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Register a new userAn Almost Optimal Rank Bound for Depth-3 Identities
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Added by
Nitin Saxena
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k^3log d. The previous best rank bound known was 2^{O(k^2)}(log d)^{k-2} by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Omega(klog d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic black-box identity tests for depth-3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-3 circuit (over any field) is at most k^3log d. The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.
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Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error $epsilon$ and has an almost optimal seed-length of $O(log n + log(1/epsilon) cdot loglog(1/epsilon))$. For an inverse-polynomially growing error $epsilon$, our generator has a seed-length optimal up to a factor of $O( log log {(n)})$. The most efficient PRG previously known (due to Kane, 2012) requires a seed-length of $Omega(log^{3/2}{(n)})$ in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. (2011) and Celis et. al. (2013), the emph{classical moment problem} from probability theory and explicit constructions of emph{orthogonal designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion in Lie groups.
One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.
We prove upper bounds on deterministic communication complexity in terms of log of the rank and simp
Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f in mathbb{F}[x_1,ldots, x_n] $ (where $mathbb{F}$ = $mathbb{Q}$ or $mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We give a deterministic polynomial identity testing algorithm to check whether $fequiv 0$ or not in time $ 2^d text{ poly}(n,s) $.
The problem of finding large cliques in random graphs and its planted variant, where one wants to recover a clique of size $omega gg log{(n)}$ added to an Erdos-Renyi graph $G sim G(n,frac{1}{2})$, have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size $omega = Omega(sqrt{n})$. By contrast, information theoretically, one can recover planted cliques so long as $omega gg log{(n)}$. In this work, we continue the investigation of algorithms from the sum of squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson (MPW, 2015) and Deshpande and Montanari (DM,2015). Our main results improve upon both these previous works by showing: 1. Degree four SoS does not recover the planted clique unless $omega gg sqrt n poly log n$, improving upon the bound $omega gg n^{1/3}$ due to DM. A similar result was obtained independently by Raghavendra and Schramm (2015). 2. For $2 < d = o(sqrt{log{(n)}})$, degree $2d$ SoS does not recover the planted clique unless $omega gg n^{1/(d + 1)} /(2^d poly log n)$, improving upon the bound due to MPW. Our proof for the second result is based on a fine spectral analysis of the certificate used in the prior works MPW,DM and Feige and Krauthgamer (2003) by decomposing it along an appropriately chosen basis. Along the way, we develop combinatorial tools to analyze the spectrum of random matrices with dependent entries and to understand the symmetries in the eigenspaces of the set symmetric matrices inspired by work of Grigoriev (2001). An argument of Kelner shows that the first result cannot be proved using the same certificate. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by correcting the certificate of previous works.
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