Federated Low-Rank Koopman Learning for Multivariate Time-Series Anomaly Detection in IoT Systems

# Federated Low-Rank Koopman Learning for Multivariate Time-Series Anomaly Detection in IoT Systems
Source: [https://arxiv.org/html/2607.08978](https://arxiv.org/html/2607.08978)
Tung\-Anh Nguyen, Van\-Phuc Bui, Anh Tuyen Le, Kim Hue Ta, Minh Thuy Le, J\.Andrew Zhang, and Xiaojing HuangT\.\-A Nguyen, A\. T\. Le, J\. A\. Zhang, and X\. Huang are with Faculty of Engineering and IT, University of Technology Sydney, Ultimo, NSW, 2007, Australia \(emails: tunganh\.nguyen@uts\.edu\.au, anhtuyen\.le@uts\.edu\.au, andrew\.zhang@uts\.edu\.au, xiaojing\.huang@uts\.edu\.au\)\. V\.\-P Bui is with the Faculty of Information Technology, FPT University, Vietnam \(email: phucbv5@fe\.edu\.vn\)\. Kim Hue Ta and Minh Thuy Le are with School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Vietnam \(email: hue\.tathikim@hust\.edu\.vn, minhthuy\.le@hust\.edu\.vn\)\.

###### Abstract

Distributed IoT systems generate multivariate time\-series streams for monitoring physical assets, servers, and embedded sensing platforms\. Detecting abnormal temporal behavior is critical for fault diagnosis, predictive maintenance, and security\. However, practical IoT anomaly detection is hindered by decentralized and non\-IID data, limited bandwidth, and the constrained computation and memory of edge devices\. This paper proposes FedKAD, a resource\-efficient federated Koopman anomaly detection framework for distributed IoT multivariate time series\. Unlike deep\-learning\-based anomaly detectors that require training and communicating large neural models, FedKAD learns normal temporal dynamics through lightweight sliding\-window Koopman representations\. Federated training is formulated as a low\-rank consensus problem, where raw sensor streams and local reduced dynamics remain on device while only compact subspace variables are exchanged with the server\. To optimize the shared representation under orthonormality constraints, we develop a federated Stiefel\-ADMM algorithm and provide convergence and stationarity analysis under partial client participation\. During inference, each client detects anomalies locally by measuring the prediction residual between observed future trajectories and the learned Koopman dynamics\. Experiments on four widely used multivariate time\-series anomaly detection benchmarks show that FedKAD maintains or improves detection performance compared with federated deep\-learning baselines\. More importantly for IoT deployment, FedKAD provides up to2\.1×1032\.1\\times 10^\{3\}faster training,80×80\\timeslower communication, and79×79\\timeslower inference latency than neural baselines, confirming its suitability for resource\-constrained edge devices\.

## IIntroduction

Distributed Internet\-of\-Things \(IoT\) systems generate multivariate time\-series \(MVTS\) streams for monitoring physical assets, industrial processes, servers, spacecraft subsystems, and embedded sensing platforms\. Detecting abnormal temporal behavior in these streams is essential for fault diagnosis, predictive maintenance, and security monitoring\[chandola2009anomaly,cook2020anomaly\]\. Unlike conventional offline anomaly detection, IoT anomaly detection must often operate continuously at the network edge, where sensor streams are produced by geographically distributed devices with limited communication, computation, memory, and energy resources\. These constraints make it difficult to rely on centralized data collection or computationally expensive models, especially when raw sensor measurements are privacy\-sensitive or operationally restricted\.

A fundamental challenge in IoT MVTS anomaly detection is that abnormal behavior is rarely limited to an isolated point outlier\. In many practical systems, faults, cyberattacks, and degradation patterns appear as abnormal temporal evolution, sensor drift, delayed responses, changes in cross\-channel dependencies, or deviations from the normal dynamical behavior of the monitored process\. For example, a machine fault may first manifest as a subtle change in vibration and temperature dynamics before a large point\-wise spike appears, while a compromised IoT device may exhibit abnormal temporal communication patterns rather than a single abnormal packet\. Therefore, an effective detector should capture both temporal dependencies and inter\-variable relationships, and should identify when the observed trajectory becomes inconsistent with learned normal dynamics\.

Early time\-series anomaly detection methods relied on statistical models, distance\-based detectors, density estimation, or forecasting residuals\[chandola2009anomaly,cook2020anomaly\]\. These approaches are attractive because they are often interpretable and computationally lightweight\. However, they usually require strong assumptions about stationarity, linearity, or simple distributional forms, and they struggle to model nonlinear dependencies in high\-dimensional MVTS\. Thresholding methods based on extreme value theory, such as \(streaming\) peaks\-over\-threshold, can reduce the need for manually chosen thresholds in streaming settings\[siffer2017pot\], but they are typically applied on top of a precomputed anomaly score and do not by themselves solve the problem of learning complex normal dynamics from distributed multivariate sensor streams\. Deep\-learning\-based anomaly detectors have been developed to address the nonlinear and high\-dimensional nature of MVTS\[xu2018unsupervised,zong2018deep,malhotra2016lstm,su2019robust,zhao2020multivariate,deng2021graph,xu2022anomaly,tuli2022tranad\]\. Reconstruction\-based models, including autoencoder and variational\-autoencoder approaches, learn compact representations of normal data and flag samples with large reconstruction errors\[xu2018donut,zong2018dagmm\]\. Recurrent models, such as Long Short\-Term Memory \(LSTM\) encoder\-decoders and stochastic recurrent networks, further capture temporal dependencies and have been applied to multi\-sensor and spacecraft telemetry anomaly detection\[malhotra2016lstm,hundman2018spacecraft,su2019omni\]\. More recent methods combine adversarial training, graph neural networks, attention mechanisms, and transformer architectures to model complex temporal and inter\-sensor dependencies\[zhao2020mtadgat,deng2021gdn,xu2022anomalytransformer,tuli2022tranad\]\. These methods have achieved strong detection accuracy in centralized benchmarks\.

Despite their success, deep anomaly detectors are not always suitable for federated IoT deployment\. First, many of them rely on heavy neural reconstruction, recurrent, graph, or attention modules, requiring repeated backpropagation over local windows\. Second, when used in federated learning \(FL\), the clients must repeatedly communicate neural parameters or gradients, which can become expensive for bandwidth\-limited IoT links\. In addition, their memory and computation costs can be prohibitive for weak edge devices such as low\-cost gateways, Raspberry Pi\-class nodes, and embedded platforms\. Finally, deep models are often sensitive to non\-independent and non\-identically distributed \(non\-IID\) client data, which is common in IoT systems because devices operate under different environments, workloads, anomaly rates, and sensing configurations\.

FL offers a natural way to train models across distributed clients without uploading raw data to a server\[mcmahan2017communication,kairouz2021advances\]\. In IoT systems, FL has been explored for privacy\-preserving analytics, intrusion detection, and decentralized spatio\-temporal modeling\[nguyen2021fliot,nguyen2019diot,meng2021crossnode\]\. For example, DIoT uses federated learning to aggregate device\-type\-specific behavior profiles for detecting compromised IoT devices\[nguyen2019diot\], while cross\-node federated graph neural networks model decentralized spatio\-temporal sensor data without direct data sharing\[meng2021crossnode\]\. However, existing FL\-based approaches usually inherit the computational and communication burden of the underlying model class\. When the local model is a deep neural anomaly detector, FL protects raw data but does not remove the need for local neural training and repeated communication of large model parameters\. These limitations suggest that federated IoT anomaly detection should not simply wrap existing deep detectors in an FL protocol\. Instead, the anomaly detector should be lightweight by construction, requiring low local computation, low memory, and compact communication while allowing client\-specific dynamics to remain on device\.

Koopman operator theory provides a principled framework for modeling nonlinear dynamical systems through linear evolution in a lifted observable space\[koopman1931hamiltonian,rowley2009spectral,williams2015edmd\]\. This perspective is well suited to MVTS because their variables are often coupled components of an underlying dynamical process rather than independent scalar streams\. In many IoT monitoring systems, a change in one variable can be an early indicator of abnormal behavior and may also affect the evolution of other variables\. For example, in physiological sensing, abnormal patterns in blood pressure and heart rate are more informative when analyzed jointly than when each signal is considered in isolation; similarly, in industrial monitoring, changes in vibration, temperature, and power consumption may jointly indicate machine degradation\. For MVTS data, sliding\-window or delay\-coordinate observables can represent a temporal segment as a dynamical state that jointly contains recent history and cross\-channel dependencies\. Data\-driven Koopman approximations, such as dynamic mode decomposition \(DMD\) and optimal mode decomposition \(OMD\), then learn a compact linear evolution model from snapshot pairs\[schmid2010dmd,williams2015edmd,wynn2013optimal\]\. The resulting low\-rank Koopman subspace captures dominant coherent modes of normal behavior, while the reduced operator predicts their temporal evolution\. Anomalies are therefore detected as deviations from these learned normal dynamics: when faults, attacks, or sensor degradation disturb either the temporal evolution of individual variables or their inter\-variable coupling, the Koopman prediction residual becomes large\.

In this paper, we propose*FedKAD*, a federated low\-rank Koopman anomaly detection framework for distributed IoT MVTS\. The key idea is to replace heavy federated neural reconstruction with lightweight federated low\-rank dynamical modeling\. Each edge device transforms its local MVTS into sliding\-window snapshots and learns a Koopman\-style model of normal temporal evolution\. Instead of learning a full high\-dimensional Koopman operator, FedKAD approximates the local dynamics using an orthonormal low\-rank subspace and a private reduced operator\. During inference, each device detects anomalies locally by comparing the observed future trajectory with the trajectory predicted by its learned normal dynamics\.

FedKAD is designed to match the constraints of federated IoT systems\. Raw time\-series snapshots never leave the device\. Each client keeps its local data matrices and reduced Koopman operator private, while the server coordinates only a shared low\-rank subspace through consensus optimization\. This design substantially reduces communication compared with federated deep\-learning baselines, because clients exchange compact subspace variables rather than full neural models\. At the same time, the local reduced dynamics remain client\-specific, allowing FedKAD to adapt to heterogeneous local behavior while still benefiting from a shared federated representation\. The resulting optimization problem is a low\-rank consensus formulation with orthonormal subspace constraints\. To solve it, we develop a federated Stiefel\-ADMM algorithm that alternates between local subspace updates, server\-side consensus aggregation, and local dual updates\. The orthonormality constraint is handled directly on the Stiefel manifold using projected gradient steps and QR retraction, following the principles of optimization on matrix manifolds\[absil2008optimization\], while the consensus structure follows the ADMM framework for distributed optimization\[boyd2011distributed\]\. We also provide convergence and stationarity analysis under partial client participation\.

The main contributions of this paper are summarized as follows:

- •We propose FedKAD, a federated Koopman anomaly detection framework for distributed IoT MVTS, enabling each edge device to detect abnormal temporal behavior locally without uploading raw sensor streams\.
- •We formulate federated anomaly detection as a low\-rank Koopman consensus problem, where each client keeps its local snapshots and reduced dynamics private while only compact subspace variables are exchanged with the server\.
- •We develop a federated Stiefel\-ADMM algorithm for optimizing the shared orthonormal Koopman subspace under partial client participation, and provide convergence and stationarity analysis\.
- •We evaluate FedKAD on four widely used MVTS anomaly detection benchmarks under non\-IID federated settings and multiple evaluation protocols, motivated by recent concerns about point\-adjusted time\-series anomaly detection metrics\[kim2022rigorous\]\. FedKAD achieves the best F1 score on three of four datasets while demonstrating its suitability for resource\-constrained IoT deployment\.

## IIBackground

### II\-AFederated Learning

FL allows distributed clients to collaboratively train a model while keeping raw data local\. Each client performs local optimization and sends model updates to a server, which aggregates them into a shared model for the next communication round\. FedAvg\[mcmahan2017communication\]is the standard formulation, and subsequent FL variants have enabled large\-scale privacy\-aware learning across domains\[kairouz2021advances,li2020federated\], including MVTS classification and forecasting\.

Despite these benefits, FL is limited by communication and client\-side computation\. Large parameter updates are expensive to exchange, while edge devices often have restricted memory, energy, and compute capacity\. These issues become more severe for MVTS anomaly detection, where anomalies are scarce, labels are often unavailable, and temporal patterns vary significantly across clients\. Prior work reduces communication through quantization\[reisizadeh2020fedpaq\], compression or sparsification\[sattler2019robust\], and adaptive communication\[caldas2018expanding\]\. However, such techniques can introduce extra processing and tuning overhead, motivating lightweight FL designs tailored to unsupervised anomaly detection on distributed time\-series streams\.

