arXiv subjects

Swarnalipa Datta

Publications and source records attributed to Swarnalipa Datta.

4 recordsLinked to original sources

arXiv2026

Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing

In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g : \mathbb{F}_2^n \to \{-1, +1\}$, Alice and Bob are tasked with determining whether $f$ and $g$ are equivalent up to a nonsingular linear transformation of the input variables, or far from being so. This problem has been extensively investigated in several models of computation, including standard algorithmic and property testing frameworks, owing to its fundamental connections with combinatorial circuit design, complexity theory, and cryptography. However, despite its broad relevance, it has remained unexplored in the context of communication complexity, a gap we address in this work. Our main results demonstrate that the approximate spectral norm of the input functions plays a central role in governing the communication complexity of this problem. We design a simple deterministic protocol whose communication cost is polynomial in the approximate spectral norm, and complement it with nearly matching lower bounds (up to a quadratic gap). In the randomised setting with private coins, we present an even more efficient protocol, though equally simple, that achieves a quadratically improved dependence on the approximate spectral norm compared to the deterministic case, and we prove that such a dependence is essentially unavoidable. These results identify the approximate spectral norm as a key complexity measure for testing linear invariance in the communication complexity framework. As a core technical ingredient, we establish new junta theorems for Boolean functions with small approximate spectral norm, which may be of independent interest in Fourier analysis and learning theory.

cs.DS↗

arXiv2025

Testing Isomorphism of Boolean Functions over Finite Abelian Groups

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $ε\geq 0$ and $τ> 0$, we seek to determine, with minimal queries, whether there exists an automorphism $σ$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ σ$ and $g$ is at most $ε$, or whether for all automorphisms $σ$, the distance is at least $ε+ τ$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/τ\right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.

cs.CC↗

arXiv2024

Structure of sparse Boolean functions over Abelian groups, and its application to testing

We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of Fourier coefficients and prove that every non-zero coefficient of an $s$-sparse Boolean function has magnitude at least \begin{equation*} \frac{1}{2^{\varphi(\Delta)/2} \, s^{\varphi(\Delta)/2}}, \end{equation*} where $\Delta$ denotes the exponent of the group $G$ (that is, the maximum order of an element in $G$) and $\varphi$ is the Euler's totient function. This generalizes the celebrated result of Gopalan et al. (SICOMP 2011) for $\mathbb{Z}_2^n$, extending it to all finite Abelian groups via new techniques from group theory and algebraic number theory. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean functions. The tester distinguishes whether a given function is $s$-sparse or $\epsilon$-far from every $s$-sparse Boolean function, with query complexity $poly\left((2s)^{\varphi(\Delta)},1/\epsilon \right)$. In addition, we generalize the classical notion of Boolean degree to arbitrary Abelian groups and establish an $\Omega(\sqrt{s})$ lower bound for adaptive sparsity testing.

cs.CC↗

arXiv2023

Spectral Norm, Economical Sieve, and Linear Invariance Testing of Boolean Functions

Given Boolean functions $ f, g : \mathbb{F}_2^n \to \{-1,+1\} $, we say they are {\em linearly isomorphic} if there exists $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ such that $ f(x)=g(Ax) $ for all $ x $. We study this problem in the tolerant property testing framework under the known--unknown model, where $ g $ is given explicitly and $ f $ is accessible only via oracle queries, meaning the algorithm may adaptively request the value of $ f(x) $ for inputs $ x \in \mathbb{F}_2^n $ of its choice. Given parameters $ \epsilon \ge 0 $ and $ \omega>0 $, the goal is to distinguish whether there exists $ A \in \mathrm{GL}_n(\mathbb{F}_{2})$ such that the normalized Hamming distance between $ f $ and $ g(Ax) $ is at most $ \epsilon $, or whether for every $ A \in \mathrm{GL}_n(\mathbb{F}_2) $ the distance is at least $ \epsilon+\omega $. Our main result is a tolerant tester making $ \widetilde{O} \left( \left( m/\omega \right)^4 \right) $ queries to $ f $, where $ m $ is an upper bound on the spectral norm of $ g $, improving the previous $ \widetilde{O} \left( \left( m/\omega \right)^{24} \right) $ bound of Wimmer and Yoshida. We complement this with a nearly matching lower bound of $ \Omega(m^2) $ for constant $ \omega $ (for example, $ \omega=1/4 $), improving the prior $ \Omega(\log m) $ lower bound of Grigorescu, Wimmer and Xie. A key technical ingredient on the algorithmic side is a query-efficient local list corrector. For the lower bound, we give a reduction from communication complexity using a novel subclass of Maiorana--McFarland functions from symmetric-key cryptography.

cs.CC↗