family contains every fixed\-resolution linear predictor\. Under the stated boundedness assumptions, empirical risk minimization over this family competes with the best fixed resolution selected in hindsight, with an additional estimation term of orderO​\(log⁡L/N\)O\(\\sqrt\{\\log L/N\}\)forNNlabeled downstream examples\. Uniform smoothing adds an approximation term that is at most linear inλunif\\lambda\_\{\\mathrm\{unif\}\}\.

Table 2\.Results on graph classification and regression benchmarks\. For reference only, we include the results of F\-GIN, a pioneering end\-to\-end supervised GNN\. SSL methods are grouped by pretraining paradigm in the following order: contrastive, generative, self\-predictive, and JEPA\. Baseline results are taken directly from prior work\(Hassani and Khasahmadi,[2020](https://arxiv.org/html/2608.00491#bib.bib7); Tan et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib18); Suresh et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib17); Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)and ”\-” indicates that the corresponding results are not reported\. The best SSL result isunderlined, and the second best iswavy underlined\. For a more direct comparison, HP\-JEPA results that outperform the Graph\-JEPA baseline are highlighted with ared cell background\.

## 5\.Experiments

In this section, we conduct experiments to substantiate our claims and demonstrate the effectiveness of HP\-JEPA\. Specifically, Sec\.[5\.1](https://arxiv.org/html/2608.00491#S5.SS1)evaluates downstream classification and regression performance, Sec\.[5\.2](https://arxiv.org/html/2608.00491#S5.SS2)presents a representation case study on PROTEINS, and Sec\.[5\.3](https://arxiv.org/html/2608.00491#S5.SS3)analyzes graph\-size and resolution\-bank sensitivity\.

### 5\.1\.Downstream Task Evaluation

#### 5\.1\.1\.Setup

Following established graph self\-supervised learning evaluation protocols\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15); Suresh et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib17); Tan et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib18)\), we evaluate HP\-JEPA on seven graph\-classification benchmarks from TUDataset\(Morris et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib12)\): PROTEINS, MUTAG, DD, REDDIT\-BINARY, REDDIT\-MULTI\-5K, IMDB\-BINARY, and IMDB\-MULTI\. We additionally consider the ZINC\-12K molecular graph\-regression benchmark following Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)\. Additional dataset statistics, preprocessing details, and dataset\-specific configurations are provided in Appendix[A](https://arxiv.org/html/2608.00491#A1)\.

We compare HP\-JEPA with the graph SSL baselines InfoGraph\(Sun et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib16)\), GraphCL\(You et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib25)\), MVGRL\(Hassani and Khasahmadi,[2020](https://arxiv.org/html/2608.00491#bib.bib7)\), AD\-GCL\-FIX/OPT\(Suresh et al\.,[2021](https://arxiv.org/html/2608.00491#bib.bib17)\), GraphMAE\(Hou et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib8)\), S2GAE\(Tan et al\.,[2023](https://arxiv.org/html/2608.00491#bib.bib18)\), BGRL\(Thakoor et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib19)\), LaGraph\(Xie et al\.,[2022](https://arxiv.org/html/2608.00491#bib.bib23)\), and Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)\. For graph classification, we perform stratified 10\-fold cross\-validation under five random seeds; accuracy is first averaged across the ten folds for each seed, after which the final mean and standard deviation are computed across the five seed\-level results\. For ZINC\-12K, we use the standard 10,000/1,000/1,000 train/validation/test split and report the test mean absolute error \(MAE\) averaged over ten random seeds\.

To construct the graph\-level representation, all valid tokens at each resolution are passed through the trained EMA target encoder and aggregated via mean pooling, yieldingLLresolution\-specific feature vectors\{hG\(ℓ\)\}ℓ=1L\\\{h\_\{G\}^\{\(\\ell\)\}\\\}\_\{\\ell=1\}^\{L\}, wherehG\(ℓ\)∈ℝdh\_\{G\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{d\}\. As a key innovation of HP\-JEPA, theseLLfeature vectors represent information from different resolutions by construction\. These representations are subsequently fused through a learnable resolution\-logit vectorbtask∈ℝLb^\{\\mathrm\{task\}\}\\in\\mathbb\{R\}^\{L\}, whose softmax\-normalized entriesωℓtask\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}learn the relative importance of each resolution for the downstream task\. The resulting weighted representationhGtask=∑ℓ=1Lωℓtask​hG\(ℓ\)h\_\{G\}^\{\\mathrm\{task\}\}=\\sum\_\{\\ell=1\}^\{L\}\\omega\_\{\\ell\}^\{\\mathrm\{task\}\}h\_\{G\}^\{\(\\ell\)\}is evaluated using a linear model:L2L\_\{2\}\-regularized logistic regression for the classification tasks and ridge regression for the ZINC dataset\.

#### 5\.1\.2\.Results and Discussion

The results are presented in Table[2](https://arxiv.org/html/2608.00491#S4.T2)\.HP\-JEPA outperforms the Graph\-JEPA baseline across 6 out of 8 tasks, highlighting the importance of hierarchical partitioning in JEPA\-style self\-supervised learning for capturing multi\-resolution graph structures\.In addition, HP\-JEPA is the best\-performing model overall, achieving the best performance on 5 out of 8 tasks and the second best on 1 task, while Graph\-JEPA achieves the second\-best performance on 4 out of 8 tasks and the best performance on 1 task\. Both Graph\-JEPA and HP\-JEPA underperform generative methods \(GraphMAE and S2GAE\) on the two IMDB datasets\. This is potentially because the IMDB datasets are relatively small \(e\.g\. only 13 nodes on average for IMDB\-M\), with node features consisting only of one\-hot degree encodings rather than rich semantic attributes\. Under such settings, raw signal reconstruction objectives may provide a stronger training signal by directly modeling the graph structure, whereas predictive objectives such as JEPA have less informative contextual signals to exploit\. Nevertheless, HP\-JEPA still consistently outperforms the Graph\-JEPA baseline on these datasets, demonstrating the benefit of hierarchical partitioning even in this challenging setting\.Overall, these results indicate that JEPA\-style training is effective at learning useful graph representations and that our proposed hierarchical partitioning further enhances representation quality by capturing graph structures across multiple resolutions\.

![Refer to caption](https://arxiv.org/html/2608.00491v1/Figure/Figure-3.png)Figure 3\.3D t\-SNE visualization of graph\-level embeddings on the PROTEINS dataset\. Each point represents one protein graph\. Columns compare Graph\-JEPA and HP\-JEPA; the top row colors embeddings by graph\-size quartile \(Q1–Q4\), while the bottom row colors the same embeddings by protein class \(C0/C1\)\. Compared with fixed\-resolution Graph\-JEPA, HP\-JEPA produces a more structured representation space, distinctly separating proteins\. This alignment indicates that HP\-JEPA captures graph\-size\-dependent structural and functional patterns without using class labels during pretraining\.

### 5\.2\.Case Study: Learning Multi\-Resolution Graph Features

In this section, we examine whether HP\-JEPA embeddings retain graph\-size\-related structural variation and how this variation aligns with downstream class labels\. For the graph classification task on the PROTEINS dataset, both local structural patterns \(e\.g\., functional motifs\) and higher\-level protein organization \(e\.g\. protein size/graph size and overall topologies\) contribute to the prediction, making it a representative benchmark for evaluating multi\-resolution representations\.

Specifically, the downstream task on the PROTEINS dataset is binary graph classification: determining whether a protein acts as an enzyme \(a biochemical catalyst\) or a non\-enzyme \(e\.g\., structural scaffold, signaling hormone, or toxin\)\. Biologically, enzymatic catalysis necessitates deep 3D active site clefts, buried hydrophobic cores, and domain\-level conformational flexibility, structural prerequisites that inherently require larger molecular scaffolds\. In contrast, small micro\-proteins lack the volumetric capacity to bury a hydrophobic core, operating instead through solvent\-exposed, surface\-dominated binding interfaces and localized structural constraints\. Consequently, small proteins occupy a fundamentally distinct thermodynamic and functional regime\.

To evaluate whether learned embeddings capture this biophysical transition, we partition protein graphs into node\-count quartiles \(Q​1−Q​4Q1\-Q4\)\. As shown in Fig\.[3](https://arxiv.org/html/2608.00491#S5.F3), standard Graph\-JEPA exhibits scale\-blindness, collapsing these distinct structural regimes into a single overlapping manifold\. In contrast, HP\-JEPA achieves topological disentanglement, isolatingQ​1Q1micro\-proteins into a distinct, structured subspace\. Critically, Fig\.[3](https://arxiv.org/html/2608.00491#S5.F3)confirms that theseQ​1Q1embeddings correspond overwhelmingly to non\-enzymatic proteins\. This demonstrates that HP\-JEPA naturally aligns its self\-supervised representations with multi\-resolution biological function without relying on class labels during pretraining\.