### II\-BMultivariate Time\-Series Anomaly Detection

MVTS anomaly detection aims to identify abnormal behaviors from temporally correlated and high\-dimensional observations\. Existing methods include recurrent models for temporal dependency modeling\[malhotra2015long,hundman2018detecting\], reconstruction\-based autoencoders that detect deviations from learned normal patterns\[zong2018deep,su2019robustly\], and GAN\-based approaches that model complex data distributions through adversarial learning\[li2019mad,geiger2020tadgan\]\. More recent methods use graph neural networks to capture inter\-variable dependencies\[deng2021graph,zhao2020multivariate\]or Transformers to model long\-range temporal interactions via self\-attention\[xu2022anomaly,zhou2021informer\]\.

While these models improve detection capacity, they often rely on computationally heavy architectures, centralized training, or careful optimization\. Such requirements are difficult to satisfy in edge and IoT settings, where data are distributed across devices, anomalies are rare and unlabeled, and clients have limited memory, energy, and computation\. This motivates lightweight methods that can capture temporal structure while remaining efficient under distributed and resource\-constrained deployment\.

### II\-CKoopman\-Based Anomaly Detection

Letxt∈ℳ⊂ℝnx\_\{t\}\\in\\mathcal\{M\}\\subset\\mathbb\{R\}^\{n\}be the state of a discrete\-time nonlinear system evolving asxt\+1=φ​\(xt\)x\_\{t\+1\}=\\varphi\(x\_\{t\}\)\. Koopman theory studies the evolution of observablesΦ\\Phirather than the nonlinear mapφ\\varphiitself\. The Koopman operator𝒦\\mathcal\{K\}acts linearly on observables as

𝒦​Φ​\(xt\)=Φ​\(φ​\(xt\)\)=Φ​\(xt\+1\)\.\\mathcal\{K\}\\Phi\(x\_\{t\}\)=\\Phi\(\\varphi\(x\_\{t\}\)\)=\\Phi\(x\_\{t\+1\}\)\.\(1\)
Since𝒦\\mathcal\{K\}is generally infinite\-dimensional, practical methods use a finite observable representationΦ​\(xt\)∈ℝD\\Phi\(x\_\{t\}\)\\in\\mathbb\{R\}^\{D\}and approximate

Φ​\(xt\+1\)≈K​Φ​\(xt\),\\Phi\(x\_\{t\+1\}\)\\approx K\\Phi\(x\_\{t\}\),\(2\)whereK∈ℝD×DK\\in\\mathbb\{R\}^\{D\\times D\}is a finite\-dimensional Koopman operator\.

Data\-driven Koopman methods, including DMD, OMD, and their variants, approximate nonlinear dynamics from snapshot pairs through a finite\-dimensional linear operator\[schmid2010dmd,wynn2013optimal\]\. These methods are interpretable, but their accuracy depends on whether the chosen observable space is sufficiently expressive\. Extended DMD \(EDMD\) improves expressiveness by lifting states into nonlinear observables\[williams2015edmd\], while deep Koopman methods learn the observable map and operator jointly\[lusch2018deep\]\. However, these approaches often require carefully designed dictionaries, high\-dimensional lifted spaces, or additional neural training\.

Koopman\-based anomaly detection uses this linearized dynamical representation to learn normal system evolution and detect anomalies through deviations from the learned dynamics\. Koopman operator frameworks have been used for time\-series modeling and anomaly detection, while recent Koopman predictors have been applied to anomaly detection in complex IoT systems with time\-series data\[fu2024deep\]\. Koopman\-based residual models have also been used for fault detection and isolation in nonlinear dynamical systems\[bakhtiaridoust2022model\]\. These studies show that Koopman representations are useful for structured and interpretable residual\-based anomaly scoring\. However, existing Koopman anomaly methods are mostly centralized, assuming that data are pooled before training\.

### II\-DOptimal Mode Decomposition

Based on the finite\-dimensional linear approximation in Eq\. \([2](https://arxiv.org/html/2607.08978#S2.E2)\), the goal is to estimate a linear operatorK∈ℝd×dK\\in\\mathbb\{R\}^\{d\\times d\}that advances raw data snapshots\. Given a sequence of raw data vectorsx1,x2,…,xM\+1x\_\{1\},x\_\{2\},\\ldots,x\_\{M\+1\}, wherext∈ℝdx\_\{t\}\\in\\mathbb\{R\}^\{d\}andMMis the number of one\-step training pairs, we form the paired snapshot matrices

X=\[x1,x2,…,xM\]∈ℝd×M,X=\\big\[x\_\{1\},x\_\{2\},\\ldots,x\_\{M\}\\big\]\\in\\mathbb\{R\}^\{d\\times M\},Y=\[x2,x3,…,xM\+1\]∈ℝd×M\.Y=\\big\[x\_\{2\},x\_\{3\},\\ldots,x\_\{M\+1\}\\big\]\\in\\mathbb\{R\}^\{d\\times M\}\.Here,X,Y∈ℝd×MX,Y\\in\\mathbb\{R\}^\{d\\times M\}contain consecutive raw data snapshots\. A full least\-squares estimate ofKKcan be expensive and may overfit whenddis large\. OMD addresses this by approximating the transition matrix with a low\-rank operator of the form

K≈P​Q​P⊤,K\\approx PQP^\{\\top\},whereP∈ℝd×rP\\in\\mathbb\{R\}^\{d\\times r\}has orthonormal columns,Q∈ℝr×rQ\\in\\mathbb\{R\}^\{r\\times r\}, andr≪dr\\ll d\. The OMD objective is therefore

minP,Q∥Y−PQP⊤X∥F2,s\.t\.P⊤P=I\.\\min\_\{P,Q\}\\left\\\|Y\-PQP^\{\\top\}X\\right\\\|\_\{F\}^\{2\},\\qquad\\mathrm\{s\.t\.\}\\quad P^\{\\top\}P=I\.\(3\)Here,PPspans anrr\-dimensional subspace of the raw data space, whileQQmodels the linear dynamics within that reduced subspace\. The projectionP⊤P^\{\\top\}maps each raw snapshot to reduced coordinates,QQadvances the reduced state, andPPmaps the prediction back to the original raw data space\. Hence,P​Q​P⊤∈ℝd×dPQP^\{\\top\}\\in\\mathbb\{R\}^\{d\\times d\}provides a rank\-rrapproximation of the finite\-dimensional transition operator in Eq\. \([2](https://arxiv.org/html/2607.08978#S2.E2)\)\.

### II\-EADMM\-based Koopman Learning\.

The Alternating Direction Method of Multipliers \(ADMM\) is a widely used optimization framework for solving structured problems by decomposing them into smaller subproblems coordinated through dual variables\[boyd2011distributed\]\. It is particularly well suited to distributed optimization and statistical learning because local subproblems can be solved independently while a global consensus is enforced through augmented Lagrangian updates\. This decompositional property makes ADMM attractive for federated learning, where clients collaboratively learn a shared model without exchanging raw data\.

In this work, we exploit ADMM to develop a privacy\-preserving federated Koopman learning framework for multivariate time\-series anomaly detection in IoT systems\. Each client locally estimates a low\-rank Koopman model from its own normal time\-series snapshots, while the server enforces consensus among client\-side Koopman subspaces through ADMM\-based aggregation\. The local dynamics model follows the OMD principle of jointly identifying a low\-rank subspace and the corresponding reduced linear dynamics\[wynn2013optimal\]\. Unlike centralized Koopman learning, the proposed formulation keeps raw sensor data on local devices and communicates only model\-related variables, which is consistent with the privacy\-preserving motivation of federated anomaly detection for MVTS\[zhang2021federated,zhu2022deep,liu2022fedtadbench\]\. The learned Koopman dynamics are then used to predict normal system evolution, and anomalies are detected from large prediction residuals\.

## IIISystem Model and Problem Formulation

![Refer to caption](https://arxiv.org/html/2607.08978v1/figures/jiot/figures/system_figure.png)Figure 1:System model ofFedKADfor federated IoT multivariate time\-series anomaly detection\.### III\-AFederated IoT Time\-Series Anomaly Detection

As illustrated in[Fig\.1](https://arxiv.org/html/2607.08978#S3.F1), we consider a federated IoT monitoring system consisting ofNNedge devices coordinated by a central server\. Clienti∈\{1,…,N\}i\\in\\\{1,\\ldots,N\\\}observes a local MVTSxi​\(t\)∈ℝnx\_\{i\}\(t\)\\in\\mathbb\{R\}^\{n\}, wherennis the number of monitored variables\. The objective is to collaboratively learn an anomaly detector for temporal system behavior across clients, while keeping raw time\-series data local\.

In multivariate IoT systems, anomalies rarely appear as isolated point\-wise outliers\[malhotra2015long,hundman2018detecting\]\. Instead, they often arise from abnormal temporal evolution, delayed effects between signals, or changes in cross\-variable dependencies\[deng2021graph,zhao2020multivariate\]\. Therefore, anomaly detection requires modeling not only the current observation, but also the temporal context around it\.

To capture local spatio\-temporal dependencies, each client converts its raw stream into a sliding\-window state\. For a window lengthWW, we define

zi​\(t\)=\[xi​\(t−W\+1\)⊤,…,xi​\(t\)⊤\]⊤∈ℝd,d=n​W\.z\_\{i\}\(t\)=\\big\[x\_\{i\}\(t\-W\+1\)^\{\\top\},\\ldots,x\_\{i\}\(t\)^\{\\top\}\\big\]^\{\\top\}\\in\\mathbb\{R\}^\{d\},\\qquad d=nW\.\(4\)This delay\-coordinate representation is widely used in time\-series anomaly detection, where reconstruction or prediction is performed over local temporal contexts\[tuli2022tranad\]\. By flattening the window intozi​\(t\)z\_\{i\}\(t\), the model operates jointly over theWWtime steps and thennmonitored variables, thereby capturing both temporal dependencies across time and cross\-variable dependencies within the system\.

### III\-BProblem Formulation

Given the local windowed states\{zi​\(t\)\}\\\{z\_\{i\}\(t\)\\\}, the goal of federated time\-series anomaly detection is to learn the normal temporal evolution of distributed MVTS without transmitting raw client data to the server\. In general, each client uses a prediction model to estimate the next windowed state from the current one\. Let

z^i​\(t\+1\)=gi​\(zi​\(t\);Θ,θi\),\\hat\{z\}\_\{i\}\(t\+1\)=g\_\{i\}\\big\(z\_\{i\}\(t\);\\Theta,\\theta\_\{i\}\\big\),whereΘ\\Thetadenotes the global parameters shared across clients andθi\\theta\_\{i\}denotes client\-specific parameters\. The prediction deviation is then used as the anomaly score:

si​\(t\)=‖zi​\(t\+1\)−z^i​\(t\+1\)‖2\.s\_\{i\}\(t\)=\\left\\\|z\_\{i\}\(t\+1\)\-\\hat\{z\}\_\{i\}\(t\+1\)\\right\\\|\_\{2\}\.A large value ofsi​\(t\)s\_\{i\}\(t\)indicates that the observed temporal evolution cannot be well explained by the learned normal dynamics, and is therefore more likely to be anomalous\.

The general federated learning objective can be written as

minΘ,\{θi\}i=1N\\displaystyle\\min\_\{\\Theta,\\\{\\theta\_\{i\}\\\}\_\{i=1\}^\{N\}\}∑i=1N∑t=WTi−1ℓi​\(zi​\(t\+1\),gi​\(zi​\(t\);Θ,θi\)\)\.\\displaystyle\\sum\_\{i=1\}^\{N\}\\sum\_\{t=W\}^\{T\_\{i\}\-1\}\\ell\_\{i\}\\left\(z\_\{i\}\(t\+1\),g\_\{i\}\\big\(z\_\{i\}\(t\);\\Theta,\\theta\_\{i\}\\big\)\\right\)\.\(5\)whereℓi​\(⋅\)\\ell\_\{i\}\(\\cdot\)measures the prediction error\. This formulation allows the federation to learn common normal temporal patterns across clients, while preserving local differences caused by non\-IID data\[collins2021exploiting\]\.

