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Electronic Colloquium on Computational Complexity

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All reports by Author Chandan Saha:

TR21-015 | 15th February 2021
Chandan Saha, Bhargav Thankey

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an $n$-variate polynomial $f(\mathbf{x})$ over a field $\mathbb{F}$ is the set $\mathrm{orb}(f) := \{f(A\mathbf{x}+\mathbf{b}) : A \in \mathrm{GL}(n,\mathbb{F}) \ \mathrm{and} \ \mathbf{b} \in \mathbb{F}^n\}$. This paper studies explicit hitting sets for the orbits of polynomials computable by certain well-studied circuit classes. This version of the hitting set ... more >>>

TR20-091 | 14th June 2020
Janaky Murthy, vineet nair, Chandan Saha

Randomized polynomial-time equivalence between determinant and trace-IMM equivalence tests

Equivalence testing for a polynomial family $\{g_m\}_{m \in \mathbb{N}}$ over a field F is the following problem: Given black-box access to an $n$-variate polynomial $f(\mathbb{x})$, where $n$ is the number of variables in $g_m$ for some $m \in \mathbb{N}$, check if there exists an $A \in \text{GL}(n,\text{F})$ such that $f(\mathbb{x}) ... more >>>

TR20-045 | 15th April 2020
Ankit Garg, Neeraj Kayal, Chandan Saha

Learning sums of powers of low-degree polynomials in the non-degenerate case

Revisions: 1

We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an $n$-variate degree-$d$ polynomial $f$ which can be written as
$$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$
where each $c_i\in \mathbb{F}^{\times}$, $Q_i$ is a homogeneous polynomial of degree $t$, and $t m = ... more >>>

TR20-028 | 27th February 2020
Nikhil Gupta, Chandan Saha, Bhargav Thankey

A Super-Quadratic Lower Bound for Depth Four Arithmetic Circuits

We show an $\widetilde{\Omega}(n^{2.5})$ lower bound for general depth four arithmetic circuits computing an explicit $n$-variate degree $\Theta(n)$ multilinear polynomial over any field of characteristic zero. To our knowledge, and as stated in the survey by Shpilka and Yehudayoff (FnT-TCS, 2010), no super-quadratic lower bound was known for depth four ... more >>>

TR19-042 | 18th March 2019
Ankit Garg, Nikhil Gupta, Neeraj Kayal, Chandan Saha

Determinant equivalence test over finite fields and over $\mathbf{Q}$

The determinant polynomial $Det_n(\mathbf{x})$ of degree $n$ is the determinant of a $n \times n$ matrix of formal variables. A polynomial $f$ is equivalent to $Det_n$ over a field $\mathbf{F}$ if there exists a $A \in GL(n^2,\mathbf{F})$ such that $f = Det_n(A \cdot \mathbf{x})$. Determinant equivalence test over $\mathbf{F}$ is ... more >>>

TR18-191 | 10th November 2018
Neeraj Kayal, Chandan Saha

Reconstruction of non-degenerate homogeneous depth three circuits

A homogeneous depth three circuit $C$ computes a polynomial
$$f = T_1 + T_2 + ... + T_s ,$$ where each $T_i$ is a product of $d$ linear forms in $n$ variables over some underlying field $\mathbb{F}$. Given black-box access to $f$, can we efficiently reconstruct (i.e. proper learn) a ... more >>>

TR18-164 | 18th September 2018
Nikhil Gupta, Chandan Saha

On the symmetries of design polynomials

Revisions: 1

In a Nisan-Wigderson design polynomial (in short, a design polynomial), the gcd of every pair of monomials has a low degree. A useful example of such a polynomial is the following:
$$\text{NW}_{d,k}(\mathbf{x}) = \sum_{h \in \mathbb{F}_d[z], ~\deg(h) \leq k}{~~~~\prod_{i = 0}^{d-1}{x_{i, h(i)}}},$$
where $d$ is a prime, $\mathbb{F}_d$ is the ... more >>>

TR18-029 | 9th February 2018
Neeraj Kayal, vineet nair, Chandan Saha

Average-case linear matrix factorization and reconstruction of low width Algebraic Branching Programs

Revisions: 2

Let us call a matrix $X$ as a linear matrix if its entries are affine forms, i.e. degree one polynomials. What is a minimal-sized representation of a given matrix $F$ as a product of linear matrices? Finding such a minimal representation is closely related to finding an optimal way to ... more >>>

TR17-021 | 11th February 2017
Neeraj Kayal, Vineet Nair, Chandan Saha, Sébastien Tavenas

Reconstruction of full rank Algebraic Branching Programs

An algebraic branching program (ABP) A can be modelled as a product expression $X_1\cdot X_2\cdot \dots \cdot X_d$, where $X_1$ and $X_d$ are $1 \times w$ and $w \times 1$ matrices respectively, and every other $X_k$ is a $w \times w$ matrix; the entries of these matrices are linear forms ... more >>>

