RKHS Representation of Algebraic Convolutional Filters with Integral Operators

Integral operators play a central role in signal processing, underpinning classical convolution, and filtering on continuous network models such as graphons. While these operators are traditionally analyzed through spectral decompositions, their connection to reproducing kernel Hilbert spaces (RKHS) has not been systematically explored within the algebraic signal processing framework. In this paper, we develop a comprehensive theory showing that the range of integral operators naturally induces RKHS convolutional signal models whose reproducing kernels are determined by a box product of the operator symbols. We characterize the algebraic and spectral properties of these induced RKHS and show that polynomial filtering with integral operators corresponds to iterated box products, giving rise to a unital kernel algebra. This perspective yields pointwise RKHS representations of filters via the reproducing property, providing an alternative to operator-based implementations. Our results establish precise connections between eigendecompositions and RKHS representations in graphon signal processing, extend naturally to directed graphons, and enable novel spatial--spectral localization results. Furthermore, we show that when the spectral domain is a subset of the original domain of the signals, optimal filters for regularized learning problems admit finite-dimensional RKHS representations, providing a principled foundation for learnable filters in integral-operator-based neural architectures.

Size Transferability of Graph Transformers with Convolutional Positional Encodings

Transformers have achieved remarkable success across domains, motivating the rise of Graph Transformers (GTs) as attention-based architectures for graph-structured data. A key design choice in GTs is the use of Graph Neural Network (GNN)-based positional encodings to incorporate structural information. In this work, we study GTs through the lens of manifold limit models for graph sequences and establish a theoretical connection between GTs with GNN positional encodings and Manifold Neural Networks (MNNs). Building on transferability results for GNNs under manifold convergence, we show that GTs inherit transferability guarantees from their positional encodings. In particular, GTs trained on small graphs provably generalize to larger graphs under mild assumptions. We complement our theory with extensive experiments on standard graph benchmarks, demonstrating that GTs exhibit scalable behavior on par with GNNs. To further show the efficiency in a real-world scenario, we implement GTs for shortest path distance estimation over terrains to better illustrate the efficiency of the transferable GTs. Our results provide new insights into the understanding of GTs and suggest practical directions for efficient training of GTs in large-scale settings.

Learning Policy Representations for Steerable Behavior Synthesis

Given a Markov decision process (MDP), we seek to learn representations for a range of policies to facilitate behavior steering at test time. As policies of an MDP are uniquely determined by their occupancy measures, we propose modeling policy representations as expectations of state-action feature maps with respect to occupancy measures. We show that these representations can be approximated uniformly for a range of policies using a set-based architecture. Our model encodes a set of state-action samples into a latent embedding, from which we decode both the policy and its value functions corresponding to multiple rewards. We use variational generative approach to induce a smooth latent space, and further shape it with contrastive learning so that latent distances align with differences in value functions. This geometry permits gradient-based optimization directly in the latent space. Leveraging this capability, we solve a novel behavior synthesis task, where policies are steered to satisfy previously unseen value function constraints without additional training.

Unrolled Neural Networks for Constrained Optimization

In this paper, we develop unrolled neural networks to solve constrained optimization problems, offering accelerated, learnable counterparts to dual ascent (DA) algorithms. Our framework, termed constrained dual unrolling (CDU), comprises two coupled neural networks that jointly approximate the saddle point of the Lagrangian. The primal network emulates an iterative optimizer that finds a stationary point of the Lagrangian for a given dual multiplier, sampled from an unknown distribution. The dual network generates trajectories towards the optimal multipliers across its layers while querying the primal network at each layer. Departing from standard unrolling, we induce DA dynamics by imposing primal-descent and dual-ascent constraints through constrained learning. We formulate training the two networks as a nested optimization problem and propose an alternating procedure that updates the primal and dual networks in turn, mitigating uncertainty in the multiplier distribution required for primal network training. We numerically evaluate the framework on mixed-integer quadratic programs (MIQPs) and power allocation in wireless networks. In both cases, our approach yields near-optimal near-feasible solutions and exhibits strong out-of-distribution (OOD) generalization.

A Constrained Optimization Perspective of Unrolled Transformers

We introduce a constrained optimization framework for training transformers that behave like optimization descent algorithms. Specifically, we enforce layerwise descent constraints on the objective function and replace standard empirical risk minimization (ERM) with a primal-dual training scheme. This approach yields models whose intermediate representations decrease the loss monotonically in expectation across layers. We apply our method to both unrolled transformer architectures and conventional pretrained transformers on tasks of video denoising and text classification. Across these settings, we observe constrained transformers achieve stronger robustness to perturbations and maintain higher out-of-distribution generalization, while preserving in-distribution performance.