Motivated by the ability of Koopman theory to represent temporal evolution, we reformulate \([5](https://arxiv.org/html/2607.08978#S3.E5)\) under the Koopman theory in \([1](https://arxiv.org/html/2607.08978#S2.E1)\) as follows:

min𝒦\\displaystyle\\min\_\{\\mathcal\{K\}\}∑i=1N∑t=WTi−1‖Φ​\(zi​\(t\+1\)\)−𝒦​Φ​\(zi​\(t\)\)‖22\.\\displaystyle\\sum\_\{i=1\}^\{N\}\\sum\_\{t=W\}^\{T\_\{i\}\-1\}\\left\\\|\\Phi\\big\(z\_\{i\}\(t\+1\)\\big\)\-\\mathcal\{K\}\\Phi\\big\(z\_\{i\}\(t\)\\big\)\\right\\\|\_\{2\}^\{2\}\.\(6\)However, directly solving \([6](https://arxiv.org/html/2607.08978#S3.E6)\) is not well suited to federated IoT settings\. The lifted Koopman operator can be high\-dimensional, making it costly to store, compute, and communicate on resource\-constrained edge devices\. Moreover, a single full operator may fail to capture heterogeneous client dynamics under non\-IID data\. These limitations motivateFedKAD, which uses a communication\- and memory\-efficient low\-rank shared subspace while retaining client\-specific reduced dynamics\.

## IVProposed FedKAD Framework

To solve the federated prediction problem in \([6](https://arxiv.org/html/2607.08978#S3.E6)\),FedKADcombines a common nonlinear observable map with low\-rank Koopman modeling\. The method consists of three main steps\. First, each client maps its local sliding\-window states into a common observable space\. Second, the resulting Koopman dynamics are factorized into a shared low\-rank subspace and a client\-specific reduced operator\. Finally, the shared subspace is learned through federated consensus optimization, while the local snapshots and reduced dynamics remain on device\.

### IV\-AKoopman Representation for Temporal Dynamics

To model nonlinear temporal dynamics, we use a Koopman\-inspired representation\. Koopman theory represents nonlinear state evolution as linear evolution in an observable space\[koopman1931hamiltonian,kaiser2021data\]\. Since the Koopman operator is generally infinite\-dimensional, data\-driven Koopman methods approximate it using finite\-dimensional observables and a linear operator learned from snapshot pairs\[schmid2010dmd,wynn2013optimal,williams2015edmd\]\. Thus, for a lifting mapΦ​\(⋅\)\\Phi\(\\cdot\), the local dynamics of clientiiare modeled as

Φ​\(zi​\(t\+1\)\)≈Ki​Φ​\(zi​\(t\)\),\\Phi\(z\_\{i\}\(t\+1\)\)\\approx K\_\{i\}\\Phi\(z\_\{i\}\(t\)\),whereKiK\_\{i\}is a finite\-dimensional lifted Koopman operator\.

We constructΦ​\(⋅\)\\Phi\(\\cdot\)using a fixed nonlinear lifting\. Classical DMD and OMD fit a linear transition directly in the observed state space, which is efficient but restrictive for nonlinear multivariate dynamics\. Inspired by EDMD\[williams2015data\], we augment the windowed state with fixed nonlinear observables:

Φ​\(zi​\(t\)\)=\[zi​\(t\)⏟Windowed statetanh⁡\(Wlift​zi​\(t\)\+b\)⏟Nonlinear observables\]∈ℝD,D=d\+dlift\.\\Phi\(z\_\{i\}\(t\)\)=\\begin\{bmatrix\}\\underbrace\{z\_\{i\}\(t\)\}\_\{\\text\{Windowed state\}\}\\\\ \\underbrace\{\\tanh\\\!\\left\(W\_\{\\mathrm\{lift\}\}z\_\{i\}\(t\)\+b\\right\)\}\_\{\\text\{Nonlinear observables\}\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{D\},\\qquad D=d\+d\_\{\\mathrm\{lift\}\}\.\(7\)Here,Wlift∈ℝdlift×dW\_\{\\mathrm\{lift\}\}\\in\\mathbb\{R\}^\{d\_\{\\mathrm\{lift\}\}\\times d\}andb∈ℝdliftb\\in\\mathbb\{R\}^\{d\_\{\\mathrm\{lift\}\}\}are fixed random parameters shared by all clients\. The first block preserves the original windowed state, while the second block addsdliftd\_\{\\mathrm\{lift\}\}nonlinear random observables\. Thetanh\\tanhnonlinearity is motivated by universal approximation theory:*finite linear combinations of affine projections followed by sigmoidal activations can approximate continuous functions on compact domains to arbitrary accuracy*\[cybenko1989approximation\]\. This fixed lifting enriches the Koopman observable space without hand\-designing a large dictionary or training a neural lifting network\. Since all clients use the same window length and lifting map, their lifted states share a common dimension and semantic structure, which makes federated subspace alignment well defined\.

For clientii, the lifted training pairs are formed by stacking consecutive lifted states column\-wise:

Xi=\[Φ​\(zi​\(t\)\)\]t∈𝒯i,Yi=\[Φ​\(zi​\(t\+1\)\)\]t∈𝒯i,Xi,Yi∈ℝD×Mi,X\_\{i\}=\\big\[\\Phi\(z\_\{i\}\(t\)\)\\big\]\_\{t\\in\\mathcal\{T\}\_\{i\}\},Y\_\{i\}=\\big\[\\Phi\(z\_\{i\}\(t\+1\)\)\\big\]\_\{t\\in\\mathcal\{T\}\_\{i\}\},X\_\{i\},Y\_\{i\}\\in\\mathbb\{R\}^\{D\\times M\_\{i\}\},whereMiM\_\{i\}is the number of valid one\-step training pairs\. For a local sequence of lengthTiT\_\{i\}, we use

𝒯i=\{W,…,Ti−1\},Mi=Ti−W\.\\mathcal\{T\}\_\{i\}=\\\{W,\\ldots,T\_\{i\}\-1\\\},\\qquad M\_\{i\}=T\_\{i\}\-W\.

### IV\-BLow\-Rank Local Koopman Learning Problem Formulation

The finite\-dimensional Koopman model in \([2](https://arxiv.org/html/2607.08978#S2.E2)\) requires a lifted operatorKi∈ℝD×DK\_\{i\}\\in\\mathbb\{R\}^\{D\\times D\}, which faces several limitations in federated IoT settings\. Its storage, computation, and communication costs scale quadratically with the lifted dimension, and a single full operator is poorly suited to heterogeneous client dynamics under non\-IID data\. These considerations motivate an OMD\-inspired low\-rank formulation, which separates the Koopman subspace from the reduced temporal operator\. Accordingly, we adopt the following low\-rank factorization\[wynn2013optimal\]:

Φ​\(zi​\(t\+1\)\)≈Pi​Qi​Pi⊤​Φ​\(zi​\(t\)\),\\Phi\(z\_\{i\}\(t\+1\)\)\\approx P\_\{i\}Q\_\{i\}P\_\{i\}^\{\\top\}\\Phi\(z\_\{i\}\(t\)\),where

Pi∈ℝD×r,Qi∈ℝr×r,r≪D\.P\_\{i\}\\in\\mathbb\{R\}^\{D\\times r\},\\qquad Q\_\{i\}\\in\\mathbb\{R\}^\{r\\times r\},\\qquad r\\ll D\.Here,PiP\_\{i\}spans anrr\-dimensional Koopman subspace in the lifted observable space, whileQiQ\_\{i\}describes the reduced linear evolution within that subspace\. The factorization reduces the number of parameters fromO​\(D2\)O\(D^\{2\}\)for a full lifted operator toO​\(D​r\+r2\)O\(Dr\+r^\{2\}\), making the local model more suitable for resource\-constrained IoT devices\.

Using the lifted snapshot matrices\(Xi,Yi\)\(X\_\{i\},Y\_\{i\}\), clientiilearns its local low\-rank Koopman model by solving the lifted OMD objective

minPi,Qi\\displaystyle\\min\_\{P\_\{i\},Q\_\{i\}\}‖Yi−Pi​Qi​Pi⊤​Xi‖F2,\\displaystyle\\left\\\|Y\_\{i\}\-P\_\{i\}Q\_\{i\}P\_\{i\}^\{\\top\}X\_\{i\}\\right\\\|\_\{F\}^\{2\},\(8\)s\.t\.Pi⊤​Pi=I\.\\displaystyle P\_\{i\}^\{\\top\}P\_\{i\}=I\.

### IV\-CFederated Koopman Learning for IoT Systems

The low\-rank factorization in \([IV\-B](https://arxiv.org/html/2607.08978#S4.Ex9)\) separates the Koopman subspacePiP\_\{i\}from the reduced dynamicsQiQ\_\{i\}\. This separation aligns with shared\-representation personalized federated learning, where a global representation is shared across clients while local components capture client\-specific variation\[collins2021exploiting\]\. Motivated by this view,FedKADfederates the low\-rank Koopman subspace rather than raw data or full lifted operators\. Each client keeps its snapshot matrices\(Xi,Yi\)\(X\_\{i\},Y\_\{i\}\)and reduced operatorQiQ\_\{i\}local, while the server coordinates a shared subspace through a consensus variableZ∈ℝD×rZ\\in\\mathbb\{R\}^\{D\\times r\}\. The distributed objective is

min\{Pi,Qi\},Z​∑i=1Nfi​\(Pi,Qi\),\\min\_\{\\\{P\_\{i\},Q\_\{i\}\\\},Z\}\\;\\sum\_\{i=1\}^\{N\}f\_\{i\}\(P\_\{i\},Q\_\{i\}\),\(9\)with the local loss

fi​\(Pi,Qi\):=‖Yi−Pi​Qi​Pi⊤​Xi‖F2,f\_\{i\}\(P\_\{i\},Q\_\{i\}\):=\\big\\\|Y\_\{i\}\-P\_\{i\}Q\_\{i\}P\_\{i\}^\{\\top\}X\_\{i\}\\big\\\|\_\{F\}^\{2\},subject to

Pi=Z,Pi⊤​Pi=I,i=1,…,N\.P\_\{i\}=Z,\\qquad P\_\{i\}^\{\\top\}P\_\{i\}=I,\\qquad i=1,\\ldots,N\.The first constraint forces the clients to agree on a shared low\-rank subspaceZZ, while eachQiQ\_\{i\}stays private and tracks the local dynamics of clientii\.

Since the optimalQQdepends on the chosen subspace, the solution of the inner least\-squares problem inQiQ\_\{i\}is denotedQ^i​\(Pi\)\\hat\{Q\}\_\{i\}\(P\_\{i\}\)\. Setting∂fi/∂Qi\\partial f\_\{i\}/\\partial Q\_\{i\}to zero gives the normal equations

\(Pi⊤​Xi​Xi⊤​Pi\)​Q^i​\(Pi\)⊤=Pi⊤​Xi​Yi⊤​Pi\.\\big\(P\_\{i\}^\{\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}\\big\)\\,\\hat\{Q\}\_\{i\}\(P\_\{i\}\)^\{\\top\}=P\_\{i\}^\{\\top\}X\_\{i\}Y\_\{i\}^\{\\top\}P\_\{i\}\.Whenever ther×rr\\times rGram matrixGi​\(Pi\):=Pi⊤​Xi​Xi⊤​PiG\_\{i\}\(P\_\{i\}\):=P\_\{i\}^\{\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}is invertible, this yields the closed\-form solution

Q^i​\(Pi\)=Pi⊤​Yi​Xi⊤​Pi​Gi​\(Pi\)−1\.\\hat\{Q\}\_\{i\}\(P\_\{i\}\)=P\_\{i\}^\{\\top\}Y\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}\\,G\_\{i\}\(P\_\{i\}\)^\{\-1\}\.\(10\)The matrixGi​\(Pi\)G\_\{i\}\(P\_\{i\}\)is not guaranteed to be invertible, however:Xi​Xi⊤X\_\{i\}X\_\{i\}^\{\\top\}has rank at mostmin⁡\(D,Mi\)\\min\(D,M\_\{i\}\), and when the local snapshots are limited or when the columns ofPiP\_\{i\}fall in a nearly degenerate direction ofXiX\_\{i\},Gi​\(Pi\)G\_\{i\}\(P\_\{i\}\)becomes singular or severely ill\-conditioned\. In such cases, we fall back to a standard Tikhonov regularization, replacingGi​\(Pi\)−1G\_\{i\}\(P\_\{i\}\)^\{\-1\}in \([10](https://arxiv.org/html/2607.08978#S4.E10)\) by\(Gi​\(Pi\)\+λ​I\)−1\\big\(G\_\{i\}\(P\_\{i\}\)\+\\lambda I\\big\)^\{\-1\}withλ\>0\\lambda\>0, which keeps the update well defined while preserving the singler×rr\\times rlinear\-system cost\.

