Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix $M$ as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.Comment: 21 page

Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message-passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the asymptotic support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that S(k,\ell) = \mbox{tr}(A_k^* B_\ell). The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.Comment: 9 page

A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.Comment: 40 pages, 1 figur

On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. Specifically, for a matrix which is distributed among a number of players, we consider the problem of determining its rank, of computing entries in its inverse, and of solving linear equations. We also consider related problems such as computing the generalized inner product of vectors held on different servers. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized $s$-player communication complexity of these problems is at least $s$ times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, which we formally define, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang. A common feature of our lower bounds is that they apply even to the special "threshold promise" versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds

An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

We prove an optimal $\Omega(n)$ lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive $n$-bit strings $x$ and $y$, respectively. They are promised that the Hamming distance between $x$ and $y$ is either at least $n/2+\sqrt{n}$ or at most $n/2-\sqrt{n}$, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses $n$ bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables