arXiv subjects

Madhur Tulsiani

Publications and source records attributed to Madhur Tulsiani.

At least 19 recordsLinked to original sources

arXiv2026

Sharp Phase Transition for Ellipsoid Fitting

We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for $m$ independent Gaussian points in dimension $d$, we show that with high probability, for $m \leq (1-o_d(1)) \cdot d^2/4$, there exists a centered ellipsoid passing through all $m$ points; for $m\geq (1+o_d(1) )\cdot d^2/4$, no such ellipsoid exists. This confirms that the ellipsoid fitting problem has a sharp phase transition at $d^2/4$.

math.PR↗

arXiv2026

Optimal Single-Pass Streaming Lower Bounds for Approximating CSPs

For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $\epsilon > 0$, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP$({\mathcal{F}})$ with at most $\beta+\epsilon$ fraction of satisfiable constraints from instances of with at least $\gamma-\epsilon$ fraction of satisfiable constraints, whenever Max-CSP$({\mathcal{F}})$ admits a $(\gamma,\beta)$-integrality gap instance for the basic LP. This subsumes the linear-space lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC 2022), which applies only to a special subclass of CSPs with linear-algebraic structure. (Their result itself generalizes work of Kapralov and Krachun (STOC 2019) for Max-CUT.) Our approach identifies the right ``analytic'' analogues of previously-used linear-algebraic conditions; this yields substantial simplifications while capturing a much larger class of problems. Our lower bound is essentially optimal for single-pass streaming, since: (1) All CSPs admit $(1-\epsilon)$-approximations in quasilinear space, and (2) sublinear-space streaming algorithms can simulate the LP (on bounded-degree instances), giving approximation algorithms when integrality gap instances do not exist. The starting point for our lower bound is a reduction from a "distributional implicit hidden partition'' problem defined by Fei, Minzer, and Wang (STOC 2026) in the context of multi-pass streaming. Our result is an analogue of theirs in the single-pass setting, where we obtain a much stronger (and tight) space lower bound.

cs.CC↗

arXiv2025

Sketching approximations and LP approximations for finite CSPs are related

We identify a connection between the approximability of CSPs in two models: (i) sublinear space streaming algorithms, and (ii) the basic LP relaxation. We show that whenever the basic LP admits an integrality gap, there is an $Ω(\sqrt{n})$-space sketching lower bound. We also show that all existing linear space streaming lower bounds for Max-CSPs can be lifted to integrality gap instances for basic LPs. For bounded-degree graphs, by combining the distributed algorithm of Yoshida (STOC 2011) for approximately solving the basic LP with techniques described in Saxena, Singer, Sudan, and Velusamy (SODA 2025) for simulating a distributed algorithm by a sublinear space streaming algorithm on bounded-degree instances of Max-DICUT, it appears that there are sublinear space streaming algorithms implementing the basic LP, for every CSP. Based on our results, we conjecture the following dichotomy theorem: Whenever the basic LP admits an integrality gap, there is a linear space single-pass streaming lower bound, and when the LP is roundable, there is a sublinear space streaming algorithm.

cs.CC↗

arXiv2025

List Decoding Expander-Based Codes up to Capacity in Near-Linear Time

We give a new framework based on graph regularity lemmas, for list decoding and list recovery of codes based on spectral expanders. Using existing algorithms for computing regularity decompositions of sparse graphs in (randomized) near-linear time, and appropriate choices for the constant-sized inner/base codes, we prove the following: - Expander-based codes constructed using the distance amplification technique of Alon, Edmonds and Luby [FOCS 1995] with rate $ρ$, can be list decoded to a radius $1 - ρ- ε$ in near-linear time. By known results, the output list has size $O(1/ε)$. - The above codes of Alon, Edmonds and Luby, with rate $ρ$, can also be list recovered to radius $1 - ρ- ε$ in near-linear time, with constant-sized output lists. - The Tanner code construction of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] with distance $δ$, can be list decoded to radius $δ- ε$ in near-linear time, with constant-sized output lists. Our results imply novel combinatorial as well as algorithmic bounds for each of the above explicit constructions. All of these bounds are obtained via combinatorial rigidity phenomena, proved using (weak) graph regularity. The regularity framework allows us to lift the list decoding and list recovery properties for the local base codes, to the global codes obtained via the above constructions.

