Related papers: Approximating the Inverse Frame Operator from Loca…

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…

Operator convex functions defined on the positive half-line play a prominent role in the theory of quantum information, where they are used to define quantum $f$-divergences. Such functions admit integral representations in terms of…

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…

A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added…

We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is…

We consider minimizing a function consisting of a quadratic term and a proximable term which is possibly nonconvex and nonsmooth. This problem is also known as scaled proximal operator. Despite its simple form, existing methods suffer from…

We study linear function approximation in a finite basis under finite-precision arithmetic. In a highly non-orthogonal basis, certain directions are only weakly represented, so that rounding errors can significantly distort the effectively…

Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large…

This paper focuses on investigating an inexact stochastic model-based optimization algorithm that integrates preconditioning techniques for solving stochastic composite optimization problems. The proposed framework unifies and extends the…

In ultrasound tomography, the speed of sound inside an object is estimated based on acoustic measurements carried out by sensors surrounding the object. An accurate forward model is a prominent factor for high-quality image reconstruction,…

Beyond novel view synthesis, Neural Radiance Fields are useful for applications that interact with the real world. In this paper, we use them as an implicit map of a given scene and propose a camera relocalization algorithm tailored for…

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of…

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are…

Spatial approximations have been traditionally used in spatial databases to accelerate the processing of complex geometric operations. However, approximations are typically only used in a first filtering step to determine a set of candidate…

Weighted average sampling is more practical and numerically more stable than sampling at single points as in the classical Shannon sampling framework. Using the frame theory, one can completely reconstruct a bandlimited function from its…

In literature, several algorithms for imaging based on interpolation or approximation methods are available. The implementation of theoretical processes highlighted the necessity of providing theoretical frameworks for the convergence and…

Optimization algorithms appear in the core calculations of numerous Artificial Intelligence (AI) and Machine Learning methods, as well as Engineering and Business applications. Following recent works on the theoretical deficiencies of AI, a…

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator…

This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…

Adaptive sparse coding methods learn a possibly overcomplete set of basis functions, such that natural image patches can be reconstructed by linearly combining a small subset of these bases. The applicability of these methods to visual…