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Olga Holtz

Publications and source records attributed to Olga Holtz.

At least 37 records · Page 2Linked to original sources
arXiv2008

Compressive sensing: a paradigm shift in signal processing

We survey a new paradigm in signal processing known as "compressive sensing". Contrary to old practices of data acquisition and reconstruction based on the Shannon-Nyquist sampling principle, the new theory shows that it is possible to reconstruct images or signals of scientific interest accurately and even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g., the number of pixels in the image. This new technique draws from results in several fields of mathematics, including algebra, optimization, probability theory, and harmonic analysis. We will discuss some of the key mathematical ideas behind compressive sensing, as well as its implications to other fields: numerical analysis, information theory, theoretical computer science, and engineering.

math.HO↗
arXiv2007

Accurate and Efficient Expression Evaluation and Linear Algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed answer has relative error less than 1, i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: Most of our results will use the so-called Traditional Model (TM). We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high accuracy algorithm or a proof that none exists. When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such as $x+y+z$, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case. Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.

math.NA↗
arXiv2007

Fast linear algebra is stable

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{ω+ η})$ operations for any $η> 0$, then it can be done stably in $O(n^{ω+ η})$ operations for any $η> 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{ω+ η})$ operations.

math.NA↗
arXiv2007

Functions preserving nonnegativity of matrices

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary conditions and sufficient conditions for entire functions that preserve nonnegativity of symmetric matrices. We also show that some of these latter conditions characterize the even or odd functions that preserve nonnegativity of symmetric matrices.

math.RA↗
arXiv2006

Fast matrix multiplication is stable

We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438--449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379--388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.

math.NA↗
arXiv2006

Toward accurate polynomial evaluation in rounded arithmetic

Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or $a \cdot b$, its computed value is $op(a,b) \cdot (1 + δ)$, where $| δ|$ is bounded by some constant $ε$ where $0 < ε\ll 1$, but $δ$ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any $p$ and $\cal D$, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials $p$ are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on $p$ and $\cal D$, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on $p$ for it to be accurately evaluable on open real or complex domains ${\cal D}$. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials $p$ with integer coefficients, ${\cal D} = \C^n$, and using only the arithmetic operations $+$, $-$ and $\cdot$.

math.NA↗
arXiv2005

Theorems and counterexamples on structured matrices

The subject of Chapter 1 is GKK $τ$-matrices and related topics. Chapter 2 is devoted to boundedly invertible collections of matrices, with applications to operator norms and spline approximation. Various structured matrices (Toeplitz, Hessenberg, Hankel, Cauchy, and other) are used extensively throughout the thesis.

math.RA↗
arXiv2005

On convergence of infinite matrix products

A necessary and sufficient condition for the convergence of an infinite right product of matrices of the form | I B | A = | 0 C |, with (uniformly) contracting submatrices $C$, is proven.

math.RA↗
arXiv2005

Potter, Wielandt, and Drazin on the matrix equation $AB=ωBA$, with some new answers to old questions

In this partly historical and partly research oriented note, we display a page of an unpublished mathematical diary of Helmut Wielandt's for 1951. There he gives a new proof of a theorem due to H. S. A. Potter on the matrix equation $AB = ωBA$, which is related to the $q$-binomial theorem, and asks some further questions, which we answer. We also describe results by M. P. Drazin and others on this equation.

math.RA↗
arXiv2005

Real and complex operator norms

Real and complex norms of a linear operator acting on a normed complexified space are considered. Bounds on the ratio of these norms are given. The real and complex norms are shown to coincide for four classes of operators: 1) real linear operators from $L_p(μ_1)$ to $L_q(μ_2)$, $1\leq p\leq q\leq \infty$; 2) real linear operators between inner product spaces; 3) nonnegative linear operators acting between complexified function spaces with absolute and monotonic norms; 4) real linear operators from a complexified function space with a norm satisfying $\|\Re x \|\leq \|x\|$ to $L_\infty(μ)$. The inequality $p\leq q$ in Case 1 is shown to be sharp. A class of norm extensions from a real vector space to its complexification is constructed that preserve operator norms.

math.FA↗
arXiv2005

M-matrices satisfy Newton's inequalities

Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of inequalities also valid for both of these classes. They are also used to derive some new necessary conditions on the eigenvalues of nonnegative matrices.

math.RA↗
arXiv2005

Classification of normal operators in spaces with indefinite scalar product of rank 2

A finite-dimensional complex space with indefinite scalar product [.,.] having v- = 2 negative squares and v+ >= 2 positive ones is considered. The paper presents a classification of operators that are normal with respect to this product. It is related to the study by Gohberg and Reichstein in which a similar classification was obtained for the case v = min{v-, v+} = 1.

math.FA↗
arXiv2005

Approximation orders of shift-invariant subspaces of $W^s_2({\Bbb R}^d)$

We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.

math.CA↗
arXiv2005

On classification of normal operators in real spaces with indefinite scalar product

A real finite-dimensional space with indefinite scalar product having v- negative squares and v+ positive ones is considered. The paper presents a classification of operators that are normal with respect to this product for the cases min{v-, v+} = 1, 2. The approach used here was developed in papers by Gohberg and Reichstein and by the authors, where a similar classification was obtained for complex spaces with v = min{v-,v+} = 1, 2, respectively.

math.FA↗
arXiv2005

On indecomposable normal matrices in spaces with indefinite scalar product

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [.,.] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in terms of specific functions of v = min{v-, v+}, where v-, (v+) is the number of negative (positive) squares of the form [x,x]. All the bounds except for one are proved to be strict.

math.FA↗