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Register a new userInner Rank and Lower Bounds for Matrix Multiplication
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Added by
Joel Friedman
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Joel Friedman
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We develop a notion of {em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $ntimes n$ matrix multiplication over an arbitrary field is at least $2n^2-n+1$. While inner rank does not provide improvements to currently known lower bounds, we argue that this notion merits further study.
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We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n=2,3,4,5, recovering Strassens algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassens algorithm; the same proof works for all n, to yield a decomposition with $n^3 - n + 1$ terms.
In the communication problem $mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x$ and $y$ in ${0,1}^n$ with the promise that $x eq y$. The last player to receive a message must output an index $i$ such that $x_i eq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $Theta(min{n, log(1/delta)log^2(frac{n}{log(1/delta)})})$ bits for failure probability $delta$. Our lower bound holds even if promised $mathop{support}(y)subset mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $ell_p$-sampling in strict turnstile streams for $0le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if it is promised that $xin{0,1}^n$ at all points in the stream. Our lower bound demonstrates that any algorithm $mathcal{A}$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $mathcal{A}$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoders interactions with $mathcal{A}$, which is loosely motivated by techniques in differential privacy. Our correctness analysis involves understanding the ability of $mathcal{A}$ to correctly answer adaptive queries which have positive but bounded mutual information with $mathcal{A}$s internal randomness, and may be of independent interest in the newly emerging area of adaptive data analysis with a theoretical computer science lens.
We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in ${0,ldots,q-1}$, we prove that improving over the trivial approximability by a factor of $q$ requires $Omega(n)$ space even on instances with $O(n)$ constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires $Omega(n)$ space. The key technical core is an optimal, $q^{-(k-1)}$-inapproximability for the case where every constraint is given by a system of $k-1$ linear equations $bmod; q$ over $k$ variables. Prior to our work, no such hardness was known for an approximation factor less than $1/2$ for any CSP. Our work builds on and extends the work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the max cut in graphs. This corresponds roughly to the case of Max $k$-LIN-$bmod; q$ with $k=q=2$. Each one of the extensions provides non-trivial technical challenges that we overcome in this work.
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.
In the communication problem $mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y in{0,1}^n$ with the promise that $x eq y$. The last player to receive a message must output an index $i$ such that $x_i eq y_i$. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly $Theta(min{n,log(1/delta)log^2(frac n{log(1/delta)})})$ for failure probability $delta$. Our lower bound holds even if promised $mathop{support}(y)subset mathop{support}(x)$. As a corollary, we obtain optimal lower bounds for $ell_p$-sampling in strict turnstile streams for $0le p < 2$, as well as for the problem of finding duplicates in a stream. Our lower bounds do not need to use large weights, and hold even if promised $xin{0,1}^n$ at all points in the stream. We give two different proofs of our main result. The first proof demonstrates that any algorithm $mathcal A$ solving sampling problems in turnstile streams in low memory can be used to encode subsets of $[n]$ of certain sizes into a number of bits below the information theoretic minimum. Our encoder makes adaptive queries to $mathcal A$ throughout its execution, but done carefully so as to not violate correctness. This is accomplished by injecting random noise into the encoders interactions with $mathcal A$, which is loosely motivated by techniques in differential privacy. Our second proof is via a novel randomized reduction from Augmented Indexing [MNSW98] which needs to interact with $mathcal A$ adaptively. To handle the adaptivity we identify certain likely interaction patterns and union bound over them to guarantee correct interaction on all of them. To guarantee correctness, it is important that the interaction hides some of its randomness from $mathcal A$ in the reduction.
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