arXiv subjects

Shuangping Li

Publications and source records attributed to Shuangping Li.

At least 19 recordsLinked to original sources

arXiv2026

Dense Language Generation Made Simple: Deterministic, Randomized, and Multi-Order Algorithms

Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.

cs.DS↗

arXiv2026

Rigorous Low-Degree Implications for Planted Subgraph Detection: Noise and Treewidth

The low-degree method predicts computational thresholds by asking whether low-degree polynomials can distinguish a planted distribution from a null one. Recent counterexamples by Buhai, Hsieh, Jain, and Kothari (FOCS 2025) and Mao (2026) show that low-degree indistinguishability does not, in general, rule out efficient algorithms, even on noisy inputs. Hsieh, Kane, Kothari, Li, Mohanty, and Tiegel (STOC 2026) asked what it does imply, and showed that for permutation-invariant distributions on $\{0,1\}^n$, low-degree indistinguishability from the uniform distribution implies statistical indistinguishability after noise. In this work, we study the same question for planted subgraphs in sparse random graphs. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $Γ_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and the treewidth of $Γ_n$ is $o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. In the critical and subcritical regimes $0<c\le 1$, we obtain the same conclusion without any treewidth assumption, provided $D_n=ω(\log n)$. Our proof has two main ingredients. First, we identify the subgraph-count and automorphism factors in the Fourier expansion with counts of isomorphism triples. Second, we cut a tree decomposition of $Γ_n$ into subtrees, which splits each large Fourier support into low-degree pieces that meet at only a few interface vertices, and we use noise to absorb the cost of reassembling them. At and below criticality, we use the low-degree assumption to rule out short cycles and noise to destroy the remaining long ones.

cs.CC↗

arXiv2026

State Evolution and Universality of Iterated Linear Programs for the Binary Perceptron

The binary perceptron is a random constraint satisfaction problem that asks for a sign vector satisfying a system of random linear inequalities. Its solution space has a strikingly extreme geometry: at every positive constraint density, all but a vanishing fraction of the solutions are isolated. Yet polynomial-time algorithms find solutions at small enough constraint density, which raises the question of how efficient algorithms navigate such a landscape. Linear programming is a natural algorithm to study here, but its analysis has so far been limited to a single round. Li, Schramm, and Zhou (STOC 2025) analyzed one linear program followed by iterative rounding. In this work, we overcome this limitation and give a precise analysis of iterated linear programming for the binary perceptron. (1) For Gaussian matrices and any fixed number of rounds, we characterize the iterated linear programs exactly through a deterministic state-evolution recursion. (2) Adding a final rounding step, we obtain a randomized polynomial-time algorithm for the binary perceptron. Numerical evaluations of the recursion provide evidence that it succeeds at constraint density $α=0.60$ with zero margin, well beyond the range $α\le0.1$ guaranteed by the single-LP analysis and $α\approx0.327$ (Jo and Lee, 2026), the best guarantee so far. (3) We establish universality, showing that the same characterization holds for matrices with i.i.d. entries drawn from any centered, unit-variance subgaussian law. The key idea of our proof is that, although conditioning on earlier rounds makes the matrix non-Gaussian, its conditional law can be computed explicitly and remains close to a Gaussian law, so the convex Gaussian min-max theorem applies in every round. For universality, we compare Gaussian and subgaussian matrices through the Lindeberg principle, using a regularized version of each linear program.

cs.DS↗

arXiv2026

Online Algorithms Cannot Find Isolated Perceptron Solutions

The binary perceptron problem asks for a length-$n$ sign vector in the intersection of $\lfloor αn\rfloor$-many independent random Gaussian half-spaces. A striking feature of this model is strong freezing: at every constraint density $α> 0$, all but a vanishing fraction of solutions are isolated -- $Ω(n)$-far from all others, in Hamming distance. Such solutions are believed to be algorithmically inaccessible. Nevertheless, efficient algorithms find solutions at some constraint densities $α$. One possible resolution of this apparent contradiction is that the solution space also contains rare well-connected structures, and that algorithms succeed by locating them. Gong, Huang, Li and Sellke (FOCS 2026) proved this for stable algorithms, by showing that such algorithms cannot locate isolated solutions with probability more than $\approx 0.84$. In this work, we study online algorithms, which receive the columns of the constraint matrix one by one and sign each irrevocably upon arrival. This is a widely used class of algorithms in discrepancy minimization, and it includes the algorithm of Jo and Lee (2026), which has one of the best provable guarantees for the binary perceptron. We prove that at any constraint density, any online algorithm outputs an isolated solution with probability $o(1)$. Our proof is based on the observation that the signs an online algorithm has already fixed can be completed to at most one isolated solution, so a successful algorithm must choose the unique correct sign for each of the last few columns. Using the exchangeability of these columns, we show that such an algorithm must match a suffix-parallel baseline, which signs each of them as if it were the next column to arrive. We then use an anti-concentration argument to show that such a baseline rarely produces a unique completion.

