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We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical…
We propose a hybrid stochastic method for the tensor renormalization group (TRG) approach. TRG is known as a powerful tool to study the many-body systems and quantum field theory on the lattice. It is based on a low-rank approximation of…
Boundary integral equation methods are widely used in the solution of many partial differential equations. The kernels that appear in these surface integrals are nearly singular when evaluated near the boundary, and straightforward…
A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any…
Autoencoders represent an effective approach for computing the underlying factors characterizing datasets of different types. The latent representation of autoencoders have been studied in the context of enabling interpolation between data…
The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and…
Convex sparsity-inducing regularizations are ubiquitous in high-dimensional machine learning, but solving the resulting optimization problems can be slow. To accelerate solvers, state-of-the-art approaches consist in reducing the size of…
A discrete analysis of the phase and dissipation errors of an explicit, semi-Lagrangian spectral element method is performed. The semi-Lagrangian method advects the Lagrange interpolant according the Lagrangian form of the transport…
We investigate the acceleration of stationary iterations for multi-term Sylvester equation by means of reduced rank extrapolation (RRE). Theoretical convergence results and implementations are provided for both small and large-scale…
This paper considers the analysis of partial differential equations (PDE) containing multiple random variables. Recently developed collocation methods enable the construction of high-order stochastic solutions by converting a stochastic PDE…
Frame interpolation attempts to synthesise frames given one or more consecutive video frames. In recent years, deep learning approaches, and notably convolutional neural networks, have succeeded at tackling low- and high-level computer…
In this paper, we present an advanced approach to solving the inverse rig problem in blendshape animation, using high-quality corrective blendshapes. Our algorithm introduces novel enhancements in three key areas: ensuring high data…
Machine learning techniques are used in a wide range of domains. However, machine learning models often suffer from the problem of over-fitting. Many data augmentation methods have been proposed to tackle such a problem, and one of them is…
We shall investigate randomized algorithms for solving large-scale linear inverse problems with general regularizations. We first present some techniques to transform inverse problems of general form into the ones of standard form, then…
Image interpolation is a special case of image super-resolution, where the low-resolution image is directly down-sampled from its high-resolution counterpart without blurring and noise. Therefore, assumptions adopted in super-resolution…
Many problems in fluid dynamics are effectively modeled as Stokes flows - slow, viscous flows where the Reynolds number is small. Boundary integral equations are often used to solve these problems, where the fundamental solutions for the…
Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
We study the interplay between surrogate methods for structured prediction and techniques from multitask learning designed to leverage relationships between surrogate outputs. We propose an efficient algorithm based on trace norm…
Based on the joint bidiagonalization process of a large matrix pair $\{A,L\}$, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: $\min\|Lx\| \…
Singular value decomposition (SVD) and matrix inversion are ubiquitous in scientific computing. Both tasks are computationally demanding for large scale matrices. Existing algorithms can approximatively solve these problems with a given…