Doubly stochastic matrix plays an essential role in several areas such as statistics and machine learning. In this paper we consider the optimal approximation of a square set doubly matrices. A structured BFGS method is proposed to solve dual primal problem. The resulting algorithm builds curvature information into diagonal components true Hessian, so that it takes only additional linear cost o...

We study a general class of nonlinear and shift-varying smoothing filters that operate based on averaging. This important class of filters includes many well-known examples such as the bilateral filter, nonlocal means, general adaptive moving average filters, and more. (Many linear filters such as linear minimum mean-squared error smoothing filters, Savitzky–Golay filters, smoothing splines, an...

We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length o(n) behave like independent exponentials. We determine the limiting empirical distribution of the si...

For a graph G, let fij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the fij’s can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F = (fij)n×n f , where f is the total number of spanning rooted for...

for vectors x, y ∈ rn, it is said that x is left matrix majorizedby y if for some row stochastic matrix r; x = ry. the relationx ∼` y, is defined as follows: x ∼` y if and only if x is leftmatrix majorized by y and y is left matrix majorized by x. alinear operator t : rp → rn is said to be a linear preserver ofa given relation ≺ if x ≺ y on rp implies that t x ≺ ty onrn. the linear preservers o...

The Birkhoff (permutation) polytope, Bn, consists of the n × n nonnegative doubly stochastic matrices, has dimension (n− 1)2, and has n2 facets. A new analogue, the alternating sign matrix polytope, ASMn, is introduced and characterized. Its vertices are the Qn−1 j=0 (3j+1)! (n+j)! n × n alternating sign matrices. It has dimension (n− 1)2, has 4[(n− 2)2 +1] facets, and has a simple inequality d...

Let us denote by Ωn the Birkhoff polytope of n× n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n× n permutation matrices. Here we consider a higherdimensional analog of this basic fact. Let Ω n be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entri...