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We study low rank approximation of tensors, focusing on the tensor train and Tucker decompositions, as well as approximations with tree tensor networks and more general tensor networks. For tensor train decomposition, we give a bicriteria…

A key bottleneck in quantum machine learning is the computational cost of repeated quantum circuit evaluations during the inference phase. To address this, we present a framework for constructing fast, cheap, provably accurate classical…

Markov chain Monte Carlo methods are a powerful and commonly used family of numerical methods for sampling from complex probability distributions. As applications of these methods increase in size and complexity, the need for efficient…

In this article we propose a novel MCMC method based on deterministic transformations T: X x D --> X where X is the state-space and D is some set which may or may not be a subset of X. We refer to our new methodology as Transformation-based…

We propose a new algorithm for the computation of a singular value decomposition (SVD) low-rank approximation of a matrix in the Matrix Product Operator (MPO) format, also called the Tensor Train Matrix format. Our tensor network randomized…

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…

By mixing the target posterior distribution with a surrogate distribution, of which the normalizing constant is tractable, we propose a method for estimating the marginal likelihood using the Wang-Landau algorithm. We show that a faster…

In Bayesian applications, there is a huge interest in rapid and accurate estimation of the posterior distribution, particularly for high dimensional or hierarchical models. In this article, we propose to use optimization to solve for a…

Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for…

Recent work has suggested using Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) to sample from target distributions of interest. PDMPs are non-reversible continuous-time processes endowed with momentum, and…

Numerical simulations are crucial for modeling complex systems, but calibrating them becomes challenging when data are noisy or incomplete and likelihood evaluations are computationally expensive. Bayesian calibration offers an interesting…

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some…

In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS)…

In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution.…

In parameter estimation problems one computes a posterior distribution over uncertain parameters defined jointly by a prior distribution, a model, and noisy data. Markov Chain Monte Carlo (MCMC) is often used for the numerical solution of…

In this work, we firstly apply the Train-Tensor (TT) networks to construct a compact representation of the classical Multilayer Perceptron, representing a reduction of up to 95% of the coefficients. A comparative analysis between tensor…

This work proposes an adaptive sequential Monte Carlo sampling algorithm to solve Bayesian inverse problems in scenarios where likelihood evaluations are costly but can be approximated using a surrogate model built from previous evaluations…

A number of problems in a variety of fields are characterised by target distributions with a multimodal structure in which the presence of several isolated local maxima dramatically reduces the efficiency of Markov Chain Monte Carlo…

It has become increasingly easy nowadays to collect approximate posterior samples via fast algorithms such as variational Bayes, but concerns exist about the estimation accuracy. It is tempting to build solutions that exploit approximate…

This paper considers the classical problem of sampling with Monte Carlo methods a target rare event distribution defined by a score function that is very expensive to compute. We assume we can build using evaluations of the true score, an…