In many application domains, networks are observed with node-level features. In such settings, a common problem is to assess whether or not nodal covariates are correlated with the network structure itself. Here, we present four novel methods for addressing this problem. Two of these are based on a linear model relating node-level covariates to latent node-level variables that drive network structure. The other two are based on applying canonical correlation analysis to the node features and network structure, avoiding the linear modeling assumptions. We provide theoretical guarantees for all four methods when the observed network is generated according to a low-rank latent space model endowed with node-level covariates, which we allow to be high-dimensional. Our methods are computationally cheaper and require fewer modeling assumptions than previous approaches to network dependency testing. We demonstrate and compare the performance of our novel methods on both simulated and real-world data.

To make sense of the world around us, we develop models, constructed to enable us to replicate, describe, and explain the behaviours we see. Focusing on the broad case of sequences of correlated random variables, i.e., classical stochastic processes, we tackle the question of determining whether or not two different models produce the same observable behavior. This is the problem of identifiability. Curiously, the physics of the model need not correspond to the physics of the observations; recent work has shown that it is even advantageous -- in terms of memory and thermal efficiency -- to employ quantum models to generate classical stochastic processes. We resolve the identifiability problem in this regime, providing a means to compare any two models of a classical process, be the models classical, quantum, or `post-quantum', by mapping them to a canonical `generalized' hidden Markov model. Further, this enables us to place (sometimes tight) bounds on the minimal dimension required of a quantum model to generate a given classical stochastic process.

Learning with physical systems is an emerging paradigm that seeks to harness the intrinsic nonlinear dynamics of physical substrates for learning. The impetus for a paradigm shift in how hardware is used for computational intelligence stems largely from the unsustainability of artificial neural network software implemented on conventional transistor-based hardware. This Perspective highlights one promising approach using physical networks comprised of resistive memory nanoscale components with dynamically reconfigurable, self-organising electrical circuitry. Experimental advances have revealed the non-trivial interactions within these Self-Organising Memristive Networks (SOMNs), offering insights into their collective nonlinear and adaptive dynamics, and how these properties can be harnessed for learning using different hardware implementations. Theoretical approaches, including mean-field theory, graph theory, and concepts from disordered systems, reveal deeper insights into the dynamics of SOMNs, especially during transitions between different conductance states where criticality and other dynamical phase transitions emerge in both experiments and models. Furthermore, parallels between adaptive dynamics in SOMNs and plasticity in biological neuronal networks suggest the potential for realising energy-efficient, brain-like continual learning. SOMNs thus offer a promising route toward embedded edge intelligence, unlocking real-time decision-making for autonomous systems, dynamic sensing, and personalised healthcare, by enabling embedded learning in resource-constrained environments. The overarching aim of this Perspective is to show how the convergence of nanotechnology, statistical physics, complex systems, and self-organising principles offers a unique opportunity to advance a new generation of physical intelligence technologies.

Stochastic gradient optimization is the dominant learning paradigm for a variety of scenarios, from classical supervised learning to modern self-supervised learning. We consider stochastic gradient algorithms for learning problems whose objectives rely on unknown nuisance parameters, and establish non-asymptotic convergence guarantees. Our results show that, while the presence of a nuisance can alter the optimum and upset the optimization trajectory, the classical stochastic gradient algorithm may still converge under appropriate conditions, such as Neyman orthogonality. Moreover, even when Neyman orthogonality is not satisfied, we show that an algorithm variant with approximately orthogonalized updates (with an approximately orthogonalized gradient oracle) may achieve similar convergence rates. Examples from orthogonal statistical learning/double machine learning and causal inference are discussed.

Quasi-Newton methods are widely used for solving convex optimization problems due to their ease of implementation, practical efficiency, and strong local convergence guarantees. However, their global convergence is typically established only under specific line search strategies and the assumption of strong convexity. In this work, we extend the theoretical understanding of Quasi-Newton methods by introducing a simple stepsize schedule that guarantees a global convergence rate of ${O}(1/k)$ for the convex functions. Furthermore, we show that when the inexactness of the Hessian approximation is controlled within a prescribed relative accuracy, the method attains an accelerated convergence rate of ${O}(1/k^2)$ -- matching the best-known rates of both Nesterov's accelerated gradient method and cubically regularized Newton methods. We validate our theoretical findings through empirical comparisons, demonstrating clear improvements over standard Quasi-Newton baselines. To further enhance robustness, we develop an adaptive variant that adjusts to the function's curvature while retaining the global convergence guarantees of the non-adaptive algorithm.