cs.DS↗

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 $O(\frac{1}ε)$. 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 $R, ε\in (0,1)$ and $k \in \mathbb{N}^+$, we obtain an \emph{explicit} family of codes $\mathcal{C} \subseteq Σ^n$, with rate $R$ such that, - They achieve the $ε$-relaxed generalized Singleton bound: for any $g \in Σ^n$ and any list $\mathcal{H}$ of at most $k$ codewords, we have, \[ \underset{h \in \mathcal{H}}{\mathbb{E}} [Δ(g,h)] ~\geq~ \frac{|\mathcal{H}|-1}{|\mathcal{H}|} \cdot (1 - R - ε). \] - The alphabet size is a constant depending only on $ε$ and $k$. - They can be list decoded up to radius $\frac{k-1}{k}(1-R-ε)$, in time $n^{O_{k,ε}(1)}$. 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

Simple Norm Bounds for Polynomial Random Matrices via Decoupling

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

math.PR↗

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

Efficient Certificates of Anti-Concentration Beyond Gaussians

A set of high dimensional points $X=\{x_1, x_2,\ldots, x_n\} \subset R^d$ in isotropic position is said to be $δ$-anti concentrated if for every direction $v$, the fraction of points in $X$ satisfying $|\langle x_i,v \rangle |\leq δ$ is at most $O(δ)$. Motivated by applications to list-decodable learning and clustering, recent works have considered the problem of constructing efficient certificates of anti-concentration in the average case, when the set of points $X$ corresponds to samples from a Gaussian distribution. Their certificates played a crucial role in several subsequent works in algorithmic robust statistics on list-decodable learning and settling the robust learnability of arbitrary Gaussian mixtures, yet remain limited to rotationally invariant distributions. This work presents a new (and arguably the most natural) formulation for anti-concentration. Using this formulation, we give quasi-polynomial time verifiable sum-of-squares certificates of anti-concentration that hold for a wide class of non-Gaussian distributions including anti-concentrated bounded product distributions and uniform distributions over $L_p$ balls (and their affine transformations). Consequently, our method upgrades and extends results in algorithmic robust statistics e.g., list-decodable learning and clustering, to such distributions. Our approach constructs a canonical integer program for anti-concentration and analysis a sum-of-squares relaxation of it, independent of the intended application. We rely on duality and analyze a pseudo-expectation on large subsets of the input points that take a small value in some direction. Our analysis uses the method of polynomial reweightings to reduce the problem to analyzing only analytically dense or sparse directions.

cs.DS↗

arXiv2023

List Decoding of Tanner and Expander Amplified Codes from Distance Certificates

We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semidefinite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following: - We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] and Zémor [IEEE Trans. Inf. Theory 2001] with alphabet size $q$, block-length $n$ and distance $δ$, based on an expander graph with degree $d$, can be list-decoded up to distance $\mathcal{J}_q(δ) - ε$ in time $n^{O_{d,q}(1/ε^4)}$, where $\mathcal{J}_q(δ)$ denotes the Johnson bound. - We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code. In particular, starting from \emph{any} base code unique-decodable up to distance $δ$, one can obtain near-MDS codes with rate $R$ and distance $1-R - ε$, list-decodable up to the Johnson bound in time $n^{O_{ε, δ}(1)}$. - We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size $q$, block-length $n$ and distance $δ$ based on a square Cayley complex with generator sets of size $d$, can be list-decoded up to distance $\mathcal{J}_q(δ) - ε$ in time $n^{O_{d,q}(1/ε^{4})}$, where $\mathcal{J}_q(δ)$ denotes the Johnson bound.

cs.DS↗

arXiv2023

Ellipsoid fitting up to constant via empirical covariance estimation

The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and Willsky considers the maximum number $n$ random Gaussian points in $\mathbb{R}^d$, such that with high probability, there exists an origin-symmetric ellipsoid passing through all the points. They conjectured a threshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lower bounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. We give a simple proof based on concentration of sample covariance matrices, that with probability $1 - o_d(1)$, it is possible to fit an ellipsoid through $d^2/C$ random Gaussian points. Similar results were also obtained in two recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023] and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].