BecauseQ^i​\(Pi\)\\hat\{Q\}\_\{i\}\(P\_\{i\}\)can be computed locally fromPiP\_\{i\}, the variablesQiQ\_\{i\}can be eliminated, yielding

min\{Pi\},Z\\displaystyle\\min\_\{\\\{P\_\{i\}\\\},Z\}∑i=1N‖Yi−Pi​Q^i​\(Pi\)​Pi⊤​Xi‖F2\\displaystyle\\sum\_\{i=1\}^\{N\}\\big\\\|Y\_\{i\}\-P\_\{i\}\\hat\{Q\}\_\{i\}\(P\_\{i\}\)P\_\{i\}^\{\\top\}X\_\{i\}\\big\\\|\_\{F\}^\{2\}\(11\)s\.t\.\\displaystyle\\mathrm\{s\.t\.\}Pi=Z,Pi⊤​Pi=I,i=1,…,N\.\\displaystyle P\_\{i\}=Z,\\qquad P\_\{i\}^\{\\top\}P\_\{i\}=I,\\qquad i=1,\\ldots,N\.

### IV\-DADMM\-Based Federated Optimization

The linear consensus constraintPi=ZP\_\{i\}=Zin \([11](https://arxiv.org/html/2607.08978#S4.E11)\) is well suited to ADMM\[boyd2011distributed\]\. ADMM alternates between local primal updates, a global consensus update, and local dual updates, progressively aligning the client subspacesPiP\_\{i\}with the global subspaceZZ\. The orthonormality constraintPi⊤​Pi=IP\_\{i\}^\{\\top\}P\_\{i\}=Iis handled directly in the localPiP\_\{i\}update; hence, the only dual variable is associated with the linear consensus constraint\. LetΛi∈ℝD×r\\Lambda\_\{i\}\\in\\mathbb\{R\}^\{D\\times r\}denote the dual variable for clientii, and letρ\>0\\rho\>0be the ADMM penalty parameter\. The augmented Lagrangian of \([11](https://arxiv.org/html/2607.08978#S4.E11)\) is

ℒ=∑i=1N\[fi\(Pi,Q^i\(Pi\)\)\+⟨Λi,Pi−Z⟩F\+ρ2∥Pi−Z∥F2\],\\small\\begin\{split\}\\mathcal\{L\}=\\sum\_\{i=1\}^\{N\}\\\!\\Big\[&f\_\{i\}\\\!\\big\(P\_\{i\},\\hat\{Q\}\_\{i\}\(P\_\{i\}\)\\big\)\+\\langle\\Lambda\_\{i\},P\_\{i\}\-Z\\rangle\_\{F\}\+\\tfrac\{\\rho\}\{2\}\\,\\big\\\|P\_\{i\}\-Z\\big\\\|\_\{F\}^\{2\}\\Big\],\\end\{split\}\(12\)where⟨⋅,⋅⟩F\\langle\\cdot,\\cdot\\rangle\_\{F\}denotes the Frobenius inner product\.

#### IV\-D1Primal Update \(Local Updates\)

At communication roundℓ\\ell, each client first refits its local reduced Koopman operator using the current subspace:

Q^iℓ\+1=Q^i​\(Piℓ\)\.\\hat\{Q\}\_\{i\}^\{\\ell\+1\}=\\hat\{Q\}\_\{i\}\\\!\\big\(P\_\{i\}^\{\\ell\}\\big\)\.Then, clientiiupdates its local subspace by minimizing the augmented Lagrangian over the orthonormal manifold:

Piℓ\+1=argminPi⊤​Pi=I\[fi​\(Pi,Q^iℓ\+1\)\+ρ2​‖Pi−Zℓ\+1ρ​Λiℓ‖F2\]\.P\_\{i\}^\{\\ell\+1\}=\\mathop\{\\mathrm\{argmin\}\}\\limits\_\{P\_\{i\}^\{\\top\}P\_\{i\}=I\}\\Big\[f\_\{i\}\\big\(P\_\{i\},\\hat\{Q\}\_\{i\}^\{\\ell\+1\}\\big\)\+\\tfrac\{\\rho\}\{2\}\\big\\\|P\_\{i\}\-Z^\{\\ell\}\+\\tfrac\{1\}\{\\rho\}\\Lambda\_\{i\}^\{\\ell\}\\big\\\|\_\{F\}^\{2\}\\Big\]\.

#### IV\-D2Consensus Update \(Global Update\)

The server aggregates the updated local subspaces and dual variables:

Zℓ\+1=1N​∑i=1N\(Piℓ\+1\+1ρ​Λiℓ\)\.Z^\{\\ell\+1\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\Big\(P\_\{i\}^\{\\ell\+1\}\+\\tfrac\{1\}\{\\rho\}\\Lambda\_\{i\}^\{\\ell\}\\Big\)\.\(13\)

#### IV\-D3Dual Update \(Local Updates\)

Each client then updates its dual variable locally:

Λiℓ\+1=Λiℓ\+ρ​\(Piℓ\+1−Zℓ\+1\)\.\\Lambda\_\{i\}^\{\\ell\+1\}=\\Lambda\_\{i\}^\{\\ell\}\+\\rho\\big\(P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\\big\)\.\(14\)
Summing \([14](https://arxiv.org/html/2607.08978#S4.E14)\) over all clients and substituting \([13](https://arxiv.org/html/2607.08978#S4.E13)\) gives

Zℓ\+1=1N​∑i=1NPiℓ\+1\.Z^\{\\ell\+1\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}P\_\{i\}^\{\\ell\+1\}\.\(15\)This simplified consensus step substantially reduces communication overhead in the federated setting\. Instead of exchanging raw snapshots\(Xi,Yi\)\(X\_\{i\},Y\_\{i\}\)or full Koopman operators, each client communicates only the compact subspace matrixPi∈ℝD×rP\_\{i\}\\in\\mathbb\{R\}^\{D\\times r\}, withr≪Dr\\ll D\. The data matrices and the local reduced Koopman operatorQ^i\\hat\{Q\}\_\{i\}remain stored and updated entirely on device\.

When only a subset𝒮ℓ⊆\{1,…,N\}\\mathcal\{S\}^\{\\ell\}\\subseteq\\\{1,\\ldots,N\\\}of clients is available at communication roundℓ\\ell, the updates are restricted toi∈𝒮ℓi\\in\\mathcal\{S\}^\{\\ell\}\. In this case, the consensus rule retains its full ADMM form:

Zℓ\+1=1\|𝒮ℓ\|​∑i∈𝒮ℓ\(Piℓ\+1\+1ρ​Λiℓ\),Z^\{\\ell\+1\}=\\frac\{1\}\{\|\\mathcal\{S\}^\{\\ell\}\|\}\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\Big\(P\_\{i\}^\{\\ell\+1\}\+\\tfrac\{1\}\{\\rho\}\\Lambda\_\{i\}^\{\\ell\}\\Big\),\(16\)because the \([15](https://arxiv.org/html/2607.08978#S4.E15)\) does not generally hold when some client dual variables are frozen\. The convergence analysis in Sec\.[V](https://arxiv.org/html/2607.08978#S5)covers this partial\-participation setting\.

### IV\-EFederated Koopman Learning on the Stiefel Manifold

The constraintPi⊤​Pi=IP\_\{i\}^\{\\top\}P\_\{i\}=Iin \([11](https://arxiv.org/html/2607.08978#S4.E11)\) places the local subspace variablePiP\_\{i\}on the Stiefel manifold

St​\(D,r\)=\{P∈ℝD×r:P⊤​P=I\}\.\\mathrm\{St\}\(D,r\)=\\\{P\\in\\mathbb\{R\}^\{D\\times r\}:P^\{\\top\}P=I\\\}\.Rather than adding a penalty for this constraint,FedKADupdatesPiP\_\{i\}directly on the manifold\. At communication roundℓ\\ell, after computingQ^iℓ\+1\\hat\{Q\}\_\{i\}^\{\\ell\+1\}, clientiiapproximately solves

Piℓ\+1=argminP∈St​\(D,r\)ℱiℓ​\(P\),P\_\{i\}^\{\\ell\+1\}=\\mathop\{\\mathrm\{argmin\}\}\_\{P\\in\\mathrm\{St\}\(D,r\)\}\\mathcal\{F\}\_\{i\}^\{\\ell\}\(P\),\(17\)where

ℱiℓ\(P\):=fi\(P,Q^iℓ\+1\)\+ρ2∥P−Zℓ\+1ρΛiℓ∥F2−12​ρ∥Λi∥F2\.\\mathcal\{F\}\_\{i\}^\{\\ell\}\(P\):=f\_\{i\}\\big\(P,\\hat\{Q\}\_\{i\}^\{\\ell\+1\}\\big\)\+\\frac\{\\rho\}\{2\}\\left\\\|P\-Z^\{\\ell\}\+\\frac\{1\}\{\\rho\}\\Lambda\_\{i\}^\{\\ell\}\\right\\\|\_\{F\}^\{2\}\-\\tfrac\{1\}\{2\\rho\}\\\|\\Lambda\_\{i\}\\\|\_\{F\}^\{2\}\.\(18\)To approximate \([17](https://arxiv.org/html/2607.08978#S4.E17)\), each client performs projected gradient descent onSt​\(D,r\)\\mathrm\{St\}\(D,r\)\. Starting from the current local subspacePiℓ,0=PiℓP\_\{i\}^\{\\ell,0\}=P\_\{i\}^\{\\ell\}, the client appliesCCinner steps\. Forc=0,…,C−1c=0,\\ldots,C\-1, the Euclidean gradient ofℱiℓ\\mathcal\{F\}\_\{i\}^\{\\ell\}is first projected onto a feasible tangent direction,

Giℓ,c=\(I−Piℓ,c​Piℓ,c⊤\)​∇Pℱiℓ​\(Piℓ,c\)\.G\_\{i\}^\{\\ell,c\}=\\big\(I\-P\_\{i\}^\{\\ell,c\}P\_\{i\}^\{\\ell,c\\top\}\\big\)\\nabla\_\{P\}\\mathcal\{F\}\_\{i\}^\{\\ell\}\(P\_\{i\}^\{\\ell,c\}\)\.\(19\)The client then takes a gradient step and retracts the result back to the Stiefel manifold using QR decomposition:

Piℓ,c\+1=QR⁡\(Piℓ,c−η​Giℓ,c\),P\_\{i\}^\{\\ell,c\+1\}=\\operatorname\{QR\}\\left\(P\_\{i\}^\{\\ell,c\}\-\\eta G\_\{i\}^\{\\ell,c\}\\right\),\(20\)whereη\\etais the local step size\. The retraction keeps every inner iterate feasible, i\.e\.,Piℓ,c⊤​Piℓ,c=IP\_\{i\}^\{\\ell,c\\top\}P\_\{i\}^\{\\ell,c\}=I\. AfterCCinner steps, the final inner iterate is used as the local ADMM update:

Piℓ\+1=Piℓ,C\.P\_\{i\}^\{\\ell\+1\}=P\_\{i\}^\{\\ell,C\}\.
Algorithm 1Federated Stiefel ADMM forFedKAD1:Server initializes the

Z0Z^\{0\}and

dliftd\_\{\\mathrm\{lift\}\}lifting functions randomly

2:Each client initializes

Pi0P\_\{i\}^\{0\}with

Pi0⊤​Pi0=IP\_\{i\}^\{0\\top\}P\_\{i\}^\{0\}=Iand

Λi0=0\\Lambda\_\{i\}^\{0\}=0
3:for

ℓ=0,…,L−1\\ell=0,\\ldots,L\-1do⊳\\trianglerightGlobal rounds

4:Server broadcasts

ZℓZ^\{\\ell\}to the clients

5:forclient

i=1,…,Ni=1,\\ldots,Nin paralleldo

6:Refit

Q^iℓ\+1=Q^i​\(Piℓ\)\\hat\{Q\}\_\{i\}^\{\\ell\+1\}=\\hat\{Q\}\_\{i\}\(P\_\{i\}^\{\\ell\}\)by Eq\. \([10](https://arxiv.org/html/2607.08978#S4.E10)\)

7:Set

Piℓ,0=PiℓP\_\{i\}^\{\\ell,0\}=P\_\{i\}^\{\\ell\}
8:for

c=0,…,C−1c=0,\\ldots,C\-1do⊳\\trianglerightLocal rounds

9:Compute

Giℓ,cG\_\{i\}^\{\\ell,c\}by Eq\. \([19](https://arxiv.org/html/2607.08978#S4.E19)\)

10:Retract

Piℓ,c\+1P\_\{i\}^\{\\ell,c\+1\}by Eq\. \([20](https://arxiv.org/html/2607.08978#S4.E20)\)

11:endfor

12:Set

Piℓ\+1=Piℓ,CP\_\{i\}^\{\\ell\+1\}=P\_\{i\}^\{\\ell,C\}and send

Piℓ\+1P\_\{i\}^\{\\ell\+1\}to the server

13:endfor

14:Server computes

Zℓ\+1Z^\{\\ell\+1\}by Eq\. \([16](https://arxiv.org/html/2607.08978#S4.E16)\)

15:Server broadcasts

Zℓ\+1Z^\{\\ell\+1\}to the clients

16:forclient

i=1,…,Ni=1,\\ldots,Nin paralleldo

17:Update

Λiℓ\+1\\Lambda\_\{i\}^\{\\ell\+1\}by Eq\. \([14](https://arxiv.org/html/2607.08978#S4.E14)\)

18:endfor

19:endfor

The complete update sequence is summarized in Algorithm[1](https://arxiv.org/html/2607.08978#alg1)\.

### IV\-FFederated Koopman Anomaly Detection

After federated training in Algorithm[1](https://arxiv.org/html/2607.08978#alg1), each clientiistores global Koopman subspaceZZand local operatorQ^i\\hat\{Q\}\_\{i\}\.

##### Prediction\.

At each valid timett, clientiiforms the windowed statezi​\(t\)z\_\{i\}\(t\)as in Eq\. \([4](https://arxiv.org/html/2607.08978#S3.E4)\) and maps it to the lifted observableΦ​\(zi​\(t\)\)\\Phi\(z\_\{i\}\(t\)\)\. Given a prediction horizonH≥1H\\geq 1, thehh\-step prediction in the lifted space is

Φ^i​\(t\+h\)=Z​Q^ih​Z⊤​Φ​\(zi​\(t\)\),h=1,…,H\.\\widehat\{\\Phi\}\_\{i\}\(t\+h\)=Z\\hat\{Q\}\_\{i\}^\{\\,h\}Z^\{\\top\}\\Phi\(z\_\{i\}\(t\)\),\\qquad h=1,\\ldots,H\.

##### Anomaly score\.

Once the future observations become available, clientiiconstructs the realized lifted statesΦ​\(zi​\(t\+h\)\)\\Phi\(z\_\{i\}\(t\+h\)\)and compares them with the predictions\. The anomaly score is the average prediction residual

si​\(t\)=1H​∑h=1H‖Φ​\(zi​\(t\+h\)\)−Φ^i​\(t\+h\)‖22\.s\_\{i\}\(t\)=\\frac\{1\}\{H\}\\sum\_\{h=1\}^\{H\}\\left\\\|\\Phi\(z\_\{i\}\(t\+h\)\)\-\\widehat\{\\Phi\}\_\{i\}\(t\+h\)\\right\\\|\_\{2\}^\{2\}\.A largesi​\(t\)s\_\{i\}\(t\)indicates that the observed future evolution is poorly explained by the learned normal Koopman dynamics, and is therefore treated as an anomalous event\.

## VFedKAD: Convergence Analysis

We analyze the ADMM iterates defined in[Sec\.IV\-D](https://arxiv.org/html/2607.08978#S4.SS4)\. Letℓ\\elldenote the communication round and let𝒮ℓ⊆\{1,…,N\}\\mathcal\{S\}^\{\\ell\}\\subseteq\\\{1,\\ldots,N\\\}denote the subset of clients participating in roundℓ\\ell\. Letℒℓ\\mathcal\{L\}^\{\\ell\}denote the augmented Lagrangian in \([12](https://arxiv.org/html/2607.08978#S4.E12)\) evaluated at\(\{Piℓ\},Zℓ,\{Λiℓ\}\)\(\\\{P\_\{i\}^\{\\ell\}\\\},Z^\{\\ell\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\), after each participating client has refittedQ^iℓ\+1=Q^i​\(Piℓ\)\\hat\{Q\}\_\{i\}^\{\\ell\+1\}=\\hat\{Q\}\_\{i\}\(P\_\{i\}^\{\\ell\}\)\. We use∇St\\nabla\_\{\\mathrm\{St\}\}to denote the Riemannian gradient with respect toPiP\_\{i\}on the Stiefel manifold\.

Assumptions\.We impose the following assumptions:

- •A1: For each clientii, the local objectivefi​\(Pi\)f\_\{i\}\(P\_\{i\}\)has anLiL\_\{i\}\-Lipschitz Riemannian gradient along the generated Stiefel iterates: ‖∇Stfi​\(Piℓ\+1\)−∇Stfi​\(Piℓ\)‖F≤Li​‖Piℓ\+1−Piℓ‖F\.\\left\\\|\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\+1\}\)\-\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\}\)\\right\\\|\_\{F\}\\leq L\_\{i\}\\left\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\right\\\|\_\{F\}\.
- •A2: For allii, the penalty parameterρ\\rhois chosen large enough such that the localPiP\_\{i\}\-subproblem \([18](https://arxiv.org/html/2607.08978#S4.E18)\) is strongly convex with modulusμi​\(ρ\)\\mu\_\{i\}\(\\rho\)\.
- •A3: The augmented Lagrangian is bounded from below along the generated iterates: infℒℓ\>−∞\.\\inf\\mathcal\{L\}^\{\\ell\}\>\-\\infty\.\(21\)
- •A4: The localPiP\_\{i\}\-subproblems in \([17](https://arxiv.org/html/2607.08978#S4.E17)\) are solved inexactly with summable errors\. Specifically, there exists a nonnegative sequenceεℓ\{\\varepsilon\_\{\\ell\}\}such that εℓ:=∑i∈𝒮ℓ\[ℱiℓ\(Piℓ\+1\)−minP∈St​\(D,r\)ℱiℓ\(P\)\],∑ℓ=0∞εℓ<∞\.\\varepsilon\_\{\\ell\}:=\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\left\[\\mathcal\{F\}\_\{i\}^\{\\ell\}\(P\_\{i\}^\{\\ell\+1\}\)\-\\min\_\{P\\in\\mathrm\{St\}\(D,r\)\}\\mathcal\{F\}\_\{i\}^\{\\ell\}\(P\)\\right\],\\sum\_\{\\ell=0\}^\{\\infty\}\\varepsilon\_\{\\ell\}<\\infty\.
- •A5: The cumulative objective variationΔQ\\Delta\_\{Q\}induced by the reduced Koopman\-operator updates is finite\. ΔQℓ:=∑i=1N\[fi​\(Piℓ\+1,Q^iℓ\+1\)−fi​\(Piℓ\+1,Q^iℓ\)\],∑ℓ=0∞ΔQℓ<∞\.\\footnotesize\\Delta\_\{Q\}^\{\\ell\}:=\\sum\_\{i=1\}^\{N\}\\left\[f\_\{i\}\(P\_\{i\}^\{\\ell\+1\},\\hat\{Q\}\_\{i\}^\{\\ell\+1\}\)\-f\_\{i\}\(P\_\{i\}^\{\\ell\+1\},\\hat\{Q\}\_\{i\}^\{\\ell\}\)\\right\],\\;\\sum\_\{\\ell=0\}^\{\\infty\}\\Delta\_\{Q\}^\{\\ell\}<\\infty\.

To establish the convergence guarantee ofFedKAD, we analyze the generated sequence\(\{Piℓ\},Zℓ,\{Λiℓ\}\)\\big\(\\\{P\_\{i\}^\{\\ell\}\\\},Z^\{\\ell\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\\big\)\. The analysis proceeds in four steps\. First, we bound the successive difference of the dual variables in terms of the change in the local subspace variables\. Next, we derive a perturbed descent bound for the augmented Lagrangian\. We then show that the augmented Lagrangian sequence converges\. Finally, we use these results to prove that the consensus residuals vanish and that the generated sequence converges to a stationary point\. The empirical convergence behavior is further evaluated in[VI\-B5](https://arxiv.org/html/2607.08978#S6.SS2.SSS5)\.

### V\-ABound on the Successive Difference of Dual Variables

We first establish an upper bound for the successive difference of the dual variablesΛi\\Lambda\_\{i\}in terms of the local subspace variablesPiP\_\{i\}\.

###### Lemma V\.2\.

Suppose Assumption A1 holds\. Then, for each participating clienti∈𝒮ℓi\\in\\mathcal\{S\}^\{\\ell\},

‖Λiℓ\+1−Λiℓ‖F2≤Li2​‖Piℓ\+1−Piℓ‖F2\.\\\|\\Lambda\_\{i\}^\{\\ell\+1\}\-\\Lambda\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\\leq L\_\{i\}^\{2\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\.\(22\)

Proof\.For each participating clienti∈𝒮ℓi\\in\\mathcal\{S\}^\{\\ell\}, the optimality condition of the localPiP\_\{i\}\-subproblem gives

∇Stfi​\(Piℓ\+1\)\+Λiℓ\+ρ​\(Piℓ\+1−Zℓ\+1\)=0\.\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\+1\}\)\+\\Lambda\_\{i\}^\{\\ell\}\+\\rho\(P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\)=0\.Using the dual update

Λiℓ\+1=Λiℓ\+ρ​\(Piℓ\+1−Zℓ\+1\),\\Lambda\_\{i\}^\{\\ell\+1\}=\\Lambda\_\{i\}^\{\\ell\}\+\\rho\(P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\),we obtain

Λiℓ\+1=−∇Stfi​\(Piℓ\+1\)\.\\Lambda\_\{i\}^\{\\ell\+1\}=\-\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\+1\}\)\.Similarly, at the previous local iterate,

Λiℓ=−∇Stfi​\(Piℓ\)\.\\Lambda\_\{i\}^\{\\ell\}=\-\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\}\)\.Therefore,

‖Λiℓ\+1−Λiℓ‖F=‖∇Stfi​\(Piℓ\+1\)−∇Stfi​\(Piℓ\)‖F≤Li​‖Piℓ\+1−Piℓ‖F,\\begin\{split\}\\\|\\Lambda\_\{i\}^\{\\ell\+1\}\-\\Lambda\_\{i\}^\{\\ell\}\\\|\_\{F\}&=\\left\\\|\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\+1\}\)\-\\nabla\_\{\\mathrm\{St\}\}f\_\{i\}\(P\_\{i\}^\{\\ell\}\)\\right\\\|\_\{F\}\\\\ &\\leq L\_\{i\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\},\\end\{split\}where the last inequality follows from Assumption A1\. Squaring both sides completes the proof\. ∎

### V\-BBound for the Augmented Lagrangian

Next, we show that the augmented Lagrangian decreases sufficiently up to the perturbation terms caused by inexact local optimization and Koopman operator refitting\.

###### Lemma V\.3\.

Suppose Assumptions A1–A5 hold\. Then the augmented Lagrangian satisfies

ℒℓ\+1−ℒℓ≤−\|𝒮ℓ\|​ρ2​‖Zℓ\+1−Zℓ‖F2\+∑i∈𝒮ℓ\(Li2ρ−μi​\(ρ\)2\)​‖Piℓ\+1−Piℓ‖F2\+εℓ\+ΔQℓ\.\\begin\{split\}\\mathcal\{L\}^\{\\ell\+1\}\-\\mathcal\{L\}^\{\\ell\}\\leq&\-\\frac\{\|\\mathcal\{S\}^\{\\ell\}\|\\rho\}\{2\}\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\+\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\left\(\\frac\{L\_\{i\}^\{2\}\}\{\\rho\}\-\\frac\{\\mu\_\{i\}\(\\rho\)\}\{2\}\\right\)\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\+\\varepsilon\_\{\\ell\}\+\\Delta\_\{Q\}^\{\\ell\}\.\\end\{split\}Moreover, ifρ\\rhois chosen such that

ρ​μi​\(ρ\)≥2​Li2\.\\rho\\mu\_\{i\}\(\\rho\)\\geq 2L\_\{i\}^\{2\}\.then there exist constantsa,b\>0a,b\>0such that

ℒℓ\+1−ℒℓ≤−a​∑i∈𝒮ℓ‖Piℓ\+1−Piℓ‖F2−b​‖Zℓ\+1−Zℓ‖F2\+εℓ\+ΔQℓ\.\\begin\{split\}\\mathcal\{L\}^\{\\ell\+1\}\-\\mathcal\{L\}^\{\\ell\}\\leq&\-a\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\-b\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\+\\varepsilon\_\{\\ell\}\+\\Delta\_\{Q\}^\{\\ell\}\.\\end\{split\}\(23\)

Proof\.For one communication roundℓ\\ell, we split the successive difference of the augmented Lagrangian into four terms:

ℒℓ\+1−ℒℓ\\displaystyle\\mathcal\{L\}^\{\\ell\+1\}\-\\mathcal\{L\}^\{\\ell\}=ℒℓ​\(\{Piℓ\+1\},Zℓ,\{Λiℓ\}\)−ℒℓ​\(\{Piℓ\},Zℓ,\{Λiℓ\}\)⏟A:local​Pi​\-update\\displaystyle=\\underbrace\{\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\)\-\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\}\\\},Z^\{\\ell\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\)\}\_\{A:\\ \\text\{local \}P\_\{i\}\\text\{\-update\}\}\+ℒℓ​\(\{Piℓ\+1\},Zℓ\+1,\{Λiℓ\}\)−ℒℓ​\(\{Piℓ\+1\},Zℓ,\{Λiℓ\}\)⏟B:Z​\-update\\displaystyle\\quad\+\\underbrace\{\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\+1\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\)\-\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\)\}\_\{B:\\ Z\\text\{\-update\}\}\+ℒℓ​\(\{Piℓ\+1\},Zℓ\+1,\{Λiℓ\+1\}\)−ℒℓ​\(\{Piℓ\+1\},Zℓ\+1,\{Λiℓ\}\)⏟C:dual update\\displaystyle\\quad\+\\underbrace\{\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\+1\},\\\{\\Lambda\_\{i\}^\{\\ell\+1\}\\\}\)\-\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\+1\},\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}\)\}\_\{C:\\ \\text\{dual update\}\}\+ℒℓ\+1−ℒℓ​\(\{Piℓ\+1\},Zℓ\+1,\{Λiℓ\+1\}\)⏟D:Koopman refitting perturbation\.\\displaystyle\\quad\+\\underbrace\{\\mathcal\{L\}^\{\\ell\+1\}\-\\mathcal\{L\}^\{\\ell\}\(\\\{P\_\{i\}^\{\\ell\+1\}\\\},Z^\{\\ell\+1\},\\\{\\Lambda\_\{i\}^\{\\ell\+1\}\\\}\)\}\_\{D:\\ \\text\{Koopman refitting perturbation\}\}\.Here, termAAcorresponds to the localPiP\_\{i\}\-updates withZℓZ^\{\\ell\}andΛiℓ\\Lambda\_\{i\}^\{\\ell\}fixed; termBBcorresponds to the global consensusZZ\-update; termCCcorresponds to the dual\-variable update; and termDDcaptures the perturbation caused by refitting the reduced Koopman operators\.

