相关论文: Deformation techniques for sparse systems

We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…

Sparse representation has attracted much attention from researchers in fields of signal processing, image processing, computer vision and pattern recognition. Sparse representation also has a good reputation in both theoretical research and…

Sparse coding algorithms are about finding a linear basis in which signals can be represented by a small number of active (non-zero) coefficients. Such coding has many applications in science and engineering and is believed to play an…

We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's flat degeneration to a toric variety. When Anderson's…

A new algorithm for the efficient numerical approximation of weakly singular integrals over convex polytopes is introduced. Such integrals appear in the Galerkin discretizations of integral equations and nonlocal partial differential…

We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form \[ y = Ax + e, \] where $x$ is a random vector in $\mathbb…

By using the squared slack variables technique, we demonstrate that the solution set of a general polynomial complementarity problem is the image, under a specific projection, of the set of real zeroes of a system of polynomials. This paper…

This paper demonstrates a practical method that can correct spatial varying blur from a set of images of the same object. The algorithm jointly estimates the object and local point spread functions~(PSF). The method prioritizes sections…

The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been…

Tomographic imaging reveals internal structures of 3D objects and is crucial for medical diagnoses. Visualizing the morphology and appearance of non-planar sparse anatomical structures that extend over multiple 2D slices in tomographic…

Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in…

This paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the…

The paper presents two-scale numerical algorithms for stress-strain analysis of porous media featured by self-contact at pore level. The porosity is constituted as a periodic lattice generated by a representative cell consisting of elastic…

Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transform. Our key…

A fast algorithm for the approximation of a low rank LU decomposition is presented. In order to achieve a low complexity, the algorithm uses sparse random projections combined with FFT-based random projections. The asymptotic approximation…

Motivated by the question of optimal functional approximation via compressed sensing, we propose generalizations of the Iterative Hard Thresholding and the Compressive Sampling Matching Pursuit algorithms able to promote sparse in levels…

In this study, the problem of computing a sparse representation of multi-dimensional visual data is considered. In general, such data e.g., hyperspectral images, color images or video data consists of signals that exhibit strong local…

A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…

A widely used method for solving SOS (Sum Of Squares) decomposition problem is to reduce it to the problem of semi-definite programs (SDPs) which can be efficiently solved in theory. In practice, although many SDP solvers can work out some…