9 research outputs found
Locally decodable codes and the failure of cotype for projective tensor products
It is shown that for every there exists a Banach space
of finite cotype such that the projective tensor product \ell_p\tp X fails to
have finite cotype. More generally, if satisfy
then
\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3} does not have finite cotype. This is a
proved via a connection to the theory of locally decodable codes
The positive semidefinite Grothendieck problem with rank constraint
Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of
size m x m, the positive semidefinite Grothendieck problem with
rank-n-constraint (SDP_n) is
maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m
\in S^{n-1}.
In this paper we design a polynomial time approximation algorithm for SDP_n
achieving an approximation ratio of
\gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 -
\Theta(1/n).
We show that under the assumption of the unique games conjecture the achieved
approximation ratio is optimal: There is no polynomial time algorithm which
approximates SDP_n with a ratio greater than \gamma(n). We improve the
approximation ratio of the best known polynomial time algorithm for SDP_1 from
2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter
approximation ratio for SDP_n when A is the Laplacian matrix of a graph with
nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on
Automata, Languages and Programming, 12 page
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
Pure state entaglement and stabilizer representations
Bibliography: p. 94-9
Revisiting the Sanders-Bogolyubov-Ruzsa Theorem in F-p(n) and its Application to Non-malleable Codes
Non-malleable codes (NMCs) protect sensitive data against degrees of corruption that prohibit error detection, ensuring instead that a corrupted codeword decodes correctly or to something that bears little relation to the original message. The split-state model, in which codewords consist of two blocks, considers adversaries who tamper with either block arbitrarily but independently of the other. The simplest construction in this model, due to Aggarwal, Dodis, and Lovett (STOC'14), was shown to give NMCs sending k-bit messages to O(k(7))-bit codewords. It is conjectured, however, that the construction allows linear-length codewords. Towards resolving this conjecture, we show that the construction allows for code-length O(k(5)). This is achieved by analysing a special case of Sanders's Bogolyubov-Ruzsa theorem for general Abelian groups. Closely following the excellent exposition of this result for the group F-2(n) by Lovett, we expose its dependence on p for the group F-p(n), where p is a prime