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Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical…
Analysing statistical properties of neural networks is a central topic in statistics and machine learning. However, most results in the literature focus on the properties of the neural network minimizing the training error. The goal of this…
We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model…
In this article, we focus on the error that is committed when computing the matrix logarithm using the Gauss--Legendre quadrature rules. These formulas can be interpreted as Pad\'e approximants of a suitable Gauss hypergeometric function.…
Stochastic-approximation gradient methods are attractive for large-scale convex optimization because they offer inexpensive iterations. They are especially popular in data-fitting and machine-learning applications where the data arrives in…
Computing the probability of evidence even with known error bounds is NP-hard. In this paper we address this hard problem by settling on an easier problem. We propose an approximation which provides high confidence lower bounds on…
This article is concerned with Monte-Carlo methods for the estimation of the trace of an implicitly given matrix $A$ whose information is only available through matrix-vector products. Such a method approximates the trace by an average of…
We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We…
We study matrix sketching methods for regularized variants of linear regression, low rank approximation, and canonical correlation analysis. Our main focus is on sketching techniques which preserve the objective function value for…
A new algorithm to approximate Hermitian matrices by positive semidefinite Hermitian matrices based on modified Cholesky decompositions is presented. In contrast to existing algorithms, this algorithm allows to specify bounds on the…
We extend the theory of low-rank matrix recovery and completion to the case when Poisson observations for a linear combination or a subset of the entries of a matrix are available, which arises in various applications with count data. We…
Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the maximum absolute deviation of errors. Such problems find application in the solution of…
The Nystr\"om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In…
The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper…
We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$…
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…