Related papers: Statistical exploration of the Manifold Hypothesis

The concepts of spread and spread dimension of a metric space were introduced by Willerton in the context of quantifying biodiversity of ecosystems. This paper develops practical applications of spread dimension in the context of machine…

System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep…

Recent work on mode connectivity in the loss landscape of deep neural networks has demonstrated that the locus of (sub-)optimal weight vectors lies on continuous paths. In this work, we train a neural network that serves as a hypernetwork,…

Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…

Analyzing signals arising from dynamical systems typically requires many modeling assumptions and parameter estimation. In high dimensions, this modeling is particularly difficult due to the "curse of dimensionality". In this paper, we…

This paper introduces a new probabilistic framework for supervised learning in neural systems. It is designed to model complex, uncertain systems whose random outputs are strongly non-Gaussian given deterministic inputs. The architecture…

A central problem in machine learning is often formulated as follows: Given a dataset $\{(x_j, y_j)\}_{j=1}^M$, which is a sample drawn from an unknown probability distribution, the goal is to construct a functional model $f$ such that…

Unsupervised learning aims at the discovery of hidden structure that drives the observations in the real world. It is essential for success in modern machine learning. Latent variable models are versatile in unsupervised learning and have…

The ability to represent and compare machine learning models is crucial in order to quantify subtle model changes, evaluate generative models, and gather insights on neural network architectures. Existing techniques for comparing data…

High-dimensional generative modeling is fundamentally a manifold-learning problem: real data concentrate near a low-dimensional structure embedded in the ambient space. Effective generators must therefore balance support fidelity -- placing…

The traditional approach of statistical physics to supervised learning routinely assumes unrealistic generative models for the data: usually inputs are independent random variables, uncorrelated with their labels. Only recently, statistical…

The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this…

Asymptotic efficiency theory is one of the pillars in the foundations of modern mathematical statistics. Not only does it serve as a rigorous theoretical benchmark for evaluating statistical methods, but it also sheds light on how to…

Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…

Distinguishing between real and AI-generated images, commonly referred to as 'image detection', presents a timely and significant challenge. Despite extensive research in the (semi-)supervised regime, zero-shot and few-shot solutions have…

The communications and interrelations between different locations on the Earth's surface have far-reaching implications for both social and natural systems. Effective spatial analytics ideally require a spatial representation, where…

Large Language Models (LLMs) perform internal computations in continuous vector spaces yet produce discrete tokens -- a fundamental mismatch whose geometric consequences remain poorly understood. We develop a mathematical framework that…

Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a…

When mining large datasets in order to predict new data, limitations of the principles behind statistical machine learning pose a serious challenge not only to the Big Data deluge, but also to the traditional assumptions that data…

As deep generative models have progressed, recent work has shown them to be capable of memorizing and reproducing training datapoints when deployed. These findings call into question the usability of generative models, especially in light…