矩阵超压缩性、流式算法与LDC：大字母表情形
Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case
AI总结：
本文证明大字母表上矩阵值函数的超压缩不等式，并应用于隐藏超匹配通信复杂性、超图Unique Games流式算法的量子空间下界及局部可解码码的指数下界。
摘要：
我们证明了大字母表上矩阵值函数的超压缩不等式。为此，我们推广了Ball、Carlen和Lieb（Inventiones Mathematicae '94）关于迹范数的强大的$2$-一致凸性不等式。利用我们的超压缩不等式，我们给出了大字母表上隐藏超匹配问题的通信复杂性的上下界。然后我们考虑流式算法，用于近似超图中$t$大小超边的Unique Games值。通过使用我们的通信下界，我们证明在对抗模型中，每个达到该值$(r-\varepsilon)$-近似的流式算法需要$\Omega(n^{1-2/t})$量子空间，其中$r$是字母表大小。接下来，我们给出大字母表上局部可解码码（LDC）$\mathbb{Z}_r^n\to \mathbb{Z}_r^N$的下界，其可恢复概率至少为$1/r + \varepsilon$。利用超压缩性，我们给出了$\mathbb{Z}_r$上$2$-查询（可能非线性）LDC的指数下界$N = 2^{\Omega(\varepsilon^4 n/r^4)}$，并利用非交换Khintchine不等式，我们证明了改进的下界$N = 2^{\Omega(\varepsilon^2 n/r^2)}$。
英文摘要：
We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful $2$-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae'94). Using our hypercontractive~inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with $t$-size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an $(r-\varepsilon)$-approximation of this value requires $Ω(n^{1-2/t})$ quantum space, where $r$ is the alphabet size. We next present a lower bound for locally decodable codes (LDC) $\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ over large alphabets with recoverability probability at least $1/r + \varepsilon$. Using hypercontractivity, we give an exponential lower bound $N = 2^{Ω(\varepsilon^4 n/r^4)}$ for $2$-query (possibly non-linear) LDCs over $\mathbb{Z}_r$ and using the non-commutative Khintchine inequality we prove an improved lower bound of $N = 2^{Ω(\varepsilon^2 n/r^2)}$.
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