math.PR↗

arXiv2023

Concentration of polynomial random matrices via Efron-Stein inequalities

Analyzing concentration of large random matrices is a common task in a wide variety of fields. Given independent random variables, many tools are available to analyze random matrices whose entries are linear in the variables, e.g. the matrix-Bernstein inequality. However, in many applications, we need to analyze random matrices whose entries are polynomials in the variables. These arise naturally in the analysis of spectral algorithms, e.g., Hopkins et al. [STOC 2016], Moitra-Wein [STOC 2019]; and in lower bounds for semidefinite programs based on the Sum of Squares hierarchy, e.g. Barak et al. [FOCS 2016], Jones et al. [FOCS 2021]. In this work, we present a general framework to obtain such bounds, based on the matrix Efron-Stein inequalities developed by Paulin-Mackey-Tropp [Annals of Probability 2016]. The Efron-Stein inequality bounds the norm of a random matrix by the norm of another simpler (but still random) matrix, which we view as arising by "differentiating" the starting matrix. By recursively differentiating, our framework reduces the main task to analyzing far simpler matrices. For Rademacher variables, these simpler matrices are in fact deterministic and hence, analyzing them is far easier. For general non-Rademacher variables, the task reduces to scalar concentration, which is much easier. Moreover, in the setting of polynomial matrices, our results generalize the work of Paulin-Mackey-Tropp. Using our basic framework, we recover known bounds in the literature for simple "tensor networks" and "dense graph matrices". Using our general framework, we derive bounds for "sparse graph matrices", which were obtained only recently by Jones et al. [FOCS 2021] using a nontrivial application of the trace power method, and was a core component in their work. We expect our framework to be helpful for other applications involving concentration phenomena for nonlinear random matrices.

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↗

arXiv2021

Sum-of-Squares Lower Bounds for Sparse Independent Set

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the "dense" input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdos-Renyi random graph $G\sim G_{n,\frac{d}{n}}$ with average degree $d>\log^2 n$, degree-$D_{SoS}$ SoS fails to refute the existence of an independent set of size $k = Ω\left(\frac{n}{\sqrt{d}(\log n)(D_{SoS})^{c_0}} \right)$ in $G$ (where $c_0$ is an absolute constant), whereas the true size of the largest independent set in $G$ is $O\left(\frac{n\log d}{d}\right)$. Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting.

cs.CC↗

arXiv2020

List Decoding of Direct Sum Codes

We consider families of codes obtained by "lifting" a base code $\mathcal{C}$ through operations such as $k$-XOR applied to "local views" of codewords of $\mathcal{C}$, according to a suitable $k$-uniform hypergraph. The $k$-XOR operation yields the direct sum encoding used in works of [Ta-Shma, STOC 2017] and [Dinur and Kaufman, FOCS 2017]. We give a general framework for list decoding such lifted codes, as long as the base code admits a unique decoding algorithm, and the hypergraph used for lifting satisfies certain expansion properties. We show that these properties are satisfied by the collection of length $k$ walks on an expander graph, and by hypergraphs corresponding to high-dimensional expanders. Instantiating our framework, we obtain list decoding algorithms for direct sum liftings on the above hypergraph families. Using known connections between direct sum and direct product, we also recover the recent results of Dinur et al. [SODA 2019] on list decoding for direct product liftings. Our framework relies on relaxations given by the Sum-of-Squares (SOS) SDP hierarchy for solving various constraint satisfaction problems (CSPs). We view the problem of recovering the closest codeword to a given word, as finding the optimal solution of a CSP. Constraints in the instance correspond to edges of the lifting hypergraph, and the solutions are restricted to lie in the base code $\mathcal{C}$. We show that recent algorithms for (approximately) solving CSPs on certain expanding hypergraphs also yield a decoding algorithm for such lifted codes. We extend the framework to list decoding, by requiring the SOS solution to minimize a convex proxy for negative entropy. We show that this ensures a covering property for the SOS solution, and the "condition and round" approach used in several SOS algorithms can then be used to recover the required list of codewords.

cs.DS↗

arXiv2020

Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound

The Gilbert-Varshamov bound (non-constructively) establishes the existence of binary codes of distance $1/2 -ε$ and rate $Ω(ε^2)$ (where an upper bound of $O(ε^2\log(1/ε))$ is known). Ta-Shma [STOC 2017] gave an explicit construction of $ε$-balanced binary codes, where any two distinct codewords are at a distance between $1/2 -ε/2$ and $1/2+ε/2$, achieving a near optimal rate of $Ω(ε^{2+β})$, where $β\to 0$ as $ε\to 0$. We develop unique and list decoding algorithms for (essentially) the family of codes constructed by Ta-Shma. We prove the following results for $ε$-balanced codes with block length $N$ and rate $Ω(ε^{2+β})$ in this family: - For all $ε, β> 0$ there are explicit codes which can be uniquely decoded up to an error of half the minimum distance in time $N^{O_{ε, β}(1)}$. - For any fixed constant $β$ independent of $ε$, there is an explicit construction of codes which can be uniquely decoded up to an error of half the minimum distance in time $(\log(1/ε))^{O(1)} \cdot N^{O_β(1)}$. - For any $ε> 0$, there are explicit $ε$-balanced codes with rate $Ω(ε^{2+β})$ which can be list decoded up to error $1/2 - ε'$ in time $N^{O_{ε,ε',β}(1)}$, where $ε', β\to 0$ as $ε\to 0$. The starting point of our algorithms is the list decoding framework from Alev et al. [SODA 2020], which uses the Sum-of-Squares SDP hierarchy. The rates obtained there were quasipolynomial in $ε$. Here, we show how to overcome the far from optimal rates of this framework obtaining unique decoding algorithms for explicit binary codes of near optimal rate. These codes are based on simple modifications of Ta-Shma's construction.