Bounding Term A\.By Assumption A2, the localPiP\_\{i\}\-subproblem is strongly convex with modulusμi​\(ρ\)\\mu\_\{i\}\(\\rho\)\. Since the local subproblem is solved inexactly, the error is captured byεℓ\\varepsilon\_\{\\ell\}in Assumption A4\. Therefore,

A≤−∑i∈𝒮ℓμi​\(ρ\)2​‖Piℓ\+1−Piℓ‖F2\+εℓ\.A\\leq\-\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\frac\{\\mu\_\{i\}\(\\rho\)\}\{2\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\+\\varepsilon\_\{\\ell\}\.\(24\)
Bounding Term B\.With\{Piℓ\+1\}\\\{P\_\{i\}^\{\\ell\+1\}\\\}and\{Λiℓ\}\\\{\\Lambda\_\{i\}^\{\\ell\}\\\}fixed, theZZ\-update is a quadratic minimization with strong\-convexity modulus\|𝒮ℓ\|​ρ\|\\mathcal\{S\}^\{\\ell\}\|\\rho\. Hence,

B≤−\|𝒮ℓ\|​ρ2​‖Zℓ\+1−Zℓ‖F2\.B\\leq\-\\frac\{\|\\mathcal\{S\}^\{\\ell\}\|\\rho\}\{2\}\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\.\(25\)
Bounding Term C\.The dual update changes only the linear dual term of the augmented Lagrangian\. Thus,

C=1ρ​∑i∈𝒮ℓ‖Λiℓ\+1−Λiℓ‖F2\.C=\\frac\{1\}\{\\rho\}\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\\|\\Lambda\_\{i\}^\{\\ell\+1\}\-\\Lambda\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\.\(26\)Using Lemma[V\.2](https://arxiv.org/html/2607.08978#S5.Thmtheorem2), we obtain

C≤∑i∈𝒮ℓLi2ρ​‖Piℓ\+1−Piℓ‖F2\.C\\leq\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\frac\{L\_\{i\}^\{2\}\}\{\\rho\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\.\(27\)
Bounding Term D\.TermDDis the perturbation introduced by refitting the local reduced Koopman operators\. By Assumption A5,

D=ΔQℓ\.D=\\Delta\_\{Q\}^\{\\ell\}\.\(28\)
Sufficient decrease\.Combining \([24](https://arxiv.org/html/2607.08978#S5.E24)\)–\([28](https://arxiv.org/html/2607.08978#S5.E28)\), we obtain

ℒℓ\+1−ℒℓ≤−\|𝒮ℓ\|​ρ2​‖Zℓ\+1−Zℓ‖F2\+∑i∈𝒮ℓ\(Li2ρ−μi​\(ρ\)2\)​‖Piℓ\+1−Piℓ‖F2\+εℓ\+ΔQℓ\.\\begin\{split\}\\mathcal\{L\}^\{\\ell\+1\}\-\\mathcal\{L\}^\{\\ell\}\\leq&\-\\frac\{\|\\mathcal\{S\}^\{\\ell\}\|\\rho\}\{2\}\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\+\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\left\(\\frac\{L\_\{i\}^\{2\}\}\{\\rho\}\-\\frac\{\\mu\_\{i\}\(\\rho\)\}\{2\}\\right\)\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\+\\varepsilon\_\{\\ell\}\+\\Delta\_\{Q\}^\{\\ell\}\.\\end\{split\}This implies that the augmented Lagrangian has sufficient descent up to the summable perturbation termsεℓ\\varepsilon\_\{\\ell\}andΔQℓ\\Delta\_\{Q\}^\{\\ell\}if the following condition is satisfied:

ρ​μi​\(ρ\)≥2​Li2\\rho\\mu\_\{i\}\(\\rho\)\\geq 2L\_\{i\}^\{2\}

### V\-CConvergence of the Augmented Lagrangian

Here, we combine Lemma[V\.2](https://arxiv.org/html/2607.08978#S5.Thmtheorem2)and Lemma[V\.3](https://arxiv.org/html/2607.08978#S5.Thmtheorem3)to establish the convergence of the augmented Lagrangian\.

###### Theorem V\.5\.

Suppose Assumptions A1–A5 hold\. Further suppose that afterLLcommunication rounds, every client has participated at least once\. Then, by choosingρ\\rhosufficiently large, the augmented Lagrangian sequence\{ℒℓ\}\\\{\\mathcal\{L\}^\{\\ell\}\\\}converges to a finite value\. Moreover,

‖Zℓ\+1−Zℓ‖F→0,‖Piℓ\+1−Piℓ‖F→0,∀i,\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}\\to 0,\\qquad\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}\\to 0,\\quad\\forall i,and

‖Piℓ\+1−Zℓ\+1‖F→0,∀i\.\\\|P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\\\|\_\{F\}\\to 0,\\quad\\forall i\.

Proof\.Summing \([23](https://arxiv.org/html/2607.08978#S5.E23)\) fromℓ=0\\ell=0toL−1L\-1yields

ℒL−ℒ0≤−a​∑ℓ=0L−1∑i∈𝒮ℓ‖Piℓ\+1−Piℓ‖F2−b​∑ℓ=0L−1‖Zℓ\+1−Zℓ‖F2\+∑ℓ=0L−1εℓ\+∑ℓ=0L−1ΔQℓ\.\\begin\{split\}\\mathcal\{L\}^\{L\}\-\\mathcal\{L\}^\{0\}\\leq&\-a\\sum\_\{\\ell=0\}^\{L\-1\}\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}\\\\ &\-b\\sum\_\{\\ell=0\}^\{L\-1\}\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\+\\sum\_\{\\ell=0\}^\{L\-1\}\\varepsilon\_\{\\ell\}\+\\sum\_\{\\ell=0\}^\{L\-1\}\\Delta\_\{Q\}^\{\\ell\}\.\\end\{split\}\(29\)By Assumption A3,ℒL\\mathcal\{L\}^\{L\}is bounded from below\. By Assumptions A4 and A5,

∑ℓ=0∞εℓ<∞,∑ℓ=0∞ΔQℓ<∞\.\\sum\_\{\\ell=0\}^\{\\infty\}\\varepsilon\_\{\\ell\}<\\infty,\\qquad\\sum\_\{\\ell=0\}^\{\\infty\}\\Delta\_\{Q\}^\{\\ell\}<\\infty\.LettingL→∞L\\to\\inftyin \([29](https://arxiv.org/html/2607.08978#S5.E29)\), we obtain

∑ℓ=0∞∑i∈𝒮ℓ‖Piℓ\+1−Piℓ‖F2​<∞,∑ℓ=0∞∥​Zℓ\+1−Zℓ∥F2<∞\.\\sum\_\{\\ell=0\}^\{\\infty\}\\sum\_\{i\\in\\mathcal\{S\}^\{\\ell\}\}\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}^\{2\}<\\infty,\\qquad\\sum\_\{\\ell=0\}^\{\\infty\}\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}<\\infty\.Therefore,

‖Zℓ\+1−Zℓ‖F→0,‖Piℓ\+1−Piℓ‖F→0\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}\\to 0,\\qquad\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}\\to 0along the rounds in which clientiiparticipates\. Since every client participates at least once within anyLLconsecutive communication rounds, the latter limit holds for all clients\.

From the dual update,

Piℓ\+1−Zℓ\+1=ρ−1​\(Λiℓ\+1−Λiℓ\)\.P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}=\\rho^\{\-1\}\\left\(\\Lambda\_\{i\}^\{\\ell\+1\}\-\\Lambda\_\{i\}^\{\\ell\}\\right\)\.Combining this with Lemma[V\.2](https://arxiv.org/html/2607.08978#S5.Thmtheorem2), we have

‖Λiℓ\+1−Λiℓ‖F→0\.\\\|\\Lambda\_\{i\}^\{\\ell\+1\}\-\\Lambda\_\{i\}^\{\\ell\}\\\|\_\{F\}\\to 0\.Therefore,

‖Piℓ\+1−Zℓ\+1‖F→0\.\\\|P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\\\|\_\{F\}\\to 0\.Finally, \([23](https://arxiv.org/html/2607.08978#S5.E23)\) shows that all possible increases ofℒℓ\\mathcal\{L\}^\{\\ell\}are controlled by the summable sequences\{εℓ\}\\\{\\varepsilon\_\{\\ell\}\\\}and\{ΔQℓ\}\\\{\\Delta\_\{Q\}^\{\\ell\}\\\}\. Sinceℒℓ\\mathcal\{L\}^\{\\ell\}is bounded from below, the sequence\{ℒℓ\}\\\{\\mathcal\{L\}^\{\\ell\}\\\}converges to a finite value\. ∎

### V\-DConvergence to a Stationary Point

In the following theorem, we show that every limit point of the sequence generated byFedKADsatisfies the first\-order stationarity conditions\.

###### Theorem V\.7\.

Suppose the conditions in Theorem[V\.5](https://arxiv.org/html/2607.08978#S5.Thmtheorem5)hold and the localPiP\_\{i\}\-subproblems are solved to first\-order stationarity\. Then every limit point\(\{Pi⋆\},Z⋆,\{Λi⋆\}\)\(\\\{P\_\{i\}^\{\\star\}\\\},Z^\{\\star\},\\\{\\Lambda\_\{i\}^\{\\star\}\\\}\)of the sequence generated byFedKADis a stationary solution\. That is, for alli=1,…,Ni=1,\\ldots,N,

Pi⋆\\displaystyle P\_\{i\}^\{\\star\}=Z⋆,\\displaystyle=Z^\{\\star\},\(30\)∇St\[fi​\(Pi⋆,Qi⋆\)\+⟨Λi⋆,Pi⋆−Z⋆⟩F\]\\displaystyle\\nabla\_\{\\mathrm\{St\}\}\\left\[f\_\{i\}\(P\_\{i\}^\{\\star\},Q\_\{i\}^\{\\star\}\)\+\\langle\\Lambda\_\{i\}^\{\\star\},P\_\{i\}^\{\\star\}\-Z^\{\\star\}\\rangle\_\{F\}\\right\]=0\.\\displaystyle=0\.\(31\)Moreover,

Qi⋆∈arg⁡minQi⁡fi​\(Pi⋆,Qi\)\.Q\_\{i\}^\{\\star\}\\in\\arg\\min\_\{Q\_\{i\}\}f\_\{i\}\(P\_\{i\}^\{\\star\},Q\_\{i\}\)\.IfPi⋆⊤​Xi​Xi⊤​Pi⋆P\_\{i\}^\{\\star\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}is nonsingular, then

Qi⋆=Pi⋆⊤​Yi​Xi⊤​Pi⋆​\(Pi⋆⊤​Xi​Xi⊤​Pi⋆\)−1\.Q\_\{i\}^\{\\star\}=P\_\{i\}^\{\\star\\top\}Y\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}\\left\(P\_\{i\}^\{\\star\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}\\right\)^\{\-1\}\.