### 5\.3\.Sensitivity Analysis

![Refer to caption](https://arxiv.org/html/2608.00491v1/Figure/RP-JEPA-Figure4.png)Figure 4\.Sensitivity analysis on IMDB\-BINARY, IMDB\-MULTI, and PROTEINS\.Panels \(a\), \(c\), and \(e\) report Graph\-JEPA and HP\-JEPA accuracy across node\-count quartilesQ​1Q1–Q​4Q4\. Panels \(b\), \(d\), and \(f\) compare HP\-JEPA using resolution banks\{2,4,8\}\\\{2,4,8\\\},\{2,4,8,16\}\\\{2,4,8,16\\\}, and the full bank, with Graph\-JEPA shown as a dashed reference\. Results are the mean and standard deviation over ten cross\-validation folds from one pretrained seed\.Figure[4](https://arxiv.org/html/2608.00491#S5.F4)evaluates the sensitivity of HP\-JEPA to graph size and resolution\-bank configuration on IMDB\-BINARY, IMDB\-MULTI, and PROTEINS\. Across the graph\-size analysis, HP\-JEPA achieves higher mean accuracy than Graph\-JEPA in most of quartiles\. Similarly, HP\-JEPA outperforms Graph\-JEPA under all nine tested resolution\-bank configurations\. Although performance varies moderately across resolution banks, the improvement over the fixed\-resolution Graph\-JEPA remains consistent\. By representing each graph at multiple structural granularities, HP\-JEPA is less dependent on a single partition resolution that may be overly coarse for large graphs or overly fine for small graphs\. Moreover, combining resolution\-specific embeddings allows complementary local, regional, and global information to compensate when an individual resolution is suboptimal\. In contrast, Graph\-JEPA relies on one fixed partition resolution and is therefore more susceptible to mismatches between its predefined granularity and the graph\-size distribution\. Overall, within these datasets and the evaluated configurations, HP\-JEPA retains an advantage over Graph\-JEPA in most graph\-size quartiles and under all tested resolution banks\.

## 6\.Conclusion

JEPAs provide a latent predictive approach to graph self\-supervised learning without explicit negative pairs or raw\-input reconstruction\. However, existing graph JEPAs rely on a single partition resolution, restricting the predictive objective to one structural granularity\. We proposed HP\-JEPA, which constructs an ordered bank of coarse\-to\-fine partition resolutions, performs context–target prediction independently within each resolution, and integrates the resulting resolution\-specific representations during downstream readout\. HP\-JEPA improves over Graph\-JEPA on six of the eight evaluated tasks, including six of seven graph\-classification benchmarks\. Size\-stratified and resolution\-bank analyses further show that HP\-JEPA retains its advantage in most evaluated graph\-size quartiles and across the tested partition banks\. These results support multi\-resolution partitioning as a flexible extension of fixed\-resolution graph joint\-embedding prediction\.

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## Appendix ADataset and Training Details

We evaluate HP\-JEPA on seven graph\-classification benchmarks from TUDataset\(Morris et al\.,[2020](https://arxiv.org/html/2608.00491#bib.bib12)\), including PROTEINS, MUTAG, DD, REDDIT\-BINARY, REDDIT\-MULTI\-5K, IMDB\-BINARY, and IMDB\-MULTI, together with the ZINC\-12K molecular graph\-regression benchmark\. These benchmarks cover biological, molecular, and social\-network domains and exhibit substantial variation in graph size, structural complexity, and label cardinality\. Their principal statistics are summarized in Table[3](https://arxiv.org/html/2608.00491#A1.T3), following the dataset statistics reported by Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)\.

Table 3\.Descriptive statistics of the datasets used in the main experiments\. The statistics are adapted from Graph\-JEPA\(Skenderi et al\.,[2025](https://arxiv.org/html/2608.00491#bib.bib15)\)\. The symbol “–” indicates that ZINC\-12K is a graph\-regression benchmark and therefore has no discrete class labels\.For PROTEINS, MUTAG, and DD, we use the node attributes or discrete node labels provided by the corresponding datasets\. MUTAG additionally contains categorical edge labels representing chemical bond types\. For ZINC\-12K, categorical atom types and bond types are used as node and edge inputs, respectively\. Graph\-level labels are used only for downstream evaluation and are never provided to HP\-JEPA during self\-supervised pretraining\.

For each graph\-classification dataset, we perform stratified 10\-fold cross\-validation under five random seeds\. Under each seed, the dataset is divided into 10 approximately class\-balanced folds\. For each fold, one fold is reserved exclusively for testing, while the remaining folds are used for self\-supervised pretraining and downstream classifier fitting\. HP\-JEPA is pretrained independently for every fold without access to any graph in the corresponding test fold\. After pretraining, graph representations are extracted using the frozen EMA target token encoder, and a linear classifier is fitted using only the labeled training representations\. The classification accuracy is first averaged across the 10 folds for each seed, after which the mean and standard deviation are calculated across the five seed\-level results\.

For ZINC\-12K, we use the standard fixed split containing 10,000 training graphs, 1,000 validation graphs, and 1,000 test graphs\. The training set is used for self\-supervised pretraining and downstream regressor fitting, the validation set is used for model and regularization selection, and the test set is reserved for final evaluation\. We report the test mean absolute error \(MAE\) averaged over 10 random seeds, where a lower value indicates better performance\.

### A\.1\.Pretraining Configuration

Table[4](https://arxiv.org/html/2608.00491#A1.T4)reports the selected dataset\-specific pretraining configurations\. The configured region capacities follow the notationKℓK\_\{\\ell\}introduced in Section[4\.1](https://arxiv.org/html/2608.00491#S4.SS1), whileMℓM\_\{\\ell\}denotes the number of target regions sampled at the corresponding resolution\. All configurations use a latent dimension of 512 and four Transformer blocks\. The remaining architectural and optimization settings are selected separately for each benchmark\.

Table 4\.Dataset\-specific HP\-JEPA pretraining configurations\.KℓK\_\{\\ell\}lists the configured region capacities across resolutions,MℓM\_\{\\ell\}lists the corresponding numbers of selected target regions,dRWd\_\{\\mathrm\{RW\}\}is the random\-walk positional\-encoding dimension,Dropis module dropout,Tok\.is token dropout, andClipis the maximum gradient norm\. All configurations use latent dimension 512 and four Transformer blocks\.Following self\-supervised pretraining, graph representations are extracted using the frozen patch\-token encoder, structural\-query encoder, and EMA target token encoder\. At each active resolutionℓ\\ell, valid token representations are mean\-pooled to obtainhG\(ℓ\)h\_\{G\}^\{\(\\ell\)\}\. The resolution\-specific representations are then combined using the dataset\-specific readout reported in Table[5](https://arxiv.org/html/2608.00491#A1.T5)\. For a uniformly smoothed task\-specific readout, the coefficientλunif\\lambda\_\{\\mathrm\{unif\}\}follows the definition in Section[4\.3](https://arxiv.org/html/2608.00491#S4.SS3):λunif=1\\lambda\_\{\\mathrm\{unif\}\}=1gives uniform averaging, whereasλunif=0\.75\\lambda\_\{\\mathrm\{unif\}\}=0\.75gives a mixture containing0\.750\.75uniform weight and0\.250\.25learned weight\. A linear classifier or regressor is fitted using only the labeled training representations, and its regularization coefficient is selected without using the corresponding test data\.

Table 5\.Dataset\-specific downstream readout and linear\-probe settings\. The readout column reports the mixture of uniform and learned resolution weights, andαprobe\\alpha\_\{\\mathrm\{probe\}\}denotes the recorded regularization parameter of the downstream linear probe\.##### Reproducibility and leakage prevention\.

Dataset labels are used only when constructing stratified classification splits, estimating task\-specific resolution weights, and fitting the downstream linear probe\. They are not provided to HP\-JEPA during self\-supervised pretraining\. For every classification fold, the held\-out fold is excluded from self\-supervised optimization, early stopping, resolution\-weight estimation, and downstream probe fitting\. For ZINC\-12K, the test set is likewise excluded from pretraining, early stopping, readout selection, and probe\-regularization selection\.