cs.CC↗

arXiv2026

The lower tail of triangle counts in sparse random graphs

Let $T_n$ be the number of triangles in the Erdős-Rényi random graph $G(n,p)$ with $n^{-1/2}\ll p\ll1$ and fix $η\in(0,1)$. We study the large deviation rate of the lower tail event $T_n\leη\mathbb{E}[T_n]$, and the typical structure of $G(n,p)$ conditioned on this event. By earlier work [16,19,21,49], both the rate and the structure are governed by a variational problem over graphons. In this work we solve this variational problem for every $η\in(0,1)$. We show that every minimizer is either a constant graphon or a step graphon with two blocks of equal size. Correspondingly, the conditioned graph is close to an Erdős-Rényi graph or to a symmetric stochastic block model with two blocks. These two regimes are separated by a first order phase transition at the threshold $η_*=0.009217\ldots$. This resolves a conjecture of Zhao [78] and an open problem of Janson and Warnke [39] on triangles. It thereby gives a complete description of the lower tail of triangle counts in this range of $p$.

math.PR↗

arXiv2026

Strong freezing in binary perceptron models

The binary perceptron model concerns the set of solutions obtained by intersecting random half-spaces. One of the most striking physics predictions for this model, going back to Krauth and Mézard [KM89] and sharpened in [HWK13, HK14], is strong freezing: typical solutions are at Hamming distance of order $n$ from every other solution. This phenomenon has attracted a lot of attention over the years, partly because of its intricate connections to statistical physics and computational complexity. We prove strong freezing at every positive constraint density for a general class of perceptron models and for any disorder with independent entries of mean zero and unit variance. More precisely, with probability at least $1-e^{-c\sqrt n}$, all but an $e^{-c\sqrt n}$ fraction of the solutions have no other solution within Hamming distance $δn$. In particular, our result includes the asymmetric binary perceptron at every margin, for which strong freezing has been a long-standing open problem [KM89,HWK13,HK14], and the symmetric binary perceptron, for which we recover the strong freezing results of Perkins and Xu [PX24] and of Abbe, Li and Sly [ALS21]. Our proof crucially relies on replacing the number of constraints by a Poisson variable, which makes the numbers of constraints cutting out different subsets of the hypercube independent. Since each constraint can only remove solutions, the FKG inequality then decouples the partition function from the event that two nearby configurations are both solutions.

math.PR↗

arXiv2026

Graph Matching in Random Geometric Graphs I: Efficient Algorithms

We study the recovery of the hidden vertex correspondence between two correlated dense random geometric graphs. Each graph is generated from $n$ latent unit vectors in $\mathbb R^d$; the vectors $u_i^A$ and $u_i^B$ of corresponding vertices satisfy $\langle u_i^A,u_i^B\rangle=ρ$, and the vertex labels of the second graph are permuted by an unknown permutation $π$. For every fixed correlation $ρ\in(0,1]$ and every edge density bounded away from zero and one, we give a randomized polynomial time algorithm that recovers $π$ exactly with probability tending to one whenever $d\ge(\log n)^L$, where $L$ depends only on the correlation and the density bounds. The algorithm assigns to each vertex a signature formed by counting decorated paths with a constant number of seeds. The number of such seeded templates grows exponentially with the path length, with an arbitrarily large constant base, so the resulting statistics amplify an arbitrarily small constant correlation. We control the dependencies created by the latent geometry in two ways: when $d$ is larger than $n/(\log n)^{L_0}$ for a suitable constant $L_0$, by comparing local moments with those of correlated Erdős--Rényi graphs; for smaller $d$, by spectral estimation of the latent geometry combined with a harmonic correction. We also provide low-degree evidence for a tradeoff between the correlation and the computational cost.