Proof\.Let\(\{Pi⋆\},Z⋆,\{Λi⋆\}\)\(\\\{P\_\{i\}^\{\\star\}\\\},Z^\{\\star\},\\\{\\Lambda\_\{i\}^\{\\star\}\\\}\)be any limit point and consider a convergent subsequence\. From Theorem[V\.5](https://arxiv.org/html/2607.08978#S5.Thmtheorem5),

‖Piℓ\+1−Zℓ\+1‖F→0\.\\\|P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\\\|\_\{F\}\\to 0\.Taking the limit along the convergent subsequence givesPi⋆=Z⋆P\_\{i\}^\{\\star\}=Z^\{\\star\}, which proves \([30](https://arxiv.org/html/2607.08978#S5.E30)\)\.

Next, since the localPiP\_\{i\}\-subproblem is solved to first\-order stationarity on the Stiefel manifold, we have

∇St\[fi​\(Piℓ\+1,Qiℓ\+1\)\+⟨Λiℓ,Piℓ\+1−Zℓ⟩F\+ρ2​‖Piℓ\+1−Zℓ‖F2\]→0\.\\nabla\_\{\\mathrm\{St\}\}\\left\[f\_\{i\}\(P\_\{i\}^\{\\ell\+1\},Q\_\{i\}^\{\\ell\+1\}\)\+\\langle\\Lambda\_\{i\}^\{\\ell\},P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\}\\rangle\_\{F\}\+\\frac\{\\rho\}\{2\}\\\|P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}^\{2\}\\right\]\\to 0\.Moreover,

Piℓ\+1−Zℓ=\(Piℓ\+1−Zℓ\+1\)\+\(Zℓ\+1−Zℓ\)→0\.P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\}=\(P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\)\+\(Z^\{\\ell\+1\}\-Z^\{\\ell\}\)\\to 0\.Thus, the gradient contribution of the quadratic consensus penalty vanishes in the limit\. Passing to the limit gives

∇St\[fi​\(Pi⋆,Qi⋆\)\+⟨Λi⋆,Pi⋆−Z⋆⟩F\]=0,\\nabla\_\{\\mathrm\{St\}\}\\left\[f\_\{i\}\(P\_\{i\}^\{\\star\},Q\_\{i\}^\{\\star\}\)\+\\langle\\Lambda\_\{i\}^\{\\star\},P\_\{i\}^\{\\star\}\-Z^\{\\star\}\\rangle\_\{F\}\\right\]=0,which proves \([31](https://arxiv.org/html/2607.08978#S5.E31)\)\.

Finally, for each fixedPiℓ\+1P\_\{i\}^\{\\ell\+1\}, the reduced Koopman operatorQiℓ\+1Q\_\{i\}^\{\\ell\+1\}is computed as a minimizer of the local least\-squares objective\. By continuity,

Qi⋆∈arg⁡minQi⁡fi​\(Pi⋆,Qi\)\.Q\_\{i\}^\{\\star\}\\in\\arg\\min\_\{Q\_\{i\}\}f\_\{i\}\(P\_\{i\}^\{\\star\},Q\_\{i\}\)\.IfPi⋆⊤​Xi​Xi⊤​Pi⋆P\_\{i\}^\{\\star\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}is nonsingular, the minimizer is unique\. Solving the normal equations gives

Qi⋆=Pi⋆⊤​Yi​Xi⊤​Pi⋆​\(Pi⋆⊤​Xi​Xi⊤​Pi⋆\)−1\.Q\_\{i\}^\{\\star\}=P\_\{i\}^\{\\star\\top\}Y\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}\\left\(P\_\{i\}^\{\\star\\top\}X\_\{i\}X\_\{i\}^\{\\top\}P\_\{i\}^\{\\star\}\\right\)^\{\-1\}\.Therefore, every limit point satisfies the stated stationarity conditions\. ∎

## VIExperimental Results

We evaluateFedKADon four widely used MVTS anomaly\-detection benchmarks summarised in[TableI](https://arxiv.org/html/2607.08978#S6.T1): \(1\)Pool Server Metrics \(PSM\)\[abdulaal2021practical\], a2525\-dimensional dataset from eBay’s pooled server resources with132,481132\{,\}481training and87,84187\{,\}841testing samples and a27\.75%27\.75\\%anomaly ratio; \(2\)Server Machine Dataset \(SMD\)\[su2019robust\], five weeks of resource\-utilisation traces from2828servers, each with3838metrics \(CPU, memory, network\) and a4\.16%4\.16\\%anomaly ratio; \(3\)Soil Moisture Active Passive \(SMAP\)and \(4\)Mars Science Laboratory \(MSL\)\[hundman2018detecting\], NASA spacecraft telemetry with5454and2727entities and2525and5555channels respectively \(anomaly ratios12\.85%12\.85\\%and10\.53%10\.53\\%\)\.[TableI](https://arxiv.org/html/2607.08978#S6.T1)lists the per\-dataset train/test counts, anomaly ratio, channel count, and the number of FL clients induced by the partitioning scheme described in[Sec\.VI\-A](https://arxiv.org/html/2607.08978#S6.SS1)\.

TABLE I:Dataset statistics\.*NS*: number of channels per entity;*NN*: number of FL clients \(one per entity for SMD/SMAP/MSL; Dirichlet\-partitioned for PSM\)\.![Refer to caption](https://arxiv.org/html/2607.08978v1/x1.png)Figure 2:Non\-IID heterogeneity across the four benchmarks, covering feature distribution skew, label distribution skew, and quantity skew\.### VI\-AFederated learning settings

#### VI\-A1Dataset

For SMD, SMAP and MSL we use the*natural per\-entity partition*: one physical entity \(server or spacecraft subsystem\) becomes one FL client, so its train and test streams remain entity\-specific and disjoint and no cross\-entity leakage is possible\. This yields2828,5454and2727clients respectively\. PSM provides a single shared stream, so we induce non\-IID\-ness by Dirichlet partitioning following\[mcmahan2017communication\]: the132,481132\{,\}481\-sample training stream is sliced into2424contiguous chunks of mean length5,5205\{,\}520which are then assigned to clients via Dirichlet\(α=0\.5\)\(\\alpha\{=\}0\.5\)over the2525feature dimensions, so each client predominantly observes a different subset of channels\.

Three independent sources of non\-IID heterogeneity co\-occur in our federations \([Fig\.2](https://arxiv.org/html/2607.08978#S6.F2)\): \(i\)*feature\-distribution skew*— per\-client std of the per\-feature mean spans0\.140\.14–2\.472\.47on SMD,0\.0110\.011–1\.961\.96on SMAP and0\.0150\.015–1\.331\.33on MSL — roughly two orders of magnitude across the federation; \(ii\)*label\-distribution skew*— the per\-client anomaly ratio spans0\.40\.4–27%27\\%on SMD and0–98%98\\%on SMAP/MSL, while PSM is essentially uniform at≈30%\{\\approx\}30\\%by construction; and \(iii\)*quantity skew*— per\-client training length varies1\.2×1\.2\{\\times\}on SMD,≈10×\{\\approx\}10\{\\times\}on SMAP/MSL and up to≈25×\{\\approx\}25\{\\times\}on PSM\.

All data arezz\-scored per client \(SMD/SMAP/MSL: each client is its own physical entity, zero leakage\) or globally on PSM \(a single shared test stream forces a single scaler\), then clipped to±10​σ\\pm 10\\sigmato remove training\-stream spikes that destabilise reconstruction baselines\. We use a sliding window of lengthw=20w\{=\}20with stride11, matching USAD/TranAD\. The shared FedAvg envelope \([TableII](https://arxiv.org/html/2607.08978#S6.T2)\) trains all five methods for3030global rounds with25%25\\%client sampling per round,55local epochs,Adamatlr=1×10−3\\text\{lr\}\{=\}1\{\\times\}10^\{\-3\}, batch size128128, and a temporal85%/15%85\\%/15\\%per\-client train/validation split\. ForFedKAD, we use the same default hyperparameter setting on SMD, SMAP, and MSL: Koopman lifting dimensiondlift=128d\_\{\\mathrm\{lift\}\}=128, subspace rankr=24r=24, ADMM penaltyρ=1\.0\\rho=1\.0, Stiefel\-ADMM step size0\.050\.05, ridge regularisationλ=10−2\\lambda=10^\{\-2\}, spectral cap0\.9950\.995, score\-smoothing window of55steps, and prediction horizonH=1H=1\. For PSM, all hyperparameters are kept unchanged except for the subspace rank, which is increased fromr=24r=24tor=32r=32based on a held\-out validation sweep without using test labels\. Every\(dataset, method\)configuration is repeated for three seeds\{0,1,2\}\\\{0,1,2\\\}; we report mean±\\pmstd across seeds\. Implementation: Python 3\.10, PyTorch 2\.1\.0, CUDA 12\.1; experiments run on4×4\{\\times\}NVIDIA A40 \(4545GB\) GPUs\.

#### VI\-A2Baselines

We compareFedKADagainst four representative neural baselines for MTAD: \(1\)DeepSVDD\[ruff2018deep\], a one\-class deep network with hypersphere objective; \(2\)LSTM\-AE\[malhotra2016lstm\], an autoencoder of stacked LSTM units; \(3\)USAD\[audibert2020usad\], an adversarial encoder–decoder for unsupervised MTAD; and \(4\)TranAD\[tuli2022tranad\], a transformer\-based reconstructor with an adversarial decoder\. Because these methods are originally designed for centralised training, we wrap them inFedAvg\[mcmahan2017communication\]with the identical federated envelope used forFedKAD, so the only dimension of variation is the local model class\. LSTM\-AE, USAD and TranAD use the default architectures of the original papers; for DeepSVDD we disable the autoencoder pretraining and centralised hypersphere\-centre initialisation under our FL\-faithful evaluation because they would require server access to the union of every client’s raw training data, violating the FL privacy constraint\.

TABLE II:Federated training configuration shared by all methods\.
#### VI\-A3Evaluation metrics

We report Precision, Recall, F1, and AUC under a per\-client macro aggregation protocol\. Specifically, each metric is first computed separately on each client’s test stream using its local ground\-truth labels, and the resulting scores are then averaged uniformly across clients\. This protocol prevents clients with longer streams from dominating the aggregate result and therefore reflects performance at the federation level rather than at the pooled\-sample level\.

To facilitate comparison with prior work while avoiding over\-reliance on the known optimistic bias of point adjustment\[kim2022rigorous\], we evaluate four complementary thresholding and adjustment protocols\. First, our primary headline metric uses segment\-aware PA%K withk=0\.01k\{=\}0\.01\[kim2022rigorous\], where the detection threshold is tuned on each client’s validation split\. Under PA%K, an anomalous segment is counted as detected only if at least a fractionkkof its points are predicted as anomalous; thus,k=0\.01k\{=\}0\.01corresponds to requiring at least1%1\\%of the segment to be detected\. This criterion mitigates the classic any\-hit pathology of point adjustment, where a single isolated alarm inside an anomalous segment is sufficient to mark the entire segment as correctly detected\. Second, we report the conventional blind\-threshold setting based on POT\[siffer2017anomaly\]with any\-hit point adjustment \(k=0k\{=\}0\)\[xu2018unsupervised\]; here, the threshold is selected without access to test labels and an anomalous segment is considered detected if at least one point within the segment is flagged\. Third, we evaluate a stricter blind setting that combines POT thresholding with PA%K atk=0\.1k\{=\}0\.1, corresponding to requiring at least10%10\\%of an anomalous segment to be detected, together with a recall cap of0\.9950\.995to reject lucky\-spike thresholds\. Finally, we report strict point\-wise F1at the POT threshold without point adjustment, which measures detection quality at the individual timestamp level\.

### VI\-BMain results

#### VI\-B1Detection performance and statistical significance

[Fig\.3](https://arxiv.org/html/2607.08978#S6.F3)summarizes the per\-client averaged F1scores ofFedKADand the four baselines under the primary PAon three of the four datasets, namely SMD, PSM, and SMAP, while showing only a small degradation on MSL\. On SMD,FedKADobtains72\.20±1\.2872\.20\{\\pm\}1\.28, outperforming the strongest baseline, LSTM\-AE, by\+7\.69\+7\.69F1\(p=0\.038p\{=\}0\.038,dz=\+2\.86d\_\{z\}\{=\}\+2\.86\)\. On PSM,FedKADachieves69\.87±0\.2169\.87\{\\pm\}0\.21, improving over LSTM\-AE by\+1\.29\+1\.29F1\(p=0\.0092p\{=\}0\.0092,dz=\+5\.97d\_\{z\}\{=\}\+5\.97\)\. On SMAP,FedKADreaches53\.32±0\.5053\.32\{\\pm\}0\.50, giving a positive margin of\+1\.76\+1\.76F1over LSTM\-AE\. Although this gain is not statistically significant at thep<0\.10p\{<\}0\.10level, it exceeds the predefined verdict margin of0\.50\.5F1\. On MSL,FedKADobtains39\.55±0\.0239\.55\{\\pm\}0\.02, compared with40\.50±4\.6340\.50\{\\pm\}4\.63for LSTM\-AE, corresponding to a small loss of0\.950\.95F1\.