## Appendix BGeometry and Population Characterization of the Latent Target

This section characterizes the geometric information retained by the latent\-coordinate target and the population quantity recovered by the coordinate\-wise Smooth L1 objective\. The results concern the target construction itself and do not imply that the unconstrained predictor output necessarily lies on the Lorentz hyperboloid\.

Foru∈ℝdu\\in\\mathbb\{R\}^\{d\}, define

\(1\)m​\(u\):=1d​𝟏d⊤​u,Φ​\(u\):=\[cosh⁡\(m​\(u\)\)sinh⁡\(m​\(u\)\)\],m\(u\):=\\frac\{1\}\{d\}\\mathbf\{1\}\_\{d\}^\{\\top\}u,\\qquad\\Phi\(u\):=\\begin\{bmatrix\}\\cosh\(m\(u\)\)\\\\ \\sinh\(m\(u\)\)\\end\{bmatrix\},where𝟏d∈ℝd\\mathbf\{1\}\_\{d\}\\in\\mathbb\{R\}^\{d\}denotes the all\-ones vector\. Let

\(2\)JL:=\[−1001\]J\_\{\\mathrm\{L\}\}:=\\begin\{bmatrix\}\-1&0\\\\ 0&1\\end\{bmatrix\}and define the positive branch of the one\-dimensional Lorentz hyperboloid as

\(3\)ℍ\+1:=\{y∈ℝ2:y⊤​JL​y=−1,y1\>0\}\.\\mathbb\{H\}\_\{\+\}^\{1\}:=\\left\\\{y\\in\\mathbb\{R\}^\{2\}:y^\{\\top\}J\_\{\\mathrm\{L\}\}y=\-1,\\;y\_\{1\}\>0\\right\\\}\.Fory,y′∈ℍ\+1y,y^\{\\prime\}\\in\\mathbb\{H\}\_\{\+\}^\{1\}, their Lorentz geodesic distance is

\(4\)dℍ​\(y,y′\):=arcosh⁡\(−y⊤​JL​y′\)\.d\_\{\\mathbb\{H\}\}\(y,y^\{\\prime\}\):=\\operatorname\{arcosh\}\\left\(\-y^\{\\top\}J\_\{\\mathrm\{L\}\}y^\{\\prime\}\\right\)\.
###### Proposition 0 \(Geometry and information content of the latent\-coordinate map\)\.

For everyu,v∈ℝdu,v\\in\\mathbb\{R\}^\{d\}, the coordinate map in Equation \([1](https://arxiv.org/html/2608.00491#A2.E1)\) satisfies

\(5\)Φ​\(u\)∈ℍ\+1\.\\Phi\(u\)\\in\\mathbb\{H\}\_\{\+\}^\{1\}\.Moreover,

\(6\)dℍ​\(Φ​\(u\),Φ​\(v\)\)=\|m​\(u\)−m​\(v\)\|=1d​\|𝟏d⊤​\(u−v\)\|\.d\_\{\\mathbb\{H\}\}\\left\(\\Phi\(u\),\\Phi\(v\)\\right\)=\|m\(u\)\-m\(v\)\|=\\frac\{1\}\{d\}\\left\|\\mathbf\{1\}\_\{d\}^\{\\top\}\(u\-v\)\\right\|\.Consequently,

\(7\)dℍ​\(Φ​\(u\),Φ​\(v\)\)≤1d​‖u−v‖2\.d\_\{\\mathbb\{H\}\}\\left\(\\Phi\(u\),\\Phi\(v\)\\right\)\\leq\\frac\{1\}\{\\sqrt\{d\}\}\\\|u\-v\\\|\_\{2\}\.Finally,

\(8\)Φ​\(u\)=Φ​\(v\)⟺u−v∈𝟏d⟂,\\Phi\(u\)=\\Phi\(v\)\\quad\\Longleftrightarrow\\quad u\-v\\in\\mathbf\{1\}\_\{d\}^\{\\perp\},where

\(9\)𝟏d⟂:=\{a∈ℝd:𝟏d⊤​a=0\}\.\\mathbf\{1\}\_\{d\}^\{\\perp\}:=\\left\\\{a\\in\\mathbb\{R\}^\{d\}:\\mathbf\{1\}\_\{d\}^\{\\top\}a=0\\right\\\}\.Therefore,Φ\\Phipreserves exactly the feature\-mean projection of the target\-token state and is invariant to all latent components orthogonal to𝟏d\\mathbf\{1\}\_\{d\}\.

###### Proof\.

Let

\(10\)a:=m​\(u\),b:=m​\(v\)\.a:=m\(u\),\\qquad b:=m\(v\)\.Using the identitycosh2⁡\(a\)−sinh2⁡\(a\)=1\\cosh^\{2\}\(a\)\-\\sinh^\{2\}\(a\)=1, we obtain

Φ​\(u\)⊤​JL​Φ​\(u\)\\displaystyle\\Phi\(u\)^\{\\top\}J\_\{\\mathrm\{L\}\}\\Phi\(u\)=−cosh2⁡\(a\)\+sinh2⁡\(a\)\\displaystyle=\-\\cosh^\{2\}\(a\)\+\\sinh^\{2\}\(a\)\(11\)=−1\.\\displaystyle=\-1\.The first coordinate ofΦ​\(u\)\\Phi\(u\)iscosh⁡\(a\)\>0\\cosh\(a\)\>0\. Hence,Φ​\(u\)∈ℍ\+1\\Phi\(u\)\\in\\mathbb\{H\}\_\{\+\}^\{1\}\.

Next, the Lorentz inner product betweenΦ​\(u\)\\Phi\(u\)andΦ​\(v\)\\Phi\(v\)satisfies

−Φ​\(u\)⊤​JL​Φ​\(v\)\\displaystyle\-\\Phi\(u\)^\{\\top\}J\_\{\\mathrm\{L\}\}\\Phi\(v\)=cosh⁡\(a\)​cosh⁡\(b\)−sinh⁡\(a\)​sinh⁡\(b\)\\displaystyle=\\cosh\(a\)\\cosh\(b\)\-\\sinh\(a\)\\sinh\(b\)\(12\)=cosh⁡\(a−b\),\\displaystyle=\\cosh\(a\-b\),where the second equality follows from the hyperbolic difference identity\. Therefore,

dℍ​\(Φ​\(u\),Φ​\(v\)\)\\displaystyle d\_\{\\mathbb\{H\}\}\\left\(\\Phi\(u\),\\Phi\(v\)\\right\)=arcosh⁡\(cosh⁡\(a−b\)\)\\displaystyle=\\operatorname\{arcosh\}\\left\(\\cosh\(a\-b\)\\right\)\(13\)=\|a−b\|\.\\displaystyle=\|a\-b\|\.The last equality follows fromarcosh⁡\(cosh⁡r\)=\|r\|\\operatorname\{arcosh\}\(\\cosh r\)=\|r\|for everyr∈ℝr\\in\\mathbb\{R\}\. Substituting the definition ofmmyields

dℍ​\(Φ​\(u\),Φ​\(v\)\)\\displaystyle d\_\{\\mathbb\{H\}\}\\left\(\\Phi\(u\),\\Phi\(v\)\\right\)=\|1d​𝟏d⊤​u−1d​𝟏d⊤​v\|\\displaystyle=\\left\|\\frac\{1\}\{d\}\\mathbf\{1\}\_\{d\}^\{\\top\}u\-\\frac\{1\}\{d\}\\mathbf\{1\}\_\{d\}^\{\\top\}v\\right\|\(14\)=1d​\|𝟏d⊤​\(u−v\)\|\.\\displaystyle=\\frac\{1\}\{d\}\\left\|\\mathbf\{1\}\_\{d\}^\{\\top\}\(u\-v\)\\right\|\.
By the Cauchy–Schwarz inequality,

1d​\|𝟏d⊤​\(u−v\)\|\\displaystyle\\frac\{1\}\{d\}\\left\|\\mathbf\{1\}\_\{d\}^\{\\top\}\(u\-v\)\\right\|≤1d​‖𝟏d‖2​‖u−v‖2\\displaystyle\\leq\\frac\{1\}\{d\}\\\|\\mathbf\{1\}\_\{d\}\\\|\_\{2\}\\\|u\-v\\\|\_\{2\}\(15\)=1d​‖u−v‖2,\\displaystyle=\\frac\{1\}\{\\sqrt\{d\}\}\\\|u\-v\\\|\_\{2\},because‖𝟏d‖2=d\\\|\\mathbf\{1\}\_\{d\}\\\|\_\{2\}=\\sqrt\{d\}\. This proves Equation \([7](https://arxiv.org/html/2608.00491#A2.E7)\)\.