TR16-006 | 22nd January 2016
Neeraj Kayal, Chandan Saha, Sébastien Tavenas

An almost Cubic Lower Bound for Depth Three Arithmetic Circuits

Revisions: 2

We show an $\Omega \left(\frac{n^3}{(\ln n)^2}\right)$ lower bound on the size of any depth three ($\SPS$) arithmetic circuit computing an explicit multilinear polynomial in $n$ variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson.

more >>>

TR15-181 | 13th November 2015
Neeraj Kayal, Chandan Saha, Sébastien Tavenas

On the size of homogeneous and of depth four formulas with low individual degree

Let $r \geq 1$ be an integer. Let us call a polynomial $f(x_1, x_2,\ldots, x_N) \in \mathbb{F}[\mathbf{x}]$ as a multi-$r$-ic polynomial if the degree of $f$ with respect to any variable is at most $r$ (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output ... more >>>

TR15-154 | 22nd September 2015
Neeraj Kayal, Vineet Nair, Chandan Saha

Separation between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

We show an exponential separation between two well-studied models of algebraic computation, namely read-once oblivious algebraic branching programs (ROABPs) and multilinear depth three circuits. In particular we show the following:

1. There exists an explicit $n$-variate polynomial computable by linear sized multilinear depth three circuits (with only two product gates) ... more >>>

TR15-073 | 25th April 2015
Neeraj Kayal, Chandan Saha

Lower Bounds for Sums of Products of Low arity Polynomials

We prove an exponential lower bound for expressing a polynomial as a sum of product of low arity polynomials. Specifically, we show that for the iterated matrix multiplication polynomial, $IMM_{d, n}$ (corresponding to the product of $d$ matrices of size $n \times n$ each), any expression of the form
more >>>

TR15-015 | 30th January 2015
Neeraj Kayal, Chandan Saha

Multi-$k$-ic depth three circuit lower bound

In a multi-$k$-ic depth three circuit every variable appears in at most $k$ of the linear polynomials in every product gate of the circuit. This model is a natural generalization of multilinear depth three circuits that allows the formal degree of the circuit to exceed the number of underlying variables ... more >>>

TR14-089 | 16th July 2014
Neeraj Kayal, Chandan Saha

Lower Bounds for Depth Three Arithmetic Circuits with small bottom fanin

Revisions: 1

Shpilka and Wigderson (CCC 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field $\mathbb{F}$ of characteristic zero. We resolve this problem by proving a $N^{\Omega(\frac{d}{\tau})}$ lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin ... more >>>

TR14-005 | 14th January 2014
Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas. That is, we give
an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with $0, 1$-coefficients such that
for any representation of ... more >>>

TR13-091 | 17th June 2013
Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

A super-polynomial lower bound for regular arithmetic formulas.

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>

TR12-113 | 7th September 2012
Manindra Agrawal, Chandan Saha, Nitin Saxena

Quasi-polynomial Hitting-set for Set-depth-$\Delta$ Formulas

We call a depth-$4$ formula $C$ $\textit{ set-depth-4}$ if there exists a (unknown) partition $X_1\sqcup\cdots\sqcup X_d$ of the variable indices $[n]$ that the top product layer respects, i.e. $C(\mathbf{x})=\sum_{i=1}^k {\prod_{j=1}^{d} {f_{i,j}(\mathbf{x}_{X_j})}}$ $ ,$ where $f_{i,j}$ is a $\textit{sparse}$ polynomial in $\mathbb{F}[\mathbf{x}_{X_j}]$. Extending this definition to any depth - we call ... more >>>

TR11-143 | 2nd November 2011
Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied ... more >>>

TR11-021 | 13th February 2011
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

Finding an efficient solution to the general problem of polynomial identity testing (PIT) is a challenging task. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-$3$ PIT, the other of depth-$4$ PIT.

Our first problem is ... more >>>

TR10-189 | 8th December 2010
Neeraj Kayal, Chandan Saha

On the Sum of Square Roots of Polynomials and related problems

The sum of square roots problem over integers is the task of deciding the sign of a nonzero sum, $S = \Sigma_{i=1}^{n}{\delta_i}$ . \sqrt{$a_i$}, where $\delta_i \in$ { +1, -1} and $a_i$'s are positive integers that are upper bounded by $N$ (say). A fundamental open question in numerical analysis and ... more >>>

TR09-036 | 14th April 2009
Chandan Saha, Ramprasad Saptharishi, Nitin Saxena

The Power of Depth 2 Circuits over Algebras

We study the problem of polynomial identity testing (PIT) for depth
2 arithmetic circuits over matrix algebra. We show that identity
testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field
F is polynomial time equivalent to identity testing of depth 2
(Pi-Sigma) arithmetic circuits over U_2(F), the ... 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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