# Matrix Discrepancy from Quantum Communication

@article{Hopkins2021MatrixDF, title={Matrix Discrepancy from Quantum Communication}, author={Samuel B. Hopkins and Prasad Raghavendra and Abhishek Shetty}, journal={ArXiv}, year={2021}, volume={abs/2110.10099} }

We develop a novel connection between discrepancy minimization and (quantum) communication complexity. As an application, we resolve a substantial special case of theMatrix Spencer conjecture. In particular, we show that for every collection of symmetric = × = matrices 1 , . . . , = with ‖ 8 ‖ 6 1 and ‖ 8 ‖ 6 =1/4 there exist signs G ∈ {±1}= such that the maximum eigenvalue of ∑ 86= G8 8 is at most $( √ =). We give a polynomial-time algorithm based on partial coloring and semidefinite…

## Figures from this paper

## 2 Citations

A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent

- Computer Science, MathematicsArXiv
- 2021

The matrix Spencer conjecture is reduced to the existence of a O(log(m/n) quantum relative entropy net on the spectraplex by improving upon the naive O( √ n log r) bound for random coloring and proving the Matrix Spencer conjecture when rm ≤ n.

Solving SDP Faster: A Robust IPM Framework and Efficient Implementation

- Mathematics
- 2021

This paper introduces a new robust interior point method analysis for semidefinite programming (SDP) and shows that for the case m = Ω(n), it can solve SDPs in m time, suggesting solving SDP is nearly as fast as solving the linear system with equal number of variables and constraints.

## References

SHOWING 1-10 OF 34 REFERENCES

The Church of the Symmetric Subspace

- Computer Science
- 2013

The purpose of the article is to collect in one place many, if not all, of the quantum information applications of the symmetric subspace, and to collect some new proofs of existing results, such as a variant of the exponential de Finetti theorem.

Dense quantum coding and quantum finite automata

- Computer Science, PhysicsJACM
- 2002

The technique is applied to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata.

Efficient algorithms for discrepancy minimization in convex sets

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2018

A constructive version of the result of Gluskin and Giannopoulos, in which the coloring is attained via the optimization of a linear function is proved, which implies a linear programming based algorithm for combinatorial discrepancy obtaining the same result as Spencer.

Constructive Discrepancy Minimization for Convex Sets

- Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

It is shown that for any symmetric convex set K with measure at least e-n/500, the following algorithm finds a point y ∈ K ∩ [-1, 1]n with Ω(n) coordinates in ±1: (1) take a random Gaussian vector x, (2) compute the point y in K∩ [- 1, 1)n that is closest to x.

Algorithms for combinatorial discrepancy

- Mathematics, Computer Science
- 2018

This thesis first attempted to solve the problem of discrepancy minimization using a convex geometric approach which led to improved algorithms in some regimes, and developed an algorithm for the most important special case when the convex body K is a scaling of the hypercube.

Linear Size Sparsifier and the Geometry of the Operator Norm Ball

- MathematicsSODA
- 2020

The Matrix Spencer Conjecture can indeed prove that this body is close to most of the Gaussian measure, which implies that a discrepancy algorithm by the second author can be used to sample a linear size sparsifer.

Constructive Discrepancy Minimization by Walking on the Edges

- Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

A new randomized algorithm to find a coloring as in Spencer's result based on a restricted random walk which is “truly” constructive in that it does not appeal to the existential arguments, giving a new proof of Spencer's theorem and the partial coloring lemma.

Strong converse for identification via quantum channels

- MathematicsIEEE Trans. Inf. Theory
- 2002

We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum…

A Matrix Hyperbolic Cosine Algorithm and Applications

- Computer Science, MathematicsICALP
- 2012

This paper generalizes Spencer's hyperbolic cosine algorithm to the matrix-valued setting, and gives an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrixSparsification, which implies an improved deterministic algorithm for spectral graph sparsify of dense graphs.

Optimal lower bounds for quantum automata and random access codes

- Computer Science40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

Holevo's theorem is turned to, and it is shown that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.