![Refer to caption](https://arxiv.org/html/2607.08978v1/x2.png)Figure 3:Performance under the primary PA%K protocol withk=0\.01k=0\.01\.TABLE III:Paired\-tttest ofFedKADversus the strongest baseline per dataset, with Cohendzd\_\{z\}effect size and F1margin\. Verdict gate:margin\>0\.5\\text\{margin\}\{\>\}0\.5F1\.[TableIII](https://arxiv.org/html/2607.08978#S6.T3)reports the paired\-tttest betweenFedKADand the strongest baseline on each dataset\. The results confirm that the F1improvements on SMD and PSM are statistically supported, with margins of\+7\.69\+7\.69and\+1\.29\+1\.29F1, respectively, and large effect sizes \(dz=\+2\.86d\_\{z\}\{=\}\+2\.86anddz=\+5\.97d\_\{z\}\{=\}\+5\.97\)\. On SMAP,FedKADstill improves over the strongest baseline by\+1\.76\+1\.76F1, although the paired test is not significant because of seed\-level variability\. On MSL,FedKADis below LSTM\-AE by only0\.950\.95F1, with a negligible effect size \(dz=−0\.17d\_\{z\}\{=\}\-0\.17\)\. It is easily to see thatFedKADsatisfies the predefined verdict gate on three of the four datasets, while its only loss remains small\.

#### VI\-B2Robustness across evaluation protocols

[TableIV](https://arxiv.org/html/2607.08978#S6.T4)evaluate FedKAD under three supplementary protocols\. Under the classic POT threshold with any\-hit point adjustment, FedKAD remains highly competitive and achieves the best F1on SMD and PSM, while tying LSTM\-AE on MSL\. However, this protocol saturates recall on long anomaly segments, so the ranking mainly reflects precision\. When the PA%K requirement is tightened tok=0\.10k=0\.10, FedKAD keeps a clear advantage on SMD but becomes less competitive on PSM, SMAP, and MSL\. This indicates that FedKAD detects anomalous segments reliably on server\-machine streams but is more sensitive to stricter segment\-coverage requirements on spacecraft and PSM streams\. Finally, under the strict point\-wise no\-PA protocol, all methods obtain substantially lower F1scores, confirming the difficulty of exact point\-level localization\. FedKAD remains competitive on MSL but does not dominate this strict setting\. Overall, the multi\-protocol results show that FedKAD’s main advantage is strongest under segment\-level anomaly detection protocols\.

![Refer to caption](https://arxiv.org/html/2607.08978v1/x3.png)Figure 4:Federated efficiency: training wall\-clock \(s\), per\-round communication \(KB\) and inference latency \(μ\\mus/step\), log scale\.FedKADis two to three orders of magnitude faster to train and one to two orders cheaper in communication than the next\-best neural baseline, with comparable or lower inference latency\.TABLE IV:FedKAD robustness under supplementary evaluation protocols\. The best baseline is selected per dataset and protocol\.
#### VI\-B3Federated efficiency

[Fig\.4](https://arxiv.org/html/2607.08978#S6.F4)reports end\-to\-end training wall\-clock, per\-round communication and inference latency\. On all four datasets,FedKADtrains in1\.71\.7–4\.84\.8s versus7575–40204020s for the neural baselines \(a200×200\{\\times\}–2000×2000\{\\times\}speed\-up\), exchanges1\.01\.0–1\.71\.7MB per round versus2\.72\.7–129129MB \(a3×3\{\\times\}–40×40\{\\times\}reduction\), and incurs1\.91\.9–5\.85\.8μ\\mus per inference step, comparable to or below every neural baseline\. Both savings stem fromFedKAD’s two architectural commitments: closed\-form local fitting, which avoids local\-SGD drift under non\-IID data, and low\-rank subspace consensus, where communication scales with the subspace sizeO​\(L​r\)O\(Lr\)rather than the number of trainable model parametersO​\(\|θ\|\)O\(\|\\theta\|\)\. In a federated deployment, TranAD’s hour\-long per\-client round and USAD’s6060–130130MB per\-round upload are prohibitive on edge hardware, whereasFedKADfits in single\-second rounds with sub\-22MB uplinks\. Combining the accuracy verdict of[TableIII](https://arxiv.org/html/2607.08978#S6.T3)with the efficiency profile of[Fig\.4](https://arxiv.org/html/2607.08978#S6.F4),FedKADis the only method in our study that is simultaneously accurate \(on3/43/4datasets\) and FL\-deployable\.

#### VI\-B4On\-device deployment \(Raspberry Pi 4\)

To complement the data\-centre measurements of[Fig\.4](https://arxiv.org/html/2607.08978#S6.F4), we re\-evaluate every method end\-to\-end on commodity edge hardware: a Raspberry Pi 4 \(Broadcom BCM2711,44×\\timesCortex\-A72 @1\.51\.5GHz, ARMv8,44GB LPDDR4\) running6464\-bit Raspberry Pi OS, Python 3\.13 and PyTorch 2\.12 \(CPU EP\)\. The setting mirrors SMAP \(nfeatures=25n\_\{\\text\{features\}\}\{=\}25,w=20w\{=\}20, FedKADr=24r\{=\}24,dlift=128d\_\{\\mathrm\{lift\}\}\{=\}128\), with100100warm\-started repeats per measurement and the median reported\.[Fig\.6](https://arxiv.org/html/2607.08978#S6.F6)plots training time per global round \(s,log\\log\), inference latency \(μ\\mus/step\) and per\-round communication: the only quantity uploaded each FedAvg round is the Stiefel basisPi∈ℝD×rP\_\{i\}\\in\\mathbb\{R\}^\{D\\times r\}, so for FedKAD the bubble area encodes\|Pi\|=59\|P\_\{i\}\|\{=\}59KB, an order of magnitude below LSTM\-AE \(227227KB\) and USAD \(1\.91\.9MB\)\.FedKADreaches the lower\-left efficient corner on the Pi:0\.230\.23s/round training \(vs\.0\.320\.32s for the next\-best DeepSVDD,0\.870\.87s for LSTM\-AE, and5\.665\.66s for TranAD\) and0\.790\.79μ\\mus/step inference \(vs\.2\.032\.03μ\\mus for DeepSVDD and24\.524\.5μ\\mus for TranAD\) — a24×24\{\\times\}training speed\-up and31×31\{\\times\}inference speed\-up over the heaviest baseline, on a $3535device\. Combined with the5959KB per\-round upload, this establishes thatFedKADis the only method in our study that fits the compute, memory and bandwidth budget of a real edge deployment\.

![Refer to caption](https://arxiv.org/html/2607.08978v1/x4.png)Figure 5:Empirical convergence diagnostics on SMD \(N=28N\{=\}28clients,r=24r\{=\}24,3030rounds\)\.\(A\)Mean consensus residual1N​∑i‖Piℓ−Zℓ‖F\\frac\{1\}\{N\}\\sum\_\{i\}\\\|P\_\{i\}^\{\\ell\}\{\-\}Z^\{\\ell\}\\\|\_\{F\}: decays atρ≥1\\rho\{\\geq\}1\([TheoremV\.5](https://arxiv.org/html/2607.08978#S5.Thmtheorem5)\)\.\(B\)Augmented Lagrangianℒℓ\\mathcal\{L\}^\{\\ell\}: descends toward a bounded neighbourhood atρ=1\\rho\{=\}1; the late\-round oscillation is consistent with[LemmaV\.3](https://arxiv.org/html/2607.08978#S5.Thmtheorem3)under partial client participation and theΔQℓ\\Delta\_\{Q\}^\{\\ell\}perturbation from theQiQ\_\{i\}\-refit \(A5\)\.\(C\)Local objective∑ifiℓ​\(Piℓ\)\\sum\_\{i\}f\_\{i\}^\{\\ell\}\(P\_\{i\}^\{\\ell\}\): decreases over training\.![Refer to caption](https://arxiv.org/html/2607.08978v1/x5.png)Figure 6:On\-device benchmark on a Raspberry Pi 4 under the SMAP setting\.FedKADachieves the best edge\-deployment trade\-off, requiring only 59 KB upload per round while reducing training and inference time by up to 25×\\timesand 31×\\timesover the slowest baseline\.
#### VI\-B5Convergence guarantee \(theoretical companion\)

The federated training procedure ofFedKADis cast as a non\-convex consensus ADMM on the Stiefel manifold and admits the convergence guarantee proved in[TheoremV\.5](https://arxiv.org/html/2607.08978#S5.Thmtheorem5)shows that for any sufficiently large penaltyρ\\rhosatisfyingρ​μi​\(ρ\)\>2​Li2\\rho\\mu\_\{i\}\(\\rho\)\>2L\_\{i\}^\{2\}, the augmented Lagrangian\{ℒℓ\}\\\{\\mathcal\{L\}^\{\\ell\}\\\}converges to a finite value, the successive differences‖Piℓ\+1−Piℓ‖F\\\|P\_\{i\}^\{\\ell\+1\}\-P\_\{i\}^\{\\ell\}\\\|\_\{F\}and‖Zℓ\+1−Zℓ‖F\\\|Z^\{\\ell\+1\}\-Z^\{\\ell\}\\\|\_\{F\}vanish, and the consensus residual‖Piℓ\+1−Zℓ\+1‖F→0\\\|P\_\{i\}^\{\\ell\+1\}\-Z^\{\\ell\+1\}\\\|\_\{F\}\\to 0for every client\. The empirical convergence behaviour is further examined below by tracking the consensus residual, augmented Lagrangian, and local data\-fit term under different choices ofρ\\rho\.

[Fig\.5](https://arxiv.org/html/2607.08978#S6.F5)visualises the three convergence indicators on SMD \(N=28N\{=\}28clients\) forρ∈\{10−3,10−2,10−1,1\}\\rho\\in\\\{10^\{\-3\},10^\{\-2\},10^\{\-1\},1\\\}\. Atρ=1\\rho\{=\}1\(the default operating point\), the mean consensus residual1N​∑i‖Piℓ−Zℓ‖F\\frac\{1\}\{N\}\\sum\_\{i\}\\\|P\_\{i\}^\{\\ell\}\-Z^\{\\ell\}\\\|\_\{F\}drops from6\.56\.5to2\.12\.1over3030rounds \(panel A\), the augmented Lagrangianℒℓ\\mathcal\{L\}^\{\\ell\}descends from670670to a neighbourhood of≈90\{\\approx\}90–160160\(panel B\), and the local data\-fit term∑ifiℓ​\(Piℓ\)\\sum\_\{i\}f\_\{i\}^\{\\ell\}\(P\_\{i\}^\{\\ell\}\)decreases from15\.715\.7to6\.26\.2\(panel C\)\. The Lagrangian oscillates mildly in later rounds rather than descending strictly; this is consistent with[LemmaV\.3](https://arxiv.org/html/2607.08978#S5.Thmtheorem3), which permits per\-round increases when the finite\-inner\-loop errorεℓ\\varepsilon\_\{\\ell\}, the Koopman refitting perturbationΔQℓ\\Delta\_\{Q\}^\{\\ell\}, or partial\-participation effects exceed the descent terms, while still guaranteeing convergence to a finite limit under A2 and A5\. At smallerρ\\rho, the penalty is insufficient: the consensus residual stagnates andℒℓ\\mathcal\{L\}^\{\\ell\}grows because the dual accumulation exceeds the descent\. The transition occurs betweenρ=0\.1\\rho\{=\}0\.1andρ=1\\rho\{=\}1, confirming the “sufficiently largeρ\\rho” condition of the theorem and validating the default choice\.

## VIIConclusion

This paper presentedFedKAD, a federated low\-rank Koopman framework for anomaly detection in resource\-constrained IoT multivariate time series\. By learning compact Koopman dynamics and exchanging only low\-rank subspace variables,FedKADavoids the heavy training and communication cost of federated neural baselines while keeping raw data and local dynamics on device\. Experiments on four benchmarks show thatFedKADachieves the best F1score on three datasets under the primary PA%K protocol and provides substantial reductions in training time, communication, and inference latency\. The on\-device Raspberry Pi 4 benchmark further demonstrates its practicality for commodity edge hardware with limited compute, memory, and bandwidth\. Future work will study adaptive rank selection, personalized thresholding, and stronger privacy mechanisms\.

## References