Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

by   Srinivasan Arunachalam, et al.

In this work, we prove a hypercontractive inequality for matrix-valued functions defined over large alphabets, generalizing a result of Ben-Aroya, Regev, de Wolf (FOCS'08) for the Boolean alphabet. To obtain our result we generalize the powerful 2-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae'94). We give two applications of this hypercontractive inequality. Locally decodable codes (LDC): we present a lower bound for LDCs over large alphabets. An LDC C:ℤ_r^n→ℤ_r^N encodes x∈ℤ_r^n into a codeword C(x) such that one can recover any x_i (with probability at least 1/r+ε) by making a few queries to a corrupted codeword. The main question is the trade-off between N and n. By using hypercontractivity, we prove that N=2^Ω(ε^4 n/r^4) for 2-query (possibly non-linear) LDCs over ℤ_r. Previously exponential lower bounds were known for r=2 (Kerenidis and de Wolf (JCSS'04)) and for linear codes (Dvir and Shpilka (SICOMP'07)). Streaming algorithms: we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem when defined over large alphabets, which generalizes the well-known Boolean Hidden Matching problem. We then consider streaming algorithms for approximating the value of Unique Games on a t-hyperedge hypergraph: a simple edge-counting argument gives an r-approximation with O(logn) space. On the other hand, we use our communication lower bound to show that any streaming algorithm in the adversarial model achieving a (r-ε)-approximation requires Ω(n^1-1/t) classical or Ω(n^1-2/t) quantum space. In this setting, these results simplify and generalize the seminal work of Kapralov, Khanna and Sudan (SODA'15) and Kapravol and Krachun (STOC'19) when r=2.



There are no comments yet.


page 1

page 2

page 3

page 4


Classical lower bounds from quantum upper bounds

We prove lower bounds on complexity measures, such as the approximate de...

Streaming Hardness of Unique Games

We study the problem of approximating the value of a Unique Game instanc...

Optimal Streaming Approximations for all Boolean Max-2CSPs

We prove tight upper and lower bounds on approximation ratios of all Boo...

Factorial Lower Bounds for (Almost) Random Order Streams

In this paper we introduce and study the StreamingCycles problem, a rand...

The aBc Problem and Equator Sampling Renyi Divergences

We investigate the problem of approximating the product a^TBc, where a,c...

On the streaming complexity of fundamental geometric problems

In this paper, we focus on lower bounds and algorithms for some basic ge...

Finding Skewed Subcubes Under a Distribution

Say that we are given samples from a distribution ψ over an n-dimensiona...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.