--Graph signal processing has become an essential tool for analyzing data structured on irregular domains. While conventional graph shift operators (GSOs) are effective for certain tasks, they inherently lack flexibility in modeling dependencies between non-adjacent nodes, limiting their ability to represent complex graph structures. T o address this limitation, this paper proposes the unified extended matrix (UEM) framework, which integrates the extended-adjacency matrix and the unified graph representation matrix through parametric design, so as to be able to flexibly adapt to different graph structures and reveal more graph signal information. Theoretical analysis of the UEM is conducted, demonstrating positive semi-definiteness and eigenvalue monotonicity under specific conditions. Then, we propose graph Fourier transform based on UEM (UEM-GFT), which can adaptively tune spectral properties to enhance signal processing performance. Experimental results on synthetic and real-world datasets demonstrate that the UEM-GFT outperforms existing GSO-based methods in anomaly detection tasks, achieving superior performance across varying network topologies. Index T erms --Graph shift operator, unified extended matrix, graph signal processing, graph Fourier transform based on unified extended matrix.

Large Neighborhood Search (LNS) is a common heuristic in combinatorial optimization that iteratively searches over a large neighborhood of the current solution for a better one. Recently, neural network-based LNS solvers have achieved great success in solving Integer Linear Programs (ILPs) by learning to greedily predict the locally optimal solution for the next neighborhood proposal. However, this greedy approach raises two key concerns: (1) to what extent this greedy proposal suffers from local optima, and (2) how can we effectively improve its sample efficiency in the long run . To address these questions, this paper first formulates LNS as a stochastic process, and then introduces SPL-LNS, a sampling-enhanced neural LNS solver that leverages locally-informed proposals to escape local optima. We also develop a novel hindsight relabeling method to efficiently train SPL-LNS on self-generated data. Experimental results demonstrate that SPL-LNS substantially surpasses prior neural LNS solvers for various ILP problems of different sizes.

This paper studies the interplay between learning algorithms and graph structure for graph neural networks (GNNs). Existing theoretical studies on the learning dynamics of GNNs primarily focus on the convergence rates of learning algorithms under the interpolation regime (noise-free) and offer only a crude connection between these dynamics and the actual graph structure (e.g., maximum degree). This paper aims to bridge this gap by investigating the excessive risk (generalization performance) of learning algorithms in GNNs within the generalization regime (with noise). Specifically, we extend the conventional settings from the learning theory literature to the context of GNNs and examine how graph structure influences the performance of learning algorithms such as stochastic gradient descent (SGD) and Ridge regression. Our study makes several key contributions toward understanding the interplay between graph structure and learning in GNNs. First, we derive the excess risk profiles of SGD and Ridge regression in GNNs and connect these profiles to the graph structure through spectral graph theory. With this established framework, we further explore how different graph structures (regular vs. power-law) impact the performance of these algorithms through comparative analysis. Additionally, we extend our analysis to multi-layer linear GNNs, revealing an increasing non-isotropic effect on the excess risk profile, thereby offering new insights into the over-smoothing issue in GNNs from the perspective of learning algorithms. Our empirical results align with our theoretical predictions, \emph{collectively showcasing a coupling relation among graph structure, GNNs and learning algorithms, and providing insights on GNN algorithm design and selection in practice.}

--We investigate the performance of a lightweight tracking controller, based on a flow version of the Newton-Raphson method, applied to a miniature blimp and a mid-size quadrotor . This tracking technique has been shown to enjoy theoretical guarantees of performance and has been applied with success in simulation studies and on mobile robots with simple motion models. This paper investigates the technique through real-world flight experiments on aerial hardware platforms subject to realistic deployment and onboard computational constraints. The technique's performance is assessed in comparison with the established control frameworks of feedback linearization for the blimp, and nonlinear model predictive control for both quadrotor and blimp. The performance metrics under consideration are (i) root mean square error of flight trajectories with respect to target trajectories, (ii) algorithms' computation times, and (iii) CPU energy consumption associated with the control algorithms. The experimental findings show that the Newton-Raphson flow-based tracking controller achieves comparable or superior tracking performance to the baseline methods with substantially reduced computation time and energy expenditure. HE past two decades have seen a significant shift in the nature of hardware research for trajectory control of aerial platforms like quadrotors. First, testing and verification of novel techniques relied heavily on numerical simulators, later transitioning to real-world deployments that depended on ground station computers and simplified models (e.g. Today, powerful single-board computers (SBCs) have enabled research to shift toward onboard execution even for computationally intensive control methods [2]-[4].

Predict+Optimize, a novel extension catering to applications where unknown parameters are instead revealed in sequential stages, with optimization decisions made in between. We further develop three training algorithms for neural networks (NNs) for our framework as proof of concept, all of which can handle mixed integer linear programs.