cs.DS↗

arXiv2020

Explicit SoS lower bounds from high-dimensional expanders

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

cs.CC↗

arXiv2019

Approximating Operator Norms via Generalized Krivine Rounding

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^*$ operator norm of $A$. The case $p=r=\infty$ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary $p,r \ge 2$ with approximation ratio $(1+ε_0)/(\sinh^{-1}(1)\cdot γ_{p^*} \,γ_{r^*})$ for some fixed $ε_0 \le 0.00863$. Comparing this with Krivine's approximation ratio of $(π/2)/\sinh^{-1}(1)$ for the original Grothendieck problem, our guarantee is off from the best known hardness factor of $(γ_{p^*} γ_{r^*})^{-1}$ for the problem by a factor similar to Krivine's defect. Our approximation follows by bounding the value of the natural vector relaxation for the problem which is convex when $p,r \ge 2$. We give a generalization of random hyperplane rounding and relate the performance of this rounding to certain hypergeometric functions, which prescribe necessary transformations to the vector solution before the rounding is applied. Unlike Krivine's Rounding where the relevant hypergeometric function was $\arcsin$, we have to study a family of hypergeometric functions. The bulk of our technical work then involves methods from complex analysis to gain detailed information about the Taylor series coefficients of the inverses of these hypergeometric functions, which then dictate our approximation factor. Our result also implies improved bounds for "factorization through $\ell_{2}^{\,n}$" of operators from $\ell_{p}^{\,n}$ to $\ell_{q}^{\,m}$ (when $p\geq 2 \geq q$)--- such bounds are of significant interest in functional analysis and our work provides modest supplementary evidence for an intriguing parallel between factorizability, and constant-factor approximability.

cs.DS↗

arXiv2019

Approximating Constraint Satisfaction Problems on High-Dimensional Expanders

We consider the problem of approximately solving constraint satisfaction problems with arity $k > 2$ ($k$-CSPs) on instances satisfying certain expansion properties, when viewed as hypergraphs. Random instances of $k$-CSPs, which are also highly expanding, are well-known to be hard to approximate using known algorithmic techniques (and are widely believed to be hard to approximate in polynomial time). However, we show that this is not necessarily the case for instances where the hypergraph is a high-dimensional expander. We consider the spectral definition of high-dimensional expansion used by Dinur and Kaufman [FOCS 2017] to construct certain primitives related to PCPs. They measure the expansion in terms of a parameter $γ$ which is the analogue of the second singular value for expanding graphs. Extending the results by Barak, Raghavendra and Steurer [FOCS 2011] for 2-CSPs, we show that if an instance of MAX k-CSP over alphabet $[q]$ is a high-dimensional expander with parameter $γ$, then it is possible to approximate the maximum fraction of satisfiable constraints up to an additive error $ε$ using $q^{O(k)} \cdot (k/ε)^{O(1)}$ levels of the sum-of-squares SDP hierarchy, provided $γ\leq ε^{O(1)} \cdot (1/(kq))^{O(k)}$. Based on our analysis, we also suggest a notion of threshold-rank for hypergraphs, which can be used to extend the results for approximating 2-CSPs on low threshold-rank graphs. We show that if an instance of MAX k-CSP has threshold rank $r$ for a threshold $τ= (ε/k)^{O(1)} \cdot (1/q)^{O(k)}$, then it is possible to approximately solve the instance up to additive error $ε$, using $r \cdot q^{O(k)} \cdot (k/ε)^{O(1)}$ levels of the sum-of-squares hierarchy. As in the case of graphs, high-dimensional expanders (with sufficiently small $γ$) have threshold rank 1 according to our definition.

cs.DS↗