We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positiv...

A Room square of side n is an n by n array of cells, whose entries are chosen from a setS of n + 1 objects called symbols, which satisfies the following conditions: (i) every cell of the array is either empty or contains an unordered pair of distinct symbols from S; (ii) each symbol occurs in every row and in every column of the array; (iii) every unordered pair of symbols occurs precisely once...

Within the tensor network framework, (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operator (iPEPO) coupled via ancilla degrees freedom. To investigate properties spin-1/2 Heisenberg model on square lattice, we introduce family fully spin-$SU(2)$ and lattice-$C_{4v}$ symmetric on-site tensors (of bond dimensions $D=4$ or $D=7$) pl...

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is  a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigr...