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Jun-Ting Hsieh

Publications and source records attributed to Jun-Ting Hsieh.

At least 19 recordsLinked to original sources
arXiv2026

Explicit Constant-Alphabet Subspace Design Codes

The subspace design property for additive codes is a higher-dimensional generalization of the minimum distance property. As shown recently by Brakensiek, Chen, Dhar and Zhang, it implies that the code has similar performance as random linear codes with respect to all "local properties". Explicit algebraic codes, such as folded Reed-Solomon and multiplicity codes, are known to have the subspace design property, but they need alphabet sizes that grow as a large polynomial in the block length. Constructing explicit constant-alphabet subspace design codes was subsequently posed as an open question in Brakensiek, Chen, Dhar and Zhang. In this work, we answer their question and give explicit constructions of subspace design codes over constant-sized alphabets, using the expander-based Alon-Edmonds-Luby (AEL) framework. This generalizes the recent work of Jeronimo and Shagrithaya, which showed that such codes share local properties of random linear codes. Our work obtains this consequence in a unified manner via the subspace design property. In addition, our approach yields some improvements in parameters for list-recovery.

cs.IT↗
arXiv2026

Explicit Almost-Optimal $\varepsilon$-Balanced Codes via Free Expander Walks

We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $\Omega(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. In this work, we give a simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu. We additionally discuss some additional applications of near-$X$-Ramanujan graphs to "on average" lossless expansion and rotating expanders.

cs.CC↗
arXiv2026

Rigorous Implications of the Low-Degree Heuristic

Over the past decade, the low-degree heuristic has been used to estimate the algorithmic thresholds for a wide range of average-case planted vs null distinguishing problems. Such results rely on the hypothesis that if the low-degree moments of the planted and null distributions are sufficiently close, then no efficient (noise-tolerant) algorithm can distinguish between them. This hypothesis is appealing due to the simplicity of calculating the low-degree likelihood ratio (LDLR) -- a quantity that measures the similarity between low-degree moments. However, despite sustained interest in the area, it remains unclear whether low-degree indistinguishability actually rules out any interesting class of algorithms. In this work, we initiate the study and develop technical tools for translating LDLR upper bounds to rigorous lower bounds against concrete algorithms. As a consequence, we prove: for any permutation-invariant distribution $\mathsf{P}$, 1. If $\mathsf{P}$ is over $\{0,1\}^n$ and is low-degree indistinguishable from $U = \mathrm{Unif}(\{0,1\}^n)$, then a noisy version of $\mathsf{P}$ is statistically indistinguishable from $U$. 2. If $\mathsf{P}$ is over $\mathbb{R}^n$ and is low-degree indistinguishable from the standard Gaussian ${N}(0, 1)^n$, then no statistic based on symmetric polynomials of degree at most $O(\log n/\log \log n)$ can distinguish between a noisy version of $\mathsf{P}$ from ${N}(0, 1)^n$. 3. If $\mathsf{P}$ is over $\mathbb{R}^{n\times n}$ and is low-degree indistinguishable from ${N}(0,1)^{n\times n}$, then no constant-sized subgraph statistic can distinguish between a noisy version of $\mathsf{P}$ and ${N}(0, 1)^{n\times n}$.

cs.CC↗
arXiv2025

Sparsifying Cayley Graphs on Every Group

A classic result in graph theory, due to Batson, Spielman, and Srivastava (STOC 2009) shows that every graph admits a $(1 \pm \varepsilon)$ cut (or spectral) sparsifier which preserves only $O(n / \varepsilon^2)$ reweighted edges. However, when applying this result to \emph{Cayley graphs}, the resulting sparsifier is no longer necessarily a Cayley graph -- it can be an arbitrary subset of edges. Thus, a recent line of inquiry, and one which has only seen minor progress, asks: for any group $G$, do all Cayley graphs over the group $G$ admit sparsifiers which preserve only $\mathrm{polylog}(|G|)/\varepsilon^2$ many re-weighted generators? As our primary contribution, we answer this question in the affirmative, presenting a proof of the existence of such Cayley graph spectral sparsifiers, along with an efficient algorithm for finding them. Our algorithm even extends to \emph{directed} Cayley graphs, if we instead ask only for cut sparsification instead of spectral sparsification. We additionally study the sparsification of linear equations over non-abelian groups. In contrast to the abelian case, we show that for non-abelian valued equations, super-polynomially many linear equations must be preserved in order to approximately preserve the number of satisfied equations for any input. Together with our Cayley graph sparsification result, this provides a formal separation between Cayley graph sparsification and sparsifying linear equations.

cs.DS↗
arXiv2025

Coloring 3-Colorable Graphs with Low Threshold Rank

We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues larger than $\varepsilon$, our algorithm finds a proper $3$-coloring on at least $(\frac{1}{2}-O(\varepsilon))n$ vertices in time $n^{O(r/\varepsilon^2)}$. This extends and improves upon the result of Bafna, Hsieh, and Kothari on $1$-sided expanders. Furthermore, an independent work by Buhai, Hua, Steurer, and V\'ari-Kakas shows that it is UG-hard to properly $3$-color more than $(\frac{1}{2}+\varepsilon)n$ vertices, thus establishing the tightness of our result. Our proof is short and simple, relying on the observation that for any distribution over proper $3$-colorings, the correlation across an edge must be large if the marginals of the endpoints are not concentrated on any single color. Notably, this property fails for $4$-colorings, which is consistent with the hardness result of [BHK25] for $4$-colorable $1$-sided expanders.

cs.DS↗
arXiv2025

Solving Random Planted CSPs below the $n^{k/2}$ Threshold

We present a family of algorithms to solve random planted instances of any $k$-ary Boolean constraint satisfaction problem (CSP). A randomly planted instance of a Boolean CSP is generated by (1) choosing an arbitrary planted assignment $x^*$, and then (2) sampling constraints from a particular "planting distribution" designed so that $x^*$ will satisfy every constraint. Given an $n$ variable instance of a $k$-ary Boolean CSP with $m$ constraints, our algorithm runs in time $n^{O(\ell)}$ for a choice of a parameter $\ell$, and succeeds in outputting a satisfying assignment if $m \geq O(n) \cdot (n/\ell)^{\frac{k}{2} - 1} \log n$. This generalizes the $\mathrm{poly}(n)$-time algorithm of [FPV15], the case of $\ell = O(1)$, to larger runtimes, and matches the constraint number vs.\ runtime trade-off established for refuting random CSPs by [RRS17]. Our algorithm is conceptually different from the recent algorithm of [GHKM23], which gave a $\mathrm{poly}(n)$-time algorithm to solve semirandom CSPs with $m \geq \tilde{O}(n^{\frac{k}{2}})$ constraints by exploiting conditions that allow a basic SDP to recover the planted assignment $x^*$ exactly. Instead, we forego certificates of uniqueness and recover $x^*$ in two steps: we first use a degree-$O(\ell)$ Sum-of-Squares SDP to find some $\hat{x}$ that is $o(1)$-close to $x^*$, and then we use a second rounding procedure to recover $x^*$ from $\hat{x}$.

cs.DS↗
arXiv2025

The Quasi-Polynomial Low-Degree Conjecture is False

There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a related planted distribution. Such hardness results are now ubiquitous not only for foundational average-case problems but also central questions in statistics and cryptography. This line of work is supported by the low-degree conjecture of Hopkins, which postulates that a vanishing degree-$D$ LDA implies the absence of any noise-tolerant distinguishing algorithm with runtime $n^{\widetilde{O}(D)}$ whenever 1) the null distribution is product on $\{0,1\}^{\binom{n}{k}}$, and 2) the planted distribution is permutation invariant, that is, invariant under any relabeling $[n] \rightarrow [n]$. In this paper, we disprove this conjecture. Specifically, we show that for any fixed $\varepsilon>0$ and $k\geq 2$, there is a permutation-invariant planted distribution on $\{0,1\}^{\binom{n}{k}}$ that has a vanishing degree-$n^{1-O(\varepsilon)}$ LDA with respect to the uniform distribution on $\{0,1\}^{\binom{n}{k}}$, yet the corresponding $\varepsilon$-noisy distinguishing problem can be solved in $n^{O(\log^{1/(k-1)}(n))}$ time. Our construction relies on algorithms for list-decoding for noisy polynomial interpolation in the high-error regime. We also give another construction of a pair of planted and (non-product) null distributions on $\mathbb{R}^{n \times n}$ with a vanishing $n^{\Omega(1)}$-degree LDA while the largest eigenvalue serves as an efficient noise-tolerant distinguisher. Our results suggest that while a vanishing LDA may still be interpreted as evidence of hardness, developing a theory of average-case complexity based on such heuristics requires a more careful approach.

cs.CC↗
arXiv2025

Explicit Lossless Vertex Expanders

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where every small set $S$ of vertices has $(1-\varepsilon)d|S|$ neighbors (which implies $(1-2\varepsilon)d|S|$ unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

math.CO↗
arXiv2024

Small Even Covers, Locally Decodable Codes and Restricted Subgraphs of Edge-Colored Kikuchi Graphs

Given a $k$-uniform hypergraph $H$ on $n$ vertices, an even cover in $H$ is a collection of hyperedges that touch each vertex an even number of times. Even covers are a generalization of cycles in graphs and are equivalent to linearly dependent subsets of a system of linear equations modulo $2$. As a result, they arise naturally in the context of well-studied questions in coding theory and refuting unsatisfiable $k$-SAT formulas. Analogous to the irregular Moore bound of Alon, Hoory, and Linial (2002), in 2008, Feige conjectured an extremal trade-off between the number of hyperedges and the length of the smallest even cover in a $k$-uniform hypergraph. This conjecture was recently settled up to a multiplicative logarithmic factor in the number of hyperedges (Guruswami, Kothari, and Manohar 2022 and Hsieh, Kothari, and Mohanty 2023). These works introduce the new technique that relates hypergraph even covers to cycles in the associated Kikuchi graphs. Their analysis of these Kikuchi graphs, especially for odd $k$, is rather involved and relies on matrix concentration inequalities. In this work, we give a simple and purely combinatorial argument that recovers the best-known bound for Feige's conjecture for even $k$. We also introduce a novel variant of a Kikuchi graph which together with this argument improves the logarithmic factor in the best-known bounds for odd $k$. As an application of our ideas, we also give a purely combinatorial proof of the improved lower bounds (Alrabiah, Guruswami, Kothari and Manohar, 2023) on 3-query binary linear locally decodable codes.

cs.CC↗
arXiv2024

Improved Lower Bounds for all Odd-Query Locally Decodable Codes

We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$. Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees. We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees.

cs.CC↗
arXiv2024

Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier

We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every small subset of vertices $S$ has at least $0.5d|S|$ distinct neighbors. However, it is possible to construct Ramanujan graphs containing a small set $S$ with no more than $0.5d|S|$ neighbors. In fact, no explicit construction was known to break the $0.5 d$-barrier. In this work, we give an explicit construction of an infinite family of $d$-regular graphs (for large enough $d$) where every small set expands by a factor of $\approx 0.6d$. More generally, for large enough $d_1,d_2$, we give an infinite family of $(d_1,d_2)$-biregular graphs where small sets on the left expand by a factor of $\approx 0.6d_1$, and small sets on the right expand by a factor of $\approx 0.6d_2$. In fact, our construction satisfies an even stronger property: small sets on the left and right have unique-neighbor expansion $0.6d_1$ and $0.6d_2$ respectively. Our construction follows the tripartite line product framework of Hsieh, McKenzie, Mohanty & Paredes, and instantiates it using the face-vertex incidence of the $4$-dimensional Ramanujan clique complex as its base component. As a key part of our analysis, we derive new bounds on the triangle density of small sets in the Ramanujan clique complex.

math.CO↗
arXiv2024

Rounding Large Independent Sets on Expanders

We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has its second eigenvalue bounded away from 1. Consequently, we obtain a polynomial time algorithm to find linear-sized independent sets in one-sided expanders that are almost $3$-colorable or are promised to contain an independent set of size $(1/2-ε)n$. Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. In a surprising contrast to our algorithmic result, we observe that the analogous task of finding a linear-sized independent set in almost $4$-colorable one-sided expanders (even when the second eigenvalue is $o_n(1)$) is NP-hard, assuming the Unique Games Conjecture. All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques (following the classical spectral methods of Alon and Kahale) and require two-sided expansion, meaning a bounded number of negative eigenvalues of magnitude $Ω(1)$. Such techniques naturally extend to almost $k$-colorable graphs for any constant $k$, in contrast to analogous guarantees on one-sided expanders, which are Unique Games-hard to achieve for $k \geq 4$. Our rounding builds on the method of simulating multiple samples from a pseudo-distribution introduced by Bafna et. al. for rounding Unique Games instances. The key to our analysis is a new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system.

cs.DS↗
arXiv2024

Predicting quantum channels over general product distributions

We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an $n$-qubit channel $E$ and an observable $O$, we aim to learn the mapping \begin{equation*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most $ρ$ sampled from a distribution $D$. Previously, Huang, Chen, and Preskill proved a surprising result that even if $E$ is arbitrary, this task can be solved in time roughly $n^{O(\log(1/ε))}$, where $ε$ is the target prediction error. However, their guarantee applied only to input distributions $D$ invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states $ρ$. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution $D$, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.

quant-ph↗
arXiv2024

Certifying Euclidean Sections and Finding Planted Sparse Vectors Beyond the $\sqrt{n}$ Dimension Threshold

We consider the task of certifying that a random $d$-dimensional subspace $X$ in $\mathbb{R}^n$ is well-spread - every vector $x \in X$ satisfies $c\sqrt{n} \|x\|_2 \leq \|x\|_1 \leq \sqrt{n}\|x\|_2$. In a seminal work, Barak et. al. showed a polynomial-time certification algorithm when $d \leq O(\sqrt{n})$. On the other hand, when $d \gg \sqrt{n}$, the certification task is information-theoretically possible but there is evidence that it is computationally hard [MW21,Cd22], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the $d \gg \sqrt{n}$ regime. Our algorithm runs in time $\exp(\widetilde{O}(n^{\varepsilon}))$ when $d \leq \widetilde{O}(n^{(1+\varepsilon)/2})$, establishing a smooth trade-off between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension trade-off for this task.

cs.DS↗
arXiv2024

Explicit two-sided unique-neighbor expanders

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $Ω(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(Ω(\sqrt{\log |V(G_n)|}))$ expand losslessly.

math.CO↗
arXiv2023

New SDP Roundings and Certifiable Approximation for Cubic Optimization

We give new rounding schemes for SDP relaxations for the problems of maximizing cubic polynomials over the unit sphere and the $n$-dimensional hypercube. In both cases, the resulting algorithms yield a $O(\sqrt{n/k})$ multiplicative approximation in $2^{O(k)} \text{poly}(n)$ time. In particular, we obtain a $O(\sqrt{n/\log n})$ approximation in polynomial time. For the unit sphere, this improves on the rounding algorithms of Bhattiprolu et. al. [BGG+17] that need quasi-polynomial time to obtain a similar approximation guarantee. Over the $n$-dimensional hypercube, our results match the guarantee of a search algorithm of Khot and Naor [KN08] that obtains a similar approximation ratio via techniques from convex geometry. Unlike their method, our algorithm obtains an upper bound on the integrality gap of SDP relaxations for the problem and as a result, also yields a certificate on the optimum value of the input instance. Our results naturally generalize to homogeneous polynomials of higher degree and imply improved algorithms for approximating satisfiable instances of Max-3SAT. Our main motivation is the stark lack of rounding techniques for SDP relaxations of higher degree polynomial optimization in sharp contrast to a rich theory of SDP roundings for the quadratic case. Our rounding algorithms introduce two new ideas: 1) a new polynomial reweighting based method to round sum-of-squares relaxations of higher degree polynomial maximization problems, and 2) a general technique to compress such relaxations down to substantially smaller SDPs by relying on an explicit construction of certain hitting sets. We hope that our work will inspire improved rounding algorithms for polynomial optimization and related problems.

cs.DS↗
arXiv2023

Efficient Algorithms for Semirandom Planted CSPs at the Refutation Threshold

We present an efficient algorithm to solve semirandom planted instances of any Boolean constraint satisfaction problem (CSP). The semirandom model is a hybrid between worst-case and average-case input models, where the input is generated by (1) choosing an arbitrary planted assignment $x^*$, (2) choosing an arbitrary clause structure, and (3) choosing literal negations for each clause from an arbitrary distribution "shifted by $x^*$" so that $x^*$ satisfies each constraint. For an $n$ variable semirandom planted instance of a $k$-arity CSP, our algorithm runs in polynomial time and outputs an assignment that satisfies all but a $o(1)$-fraction of constraints, provided that the instance has at least $\tilde{O}(n^{k/2})$ constraints. This matches, up to $polylog(n)$ factors, the clause threshold for algorithms that solve fully random planted CSPs [FPV15], as well as algorithms that refute random and semirandom CSPs [AOW15, AGK21]. Our result shows that despite having worst-case clause structure, the randomness in the literal patterns makes semirandom planted CSPs significantly easier than worst-case, where analogous results require $O(n^k)$ constraints [AKK95, FLP16]. Perhaps surprisingly, our algorithm follows a significantly different conceptual framework when compared to the recent resolution of semirandom CSP refutation. This turns out to be inherent and, at a technical level, can be attributed to the need for relative spectral approximation of certain random matrices - reminiscent of the classical spectral sparsification - which ensures that an SDP can certify the uniqueness of the planted assignment. In contrast, in the refutation setting, it suffices to obtain a weaker guarantee of absolute upper bounds on the spectral norm of related matrices.

cs.CC↗
arXiv2023

Ellipsoid Fitting Up to a Constant

In [Sau11,SPW13], Saunderson, Parrilo and Willsky asked the following elegant geometric question: what is the largest $m= m(d)$ such that there is an ellipsoid in $\mathbb{R}^d$ that passes through $v_1, v_2, \ldots, v_m$ with high probability when the $v_i$s are chosen independently from the standard Gaussian distribution $N(0,I_{d})$. The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix $X$ such that $v_i^{\top}X v_i =1$ for every $1 \leq i \leq m$ - a natural example of a random semidefinite program. SPW conjectured that $m= (1-o(1)) d^2/4$ with high probability. Very recently, Potechin, Turner, Venkat and Wein and Kane and Diakonikolas proved that $m \geq d^2/\log^{O(1)}(d)$ via certain explicit constructions. In this work, we give a substantially tighter analysis of their construction to prove that $m \geq d^2/C$ for an absolute constant $C>0$. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [BHK+19]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension.

math.PR↗