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REPORTS > AUTHORS > ANINDYA DE:
All reports by Author Anindya De:

TR21-005 | 13th January 2021
Anindya De, Elchanan Mossel, Joe Neeman

Robust testing of low-dimensional functions

A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. ... more >>>


TR18-100 | 18th May 2018
Eshan Chattopadhyay, Anindya De, Rocco Servedio

Simple and efficient pseudorandom generators from Gaussian processes

Revisions: 1

We show that a very simple pseudorandom generator fools intersections of $k$ linear threshold functions (LTFs) and arbitrary functions of $k$ LTFs over $n$-dimensional Gaussian space.

The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of ... more >>>


TR16-026 | 20th February 2016
Anindya De, Michael Saks, Sijian Tang

Noisy population recovery in polynomial time

In the noisy population recovery problem of Dvir et al., the goal is to learn
an unknown distribution $f$ on binary strings of length $n$ from noisy samples. For some parameter $\mu \in [0,1]$,
a noisy sample is generated by flipping each coordinate of a sample from $f$ independently with
more >>>


TR14-125 | 9th October 2014
Anindya De

Beyond the Central Limit Theorem: asymptotic expansions and pseudorandomness for combinatorial sums

In this paper, we construct pseudorandom generators for the class of \emph{combinatorial sums}, a class of functions first studied by \cite{GMRZ13}
and defined as follows: A function $f: [m]^n \rightarrow \{0,1\}$ is said to be a combinatorial sum if there exists functions $f_1, \ldots, f_n: [m] \rightarrow \{0,1\}$ such that
more >>>


TR13-173 | 28th November 2013
Anindya De, Rocco Servedio

Efficient deterministic approximate counting for low degree polynomial threshold functions

We give a deterministic algorithm for
approximately counting satisfying assignments of a degree-$d$ polynomial threshold function
(PTF).
Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $\mathbb{R}^n$
and a parameter $\epsilon > 0$, our algorithm approximates
$
\mathbf{P}_{x \sim \{-1,1\}^n}[p(x) \geq 0]
$
to within an additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot ... more >>>


TR13-172 | 1st December 2013
Anindya De, Ilias Diakonikolas, Rocco Servedio

Deterministic Approximate Counting for Degree-$2$ Polynomial Threshold Functions

We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps > 0$, the algorithm approximates
\[
\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]
\]
to within an additive $\pm \eps$ in ... more >>>


TR13-171 | 1st December 2013
Anindya De, Ilias Diakonikolas, Rocco Servedio

Deterministic Approximate Counting for Juntas of Degree-$2$ Polynomial Threshold Functions

Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[
\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1] \]
to within an additive $\pm \eps$. For any constant ... more >>>


TR12-181 | 20th December 2012
Anindya De, Ilias Diakonikolas, Rocco Servedio

The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley ... more >>>


TR12-152 | 7th November 2012
Anindya De, Ilias Diakonikolas, Rocco Servedio

Inverse Problems in Approximate Uniform Generation

We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform random satisfying assignments of an unknown function $f$ belonging to a class $\C$ of Boolean functions ... more >>>


TR12-072 | 5th June 2012
Anindya De, Ilias Diakonikolas, Vitaly Feldman, Rocco Servedio

Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 \cite{Chow:61, Tannenbaum:61} that the (exact values of the) Chow parameters of any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean ... more >>>


TR12-016 | 24th February 2012
Anindya De, Elchanan Mossel

Explicit Optimal hardness via Gaussian stability results

Revisions: 3

The results of Raghavendra (2008) show that assuming Khot's Unique Games Conjecture (2002), for every constraint satisfaction problem there exists a generic semi-definite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate ... more >>>


TR11-037 | 18th March 2011
Anindya De, Thomas Watson

Extractors and Lower Bounds for Locally Samplable Sources

Revisions: 3

We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number $d$ of the random input bits. As our main result, we construct a deterministic extractor that, given any $d$-local source with ... more >>>


TR09-141 | 19th December 2009
Anindya De, Omid Etesami, Luca Trevisan, Madhur Tulsiani

Improved Pseudorandom Generators for Depth 2 Circuits

We prove the existence of a $poly(n,m)$-time computable
pseudorandom generator which ``$1/poly(n,m)$-fools'' DNFs with $n$ variables
and $m$ terms, and has seed length $O(\log^2 nm \cdot \log\log nm)$.
Previously, the best pseudorandom generator for depth-2 circuits had seed
length $O(\log^3 nm)$, and was due to Bazzi (FOCS 2007).

It ... more >>>


TR09-133 | 9th December 2009
Anindya De, Thomas Vidick

Near-optimal extractors against quantum storage

We give near-optimal constructions of extractors secure against quantum bounded-storage adversaries. One instantiation gives the first such extractor to achieve an output length Theta(K-b), where K is the source's entropy and b the adversary's storage, depending linearly on the adversary's amount of storage, together with a poly-logarithmic seed length. Another ... more >>>


TR09-113 | 9th November 2009
Anindya De, Luca Trevisan, Madhur Tulsiani

Non-uniform attacks against one-way functions and PRGs

We study the power of non-uniform attacks against one-way
functions and pseudorandom generators.

Fiat and Naor show that for every function
$f: [N]\to [N]$
there is an algorithm that inverts $f$ everywhere using
(ignoring lower order factors)
time, space and advice at most $N^{3/4}$.

We show that ... more >>>


TR08-023 | 10th January 2008
Anindya De, Piyush Kurur, Chandan Saha, Ramprasad Saptharishi

Fast Integer Multiplication using Modular Arithmetic

We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for
multiplying two $N$-bit integers that improves the $O(N\cdot \log
N\cdot \log\log N)$ algorithm by
Sch\"{o}nhage-Strassen. Both these algorithms use modular
arithmetic. Recently, F\"{u}rer gave an $O(N\cdot \log
N\cdot 2^{O(\log^*N)})$ algorithm which however uses arithmetic over
complex numbers as opposed to ... more >>>




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