arXiv subjects

Tushant Mittal

Publications and source records attributed to Tushant Mittal.

10 recordsLinked to original sources

arXiv2026

Low Soundness Linearity Testing on the Half-Slice

Let $f: T\to \{ 0,1 \}$ be a Boolean function on the Boolean half-slice, $T$, \ie elements of $\{0,1\}^n$ with Hamming weight $n/2$. We show that if $f(x)+f(y)=f(x+y)$ holds with probability $\frac{1+\delta}{2}$ over a uniform pair $(x,y)$ such that $x,y,x+y\in T$, then $f$ agrees with some linear function on at least $\frac{1+\delta}{2}-o(1)$ fraction of the points in $T$. More generally, we show that if $f$ passes the natural $k$-query BLR test with probability $\frac{1+\delta}{2}$ for any $k\geq3$, then it must agree with some affine function at $\frac{1+\delta^{\frac{1}{k-2}}}{2}-o(1)$ fraction of the points in $T$. The only other known linearity test for the slice in the low soundness regime (i.e., when $\delta$ can be arbitrarily small) was given by Kalai, Lifshitz, Minzer, and Ziegler [FOCS'24]. Our result improves upon this result in two significant ways: firstly, it works for $k=3$ queries, instead of requiring $k\geq4$; secondly, our result is sharper, e.g., when $k=4$, we are able to conclude an agreement of $\frac{1+\sqrt{\delta}}{2}-o(1)$ instead of $\frac{1+c\sqrt{\delta}}{2}$ for $c\approx.0035$. In particular, our result matches (up to the $o(1)$ term) the conclusion one obtains over the full hypercube via the classical BLR analysis. Our main technical contribution is a new dense model theorem using bounds on Krawtchouk polynomials. Using these Krawtchouk polynomial bounds, we also obtain a simple $k$-query test ($k\geq 5$) that avoids any use of the dense model machinery. This simplified test naturally extends to the slice over the $q$-ary hypercube, giving the first such result over larger alphabets.

cs.CC↗

arXiv2025

A General Framework for Low Soundness Homomorphism Testing

We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and $\mathbb{Z}_p$, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of $\mathrm{GL}_n(\mathbb{F}_q)$, and finite-dimensional Lie algebras over $\mathbb{F}_q$. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between $\mathbb{F}_q^n$ and $\mathbb{F}_q$. Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as $\mathbb{F}_q^n$). No tests were known for non-abelian groups. As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of $O(\varepsilon^{-2})$ (for agreement parameter $\varepsilon$). This improves upon the currently best-known bound of $O(\varepsilon^{-105})$ due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].

cs.CC↗

arXiv2025

Pseudorandomness of Expander Walks via Fourier Analysis on Groups

One approach to study the pseudorandomness properties of walks on expander graphs is to label the vertices of an expander with elements from an alphabet $\Sigma$, and study the mean of functions over $\Sigma^n$. We say expander walks $\varepsilon$-fool a function if, for any unbiased labeling of the vertices, the expander walk mean is $\varepsilon$-close to the true mean. We show that: - The class of symmetric functions is $O(|\Sigma|\cdot\lambda)$-fooled by expander walks over any generic $\lambda$-expander, and any alphabet $\Sigma$ . This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for $|\Sigma| =2$, and exponentially improves the previous bound of $O(|\Sigma|^{O(|\Sigma|)}\cdot \lambda)$, by Golowich and Vadhan [CCC'22]. Additionally, if the expander is a Cayley graph over $\mathbb{Z}_{|\Sigma|}$, we get a further improved bound of $O(\sqrt{|\Sigma|}\cdot\lambda)$. Morever, when $\Sigma$ is a finite group $G$, we show the following for functions over $G^n$: - The class of symmetric class functions is $O\Big({\frac{\sqrt{|G|}}{D}\cdot\lambda}\Big)$-fooled by expander walks over "structured" $\lambda$-expanders, if $G$ is $D$-quasirandom. - We show a lower bound of $\Omega(\lambda)$ for symmetric functions for any finite group $G$ (even for "structured" $\lambda$-expanders). - We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks. Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of $\mathbb{Z}_2$ or $\mathbb{Z}$. This enables us to get quantitatively better bounds even for unstructured sets.

