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

09/06/2021
by   Srinivasan Arunachalam, et al.
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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.

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