Leslie Hogben

Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

This paper studies the recursive robust principal components analysis problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low-dimensional subspace that is either fixed or changes slowly enough. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above problem, in recent work, we introduced a novel solution called recursive projected CS (ReProCS). In this paper, we develop a simple modification of the original ReProCS idea and analyze it. This modification assumes knowledge of a subspace change model on the Lts. Under mild assumptions and a denseness assumption on the unestimated part of the subspace of Lt at various times, we show that, with high probability, the proposed approach can exactly recover the support set of St at all times, and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value. In simulation experiments, we observe that the last assumption holds as long as there is some support change of St every few frames.

Propagation time for zero forcing on a graph

Zero forcing (also called graph infection) on a simple, undirected graph G is based on the color-change rule: if each vertex of G is colored either white or black, and vertex v is a black vertex with only one white neighbor w, then change the color of w to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color change rule. The propagation time of a zero forcing set B of graph G is the minimum number of steps that it takes to force all the vertices of G black, starting with the vertices in B black and performing independent forces simultaneously. The minimum and maximum propagation times of a graph are taken over all minimum zero forcing sets of the graph. It is shown that a connected graph of order at least two has more than one minimum zero forcing set realizing minimum propagation time. Graphs G having extreme minimum propagation times |G|-1, |G|-2, and 0 are characterized, and results regarding graphs having minimum propagation time 1 are established. It is shown that the diameter is an upper bound for maximum propagation time for a tree, but in general propagation time and diameter of a graph are not comparable.

Spectral graph theory and the inverse eigenvalue problem of a graph

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative off-diagonal entries in the positions described by the edges of the graph (and zero in every other off-diagonal position). The set of all real symmetric matrices having nonzero off-diagonal entries exactly where the graph G has edges is denoted by S(G). Given a graph G, the problem of characterizing the possible spectra of B, such that B ∈S (G), has been referred to as the Inverse Eigenvalue Problem of a Graph .I n the last fifteen years a number of papers on this problem have appeared, primarily concerning trees. The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S(G). Recent work on generalized Laplacians and Colin de Verdiere matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1-matrix or a generalized Laplacian.

Graph theoretic methods for matrix completion problems

Abstract A pattern is a list of positions in an n×n real matrix. A matrix completion problem for the class of Π -matrices asks whether every partial Π -matrix whose specified entries are exactly the positions of the pattern can be completed to a Π -matrix. We survey the current state of research on Π -matrix completion problems for many subclasses Π of P 0 -matrices, including positive definite matrices, M-matrices, inverse M-matrices, P-matrices, and matrices defined by various sign symmetry and positivity conditions on P 0 - and P-matrices. Graph theoretic techniques used to study completion problems are discussed. Several new results are also presented, including the solution to the M 0 -matrix completion problem and the sign symmetric P 0 -matrix completion problem.

A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs

For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever ij and {i, j} is an edge in G. Building upon recent workinvolving maximal corank s (or nu llities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs - the original motivation.

Zero Forcing, Linear and Quantum Controllability for Systems Evolving on Networks

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Our main result says that controllability in the quantum sense, expressed by the Lie algebra rank condition, and controllability in the sense of linear systems, expressed by the controllability matrix rank condition, are equivalent conditions. We also investigate how the graph theoretic concept of a zero forcing set impacts the controllability property; if a set of vertices is a zero forcing set, the associated dynamical system is controllable. These results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena.

ON THE MINIMUM RANK OF NOT NECESSARILY SYMMETRIC MATRICES: A PRELIMINARY STUDY ∗

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

Logic circuits from zero forcing

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of “back forcing” as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we show that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity. Moreover, in the light of applications of zero forcing in quantum mechanics, the link with Boolean functions may suggest a new directions in quantum control theory and in the study of engineered quantum spin systems. It is an open technical problem to verify whether there is a link between zero forcing and computation with contact circuits.

SIGN PATTERNS THAT REQUIRE EVENTUAL POSITIVITY OR REQUIRE EVENTUAL NONNEGATIVITY

It is shown that a square sign pattern A requires eventual positivity if and only if it is nonnegative and primitive. Let the set of vertices in the digraph of A that have access to a vertex s be denoted by In(s) and the set of vertices to which t has access denoted by Out(t). It is shown that A = (�ij) requires eventual nonnegativity if and only if for every s,t such thatst = , the two principal submatrices of A indexed by In(s) and Out(t) require nilpotence. It is shown that A requires eventual exponential positivity if and only if it requires exponential positivity, i.e., A is irreducible and its off-diagonal entries are nonnegative.

Completions of P-matrix patterns

Abstract A list of positions in an n × n real matrix (a pattern) is said to have M - completion if every partial M -matrix that specifies exactly these positions can be completed to an M -matrix. Let Q be a pattern that includes all diagonal entries and let G be its digraph. The following are equivalent. (1) the pattern Q has M -completion; (2) the pattern Q is permutation similar to a block triangular pattern with all the diagonal blocks completely specified; (3) any strongly connected subdigraph of G is complete. A pattern with some diagonal entries unspecified has M -completion if and only if the principal subpattern defined by the specified diagonal positions has M -completion.