Cross-Learning from Scarce Data via Multi-Task Constrained Optimization

A learning task, understood as the problem of fitting a parametric model from supervised data, fundamentally requires the dataset to be large enough to be representative of the underlying distribution of the source. When data is limited, the learned models fail generalize to cases not seen during training. This paper introduces a multi-task \emph{cross-learning} framework to overcome data scarcity by jointly estimating \emph{deterministic} parameters across multiple, related tasks. We formulate this joint estimation as a constrained optimization problem, where the constraints dictate the resulting similarity between the parameters of the different models, allowing the estimated parameters to differ across tasks while still combining information from multiple data sources. This framework enables knowledge transfer from tasks with abundant data to those with scarce data, leading to more accurate and reliable parameter estimates, providing a solution for scenarios where parameter inference from limited data is critical. We provide theoretical guarantees in a controlled framework with Gaussian data, and show the efficiency of our cross-learning method in applications with real data including image classification and propagation of infectious diseases.

Disentangling Neurodegeneration with Brain Age Gap Prediction Models: A Graph Signal Processing Perspective

Neurodegeneration, characterized by the progressive loss of neuronal structure or function, is commonly assessed in clinical practice through reductions in cortical thickness or brain volume, as visualized by structural MRI. While informative, these conventional approaches lack the statistical sophistication required to fully capture the spatially correlated and heterogeneous nature of neurodegeneration, which manifests both in healthy aging and in neurological disorders. To address these limitations, brain age gap has emerged as a promising data-driven biomarker of brain health. The brain age gap prediction (BAGP) models estimate the difference between a person's predicted brain age from neuroimaging data and their chronological age. The resulting brain age gap serves as a compact biomarker of brain health, with recent studies demonstrating its predictive utility for disease progression and severity. However, practical adoption of BAGP models is hindered by their methodological obscurities and limited generalizability across diverse clinical populations. This tutorial article provides an overview of BAGP and introduces a principled framework for this application based on recent advancements in graph signal processing (GSP). In particular, we focus on graph neural networks (GNNs) and introduce the coVariance neural network (VNN), which leverages the anatomical covariance matrices derived from structural MRI. VNNs offer strong theoretical grounding and operational interpretability, enabling robust estimation of brain age gap predictions. By integrating perspectives from GSP, machine learning, and network neuroscience, this work clarifies the path forward for reliable and interpretable BAGP models and outlines future research directions in personalized medicine.

Composition and Alignment of Diffusion Models using Constrained Learning

Diffusion models have become prevalent in generative modeling due to their ability to sample from complex distributions. To improve the quality of generated samples and their compliance with user requirements, two commonly used methods are: (i) Alignment, which involves fine-tuning a diffusion model to align it with a reward; and (ii) Composition, which combines several pre-trained diffusion models, each emphasizing a desirable attribute in the generated outputs. However, trade-offs often arise when optimizing for multiple rewards or combining multiple models, as they can often represent competing properties. Existing methods cannot guarantee that the resulting model faithfully generates samples with all the desired properties. To address this gap, we propose a constrained optimization framework that unifies alignment and composition of diffusion models by enforcing that the aligned model satisfies reward constraints and/or remains close to (potentially multiple) pre-trained models. We provide a theoretical characterization of the solutions to the constrained alignment and composition problems and develop a Lagrangian-based primal-dual training algorithm to approximate these solutions. Empirically, we demonstrate the effectiveness and merits of our proposed approach in image generation, applying it to alignment and composition, and show that our aligned or composed model satisfies constraints effectively, and improves on the equally-weighted approach. Our implementation can be found at https://github.com/shervinkhalafi/constrained_comp_align.

Infinity Search: Approximate Vector Search with Projections on q-Metric Spaces

Despite the ubiquity of vector search applications, prevailing search algorithms overlook the metric structure of vector embeddings, treating it as a constraint rather than exploiting its underlying properties. In this paper, we demonstrate that in $q$-metric spaces, metric trees can leverage a stronger version of the triangle inequality to reduce comparisons for exact search. Notably, as $q$ approaches infinity, the search complexity becomes logarithmic. Therefore, we propose a novel projection method that embeds vector datasets with arbitrary dissimilarity measures into $q$-metric spaces while preserving the nearest neighbor. We propose to learn an approximation of this projection to efficiently transform query points to a space where euclidean distances satisfy the desired properties. Our experimental results with text and image vector embeddings show that learning $q$-metric approximations enables classic metric tree algorithms -- which typically underperform with high-dimensional data -- to achieve competitive performance against state-of-the-art search methods.

Alignment of large language models with constrained learning

We study the problem of computing an optimal large language model (LLM) policy for a constrained alignment problem, where the goal is to maximize a primary reward objective while satisfying constraints on secondary utilities. Despite the popularity of Lagrangian-based LLM policy search in constrained alignment, iterative primal-dual methods often fail to converge, and non-iterative dual-based methods do not achieve optimality in the LLM parameter space. To address these challenges, we employ Lagrangian duality to develop an iterative dual-based alignment method that alternates between updating the LLM policy via Lagrangian maximization and updating the dual variable via dual descent. In theory, we characterize the primal-dual gap between the primal value in the distribution space and the dual value in the LLM parameter space. We further quantify the optimality gap of the learned LLM policies at near-optimal dual variables with respect to both the objective and the constraint functions. These results prove that dual-based alignment methods can find an optimal constrained LLM policy, up to an LLM parametrization gap. We demonstrate the effectiveness and merits of our approach through extensive experiments conducted on the PKU-SafeRLHF dataset.