Related papers: Fourier Sparse Leverage Scores and Approximate Ker…
It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the non-linear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation…
We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem…
We provide a methodology for learning sparse statistical models that use as features all possible multiplicative interactions among an underlying atomic set of features. While the resulting optimization problems are exponentially sized, our…
We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic…
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions…
Sparse autoencoders (SAEs) are widely used in mechanistic interpretability research for large language models; however, the state-of-the-art method of using $k$-sparse autoencoders lacks a theoretical grounding for selecting the…
We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is…
We study the problem of estimating the best B term Fourier representation for a given frequency-sparse signal (i.e., vector) $\textbf{A}$ of length $N \gg B$. More explicitly, we investigate how to deterministically identify B of the…
Slow kinetic processes of molecular systems can be analyzed by computing dominant eigenpairs of the Koopman operator or its generator. In this context, the Variational Approach to Markov Processes (VAMP) provides a rigorous way of…
We consider Gaussian mixture models in high dimensions and concentrate on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive information bounds and establish the performance…
While leverage score sampling provides powerful tools for approximating solutions to large least squares problems, the cost of computing exact scores and sampling often prohibits practical application. This paper addresses this challenge by…
Kernel methods are powerful learning methodologies that allow to perform non-linear data analysis. Despite their popularity, they suffer from poor scalability in big data scenarios. Various approximation methods, including random feature…
The sparse modeling is an evident manifestation capturing the parsimony principle just described, and sparse models are widespread in statistics, physics, information sciences, neuroscience, computational mathematics, and so on. In…
The statistical leverage scores of a matrix $A$ are the squared row-norms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recently-popular…
Recent advances in machine learning have shown promising results for financial prediction using large, over-parameterized models. This paper provides theoretical foundations and empirical validation for understanding when and how these…
Kernel methods are one of the cornerstones of learning-based control, modern system identification, surrogate modelling, and related fields. A key advantage of this class of learning and function approximation methods is the availability of…
In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em…
Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated…
We develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set $\mathcal{G}$ up to the smallest possible additive term, called the convergence rate. When the…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…