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Electronic Colloquium on Computational Complexity

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All reports by Author Mahdi Cheraghchi:

TR15-076 | 28th April 2015
Mahdi Cheraghchi, Piotr Indyk

Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform

For every fixed constant $\alpha > 0$, we design an algorithm for computing the $k$-sparse Walsh-Hadamard transform of an $N$-dimensional vector $x \in \mathbb{R}^N$ in time $k^{1+\alpha} (\log N)^{O(1)}$. Specifically, the algorithm is given query access to $x$ and computes a $k$-sparse $\tilde{x} \in \mathbb{R}^N$ satisfying $\|\tilde{x} - \hat{x}\|_1 \leq ... more >>>

TR15-030 | 6th March 2015
Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, Ning Xie

${\mathrm{AC}^{0} \circ \mathrm{MOD}_2}$ lower bounds for the Boolean Inner Product

Revisions: 1

$\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuits are $\mathrm{AC}^{0}$ circuits augmented with a layer of parity gates just above the input layer. We study the $\mathrm{AC}^{0} \circ \mathrm{MOD}_2$ circuit lower bound for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have ... more >>>

TR13-121 | 4th September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Non-Malleable Coding Against Bit-wise and Split-State Tampering

Revisions: 1

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>

TR13-118 | 2nd September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Capacity of Non-Malleable Codes

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly ... more >>>

TR12-172 | 8th December 2012
Mahdi Cheraghchi, Anna Gal, Andrew Mills

Correctness and Corruption of Locally Decodable Codes

Locally decodable codes (LDCs) are error correcting codes with the extra property that it is sufficient to read just a small number of positions of a possibly corrupted codeword in order to recover any one position of the input. To achieve this, it is necessary to use randomness in the ... more >>>

TR12-082 | 28th June 2012
Mahdi Cheraghchi, Venkatesan Guruswami, Ameya Velingker

Restricted Isometry of Fourier Matrices and List Decodability of Random Linear Codes

We prove that a random linear code over $\mathbb{F}_q$, with probability arbitrarily close to $1$, is list decodable at radius $1-1/q-\epsilon$ with list size $L=O(1/\epsilon^2)$ and rate $R=\Omega_q(\epsilon^2/(\log^3(1/\epsilon)))$. Up to the polylogarithmic factor in $1/\epsilon$ and constant factors depending on $q$, this matches the lower bound $L=\Omega_q(1/\epsilon^2)$ for the list ... more >>>

TR11-090 | 2nd June 2011
Mahdi Cheraghchi, Adam Klivans, Pravesh Kothari, Homin Lee

Submodular Functions Are Noise Stable

Revisions: 2

We show that all non-negative submodular functions have high noise-stability. As a consequence, we obtain a polynomial-time learning algorithm for this class with respect to any product distribution on $\{-1,1\}^n$ (for any constant accuracy parameter $\epsilon$ ). Our algorithm also succeeds in the agnostic setting. Previous work on learning submodular ... more >>>

TR10-132 | 18th August 2010
Mahdi Cheraghchi, Johan Hastad, Marcus Isaksson, Ola Svensson

Approximating Linear Threshold Predicates

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

TR05-070 | 6th July 2005
Mahdi Cheraghchi

On Matrix Rigidity and the Complexity of Linear Forms

The rigidity function of a matrix is defined as the minimum number of its entries that need to be changed in order to reduce the rank of the matrix to below a given parameter. Proving a strong enough lower bound on the rigidity of a matrix implies a nontrivial lower ... more >>>

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