It remains to characterize when two target\-token states have the same coordinate representation\. IfΦ​\(u\)=Φ​\(v\)\\Phi\(u\)=\\Phi\(v\), then their Lorentz distance is zero\. Equation \([6](https://arxiv.org/html/2608.00491#A2.E6)\) therefore gives

\(16\)\|m​\(u\)−m​\(v\)\|=0,\|m\(u\)\-m\(v\)\|=0,which is equivalent to

\(17\)𝟏d⊤​\(u−v\)=0\.\\mathbf\{1\}\_\{d\}^\{\\top\}\(u\-v\)=0\.Hence,u−v∈𝟏d⟂u\-v\\in\\mathbf\{1\}\_\{d\}^\{\\perp\}\. Conversely, ifu−v∈𝟏d⟂u\-v\\in\\mathbf\{1\}\_\{d\}^\{\\perp\}, thenm​\(u\)=m​\(v\)m\(u\)=m\(v\), and the definition ofΦ\\Phidirectly givesΦ​\(u\)=Φ​\(v\)\\Phi\(u\)=\\Phi\(v\)\. This proves Equation \([8](https://arxiv.org/html/2608.00491#A2.E8)\)\. ∎

We next characterize the population target of coordinate\-wise Smooth L1 prediction\. For a scalar residualr∈ℝr\\in\\mathbb\{R\}and thresholdβ\>0\\beta\>0, define

\(18\)φβ​\(r\):=\{r22​β,\|r\|≤β,\|r\|−β2,\|r\|\>β\.\\varphi\_\{\\beta\}\(r\):=\\begin\{cases\}\\dfrac\{r^\{2\}\}\{2\\beta\},&\|r\|\\leq\\beta,\\\\\[6\.0pt\] \|r\|\-\\dfrac\{\\beta\}\{2\},&\|r\|\>\\beta\.\\end\{cases\}Let

\(19\)𝖷:=scℓ\(ℓ\)\+pt\(ℓ\)\\mathsf\{X\}:=s\_\{c\_\{\\ell\}\}^\{\(\\ell\)\}\+p\_\{t\}^\{\(\\ell\)\}denote the predictor input and let

\(20\)𝖸:=Φ​\(ut\(ℓ\)\)∈ℝ2\\mathsf\{Y\}:=\\Phi\\left\(u\_\{t\}^\{\(\\ell\)\}\\right\)\\in\\mathbb\{R\}^\{2\}denote the corresponding target coordinate\. For coordinatej∈\{1,2\}j\\in\\\{1,2\\\}and a fixed input valuexx, define the conditional risk

\(21\)Qj,x\(a\):=𝔼\[φβ\(a−𝖸j\)\|𝖷=x\]\.Q\_\{j,x\}\(a\):=\\mathbb\{E\}\\left\[\\varphi\_\{\\beta\}\\left\(a\-\\mathsf\{Y\}\_\{j\}\\right\)\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\.
###### Proposition 0 \(Population target of coordinate\-wise Smooth L1\)\.

Forℙ𝖷\\mathbb\{P\}\_\{\\mathsf\{X\}\}\-almost everyxx, suppose that

\(22\)𝔼\[\|𝖸j\|\|𝖷=x\]<∞\.\\mathbb\{E\}\\left\[\|\\mathsf\{Y\}\_\{j\}\|\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]<\\infty\.ThenQj,xQ\_\{j,x\}is convex, coercive, and admits at least one minimizer\. A scalaraj,x⋆a\_\{j,x\}^\{\\star\}minimizesQj,xQ\_\{j,x\}if and only if

\(23\)𝔼\[clip\(aj,x⋆−𝖸j,−β,β\)\|𝖷=x\]=0\.\\mathbb\{E\}\\left\[\\operatorname\{clip\}\\left\(a\_\{j,x\}^\{\\star\}\-\\mathsf\{Y\}\_\{j\},\-\\beta,\\beta\\right\)\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]=0\.Thus, the population\-optimal coordinate\-wise Smooth L1 predictor returns a conditional Huber location functional of the target coordinate\.

In particular, let

\(24\)μj\(x\):=𝔼\[𝖸j\|𝖷=x\]\.\\mu\_\{j\}\(x\):=\\mathbb\{E\}\\left\[\\mathsf\{Y\}\_\{j\}\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\.If

\(25\)Pr⁡\(\|𝖸j−μj​\(x\)\|≤β\|𝖷=x\)=1,\\Pr\\left\(\|\\mathsf\{Y\}\_\{j\}\-\\mu\_\{j\}\(x\)\|\\leq\\beta\\,\\middle\|\\,\\mathsf\{X\}=x\\right\)=1,thenμj​\(x\)\\mu\_\{j\}\(x\)is a minimizer ofQj,xQ\_\{j,x\}\.

###### Proof\.

The scalar Smooth L1 function is convex and continuously differentiable\. Its derivative is

\(26\)φβ′​\(r\)=1β​clip⁡\(r,−β,β\)\.\\varphi\_\{\\beta\}^\{\\prime\}\(r\)=\\frac\{1\}\{\\beta\}\\operatorname\{clip\}\\left\(r,\-\\beta,\\beta\\right\)\.In particular,

\(27\)\|φβ′​\(r\)\|≤1\|\\varphi\_\{\\beta\}^\{\\prime\}\(r\)\|\\leq 1for everyr∈ℝr\\in\\mathbb\{R\}\. The bounded derivative permits differentiation under the conditional expectation, yielding

Qj,x′​\(a\)\\displaystyle Q\_\{j,x\}^\{\\prime\}\(a\)=𝔼\[φβ′\(a−𝖸j\)\|𝖷=x\]\\displaystyle=\\mathbb\{E\}\\left\[\\varphi\_\{\\beta\}^\{\\prime\}\\left\(a\-\\mathsf\{Y\}\_\{j\}\\right\)\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\(28\)=1β𝔼\[clip\(a−𝖸j,−β,β\)\|𝖷=x\]\.\\displaystyle=\\frac\{1\}\{\\beta\}\\mathbb\{E\}\\left\[\\operatorname\{clip\}\\left\(a\-\\mathsf\{Y\}\_\{j\},\-\\beta,\\beta\\right\)\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\.
Becauseφβ\\varphi\_\{\\beta\}is convex, its conditional expectationQj,xQ\_\{j,x\}is also convex\. Moreover, for everyr∈ℝr\\in\\mathbb\{R\},

\(29\)φβ​\(r\)≥\|r\|−β2\.\\varphi\_\{\\beta\}\(r\)\\geq\|r\|\-\\frac\{\\beta\}\{2\}\.Therefore,

Qj,x​\(a\)\\displaystyle Q\_\{j,x\}\(a\)≥𝔼\[\|a−𝖸j\|\|𝖷=x\]−β2\\displaystyle\\geq\\mathbb\{E\}\\left\[\|a\-\\mathsf\{Y\}\_\{j\}\|\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\-\\frac\{\\beta\}\{2\}\(30\)≥\|a\|−𝔼\[\|𝖸j\|\|𝖷=x\]−β2\.\\displaystyle\\geq\|a\|\-\\mathbb\{E\}\\left\[\|\\mathsf\{Y\}\_\{j\}\|\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\-\\frac\{\\beta\}\{2\}\.Under Equation \([22](https://arxiv.org/html/2608.00491#A2.E22)\), the right\-hand side approaches infinity as\|a\|→∞\|a\|\\rightarrow\\infty\. Hence,Qj,xQ\_\{j,x\}is coercive and admits at least one minimizer\.

SinceQj,xQ\_\{j,x\}is convex and differentiable, a scalaraj,x⋆a\_\{j,x\}^\{\\star\}is a minimizer if and only if

\(31\)Qj,x′​\(aj,x⋆\)=0\.Q\_\{j,x\}^\{\\prime\}\(a\_\{j,x\}^\{\\star\}\)=0\.Substituting Equation \([28](https://arxiv.org/html/2608.00491#A2.E28)\) proves Equation \([23](https://arxiv.org/html/2608.00491#A2.E23)\)\.

Now suppose that Equation \([25](https://arxiv.org/html/2608.00491#A2.E25)\) holds\. Ata=μj​\(x\)a=\\mu\_\{j\}\(x\), clipping is inactive almost surely conditional on𝖷=x\\mathsf\{X\}=x, and therefore

\(32\)clip⁡\(μj​\(x\)−𝖸j,−β,β\)=μj​\(x\)−𝖸j\.\\operatorname\{clip\}\\left\(\\mu\_\{j\}\(x\)\-\\mathsf\{Y\}\_\{j\},\-\\beta,\\beta\\right\)=\\mu\_\{j\}\(x\)\-\\mathsf\{Y\}\_\{j\}\.It follows that

𝔼\[clip\(μj\(x\)−𝖸j,−β,β\)\|𝖷=x\]\\displaystyle\\mathbb\{E\}\\left\[\\operatorname\{clip\}\\left\(\\mu\_\{j\}\(x\)\-\\mathsf\{Y\}\_\{j\},\-\\beta,\\beta\\right\)\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]=μj\(x\)−𝔼\[𝖸j\|𝖷=x\]\\displaystyle=\\mu\_\{j\}\(x\)\-\\mathbb\{E\}\\left\[\\mathsf\{Y\}\_\{j\}\\,\\middle\|\\,\\mathsf\{X\}=x\\right\]\(33\)=0\.\\displaystyle=0\.Equation \([23](https://arxiv.org/html/2608.00491#A2.E23)\) then shows thatμj​\(x\)\\mu\_\{j\}\(x\)is a minimizer\. Since the vector\-valued Smooth L1 loss used by HP\-JEPA averages the two scalar coordinate losses, its population minimization separates coordinate\-wise\. ∎

## Appendix CGeneralization of Task\-Specific Resolution Weighting

This section analyzes the task\-specific resolution\-weighting family after self\-supervised pretraining\. We condition on the pretrained encoder and treat the resolution\-specific representations as fixed feature maps\. The analysis concerns the downstream hypothesis class and does not establish convergence of the nonlinear pretraining procedure or of a particular nonconvex optimization algorithm\.

Let

\(34\)S=\{\(Gi,Yi\)\}i=1NS=\\left\\\{\(G\_\{i\},Y\_\{i\}\)\\right\\\}\_\{i=1\}^\{N\}beNNindependent labeled downstream examples drawn from a distributionℙ\\mathbb\{P\}\. For each resolutionℓ∈\{1,…,L\}\\ell\\in\\\{1,\\ldots,L\\\}, define the fixed\-resolution linear predictor class

\(35\)ℱℓ:=\{fℓ,w​\(G\)=⟨w,hG\(ℓ\)⟩:‖w‖2≤B\}\.\\mathcal\{F\}\_\{\\ell\}:=\\left\\\{f\_\{\\ell,w\}\(G\)=\\left\\langle w,h\_\{G\}^\{\(\\ell\)\}\\right\\rangle:\\\|w\\\|\_\{2\}\\leq B\\right\\\}\.Let

\(36\)ΔL:=\{α∈ℝ\+L:∑ℓ=1Lαℓ=1\}\\Delta\_\{L\}:=\\left\\\{\\alpha\\in\\mathbb\{R\}\_\{\+\}^\{L\}:\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}=1\\right\\\}denote the closed probability simplex\. The resolution\-weighted class is

\(37\)ℱmix:=\{fw,α​\(G\)=⟨w,∑ℓ=1Lαℓ​hG\(ℓ\)⟩:‖w‖2≤B,α∈ΔL\}\.\\mathcal\{F\}\_\{\\mathrm\{mix\}\}:=\\left\\\{f\_\{w,\\alpha\}\(G\)=\\left\\langle w,\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}h\_\{G\}^\{\(\\ell\)\}\\right\\rangle:\\\|w\\\|\_\{2\}\\leq B,\\;\\alpha\\in\\Delta\_\{L\}\\right\\\}\.
###### Assumption 1 \(Bounded downstream hypothesis class\)\.

There exist constantsRh,B,κ,M\>0R\_\{h\},B,\\kappa,M\>0such that the following conditions hold:

\(38\)‖hG\(ℓ\)‖2\\displaystyle\\\|h\_\{G\}^\{\(\\ell\)\}\\\|\_\{2\}≤Rh\\displaystyle\\leq R\_\{h\}for everyGandℓ,\\displaystyle\\text\{for every $G$ and $\\ell$\},\(39\)‖w‖2\\displaystyle\\\|w\\\|\_\{2\}≤B,\\displaystyle\\leq B,\(40\)0≤ℓds​\(y^,y\)\\displaystyle 0\\leq\\ell\_\{\\mathrm\{ds\}\}\(\\widehat\{y\},y\)≤M,\\displaystyle\\leq M,\(41\)\|ℓds​\(a,y\)−ℓds​\(b,y\)\|\\displaystyle\\left\|\\ell\_\{\\mathrm\{ds\}\}\(a,y\)\-\\ell\_\{\\mathrm\{ds\}\}\(b,y\)\\right\|≤κ​\|a−b\|\\displaystyle\\leq\\kappa\|a\-b\|for everya,b,y\.\\displaystyle\\text\{for every $a,b,y$\}\.

For a predictorff, define its population and empirical risks by

\(42\)ℛ​\(f\):=𝔼\(G,Y\)∼ℙ​\[ℓds​\(f​\(G\),Y\)\]\\mathcal\{R\}\(f\):=\\mathbb\{E\}\_\{\(G,Y\)\\sim\\mathbb\{P\}\}\\left\[\\ell\_\{\\mathrm\{ds\}\}\\left\(f\(G\),Y\\right\)\\right\]and

\(43\)ℛ^S​\(f\):=1N​∑i=1Nℓds​\(f​\(Gi\),Yi\)\.\\widehat\{\\mathcal\{R\}\}\_\{S\}\(f\):=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\ell\_\{\\mathrm\{ds\}\}\\left\(f\(G\_\{i\}\),Y\_\{i\}\\right\)\.Letf^mix∈ℱmix\\widehat\{f\}\_\{\\mathrm\{mix\}\}\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}be anεopt\\varepsilon\_\{\\mathrm\{opt\}\}\-approximate empirical\-risk minimizer satisfying

\(44\)ℛ^S​\(f^mix\)≤inff∈ℱmixℛ^S​\(f\)\+εopt\.\\widehat\{\\mathcal\{R\}\}\_\{S\}\\left\(\\widehat\{f\}\_\{\\mathrm\{mix\}\}\\right\)\\leq\\inf\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\}\\widehat\{\\mathcal\{R\}\}\_\{S\}\(f\)\+\\varepsilon\_\{\\mathrm\{opt\}\}\.
###### Theorem 1 \(Resolution\-adaptive oracle bound\)\.

Under Assumption[1](https://arxiv.org/html/2608.00491#Thmassumption1), there exist universal constantsC1,C2\>0C\_\{1\},C\_\{2\}\>0such that, for everyδ∈\(0,1\)\\delta\\in\(0,1\), with probability at least1−δ1\-\\delta,

ℛ​\(f^mix\)≤\\displaystyle\\mathcal\{R\}\\left\(\\widehat\{f\}\_\{\\mathrm\{mix\}\}\\right\)\\leq\\;minℓ∈\{1,…,L\}​inff∈ℱℓℛ​\(f\)\\displaystyle\\min\_\{\\ell\\in\\\{1,\\ldots,L\\\}\}\\inf\_\{f\\in\\mathcal\{F\}\_\{\\ell\}\}\\mathcal\{R\}\(f\)\(45\)\+C1​κ​B​Rh​1\+log⁡LN\+C2​M​log⁡\(2/δ\)N\+εopt\.\\displaystyle\+C\_\{1\}\\kappa BR\_\{h\}\\sqrt\{\\frac\{1\+\\log L\}\{N\}\}\+C\_\{2\}M\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{N\}\}\+\\varepsilon\_\{\\mathrm\{opt\}\}\.Thus, the closed resolution\-weighting family competes with the best fixed\-resolution linear predictor selected in hindsight, while its additional dependence on the number of configured resolutions is logarithmic inLL\.

###### Proof\.

For a scalar function classℱ\\mathcal\{F\}, define its empirical Rademacher complexity on the graph inputs inSSas

\(46\)ℜ^S​\(ℱ\):=𝔼σ​\[supf∈ℱ1N​∑i=1Nσi​f​\(Gi\)\],\\widehat\{\\mathfrak\{R\}\}\_\{S\}\(\\mathcal\{F\}\):=\\mathbb\{E\}\_\{\\sigma\}\\left\[\\sup\_\{f\\in\\mathcal\{F\}\}\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}f\(G\_\{i\}\)\\right\],whereσ1,…,σN\\sigma\_\{1\},\\ldots,\\sigma\_\{N\}are independent Rademacher random variables\.

