Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPseudorandom Generators for Width-3 Branching Programs
81
0
0.0
(
0
)
Added by
Avishay Tal
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We construct pseudorandom generators of seed length $tilde{O}(log(n)cdot log(1/epsilon))$ that $epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $tilde{O}(log(n) cdot mathrm{poly}(1/epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work of Nisan [Combinatorica, 1992]. Our constructions are based on the `iterated milder restrictions approach of Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered case, we combine iterated milder restrictions with the generator of Chattopadhyay et al. [CCC, 2018]. Two conceptual ideas that play an important role in our analysis are: (1) A relabeling technique allowing us to analyze a relabeled version of the given branching program, which turns out to be much easier. (2) Treating the number of colliding layers in a branching program as a progress measure and showing that it reduces significantly under pseudorandom restrictions. In addition, we achieve nearly optimal seed-length $tilde{O}(log(n/epsilon))$ for the classes of: (1) read-once polynomials on $n$ variables, (2) locally-monotone ROBPs of length $n$ and width $3$ (generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length $n$ having a layer of width $2$ in every consecutive $mathrm{poly}log(n)$ layers.
rate research
Read More
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.
We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit $L_1$ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the $k$-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with $k$. This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], or have polynomial dependence on the error parameter in the seed length [CHLT10], and thus answers an open question in [CHLT19]. As an example, we show that for polynomial error, Fourier bounds on the first $O(log n)$ levels is sufficient to recover the seed length in [CHHL19], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylors theorem and bounding the degree-$k$ Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the emph{level-k unsigned Fourier sum}, which is potentially a much smaller quantity than the $L_1$ notion in previous works. By generalizing a connection established in [CHH+20], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for $mathbb{F}_2$ polynomials with seed length close to the state-of-the-art construction due to Viola [Vio09], which was not known to be possible using this framework.
Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most $k$ is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width. It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most $k$ is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.
We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the emph{rectangular} symbolic matrix in both commutative and noncommutative settings. The main results are: 1. We show an explicit $O^{*}({nchoose {downarrow k/2}})$-size ABP construction for noncommutative permanent polynomial of $ktimes n$ symbolic matrix. We obtain this via an explicit ABP construction of size $O^{*}({nchoose {downarrow k/2}})$ for $S_{n,k}^*$, noncommutative symmetrized version of the elementary symmetric polynomial $S_{n,k}$. 2. We obtain an explicit $O^{*}(2^k)$-size ABP construction for the commutative rectangular determinant polynomial of the $ktimes n$ symbolic matrix. 3. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is $W[1]$-hard.
The information geometry of the 2-manifold of gamma probability density functions provides a framework in which pseudorandom number generators may be evaluated using a neighbourhood of the curve of exponential density functions. The process is illustrated using the pseudorandom number generator in Mathematica. This methodology may be useful to add to the current family of test procedures in real applications to finite sampling data.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
