Principal symmetric space analysis

Improved Convergence Speed of Fully Symmetric Learning Rules for Principal Component Analysis

Fully symmetric learning rules for principal component analysis can be derived from a novel objective function suggested in our previous work. We observed that these learning rules suffer from slow convergence for covariance matrices where some principal eigenvalues are close to each other. Here we describe a modified objective function with an additional term which mitigates this convergence problem. We show that the learning rule derived from the modified objective function inherits all fixed points from the original learning rule (but may introduce additional ones). Also the stability of th…

On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis

We examine the problem of approximating a positive, semidefinite matrix $Σ$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$. This problem arises in sparse principal component analysis, where a decomposition of $Σ$ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest eigenvalue of a symmetric matrix that is linear in the variables. This representation allows to use the techniques of robust optimization, to derive a bound based on semidefinite programming. The q…

正半定矩阵的一个更强的西尔维斯特判据

A stronger Sylvester's criterion for positive semidefinite matrices

本文提出一个更强的正半定矩阵西尔维斯特判据，将所需验证的行列式数量从指数级 $2^m-1$ 降至 $m(m+1)/2$，并基于新判据给出逐元素判据及其在矩阵填充和非线性半定规划中的应用。

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the satisfiability of the constraints over that field. This implies two complexity results about Gaussian conditional independence structures. First, all real algebraic numbers are necessary to construct inhabitants of non-empty Gaussian statistical models defined by conditional independence and dependence constraints. This gives a negative answer to a question o…

Learning Signed Determinantal Point Processes through the Principal Minor Assignment Problem

Symmetric determinantal point processes (DPP's) are a class of probabilistic models that encode the random selection of items that exhibit a repulsive behavior. They have attracted a lot of attention in machine learning, when returning diverse sets of items is sought for. Sampling and learning these symmetric DPP's is pretty well understood. In this work, we consider a new class of DPP's, which we call signed DPP's, where we break the symmetry and allow attractive behaviors. We set the ground for learning signed DPP's through a method of moments, by solving the so called principal assignment p…

Symmetric matrices, Catalan paths, and correlations

Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope…