Kernel Regression of Multi-Way Data via Tensor Trains with Hadamard Overparametrization: The Dynamic Graph Flow Case

A regression-based framework for interpretable multi-way data imputation, termed Kernel Regression via Tensor Trains with Hadamard overparametrization (KReTTaH), is introduced. KReTTaH adopts a nonparametric formulation by casting imputation as regression via reproducing kernel Hilbert spaces. Parameter efficiency is achieved through tensors of fixed tensor-train (TT) rank, which reside on low-dimensional Riemannian manifolds, and is further enhanced via Hadamard overparametrization, which promotes sparsity within the TT parameter space. Learning is accomplished by solving a smooth inverse problem posed on the Riemannian manifold of fixed TT-rank tensors. As a representative application, the estimation of dynamic graph flows is considered. In this setting, KReTTaH exhibits flexibility by seamlessly incorporating graph-based (topological) priors via its inverse problem formulation. Numerical tests on real-world graph datasets demonstrate that KReTTaH consistently outperforms state-of-the-art alternatives-including a nonparametric tensor- and a neural-network-based methods-for imputing missing, time-varying edge flows.

Nonconvex Regularization for Feature Selection in Reinforcement Learning

This work proposes an efficient batch algorithm for feature selection in reinforcement learning (RL) with theoretical convergence guarantees. To mitigate the estimation bias inherent in conventional regularization schemes, the first contribution extends policy evaluation within the classical least-squares temporal-difference (LSTD) framework by formulating a Bellman-residual objective regularized with the sparsity-inducing, nonconvex projected minimax concave (PMC) penalty. Owing to the weak convexity of the PMC penalty, this formulation can be interpreted as a special instance of a general nonmonotone-inclusion problem. The second contribution establishes novel convergence conditions for the forward-reflected-backward splitting (FRBS) algorithm to solve this class of problems. Numerical experiments on benchmark datasets demonstrate that the proposed approach substantially outperforms state-of-the-art feature-selection methods, particularly in scenarios with many noisy features.

Online reinforcement learning via sparse Gaussian mixture model Q-functions

This paper introduces a structured and interpretable online policy-iteration framework for reinforcement learning (RL), built around the novel class of sparse Gaussian mixture model Q-functions (S-GMM-QFs). Extending earlier work that trained GMM-QFs offline, the proposed framework develops an online scheme that leverages streaming data to encourage exploration. Model complexity is regulated through sparsification by Hadamard overparametrization, which mitigates overfitting while preserving expressiveness. The parameter space of S-GMM-QFs is naturally endowed with a Riemannian manifold structure, allowing for principled parameter updates via online gradient descent on a smooth objective. Numerical tests show that S-GMM-QFs match the performance of dense deep RL (DeepRL) methods on standard benchmarks while using significantly fewer parameters, and maintain strong performance even in low-parameter-count regimes where sparsified DeepRL methods fail to generalize.

Nonparametric Bellman Mappings for Value Iteration in Distributed Reinforcement Learning

This paper introduces novel Bellman mappings (B-Maps) for value iteration (VI) in distributed reinforcement learning (DRL), where multiple agents operate over a network without a centralized fusion node. Each agent constructs its own nonparametric B-Map for VI while communicating only with direct neighbors to achieve consensus. These B-Maps operate on Q-functions represented in a reproducing kernel Hilbert space, enabling a nonparametric formulation that allows for flexible, agent-specific basis function design. Unlike existing DRL methods that restrict information exchange to Q-function estimates, the proposed framework also enables agents to share basis information in the form of covariance matrices, capturing additional structural details. A theoretical analysis establishes linear convergence rates for both Q-function and covariance-matrix estimates toward their consensus values. The optimal learning rates for consensus-based updates are dictated by the ratio of the smallest positive eigenvalue to the largest one of the network's Laplacian matrix. Furthermore, each nodal Q-function estimate is shown to lie very close to the fixed point of a centralized nonparametric B-Map, effectively allowing the proposed DRL design to approximate the performance of a centralized fusion center. Numerical experiments on two well-known control problems demonstrate the superior performance of the proposed nonparametric B-Maps compared to prior methods. Notably, the results reveal a counter-intuitive finding: although the proposed approach involves greater information exchange -- specifically through the sharing of covariance matrices -- it achieves the desired performance with lower cumulative communication cost than existing DRL schemes, highlighting the crucial role of basis information in accelerating the learning process.

Model-Free Adversarial Purification via Coarse-To-Fine Tensor Network Representation

Deep neural networks are known to be vulnerable to well-designed adversarial attacks. Although numerous defense strategies have been proposed, many are tailored to the specific attacks or tasks and often fail to generalize across diverse scenarios. In this paper, we propose Tensor Network Purification (TNP), a novel model-free adversarial purification method by a specially designed tensor network decomposition algorithm. TNP depends neither on the pre-trained generative model nor the specific dataset, resulting in strong robustness across diverse adversarial scenarios. To this end, the key challenge lies in relaxing Gaussian-noise assumptions of classical decompositions and accommodating the unknown distribution of adversarial perturbations. Unlike the low-rank representation of classical decompositions, TNP aims to reconstruct the unobserved clean examples from an adversarial example. Specifically, TNP leverages progressive downsampling and introduces a novel adversarial optimization objective to address the challenge of minimizing reconstruction error but without inadvertently restoring adversarial perturbations. Extensive experiments conducted on CIFAR-10, CIFAR-100, and ImageNet demonstrate that our method generalizes effectively across various norm threats, attack types, and tasks, providing a versatile and promising adversarial purification technique.

