On Polyhedral Approximations of the Positive Semidefinite Cone

It is well known that state-of-the-art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes. This leads us to consider the question, how well can we approximate semidefinite programs with linear programs? In this paper, we prove lower bounds on the size of linear programs that approximate the positive semidefinite cone. Let D be the set of n × n positive semidefinite matrices of trace equal to one. We prove two results on the hardness of approximating D with polytopes. We show that if 0 < ε < 1and A is an arbitrary matrix of trace equal to one, any polytope P such that (1-ε) (D-A) ⊂ P ⊂ D-A must have linear programming extension complexity at least [Formula: see text], where c > 0 is a constant that depends on ε. Second, we show that any polytope P such that D ⊂ P and such that the Gaussian width of P is at most twice the Gaussian width of D must have extension complexity at least [Formula: see text]. Our bounds are both superpolynomial in n and demonstrate that there is no generic way of approximating semidefinite programs with compact linear programs. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.

On similarity of an arbitrary matrix to a block diagonal matrix

Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||T||||T-1|| is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks

Abstract—Inspired by the depth and breadth of developments on the theory of deep learning, we pose these fundamental questions: can we accurately approximate an arbitrary matrix-vector product using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so, can we bound the resulting approximation error? Attempting to answer these questions, we derive error bounds in Lebesgue and Sobolev norms for a matrix-vector product approximation with deep ReLU FNNs. Since a matrix-vector product models several problems in wireless communications and signal processing; network science and graph signal processing; and network neuroscience and brain physics, we discuss various applications that are motivated by an accurate matrix-vector product approximation with deep ReLU FNNs. Toward this end, the derived error bounds offer a theoretical insight and guarantee in the development of algorithms based on deep ReLU FNNs. <br>

Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networks

Abstract—Inspired by the depth and breadth of developments on the theory of deep learning, we pose these fundamental questions: can we accurately approximate an arbitrary matrix-vector product using deep rectified linear unit (ReLU) feedforward neural networks (FNNs)? If so, can we bound the resulting approximation error? Attempting to answer these questions, we derive error bounds in Lebesgue and Sobolev norms for a matrix-vector product approximation with deep ReLU FNNs. Since a matrix-vector product models several problems in wireless communications and signal processing; network science and graph signal processing; and network neuroscience and brain physics, we discuss various applications that are motivated by an accurate matrix-vector product approximation with deep ReLU FNNs. Toward this end, the derived error bounds offer a theoretical insight and guarantee in the development of algorithms based on deep ReLU FNNs. <br>

The Generalized Matrix Decomposition Biplot and Its Application to Microbiome Data

ABSTRACT Exploratory analysis of human microbiome data is often based on dimension-reduced graphical displays derived from similarities based on non-Euclidean distances, such as UniFrac or Bray-Curtis. However, a display of this type, often referred to as the principal-coordinate analysis (PCoA) plot, does not reveal which taxa are related to the observed clustering because the configuration of samples is not based on a coordinate system in which both the samples and variables can be represented. The reason is that the PCoA plot is based on the eigen-decomposition of a similarity matrix and not the singular value decomposition (SVD) of the sample-by-abundance matrix. We propose a novel biplot that is based on an extension of the SVD, called the generalized matrix decomposition biplot (GMD-biplot), which involves an arbitrary matrix of similarities and the original matrix of variable measures, such as taxon abundances. As in a traditional biplot, points represent the samples, and arrows represent the variables. The proposed GMD-biplot is illustrated by analyzing multiple real and simulated data sets which demonstrate that the GMD-biplot provides improved clustering capability and a more meaningful relationship between the arrows and points. IMPORTANCE Biplots that simultaneously display the sample clustering and the important taxa have gained popularity in the exploratory analysis of human microbiome data. Traditional biplots, assuming Euclidean distances between samples, are not appropriate for microbiome data, when non-Euclidean distances are used to characterize dissimilarities among microbial communities. Thus, incorporating information from non-Euclidean distances into a biplot becomes useful for graphical displays of microbiome data. The proposed GMD-biplot accounts for any arbitrary non-Euclidean distances and provides a robust and computationally efficient approach for graphical visualization of microbiome data. In addition, the proposed GMD-biplot displays both the samples and taxa with respect to the same coordinate system, which further allows the configuration of future samples.

The GMD-biplot and its application to microbiome data

Exploratory analysis of human microbiome data is often based on dimension-reduced graphical displays derived from similarities based on non-Euclidean distances, such as UniFrac or Bray-Curtis. However, a display of this type, often referred to as the principal coordinate analysis (PCoA) plot, does not reveal which taxa are related to the observed clustering because the configuration of samples is not based on a coordinate system in which both the samples and variables can be represented. The reason is that the PCoA plot is based on the eigen-decomposition of a similarity matrix and not the singular value decomposition (SVD) of the sample-by-abundance matrix. We propose a novel biplot that is based on an extension of the SVD, called the generalized matrix decomposition (GMD), which involves an arbitrary matrix of similarities and the original matrix of variable measures, such as taxon abundances. As in a traditional biplot, points represent the samples and arrows represent the variables. The proposed GMD-biplot is illustrated by analyzing multiple real and simulated data sets which demonstrate that the GMD-biplot provides improved clustering capability and a more meaningful relationship between the arrows and the points.

On idempotency of linear combinations of a quadratic or a cubic matrix and an arbitrary matrix

Let A be a quadratic or a cubic n x n nonzero matrix and B be an arbitrary n x n nonzero matrix. In this study, we have established necessary and sufficient conditions for the idempotency of the linear combinations of the form aA + bB, under the some certain conditions imposed on A and B, where a, b are nonzero complex numbers.