Hypercontractive inequalities for real-valued functions over the Boolean cube play an important role in theoretical computer science. In this work, we prove a hypercontractive inequality for matrix-valued functions defined over large alphabets, generalizing the result of Ben-Aroya, Regev, de Wolf (FOCS'08) for the Boolean alphabet. To obtain our result we prove a generalization of 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: we present a lower bound for locally decodable codes (LDC) over large alphabets. An LDC $C:\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ is an encoding of $x\in \mathbb{Z}_r^n$ into a codeword $C(x)$ in such a way that one can recover an arbitrary $x_i\in \mathbb{Z}_r$ (with probability at least $1/r+\varepsilon$) by making only a few queries to a corrupted codeword. The main question in LDCs is the trade-off between $N$ and $n$. By using hypercontractivity, we give an exponential lower bound $N= 2^{\Omega(\varepsilon^4 n/r^4)}$ 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 \emph{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: in this direction 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 every streaming algorithm in the adversarial model achieving a $(r-\varepsilon)$-approximation of this value requires $\Omega(n^{1-1/t})$ classical space 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) for the case $r=2$.