Gaussian-Mixture-Model Q-Functions for Reinforcement Learning by Riemannian Optimization

This paper establishes a novel role for Gaussian-mixture models (GMMs) as functional approximators of Q-function losses in reinforcement learning (RL). Unlike the existing RL literature, where GMMs play their typical role as estimates of probability density functions, GMMs approximate here Q-function losses. The new Q-function approximators, coined GMM-QFs, are incorporated in Bellman residuals to promote a Riemannian-optimization task as a novel policy-evaluation step in standard policy-iteration schemes. The paper demonstrates how the hyperparameters (means and covariance matrices) of the Gaussian kernels are learned from the data, opening thus the door of RL to the powerful toolbox of Riemannian optimization. Numerical tests show that with no use of experienced data, the proposed design outperforms state-of-the-art methods, even deep Q-networks which use experienced data, on benchmark RL tasks.

Imputation of Time-varying Edge Flows in Graphs by Multilinear Kernel Regression and Manifold Learning

This paper extends the recently developed framework of multilinear kernel regression and imputation via manifold learning (MultiL-KRIM) to impute time-varying edge flows in a graph. MultiL-KRIM uses simplicial-complex arguments and Hodge Laplacians to incorporate the graph topology, and exploits manifold-learning arguments to identify latent geometries within features which are modeled as a point-cloud around a smooth manifold embedded in a reproducing kernel Hilbert space (RKHS). Following the concept of tangent spaces to smooth manifolds, linear approximating patches are used to add a collaborative-filtering flavor to the point-cloud approximations. Together with matrix factorizations, MultiL-KRIM effects dimensionality reduction, and enables efficient computations, without any training data or additional information. Numerical tests on real-network time-varying edge flows demonstrate noticeable improvements of MultiL-KRIM over several state-of-the-art schemes.

Nonparametric Bellman Mappings for Reinforcement Learning: Application to Robust Adaptive Filtering

This paper designs novel nonparametric Bellman mappings in reproducing kernel Hilbert spaces (RKHSs) for reinforcement learning (RL). The proposed mappings benefit from the rich approximating properties of RKHSs, adopt no assumptions on the statistics of the data owing to their nonparametric nature, require no knowledge on transition probabilities of Markov decision processes, and may operate without any training data. Moreover, they allow for sampling on-the-fly via the design of trajectory samples, re-use past test data via experience replay, effect dimensionality reduction by random Fourier features, and enable computationally lightweight operations to fit into efficient online or time-adaptive learning. The paper offers also a variational framework to design the free parameters of the proposed Bellman mappings, and shows that appropriate choices of those parameters yield several popular Bellman-mapping designs. As an application, the proposed mappings are employed to offer a novel solution to the problem of countering outliers in adaptive filtering. More specifically, with no prior information on the statistics of the outliers and no training data, a policy-iteration algorithm is introduced to select online, per time instance, the ``optimal'' coefficient p in the least-mean-p-power-error method. Numerical tests on synthetic data showcase, in most of the cases, the superior performance of the proposed solution over several RL and non-RL schemes.

Multilinear Kernel Regression and Imputation via Manifold Learning

This paper introduces a novel nonparametric framework for data imputation, coined multilinear kernel regression and imputation via the manifold assumption (MultiL-KRIM). Motivated by manifold learning, MultiL-KRIM models data features as a point cloud located in or close to a user-unknown smooth manifold embedded in a reproducing kernel Hilbert space. Unlike typical manifold-learning routes, which seek low-dimensional patterns via regularizers based on graph-Laplacian matrices, MultiL-KRIM builds instead on the intuitive concept of tangent spaces to manifolds and incorporates collaboration among point-cloud neighbors (regressors) directly into the data-modeling term of the loss function. Multiple kernel functions are allowed to offer robustness and rich approximation properties, while multiple matrix factors offer low-rank modeling, integrate dimensionality reduction, and streamline computations with no need of training data. Two important application domains showcase the functionality of MultiL-KRIM: time-varying-graph-signal (TVGS) recovery, and reconstruction of highly accelerated dynamic-magnetic-resonance-imaging (dMRI) data. Extensive numerical tests on real and synthetic data demonstrate MultiL-KRIM's remarkable speedups over its predecessors, and outperformance over prevalent "shallow" data-imputation techniques, with a more intuitive and explainable pipeline than deep-image-prior methods.

Proximal Bellman mappings for reinforcement learning and their application to robust adaptive filtering

This paper aims at the algorithmic/theoretical core of reinforcement learning (RL) by introducing the novel class of proximal Bellman mappings. These mappings are defined in reproducing kernel Hilbert spaces (RKHSs), to benefit from the rich approximation properties and inner product of RKHSs, they are shown to belong to the powerful Hilbertian family of (firmly) nonexpansive mappings, regardless of the values of their discount factors, and possess ample degrees of design freedom to even reproduce attributes of the classical Bellman mappings and to pave the way for novel RL designs. An approximate policy-iteration scheme is built on the proposed class of mappings to solve the problem of selecting online, at every time instance, the "optimal" exponent $p$ in a $p$-norm loss to combat outliers in linear adaptive filtering, without training data and any knowledge on the statistical properties of the outliers. Numerical tests on synthetic data showcase the superior performance of the proposed framework over several non-RL and kernel-based RL schemes.