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Electronic Colloquium on Computational Complexity

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All reports by Author Pooya Hatami:

TR21-066 | 5th May 2021
Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami

Dimension-free Bounds and Structural Results in Communication Complexity

The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to ... more >>>

TR21-002 | 8th January 2021
Pooya Hatami, William Hoza, Avishay Tal, Roei Tell

Fooling Constant-Depth Threshold Circuits

Revisions: 1

We present new constructions of pseudorandom generators (PRGs) for two of the most widely-studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth $d\in\mathbb{N}$ ... more >>>

TR19-149 | 4th November 2019
Dean Doron, Pooya Hatami, William Hoza

Log-Seed Pseudorandom Generators via Iterated Restrictions

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The ``iterated restrictions'' approach, pioneered by Ajtai and Wigderson [AW89], has provided PRGs with seed length $\mathrm{polylog} n$ or even $\tilde{O}(\log n)$ for several restricted models of computation. Can this approach ever achieve the optimal seed ... more >>>

TR19-145 | 31st October 2019
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, David Zuckerman

XOR Lemmas for Resilient Functions Against Polynomials

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over $F_2$. We introduce a new technique to prove such correlation bounds with $F_2$ polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In ... more >>>

TR19-029 | 20th February 2019
Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, David Zuckerman

Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of *any* Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely ... more >>>

TR18-183 | 5th November 2018
Dean Doron, Pooya Hatami, William Hoza

Near-Optimal Pseudorandom Generators for Constant-Depth Read-Once Formulas

Revisions: 2

We give an explicit pseudorandom generator (PRG) for constant-depth read-once formulas over the basis $\{\wedge, \vee, \neg\}$ with unbounded fan-in. The seed length of our PRG is $\widetilde{O}(\log(n/\varepsilon))$. Previously, PRGs with near-optimal seed length were known only for the depth-2 case (Gopalan et al. FOCS '12). For a constant depth ... more >>>

TR18-155 | 8th September 2018
Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, Avishay Tal

Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design ... more >>>

TR18-081 | 20th April 2018
Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar

On Multilinear Forms: Bias, Correlation, and Tensor Rank

Revisions: 1

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

TR18-015 | 25th January 2018
Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

Pseudorandom Generators from Polarizing Random Walks

Revisions: 1 , Comments: 1

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that ... more >>>

TR17-171 | 6th November 2017
Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal

Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

Revisions: 1

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.

A central ingredient in our work ... more >>>

TR17-025 | 16th February 2017
Pooya Hatami, Avishay Tal

Pseudorandom Generators for Low-Sensitivity Functions

A Boolean function is said to have maximal sensitivity $s$ if $s$ is the largest number of Hamming neighbors of a point which differ from it in function value. We construct a pseudorandom generator with seed-length $2^{O(\sqrt{s})} \cdot \log(n)$ that fools Boolean functions on $n$ variables with maximal sensitivity at ... more >>>

TR14-040 | 30th March 2014
Hamed Hatami, Pooya Hatami, Shachar Lovett

General systems of linear forms: equidistribution and true complexity

Revisions: 1

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate ... more >>>

TR12-184 | 26th December 2012
Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar Lovett

Every locally characterized affine-invariant property is testable.

Revisions: 1

Let $\mathbb{F} = \mathbb{F}_p$ for any fixed prime $p \geq 2$. An affine-invariant property is a property of functions on $\mathbb{F}^n$ that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for ... more >>>

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