For a fixed resolutionℓ\\ell, Euclidean norm duality gives

ℜ^S​\(ℱℓ\)\\displaystyle\\widehat\{\\mathfrak\{R\}\}\_\{S\}\(\\mathcal\{F\}\_\{\\ell\}\)=𝔼σ​\[sup‖w‖2≤B1N​∑i=1Nσi​⟨w,hGi\(ℓ\)⟩\]\\displaystyle=\\mathbb\{E\}\_\{\\sigma\}\\left\[\\sup\_\{\\\|w\\\|\_\{2\}\\leq B\}\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}\\left\\langle w,h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\rangle\\right\]\(47\)=BN​𝔼σ​‖∑i=1Nσi​hGi\(ℓ\)‖2\.\\displaystyle=\\frac\{B\}\{N\}\\mathbb\{E\}\_\{\\sigma\}\\left\\\|\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\\|\_\{2\}\.By Jensen’s inequality,

\(48\)𝔼σ​‖∑i=1Nσi​hGi\(ℓ\)‖2\\displaystyle\\mathbb\{E\}\_\{\\sigma\}\\left\\\|\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\\|\_\{2\}≤\(𝔼σ​‖∑i=1Nσi​hGi\(ℓ\)‖22\)1/2\.\\displaystyle\\leq\\left\(\\mathbb\{E\}\_\{\\sigma\}\\left\\\|\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\\|\_\{2\}^\{2\}\\right\)^\{1/2\}\.Expanding the squared norm and using independence and zero mean of the Rademacher variables eliminates the cross terms, so

𝔼σ​‖∑i=1Nσi​hGi\(ℓ\)‖22\\displaystyle\\mathbb\{E\}\_\{\\sigma\}\\left\\\|\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\\|\_\{2\}^\{2\}=∑i=1N‖hGi\(ℓ\)‖22\\displaystyle=\\sum\_\{i=1\}^\{N\}\\left\\\|h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\right\\\|\_\{2\}^\{2\}\(49\)≤N​Rh2\.\\displaystyle\\leq NR\_\{h\}^\{2\}\.Consequently,

\(50\)ℜ^S​\(ℱℓ\)≤B​RhN\.\\widehat\{\\mathfrak\{R\}\}\_\{S\}\(\\mathcal\{F\}\_\{\\ell\}\)\\leq\\frac\{BR\_\{h\}\}\{\\sqrt\{N\}\}\.
We next consider the union of theLLfixed\-resolution classes\. Define

\(51\)Zℓ​\(σ\):=supf∈ℱℓ1N​∑i=1Nσi​f​\(Gi\)\.Z\_\{\\ell\}\(\\sigma\):=\\sup\_\{f\\in\\mathcal\{F\}\_\{\\ell\}\}\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\sigma\_\{i\}f\(G\_\{i\}\)\.For everyf∈ℱℓf\\in\\mathcal\{F\}\_\{\\ell\}, Assumption[1](https://arxiv.org/html/2608.00491#Thmassumption1)gives

\(52\)\|f​\(Gi\)\|≤‖w‖2​‖hGi\(ℓ\)‖2≤B​Rh\.\|f\(G\_\{i\}\)\|\\leq\\\|w\\\|\_\{2\}\\\|h\_\{G\_\{i\}\}^\{\(\\ell\)\}\\\|\_\{2\}\\leq BR\_\{h\}\.Changing one Rademacher sign changesZℓZ\_\{\\ell\}by at most2​B​Rh/N2BR\_\{h\}/N\. The bounded\-difference exponential\-moment inequality therefore yields

\(53\)𝔼σ​\[exp⁡\(λ​\[Zℓ​\(σ\)−𝔼σ​Zℓ​\(σ\)\]\)\]≤exp⁡\(λ2​B2​Rh22​N\)\\mathbb\{E\}\_\{\\sigma\}\\left\[\\exp\\left\(\\lambda\\left\[Z\_\{\\ell\}\(\\sigma\)\-\\mathbb\{E\}\_\{\\sigma\}Z\_\{\\ell\}\(\\sigma\)\\right\]\\right\)\\right\]\\leq\\exp\\left\(\\frac\{\\lambda^\{2\}B^\{2\}R\_\{h\}^\{2\}\}\{2N\}\\right\)for everyλ\>0\\lambda\>0\. Applying the log\-sum\-exp inequality gives

\(54\)𝔼σ​\[maxℓ∈\{1,…,L\}⁡Zℓ​\(σ\)\]≤\\displaystyle\\mathbb\{E\}\_\{\\sigma\}\\left\[\\max\_\{\\ell\\in\\\{1,\\ldots,L\\\}\}Z\_\{\\ell\}\(\\sigma\)\\right\]\\leq\\;maxℓ⁡𝔼σ​\[Zℓ​\(σ\)\]\+log⁡Lλ\+λ​B2​Rh22​N\.\\displaystyle\\max\_\{\\ell\}\\mathbb\{E\}\_\{\\sigma\}\\left\[Z\_\{\\ell\}\(\\sigma\)\\right\]\+\\frac\{\\log L\}\{\\lambda\}\+\\frac\{\\lambda B^\{2\}R\_\{h\}^\{2\}\}\{2N\}\.Optimizing the right\-hand side overλ\\lambdagives

ℜ^S​\(⋃ℓ=1Lℱℓ\)\\displaystyle\\widehat\{\\mathfrak\{R\}\}\_\{S\}\\left\(\\bigcup\_\{\\ell=1\}^\{L\}\\mathcal\{F\}\_\{\\ell\}\\right\)≤maxℓ⁡ℜ^S​\(ℱℓ\)\+B​Rh​2​log⁡LN\\displaystyle\\leq\\max\_\{\\ell\}\\widehat\{\\mathfrak\{R\}\}\_\{S\}\(\\mathcal\{F\}\_\{\\ell\}\)\+BR\_\{h\}\\sqrt\{\\frac\{2\\log L\}\{N\}\}\(55\)≤B​Rh​\(1N\+2​log⁡LN\)\.\\displaystyle\\leq BR\_\{h\}\\left\(\\frac\{1\}\{\\sqrt\{N\}\}\+\\sqrt\{\\frac\{2\\log L\}\{N\}\}\\right\)\.
For everywwandα∈ΔL\\alpha\\in\\Delta\_\{L\},

fw,α​\(G\)\\displaystyle f\_\{w,\\alpha\}\(G\)=⟨w,∑ℓ=1Lαℓ​hG\(ℓ\)⟩\\displaystyle=\\left\\langle w,\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}h\_\{G\}^\{\(\\ell\)\}\\right\\rangle\(56\)=∑ℓ=1Lαℓ​⟨w,hG\(ℓ\)⟩\.\\displaystyle=\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}\\left\\langle w,h\_\{G\}^\{\(\\ell\)\}\\right\\rangle\.Thus,

\(57\)ℱmix⊆conv⁡\(⋃ℓ=1Lℱℓ\)\.\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\\subseteq\\operatorname\{conv\}\\left\(\\bigcup\_\{\\ell=1\}^\{L\}\\mathcal\{F\}\_\{\\ell\}\\right\)\.The supremum of a linear functional over a convex hull equals its supremum over the original set\. It follows that

ℜ^S​\(ℱmix\)\\displaystyle\\widehat\{\\mathfrak\{R\}\}\_\{S\}\\left\(\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\\right\)≤ℜ^S​\(conv⁡\(⋃ℓ=1Lℱℓ\)\)\\displaystyle\\leq\\widehat\{\\mathfrak\{R\}\}\_\{S\}\\left\(\\operatorname\{conv\}\\left\(\\bigcup\_\{\\ell=1\}^\{L\}\\mathcal\{F\}\_\{\\ell\}\\right\)\\right\)\(58\)=ℜ^S​\(⋃ℓ=1Lℱℓ\)\.\\displaystyle=\\widehat\{\\mathfrak\{R\}\}\_\{S\}\\left\(\\bigcup\_\{\\ell=1\}^\{L\}\\mathcal\{F\}\_\{\\ell\}\\right\)\.Combining Equations \([55](https://arxiv.org/html/2608.00491#A3.E55)\) and \([58](https://arxiv.org/html/2608.00491#A3.E58)\), and taking expectation over the sample, gives

\(59\)ℜN​\(ℱmix\)≤B​Rh​\(1N\+2​log⁡LN\),\\mathfrak\{R\}\_\{N\}\\left\(\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\\right\)\\leq BR\_\{h\}\\left\(\\frac\{1\}\{\\sqrt\{N\}\}\+\\sqrt\{\\frac\{2\\log L\}\{N\}\}\\right\),whereℜN\\mathfrak\{R\}\_\{N\}denotes the expected Rademacher complexity\.

