Related papers: Optimizing positive maps in the matrix algebra $M

Optimizing positive maps in the matrix algebra $M_n$

URL: http://arxiv.org/abs/2309.09621v1
Date: Mon, 18 Sep 2023 09:47:55 GMT
Title: Optimizing positive maps in the matrix algebra $M_n$
Authors: Anindita Bera, Gniewomir Sarbicki and Dariusz Chru\'sci\'nski
Abstract summary: We conjecture how to optimize a map $tau_n,k$ when $GCD(n,k)=2$ or 3. For $GCD(n,k)=2$, a series of analytical results are derived and for $GCD(n,k)=3$, we provide a suitable numerical analysis.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
Abstract: We present an optimization procedure for a seminal class of positive maps $\tau_{n,k}$ in the algebra of $n \times n$ complex matrices introduced and studied by Tanahasi and Tomiyama, Ando, Nakamura and Osaka. Recently, these maps were proved to be optimal whenever the greatest common divisor $GCD(n,k)=1$. We attain a general conjecture how to optimize a map $\tau_{n,k}$ when $GCD(n,k)=2$ or 3. For $GCD(n,k)=2$, a series of analytical results are derived and for $GCD(n,k)=3$, we provide a suitable numerical analysis.

Related papers

Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
arXiv Detail & Related papers (2024-08-16T02:33:07Z)
Optimality of generalized Choi maps in $M_3$ [0.0]
We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.
arXiv Detail & Related papers (2023-12-05T14:57:11Z)
A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions [44.655316553524855]
We design algorithms for minimizing $max_iin[n] f_i(x) over a $d$-dimensional Euclidean or simplex domain. When each $f_i$ is $1$-Lipschitz and $1$-smooth, our method computes an $epsilon-approximate solution.
arXiv Detail & Related papers (2023-11-17T22:07:18Z)
Structured Semidefinite Programming for Recovering Structured Preconditioners [41.28701750733703]
We give an algorithm which, given positive definite $mathbfK in mathbbRd times d$ with $mathrmnnz(mathbfK)$ nonzero entries, computes an $epsilon$-optimal diagonal preconditioner in time. We attain our results via new algorithms for a class of semidefinite programs we call matrix-dictionary approximation SDPs.
arXiv Detail & Related papers (2023-10-27T16:54:29Z)
Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
arXiv Detail & Related papers (2023-06-02T18:55:27Z)
An Optimal Algorithm for Strongly Convex Min-min Optimization [79.11017157526815]
Existing optimal first-order methods require $mathcalO(sqrtmaxkappa_x,kappa_y log 1/epsilon)$ of computations of both $nabla_x f(x,y)$ and $nabla_y f(x,y)$. We propose a new algorithm that only requires $mathcalO(sqrtkappa_x log 1/epsilon)$ of computations of $nabla_x f(x,
arXiv Detail & Related papers (2022-12-29T19:26:12Z)
Beyond Ans\"atze: Learning Quantum Circuits as Unitary Operators [30.5744362478158]
We run gradient-based optimization in the Lie algebra $mathfrak u(2N)$. We argue that $U(2N)$ is not only more general than the search space induced by ansatz, but in ways easier to work with on classical computers.
arXiv Detail & Related papers (2022-03-01T16:40:21Z)
Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
arXiv Detail & Related papers (2022-02-10T16:10:41Z)
Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment [52.577757919003844]
We study the problem of efficiently summarizing a short video into several paragraphs, leveraging recent progress in fast graph sampling. Experimental results show that our algorithm achieves comparable video summarization as state-of-the-art methods, at a substantially reduced complexity.
arXiv Detail & Related papers (2021-10-21T18:43:00Z)
Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment [46.145969174332485]
We propose a fast general metric learning framework that is entirely projection-free. We replace the PD cone constraint in the metric learning problem with possible linear constraints per distances. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes.
arXiv Detail & Related papers (2020-06-15T23:15:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.