We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik-Ben Arous-Peche (BBP) transition in spiked models arising in random matrix theory.

This paper establishes the theoretical limits of graph clustering under the PopularityAdjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra-and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra-and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between k and k2, where k is the number of clusters. As a result, spectral embeddings based on the top k eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating k2 eigenvectors outperform traditional spectral approaches.

Graph transformers (GTs) have shown strong empirical performance, yet current architectures vary widely in their use of attention mechanisms, positional embeddings (PEs), and expressivity. Existing expressivity results are often tied to specific design choices and lack comprehensive empirical validation on large-scale data. This leaves a gap between theory and practice, preventing generalizable insights that exceed particular application domains. Here, we propose the GeneralizedDistance Transformer (GDT), a GT architecture based on standard attention that incorporates many recent advancements for GTs, and we develop a fine-grained understanding of the GDT's representation power in terms of attention and PEs. Through extensive experiments, we identify design choices that consistently perform well across various applications, tasks, and model scales, demonstrating strong performance in a few-shot transfer setting without fine-tuning. Our evaluation covers over eight million graphs with roughly 270M tokens across diverse domains, including image-based object detection, molecular property prediction, code summarization, and out-of-distribution algorithmic reasoning.

What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than π/4.

This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum.

Generative models such as Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and diffusion models often suffer from mode collapse, failing to reproduce the full diversity of their training data. While this problem has been extensively studied in image generation, it remains largely unaddressed for time series. We introduce a formal definition of mode collapse for time series and propose DMD-GEN, a geometry-aware metric that quantifies its severity. DMD-GEN leverages Dynamic Mode Decomposition (DMD) to extract coherent temporal structures and uses Optimal Transport between DMD eigenvectors to measure discrepancies in underlying dynamics. By representing the subspaces spanned by the DMD eigenvectors as point structures on a Grassmann manifold, and comparing them via Wasserstein distances computed from principal angles, DMD-GEN enables a principled geometric comparison between real and generated sequences. The metric is efficient, requiring no additional training, supports minibatch evaluation, and is easily parallelizable. Beyond quantification, DMD-GEN offers interpretability by revealing which dynamical modes are distorted or missing in the generated data.

Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm (ADASAP) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. ADASAPoutperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, ADASAPscales to a dataset with > 3 108 samples, a feat which has not been accomplished in the literature.

Efficient exploration remains one of the most important open problems in reinforcement learning. Discovering novel states or transitions requires policies that efficiently direct the agent away from the regions of the state space that are already well explored. We introduce Novel Exploration via Orthogonality (NEO), an approach that automatically uncovers not only which regions of the environment are novel but also how to reach them by leveraging Laplacian representations. NEO uses the eigenvectors of a modified graph Laplacian to induce gradient flows from states that are frequently visited (less novel) to states that are seldom visited (more novel). We show that NEO's modified Laplacian yields eigenvectors whose extreme values align with the most novel regions of the state space. We provide bounds for the eigenvalues of the modified Laplacian; and we show that the smoothest eigenvectors with real eigenvalues below certain thresholds provide guaranteed gradients to novel states for both undirected and directed graphs. In an empirical evaluation in online, incremental settings, NEO outperformed related state-of-theart approaches, including eigen-options and cover options, in a large collection of undirected and directed environments with varying connectivity structures.

We consider the problem of federated learning (FL) with graph-structured data distributed across multiple clients. In particular, we address the prevalent scenario of interconnected subgraphs, where interconnections between clients significantly influence the learning process. Existing approaches suffer from critical limitations, either requiring the exchange of sensitive node embeddings, thereby posing privacy risks, or relying on computationally-intensive steps, which hinders scalability. To tackle these challenges, we propose FEDLAP, a novel framework that leverages global structure information via Laplacian smoothing in the spectral domain to effectively capture inter-node dependencies while ensuring privacy and scalability. We provide a formal analysis of the privacy of FEDLAP, demonstrating that it preserves privacy. Notably, FEDLAP is the first subgraph FL scheme with strong privacy guarantees. Extensive experiments on benchmark datasets demonstrate that FEDLAP achieves competitive or superior utility compared to existing techniques.

Diffusion models transform random noise into images of remarkable fidelity, yet the structure of this noise-to-image map remains largely unexplored. We investigate this relationship using patch-wise Principal Component Analysis (PCA) and empirically demonstrate that low-frequency components of the initial noise predominantly influence the compositional structure of generated images. Our analyses reveal that noise seeds inherently contain universal compositional cues, evident when identical seeds produce images with similar structural attributes across different datasets and model architectures. Leveraging these insights, we develop and theoretically justify a simple yet effective Patch PCA denoiser that extracts underlying structure from noise using only generic natural image statistics. The robustness of these structural cues is observed to persist across both pixel-space models and latent diffusion models, highlighting their fundamental nature. Finally, we introduce a zero-shot editing method that enables injecting compositional control over generated images, providing an intuitive approach to guided generation without requiring model fine-tuning or additional training.