By the contraction principle and theκ\\kappa\-Lipschitz property ofℓds\\ell\_\{\\mathrm\{ds\}\}, the complexity of the induced loss class is bounded byκ​ℜN​\(ℱmix\)\\kappa\\mathfrak\{R\}\_\{N\}\(\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\)\. A standard Rademacher uniform\-convergence bound for bounded losses therefore implies that there exists a universal constantC0\>0C\_\{0\}\>0such that, with probability at least1−δ1\-\\delta,

supf∈ℱmix\|ℛ​\(f\)−ℛ^S​\(f\)\|≤\\displaystyle\\sup\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\}\\left\|\\mathcal\{R\}\(f\)\-\\widehat\{\\mathcal\{R\}\}\_\{S\}\(f\)\\right\|\\leq\\;C0​κ​B​Rh​\(1N\+2​log⁡LN\)\\displaystyle C\_\{0\}\\kappa BR\_\{h\}\\left\(\\frac\{1\}\{\\sqrt\{N\}\}\+\\sqrt\{\\frac\{2\\log L\}\{N\}\}\\right\)\(60\)\+C0​M​log⁡\(2/δ\)N\.\\displaystyle\+C\_\{0\}M\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{N\}\}\.Denote the right\-hand side byΓN​\(δ\)\\Gamma\_\{N\}\(\\delta\)\. On the event in Equation \([60](https://arxiv.org/html/2608.00491#A3.E60)\),

ℛ​\(f^mix\)\\displaystyle\\mathcal\{R\}\\left\(\\widehat\{f\}\_\{\\mathrm\{mix\}\}\\right\)≤ℛ^S​\(f^mix\)\+ΓN​\(δ\)\\displaystyle\\leq\\widehat\{\\mathcal\{R\}\}\_\{S\}\\left\(\\widehat\{f\}\_\{\\mathrm\{mix\}\}\\right\)\+\\Gamma\_\{N\}\(\\delta\)≤inff∈ℱmixℛ^S​\(f\)\+εopt\+ΓN​\(δ\)\\displaystyle\\leq\\inf\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\}\\widehat\{\\mathcal\{R\}\}\_\{S\}\(f\)\+\\varepsilon\_\{\\mathrm\{opt\}\}\+\\Gamma\_\{N\}\(\\delta\)\(61\)≤inff∈ℱmixℛ​\(f\)\+εopt\+2​ΓN​\(δ\)\.\\displaystyle\\leq\\inf\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\}\\mathcal\{R\}\(f\)\+\\varepsilon\_\{\\mathrm\{opt\}\}\+2\\Gamma\_\{N\}\(\\delta\)\.
For every resolutionℓ\\ell, the standard basis vectoreℓe\_\{\\ell\}belongs to the closed simplexΔL\\Delta\_\{L\}\. Choosingα=eℓ\\alpha=e\_\{\\ell\}gives

\(62\)fw,eℓ​\(G\)=⟨w,hG\(ℓ\)⟩\.f\_\{w,e\_\{\\ell\}\}\(G\)=\\left\\langle w,h\_\{G\}^\{\(\\ell\)\}\\right\\rangle\.Therefore,

\(63\)⋃ℓ=1Lℱℓ⊆ℱmix,\\bigcup\_\{\\ell=1\}^\{L\}\\mathcal\{F\}\_\{\\ell\}\\subseteq\\mathcal\{F\}\_\{\\mathrm\{mix\}\},and hence

\(64\)inff∈ℱmixℛ​\(f\)≤minℓ∈\{1,…,L\}​inff∈ℱℓℛ​\(f\)\.\\inf\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\}\\mathcal\{R\}\(f\)\\leq\\min\_\{\\ell\\in\\\{1,\\ldots,L\\\}\}\\inf\_\{f\\in\\mathcal\{F\}\_\{\\ell\}\}\\mathcal\{R\}\(f\)\.Substituting Equations \([60](https://arxiv.org/html/2608.00491#A3.E60)\) and \([64](https://arxiv.org/html/2608.00491#A3.E64)\) into Equation \([61](https://arxiv.org/html/2608.00491#A3.E61)\), and absorbing numerical factors into universal constantsC1C\_\{1\}andC2C\_\{2\}, proves Equation \([45](https://arxiv.org/html/2608.00491#A3.E45)\)\. ∎

##### Relation to the softmax parameterization\.

A finite softmax logit vector produces a weight vector in the interior ofΔL\\Delta\_\{L\}, whereas a fixed\-resolution predictor corresponds to a simplex vertex\. Therefore, the exact class\-containment statement in Theorem[1](https://arxiv.org/html/2608.00491#A3.Thmtheorem1)applies to the closure of the softmax\-parameterized family\. Every simplex vertex can be approached arbitrarily closely by increasing one resolution logit relative to the remaining logits\.

We next quantify the effect of the uniform smoothing used in the task\-specific readout\. Let

\(65\)uL:=1L​𝟏Lu\_\{L\}:=\\frac\{1\}\{L\}\\mathbf\{1\}\_\{L\}and, forα∈ΔL\\alpha\\in\\Delta\_\{L\}, define

\(66\)α\(λ\):=\(1−λunif\)​α\+λunif​uL,λunif∈\[0,1\]\.\\alpha^\{\(\\lambda\)\}:=\(1\-\\lambda\_\{\\mathrm\{unif\}\}\)\\alpha\+\\lambda\_\{\\mathrm\{unif\}\}u\_\{L\},\\qquad\\lambda\_\{\\mathrm\{unif\}\}\\in\[0,1\]\.
###### Lemma 0 \(Stability under uniform resolution smoothing\)\.

Forα∈ΔL\\alpha\\in\\Delta\_\{L\}, define

\(67\)hGα:=∑ℓ=1Lαℓ​hG\(ℓ\)h\_\{G\}^\{\\alpha\}:=\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}h\_\{G\}^\{\(\\ell\)\}and

\(68\)hGα\(λ\):=∑ℓ=1Lαℓ\(λ\)​hG\(ℓ\)\.h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}:=\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}^\{\(\\lambda\)\}h\_\{G\}^\{\(\\ell\)\}\.Under Assumption[1](https://arxiv.org/html/2608.00491#Thmassumption1),

\(69\)‖hGα\(λ\)−hGα‖2≤2​λunif​Rh\.\\left\\\|h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\-h\_\{G\}^\{\\alpha\}\\right\\\|\_\{2\}\\leq 2\\lambda\_\{\\mathrm\{unif\}\}R\_\{h\}\.Consequently, for every‖w‖2≤B\\\|w\\\|\_\{2\}\\leq B,

\(70\)\|⟨w,hGα\(λ\)⟩−⟨w,hGα⟩\|≤2​λunif​B​Rh\.\\left\|\\left\\langle w,h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\\right\\rangle\-\\left\\langle w,h\_\{G\}^\{\\alpha\}\\right\\rangle\\right\|\\leq 2\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\.The corresponding population risks satisfy

\(71\)ℛ​\(fw,α\(λ\)\)≤ℛ​\(fw,α\)\+2​κ​λunif​B​Rh\.\\mathcal\{R\}\\left\(f\_\{w,\\alpha^\{\(\\lambda\)\}\}\\right\)\\leq\\mathcal\{R\}\\left\(f\_\{w,\\alpha\}\\right\)\+2\\kappa\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\.

