The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology It is also clear that the nature of the data we are obtaining is significantly different For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it The data obtained is also often much noisier than in the past and has more missing information (missing data) This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data Geometry and topology are very natural tools to apply in this direction, since geometry can be regarded as the study of distance functions, and what one often works with are distance functions on large finite sets of data The mathematical formalism which has been developed for incorporating geometric and topological techniques deals with point clouds, ie finite sets of points equipped with a distance function It then adapts tools from the various branches of geometry to the study of point clouds The point clouds are intended to be thought of as finite samples taken from a geometric object, perhaps with noise Here are some of the key points which come up when applying these geometric methods to data analysis • Qualitative information is needed: One important goal of data analysis is to allow the user to obtain knowledge about the data, ie to understand how it is organized on a large scale For example, if we imagine that we are looking at a data set constructed somehow from diabetes patients, it would be important to develop the understanding that there are two types of the disease, namely the juvenile and adult onset forms Once that is established, one of course wants to develop quantitative methods for distinguishing them, but the first insight about the distinct forms of the disease is key

Topology and data

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.

/pdf/computing-persistent-homology-1tl310zxk3.pdf

Computing Persistent Homology

https://www-polsys.lip6.fr/~jcf/Papers/F99a.pdf

A new efficient algorithm for computing Gröbner bases (F4)

Douglas Wiedemann’s (1986) landmark approach to solving sparse linear systems over finite fields provides the symbolic counterpart to non-combinatorial numerical methods for solving sparse linear systems, such as the Lanczos or conjugate gradient method (see Golub and van Loan (1983)). The problem is to solve a sparse linear system, when the individual entries lie in a generic field, and the only operations possible are field arithmetic; the solution is to be exact. Such is the situation, for instance, if one works in a finite field. Wiedemann bases his approach on Krylov subspaces, but projects further to a sequence of individual field elements. By making a link to the Berlekamp/Massey problem from coding theory — the coordinate recurrences — and by using randomization an algorithm is obtained with the following property. On input of an n×n coefficient matrix A given by a so-called black box, which is a program that can multiply the matrix by a vector (see Figure 1), and of a vector b, the algorithm finds, with high probability in case the system is solvable, a random solution vector x with Ax = b. It is assumed that the field has sufficiently many elements, say no less than 50n log(n), otherwise one goes to a finite algebraic extension. The complexity of the method is in the general singular case O(n log(n)) calls to the black box for A and an additional O(n log(n)) field arithmetic operations. Note that the black box model for matrix sparsity is a significant abstraction. For a matrix that has an abundance of zero entries, multiplying the matrix by a vector may cost no more than O(n) field operations, in which case the algorithm becomes almost quadratic. However, the model also applies to structured matrices with few or no zero entries, such as Toeplitzand Vandermonde-like matrices, or matrices that correspond to resultants (Canny

On Wiedemann's Method of Solving Sparse Linear Systems

http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/valence.pdf

On Efficient Sparse Integer Matrix Smith Normal Form Computations

http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/CEKSTV01-2.pdf

Efficient matrix preconditioners for black box linear algebra

We recall that the calculation of homology with integer coefficients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.

B. David Saunders

Papers

On Wiedemann's Method of Solving Sparse Linear Systems

On Efficient Sparse Integer Matrix Smith Normal Form Computations

Efficient matrix preconditioners for black box linear algebra

Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms

Parallel algorithms for matrix normal forms