Related papers: On randomized trace estimates for indefinite matri…
The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a…
The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the…
Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and…
Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume…
An influential line of recent work has focused on the generalization properties of unregularized gradient-based learning procedures applied to separable linear classification with exponentially-tailed loss functions. The ability of such…
Some data analysis problems require the computation of (regularised) inverse traces, i.e. quantities of the form $\Tr (q \bI + \bL)^{-1}$. For large matrices, direct methods are unfeasible and one must resort to approximations, for example…
We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator…
Motivated by the goal of improving the efficiency of small sample design, we propose a novel Bayesian stochastic approximation method to estimate the root of a regression function. The method features adaptive local modelling and…
This paper addresses the advancement of probability tail bound analysis, a crucial statistical tool for assessing the probability of large deviations of random variables from their expected values. Traditional tail bounds, such as Markov's,…
This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the…
We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$,…
In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the…
We develop randomized matrix-free algorithms for estimating partial traces, a generalization of the trace arising in quantum physics and chemistry. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng,…
The trace $\tr(q(\ma{L} + q\ma{I})^{-1})$, where $\ma{L}$ is a symmetric diagonally dominant matrix, is the quantity of interest in some machine learning problems. However, its direct computation is impractical if the matrix size is large.…
The scalable calculation of matrix determinants has been a bottleneck to the widespread application of many machine learning methods such as determinantal point processes, Gaussian processes, generalised Markov random fields, graph models…
Stochastic gradient methods are among the most widely used algorithms for large-scale optimization and machine learning. A key technique for improving the statistical efficiency and stability of these methods is the use of averaging schemes…
Parametric stochastic simulators are ubiquitous in science, often featuring high-dimensional input parameters and/or an intractable likelihood. Performing Bayesian parameter inference in this context can be challenging. We present a neural…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
We study the stability of the Lanczos algorithm run on problems whose eigenvector empirical spectral distribution is near to a reference measure with well-behaved orthogonal polynomials. We give a backwards stability result which can be…
According to the concept of typicality, an ensemble average can be accurately approximated by an expectation value with respect to a single pure state drawn at random from a high-dimensional Hilbert space. This random-vector approximation,…