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 $C:\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ encodes $x\in\mathbb{Z}_r^n$ 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 $r=2$.

中文翻译：

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矩阵超收缩、流算法和 LDC：大字母表案例

在这项工作中，我们证明了在大字母表上定义的矩阵值函数的超收缩不等式，概括了 Ben-Aroya、Regev、de Wolf (FOCS'08) 对布尔字母表的结果。为了获得我们的结果，我们概括了 Ball, Carlen, Lieb (Inventiones Mathematicae'94) 的迹范数的强大的 $2$-uniform 凸度不等式。我们给出了这种超收缩不等式的两个应用。本地可解码代码 (LDC)：我们在大字母表上为 LDC 提供了一个下限。LDC $C:\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ 将 $x\in\mathbb{Z}_r^n$ 编码为代码字 $C(x)$，使得可以通过对损坏的代码字进行一些查询来恢复任何 $x_i$（概率至少为 $1/r+\varepsilon$）。主要问题是 $N$ 和 $n$ 之间的权衡。通过使用超收缩，我们证明了 $N=2^{\Omega(\varepsilon^4 n/r^4)}$ 对于 $\mathbb{Z}_r$ 上的 $2$-query（可能是非线性的）LDC。以前已知的指数下界为 $r=2$（Kerenidis 和 de Wolf（JCSS'04））和线性代码（Dvir 和 Shpilka（SICOMP'07））。流算法：当在大字母表上定义时，我们给出了隐藏超匹配问题的通信复杂度的上限和下限，这概括了众所周知的布尔隐藏匹配问题。然后，我们考虑在 $t$-hyperedge 超图上逼近 Unique Games 的价值的流算法：一个简单的边计数参数给出一个 $r$-approximation 和 $O(\log{n})$ 空间。另一方面，我们使用我们的通信下界来表明对抗模型中实现 $(r-\varepsilon)$-近似的任何流算法都需要 $\Omega(n^{1-1/t})$ classic 或 $\Omega( n^{1-2/t})$ 量子空间。在这种情况下，当 $r=2$ 时，这些结果简化和概括了 Kapralov、Khanna 和苏丹 (SODA'15) 以及 Kapravol 和 Krachun (STOC'19) 的开创性工作。