arXiv subjects

Edric Tam

Publications and source records attributed to Edric Tam.

7 recordsLinked to original sources

arXiv2025

Beyond the Laplacian: Interpolated Spectral Augmentation for Graph Neural Networks

Graph neural networks (GNNs) are fundamental tools in graph machine learning. The performance of GNNs relies crucially on the availability of informative node features, which can be limited or absent in real-life datasets and applications. A natural remedy is to augment the node features with embeddings computed from eigenvectors of the graph Laplacian matrix. While it is natural to default to Laplacian spectral embeddings, which capture meaningful graph connectivity information, we ask whether spectral embeddings from alternative graph matrices can also provide useful representations for learning. We introduce Interpolated Laplacian Embeddings (ILEs), which are derived from a simple yet expressive family of graph matrices. Using tools from spectral graph theory, we offer a straightforward interpretation of the structural information that ILEs capture. We demonstrate through simulations and experiments on real-world datasets that feature augmentation via ILEs can improve performance across commonly used GNN architectures. Our work offers a straightforward and practical approach that broadens the practitioner's spectral augmentation toolkit when node features are limited.

cs.LG↗

arXiv2025

On the Statistical Capacity of Deep Generative Models

Deep generative models are routinely used in generating samples from complex, high-dimensional distributions. Despite their apparent successes, their statistical properties are not well understood. A common assumption is that with enough training data and sufficiently large neural networks, deep generative model samples will have arbitrarily small errors in sampling from any continuous target distribution. We set up a unifying framework that debunks this belief. We demonstrate that broad classes of deep generative models, including variational autoencoders and generative adversarial networks, are not universal generators. Under the predominant case of Gaussian latent variables, these models can only generate concentrated samples that exhibit light tails. Using tools from concentration of measure and convex geometry, we give analogous results for more general log-concave and strongly log-concave latent variable distributions. We extend our results to diffusion models via a reduction argument. We use the Gromov--Levy inequality to give similar guarantees when the latent variables lie on manifolds with positive Ricci curvature. These results shed light on the limited capacity of common deep generative models to handle heavy tails. We illustrate the empirical relevance of our work with simulations and financial data.

stat.ML↗

arXiv2025

Assessing the Distributional Fidelity of Synthetic Chest X-rays using the Embedded Characteristic Score

Chest X-ray (CXR) images are among the most commonly used diagnostic imaging modalities in clinical practice. Stringent privacy constraints often limit the public dissemination of patient CXR images, contributing to the increasing use of synthetic images produced by deep generative models for data sharing and training machine learning models. Given the high-stakes downstream applications of CXR images, it is crucial to evaluate how faithfully synthetic images reflect the underlying target distribution. We propose the embedded characteristic score (ECS), a flexible evaluation procedure that compares synthetic and patient CXR samples through characteristic function transforms of feature embeddings. The choice of embedding can be tailored to the clinical or scientific context of interest. By leveraging the behavior of characteristic functions near the origin, ECS is sensitive to differences in higher moments and distribution tails, aspects that are often overlooked by commonly used evaluation metrics such as the Fr\'echet Inception Distance (FID). We establish theoretical properties of ECS and describe a calibration strategy based on a simple resampling procedure. We compare the empirical performance of ECS against FID via simulations and standard benchmark imaging datasets. Assessing synthetic CXR images with ECS uncovers clinically relevant distributional discrepancies relative to patient CXR images. These results highlight the importance of reliable evaluation of synthetic data that inform high-stakes decisions.

stat.ML↗

arXiv2024

Exact Sampling of Spanning Trees via Fast-forwarded Random Walks

Tree graphs are routinely used in statistics. When estimating a Bayesian model with a tree component, sampling the posterior remains a core difficulty. Existing Markov chain Monte Carlo methods tend to rely on local moves, often leading to poor mixing. A promising approach is to instead directly sample spanning trees on an auxiliary graph. Current spanning tree samplers, such as the celebrated Aldous--Broder algorithm, predominantly rely on simulating random walks that are required to visit all the nodes of the graph. Such algorithms are prone to getting stuck in certain sub-graphs. We formalize this phenomenon using the bottlenecks in the random walk's transition probability matrix. We then propose a novel fast-forwarded cover algorithm that can break free from bottlenecks. The core idea is a marginalization argument that leads to a closed-form expression which allows for fast-forwarding to the event of visiting a new node. Unlike many existing approximation algorithms, our algorithm yields exact samples. We demonstrate the enhanced efficiency of the fast-forwarded cover algorithm, and illustrate its application in fitting a Bayesian dendrogram model on a Massachusetts crimes and communities dataset.

stat.ME↗

arXiv2023

Spectral Gap Regularization of Neural Networks

We introduce Fiedler regularization, a novel approach for regularizing neural networks that utilizes spectral/graphical information. Existing regularization methods often focus on penalizing weights in a global/uniform manner that ignores the connectivity structure of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical motivation for this approach via spectral graph theory. We demonstrate several useful properties of the Fiedler value that make it useful as a regularization tool. We provide an approximate, variational approach for faster computation during training. We provide an alternative formulation of this framework in the form of a structurally weighted $\text{L}_1$ penalty, thus linking our approach to sparsity induction. We provide uniform generalization error bounds for Fiedler regularization via a Rademacher complexity analysis. We performed experiments on datasets that compare Fiedler regularization with classical regularization methods such as dropout and weight decay. Results demonstrate the efficacy of Fiedler regularization. This is a journal extension of the conference paper by Tam and Dunson (2020).

stat.ML↗

arXiv2022

Multiscale Graph Comparison via the Embedded Laplacian Discrepancy

Laplacian eigenvectors capture natural community structures on graphs and are widely used in spectral clustering and manifold learning. The use of Laplacian eigenvectors as embeddings for the purpose of multiscale graph comparison has however been limited. Here we propose the Embedded Laplacian Discrepancy (ELD) as a simple and fast approach to compare graphs (of potentially different sizes) based on the similarity of the graphs' community structures. The ELD operates by representing graphs as point clouds in a common, low-dimensional space, on which a natural Wasserstein-based distance can be efficiently computed. A main challenge in comparing graphs through any eigenvector-based approaches is the potential ambiguity that could arise due to sign-flips and basis symmetries. The ELD leverages a simple symmetrization trick to bypass any sign ambiguities. For comparing graphs that do not have any ambiguities due to basis symmetries (i.e. the spectrums are simple), we show that the ELD becomes a natural pseudo-metric that enjoys nice properties such as invariance under graph isomorphism. For comparing graphs with non-simple spectrums, we propose a procedure to approximate the ELD via a simple perturbation technique to resolve any ambiguity from basis symmetries. We show that such perturbations are stable using matrix perturbation theory under mild assumptions that are straightforward to verify in practice. We demonstrate the excellent applicability of the ELD approach on both simulated and real datasets.

stat.ML↗

arXiv2020

Fiedler Regularization: Learning Neural Networks with Graph Sparsity

We introduce a novel regularization approach for deep learning that incorporates and respects the underlying graphical structure of the neural network. Existing regularization methods often focus on dropping/penalizing weights in a global manner that ignores the connectivity structure of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical support for this approach via spectral graph theory. We list several useful properties of the Fiedler value that makes it suitable in regularization. We provide an approximate, variational approach for fast computation in practical training of neural networks. We provide bounds on such approximations. We provide an alternative but equivalent formulation of this framework in the form of a structurally weighted L1 penalty, thus linking our approach to sparsity induction. We performed experiments on datasets that compare Fiedler regularization with traditional regularization methods such as dropout and weight decay. Results demonstrate the efficacy of Fiedler regularization.

stat.ML↗