Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the matrix remains static but cannot be applied to problems where the matrix entries are updated or the number of rows and columns increases frequently. Such scenarios occur routinely in graph analytics when the graph is changing dynamically and either edges and/or nodes are being added and removed. This paper puts forth a new algorithmic framework to update the eigenvectors associated with the leading eigenvalues of an initial adjacency or Laplacian matrix as the graph evolves dynamically. The proposed algorithm is based on Rayleigh-Ritz projections, in which the original eigenvalue problem is projected onto a restricted subspace which ideally encapsulates the invariant subspace associated with the sought eigenvectors. Following ideas from eigenvector perturbation analysis, we present a new methodology to build the projection subspace. The proposed framework features lower computational and memory complexity with respect to competitive alternatives while empirical results show strong qualitative performance, both in terms of eigenvector approximation and accuracy of downstream learning tasks of central node identification and node clustering.

Toovercome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. Wepropose aframeworkforlearning frombiased simulations rooted in the infinitesimal generator of the process and the associated resolvent operator. Wecontrast our approach to more common ones based on the transfer operator, showing thatitcanprovably learn thespectral properties oftheunbiased system frombiaseddata.

Eigenvalue problems have a distinctive forward-inverse structure and are fundamental to characterizing a system's thermal response, stability, and natural modes. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative for solving such problems but are often orders of magnitude slower than classical numerical schemes. In this paper, we introduce a reformulated PINN approach that casts the search for eigenpairs as a biconvex optimization problem, enabling fast and provably convergent alternating convex search (ACS) over eigenvalues and eigenfunctions using analytically optimal updates. Numerical experiments show that PINN-ACS attains high accuracy with convergence speeds up to 500$\times$ faster than gradient-based PINN training. We release our codes at https://github.com/NeurIPS-ML4PS-2025/PINN_ACS_CODES.

A general idea to overcome the above problem is to perturb the system dynamics.

However, it struggles with high-dimensional data.

In many robotic systems where dynamics are best modeled as stochastic systems due to factors such as sensing noise and environmental disturbances, it is challenging for conventional methods such as Hamilton-Jacobi reachability and control barrier functions to provide a holistic measure of safety. We derive a linear operator governing the dynamic programming principle for safety probability, and find that its dominant eigenpair provides information about safety for both individual states and the overall closed-loop system. The proposed learning framework, called EigenSafe, jointly learns this dominant eigenpair and a safe backup policy in an offline manner. The learned eigenfunction is then used to construct a safety filter that detects potentially unsafe situations and falls back to the backup policy. The framework is validated in three simulated stochastic safety-critical control tasks.