###### Proof\.

Using Equation \([66](https://arxiv.org/html/2608.00491#A3.E66)\),

\(72\)hGα\(λ\)−hGα\\displaystyle h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\-h\_\{G\}^\{\\alpha\}=λunif​\(1L​∑ℓ=1LhG\(ℓ\)−∑ℓ=1Lαℓ​hG\(ℓ\)\)\.\\displaystyle=\\lambda\_\{\\mathrm\{unif\}\}\\left\(\\frac\{1\}\{L\}\\sum\_\{\\ell=1\}^\{L\}h\_\{G\}^\{\(\\ell\)\}\-\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}h\_\{G\}^\{\(\\ell\)\}\\right\)\.The triangle inequality and Assumption[1](https://arxiv.org/html/2608.00491#Thmassumption1)give

‖hGα\(λ\)−hGα‖2≤\\displaystyle\\left\\\|h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\-h\_\{G\}^\{\\alpha\}\\right\\\|\_\{2\}\\leq\\;λunif​‖1L​∑ℓ=1LhG\(ℓ\)‖2\+λunif​‖∑ℓ=1Lαℓ​hG\(ℓ\)‖2\\displaystyle\\lambda\_\{\\mathrm\{unif\}\}\\left\\\|\\frac\{1\}\{L\}\\sum\_\{\\ell=1\}^\{L\}h\_\{G\}^\{\(\\ell\)\}\\right\\\|\_\{2\}\+\\lambda\_\{\\mathrm\{unif\}\}\\left\\\|\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}h\_\{G\}^\{\(\\ell\)\}\\right\\\|\_\{2\}≤\\displaystyle\\leq\\;λunif​1L​∑ℓ=1L‖hG\(ℓ\)‖2\+λunif​∑ℓ=1Lαℓ​‖hG\(ℓ\)‖2\\displaystyle\\lambda\_\{\\mathrm\{unif\}\}\\frac\{1\}\{L\}\\sum\_\{\\ell=1\}^\{L\}\\left\\\|h\_\{G\}^\{\(\\ell\)\}\\right\\\|\_\{2\}\+\\lambda\_\{\\mathrm\{unif\}\}\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{\\ell\}\\left\\\|h\_\{G\}^\{\(\\ell\)\}\\right\\\|\_\{2\}\(73\)≤\\displaystyle\\leq\\;2​λunif​Rh\.\\displaystyle 2\\lambda\_\{\\mathrm\{unif\}\}R\_\{h\}\.This proves Equation \([69](https://arxiv.org/html/2608.00491#A3.E69)\)\.

By the Cauchy–Schwarz inequality,

\|⟨w,hGα\(λ\)−hGα⟩\|\\displaystyle\\left\|\\left\\langle w,h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\-h\_\{G\}^\{\\alpha\}\\right\\rangle\\right\|≤‖w‖2​‖hGα\(λ\)−hGα‖2\\displaystyle\\leq\\\|w\\\|\_\{2\}\\left\\\|h\_\{G\}^\{\\alpha^\{\(\\lambda\)\}\}\-h\_\{G\}^\{\\alpha\}\\right\\\|\_\{2\}\(74\)≤2​λunif​B​Rh,\\displaystyle\\leq 2\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\},which proves Equation \([70](https://arxiv.org/html/2608.00491#A3.E70)\)\.

Finally, theκ\\kappa\-Lipschitz property of the downstream loss implies

\|ℓds​\(fw,α\(λ\)​\(G\),Y\)−ℓds​\(fw,α​\(G\),Y\)\|\\displaystyle\\left\|\\ell\_\{\\mathrm\{ds\}\}\\left\(f\_\{w,\\alpha^\{\(\\lambda\)\}\}\(G\),Y\\right\)\-\\ell\_\{\\mathrm\{ds\}\}\\left\(f\_\{w,\\alpha\}\(G\),Y\\right\)\\right\|≤κ​\|fw,α\(λ\)​\(G\)−fw,α​\(G\)\|\\displaystyle\\hskip 99\.58464pt\\leq\\kappa\\left\|f\_\{w,\\alpha^\{\(\\lambda\)\}\}\(G\)\-f\_\{w,\\alpha\}\(G\)\\right\|\(75\)≤2​κ​λunif​B​Rh\.\\displaystyle\\hskip 99\.58464pt\\leq 2\\kappa\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\.Taking expectation over\(G,Y\)∼ℙ\(G,Y\)\\sim\\mathbb\{P\}proves Equation \([71](https://arxiv.org/html/2608.00491#A3.E71)\)\. ∎

Define the uniformly smoothed hypothesis class as

\(76\)ℱmix\(λ\):=\{fw,α\(λ\):‖w‖2≤B,α∈ΔL\}\.\\mathcal\{F\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}:=\\left\\\{f\_\{w,\\alpha^\{\(\\lambda\)\}\}:\\\|w\\\|\_\{2\}\\leq B,\\;\\alpha\\in\\Delta\_\{L\}\\right\\\}\.Letf^mix\(λ\)\\widehat\{f\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}be anεopt\\varepsilon\_\{\\mathrm\{opt\}\}\-approximate empirical\-risk minimizer overℱmix\(λ\)\\mathcal\{F\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}\.

###### Corollary 3 \(Oracle bound under uniform smoothing\)\.

Under Assumption[1](https://arxiv.org/html/2608.00491#Thmassumption1), there exist universal constantsC1,C2\>0C\_\{1\},C\_\{2\}\>0such that, for everyδ∈\(0,1\)\\delta\\in\(0,1\), with probability at least1−δ1\-\\delta,

ℛ​\(f^mix\(λ\)\)≤\\displaystyle\\mathcal\{R\}\\left\(\\widehat\{f\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}\\right\)\\leq\\;minℓ∈\{1,…,L\}​inff∈ℱℓℛ​\(f\)\+2​κ​λunif​B​Rh\\displaystyle\\min\_\{\\ell\\in\\\{1,\\ldots,L\\\}\}\\inf\_\{f\\in\\mathcal\{F\}\_\{\\ell\}\}\\mathcal\{R\}\(f\)\+2\\kappa\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\(77\)\+C1​κ​B​Rh​1\+log⁡LN\+C2​M​log⁡\(2/δ\)N\+εopt\.\\displaystyle\+C\_\{1\}\\kappa BR\_\{h\}\\sqrt\{\\frac\{1\+\\log L\}\{N\}\}\+C\_\{2\}M\\sqrt\{\\frac\{\\log\(2/\\delta\)\}\{N\}\}\+\\varepsilon\_\{\\mathrm\{opt\}\}\.Therefore, uniform smoothing introduces an additional approximation term that is at most linear inλunif\\lambda\_\{\\mathrm\{unif\}\}\.

###### Proof\.

Sinceα\(λ\)∈ΔL\\alpha^\{\(\\lambda\)\}\\in\\Delta\_\{L\}wheneverα∈ΔL\\alpha\\in\\Delta\_\{L\},

\(78\)ℱmix\(λ\)⊆ℱmix\.\\mathcal\{F\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}\\subseteq\\mathcal\{F\}\_\{\\mathrm\{mix\}\}\.Consequently, the Rademacher\-complexity and uniform\-convergence bounds derived in the proof of Theorem[1](https://arxiv.org/html/2608.00491#A3.Thmtheorem1)also apply toℱmix\(λ\)\\mathcal\{F\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}\.

Fix a resolutionℓ\\elland a vectorwwsatisfying‖w‖2≤B\\\|w\\\|\_\{2\}\\leq B\. Leteℓe\_\{\\ell\}denote theℓ\\ell\-th standard basis vector inℝL\\mathbb\{R\}^\{L\}\. The corresponding unsmoothed representation is

\(79\)hGeℓ=hG\(ℓ\)\.h\_\{G\}^\{e\_\{\\ell\}\}=h\_\{G\}^\{\(\\ell\)\}\.Applying Lemma[2](https://arxiv.org/html/2608.00491#A3.Thmtheorem2)withα=eℓ\\alpha=e\_\{\\ell\}gives

\(80\)ℛ​\(fw,eℓ\(λ\)\)≤ℛ​\(fℓ,w\)\+2​κ​λunif​B​Rh\.\\mathcal\{R\}\\left\(f\_\{w,e\_\{\\ell\}^\{\(\\lambda\)\}\}\\right\)\\leq\\mathcal\{R\}\\left\(f\_\{\\ell,w\}\\right\)\+2\\kappa\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\.Taking the infimum over‖w‖2≤B\\\|w\\\|\_\{2\}\\leq Band then the minimum overℓ\\ellyields

\(81\)inff∈ℱmix\(λ\)ℛ​\(f\)≤\\displaystyle\\inf\_\{f\\in\\mathcal\{F\}\_\{\\mathrm\{mix\}\}^\{\(\\lambda\)\}\}\\mathcal\{R\}\(f\)\\leq\\;minℓ∈\{1,…,L\}​inff∈ℱℓℛ​\(f\)\+2​κ​λunif​B​Rh\.\\displaystyle\\min\_\{\\ell\\in\\\{1,\\ldots,L\\\}\}\\inf\_\{f\\in\\mathcal\{F\}\_\{\\ell\}\}\\mathcal\{R\}\(f\)\+2\\kappa\\lambda\_\{\\mathrm\{unif\}\}BR\_\{h\}\.Applying the same approximate empirical\-risk\-minimization and uniform\-convergence argument as in Equation \([61](https://arxiv.org/html/2608.00491#A3.E61)\), together with Equation \([81](https://arxiv.org/html/2608.00491#A3.E81)\), proves Equation \([77](https://arxiv.org/html/2608.00491#A3.E77)\)\. ∎