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• W3198445769 abstract "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 $C:mathbb{Z}_r^nto mathbb{Z}_r^N$ encodes $xinmathbb{Z}_r^n$ into a codeword $C(x)$ such that one can recover any $x_i$ (with probability at least $1/r+varepsilon$) 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 $N=2^{Omega(varepsilon^4 n/r^4)}$ for $2$-query (possibly non-linear) LDCs over $mathbb{Z}_r$. 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 $O(log{n})$ space. On the other hand, we use our communication lower bound to show that any streaming algorithm in the adversarial model achieving a $(r-varepsilon)$-approximation requires $Omega(n^{1-1/t})$ classical or $Omega(n^{1-2/t})$ 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 $r=2$." @default.
• W3198445769 created "2021-09-13" @default.
• W3198445769 creator A5051227736 @default.
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• W3198445769 date "2021-09-06" @default.
• W3198445769 modified "2023-10-18" @default.
• W3198445769 title "Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case" @default.
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