cs.CC↗

arXiv2025

Explicit Codes approaching Generalized Singleton Bound using Expanders

We construct a new family of explicit codes that are list decodable to capacity and achieve an optimal list size of $O(\frac{1}{\epsilon})$. In contrast to existing explicit constructions of codes achieving list decoding capacity, our arguments do not rely on algebraic structure but utilize simple combinatorial properties of expander graphs. Our construction is based on a celebrated distance amplification procedure due to Alon, Edmonds, and Luby [FOCS'95], which transforms any high-rate code into one with near-optimal rate-distance tradeoff. We generalize it to show that the same procedure can be used to transform any high-rate code into one that achieves list decoding capacity. Our proof can be interpreted as a "local-to-global" phenomenon for (a slight strengthening of) the generalized Singleton bound. Using this construction, for every $R, \epsilon \in (0,1)$ and $k \in \mathbb{N}^+$, we obtain an \emph{explicit} family of codes $\mathcal{C} \subseteq \Sigma^n$, with rate $R$ such that, - They achieve the $\epsilon$-relaxed generalized Singleton bound: for any $g \in \Sigma^n$ and any list $\mathcal{H}$ of at most $k$ codewords, we have, \[ \underset{h \in \mathcal{H}}{\mathbb{E}} [\Delta(g,h)] ~\geq~ \frac{|\mathcal{H}|-1}{|\mathcal{H}|} \cdot (1 - R - \epsilon). \] - The alphabet size is a constant depending only on $\epsilon$ and $k$. - They can be list decoded up to radius $\frac{k-1}{k}(1-R-\epsilon)$, in time $n^{O_{k,\epsilon}(1)}$. As a corollary of our result, we also obtain the first explicit construction of LDPC codes achieving list decoding capacity, and in fact arbitrarily close to the generalized Singleton bound.

cs.IT↗

arXiv2024

List Decodable Quantum LDPC Codes

We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.

cs.IT↗

arXiv2024

Derandomized Non-Abelian Homomorphism Testing in Low Soundness Regime

We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized Blum--Luby--Rubinfeld (BLR) test (using small-bias sets), then (i) it correlates with a function arising from a genuine homomorphism, and (ii) it has a non-trivial Fourier mass on a low-dimensional irreducible representation. In the full randomness regime, such a test for matrix-valued functions on finite groups implicitly appears in the works of Gowers and Hatami [Sbornik: Mathematics '17], and Moore and Russell [SIAM Journal on Discrete Mathematics '15]. Thus, our work can be seen as a near-optimal derandomization of their results. Our key technical contribution is a "degree-2 expander mixing lemma'' that shows that Gowers' $\mathrm{U}^2$ norm can be efficiently estimated by restricting it to a small-bias subset. Another corollary is a "derandomized'' version of a useful lemma due to Babai, Nikolov, and Pyber [SODA'08] and Gowers [Comb. Probab. Comput.'08].

cs.CC↗

arXiv2022

Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification

We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, $d \leq 1/\lambda^{2+o(1)}$) trade-off between (any desired) spectral expansion $\lambda$ and degree $d$. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius $O(\log(1/\lambda))$. The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert--Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.

cs.DS↗

arXiv2021

Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras

One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).

cs.CC↗

arXiv2021

Explicit Abelian Lifts and Quantum LDPC Codes

For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and each edge by a matching corresponding to the action of an element of $H$. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group $H \leqslant \text{Sym}(\ell)$, constant degree $d \ge 3$ and $ε> 0$, we construct explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a (suitable) base $n$-vertex expander $G_0$ with the following parameters: (i) $λ(G) \le 2\sqrt{d-1} + ε$, for any lift size $\ell \le 2^{n^δ}$ where $δ=δ(d,ε)$, (ii) $λ(G) \le ε\cdot d$, for any lift size $\ell \le 2^{n^{δ_0}}$ for a fixed $δ_0 > 0$, when $d \ge d_0(ε)$, or (iii) $λ(G) \le \widetilde{O}(\sqrt{d})$, for lift size ``exactly'' $\ell = 2^{Θ(n)}$. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items $(i)$ and $(ii)$ above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for $2$-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing'" depth-first search traversals. Result $(iii)$ is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.

cs.DS↗

arXiv2017

The Mahler measure for arbitrary tori

We consider a variation of the Mahler measure where the defining integral is performed over a more general torus. We focus our investigation on two particular polynomials related to certain elliptic curve $E$ and we establish new formulas for this variation of the Mahler measure in terms of $L'(E,0)$.

math.NT↗