math.PR↗

arXiv2026

Detecting Random Geometric Graphs with General Kernel

We study testing a random geometric graph with a fixed kernel against an Erdős-Rényi graph with the same edge density. Under the null hypothesis, the graph has $n$ vertices and independent edges with probability $p$. Under the alternative hypothesis, latent points $X_1,\ldots,X_n$ are independent and uniform on $\mathbb{S}^{d-1}$, and edges are conditionally independent with probabilities $K(\langle X_i,X_j\rangle)$, where $p=\mathbb{E}[K(\langle X_1,X_2\rangle)]$. The goal is to determine a critical dimension $d_\star = d_\star(n,K)$: the total variation distance between the two graph laws tends to one for $d\ll d_\star$ and to zero for $d\gg d_\star$. For dense graphs, [BDER16] established $d_\star\asymp n^3$ for hard threshold kernels, whereas [MWX26] proved $d_\star\asymp n^{3/4}$ for smooth kernels with $K'(0)\ne0$. We connect these results by characterizing the detection thresholds for kernels that need not be differentiable at zero. We also determine the thresholds for smooth kernels whose first nonzero derivative at zero has finite order greater than one. Throughout these classes, signed triangle or quadrilateral counts attain the optimal detection threshold. For certain nondifferentiable kernels, quadrilateral counts are optimal while triangle counts are suboptimal.

math.ST↗

arXiv2026

Stable and Online Algorithms for Random Matrix Discrepancy

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $κ>0$, find signs $σ_1,\dots,σ_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N σ_i A_i$ is at most $κ\sqrt{N}$. Focusing on the proportional regime $N/d^2\to τ\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $κ\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time \emph{recenter-and-round} algorithm that is noise-stable and succeeds whenever $τ=Ω(\frac{1}{κ^2\log(1/κ)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the \emph{Frobenius-greedy} algorithm, establishing that it succeeds when $τ>τ_{\rm FG}(κ)\sim \fracπ{4κ^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $Θ(\frac{1}{κ^2\log(1/κ)})$ for stable offline algorithms and $Θ(\frac{1}{κ^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $Θ(\log(1/κ))$, as shown by Maillard~\cite{maillard2025}.

cs.DS↗

arXiv2026

GenLimitLib: A Formal Library for Language Generation in the Limit and AI-Assisted Mathematical Research

We present GenLimitLib, a source-aligned Lean 4 library for language generation in the limit. Introduced by Kleinberg and Mullainathan at NeurIPS 2024, language generation in the limit studies a theoretical question motivated by LLMs: how to generate valid new strings from observed examples. This young and rapidly evolving field offers a natural testbed for studying large-scale formalization. GenLimitLib contains formal developments for 30 papers. It extracts shared definitions and reusable proof components while preserving paper-specific assumptions and statements, and records relationships across papers. In this way, GenLimitLib provides a concrete and structured view of the literature. We show through mathematical case studies and LLM experiments how our library can support both human mathematical research and AI-assisted research. Our Library: https://github.com/pengzhang91/generation-in-the-limit-lib.

cs.LG↗

arXiv2026

Phylodynamic inference with the bounded coalescent: a point process perspective

The coalescent is a central framework in population genetics for modelling the ancestral relationships among sampled individuals through a genealogy, represented as a rooted and ranked binary tree. In this model, lineages coalesce at a rate inversely proportional to the effective population size, a time-varying quantity of primary interest. The bounded coalescent conditions genealogies on the time to the most recent common ancestor being bounded above by a fixed time. This model is useful in various contexts, such as phylodynamics of infectious diseases with known introduction times and single-cell lineage tracing in synthetic barcoding experiments. To our knowledge, there is no existing tool that infers variable effective population size trajectories under the bounded coalescent. We view estimation under the bounded coalescent as equivalent to estimation of the intensity function of an inhomogeneous point process. We provide an efficient algorithm for coalescent simulation under the bounded coalescent using point process methods, retaining the exactness of naive rejection sampling while substantially reducing computational cost and avoiding repeated numerical inversion of the bounded cumulative hazard. We then develop a Markov chain Monte Carlo procedure for posterior inference of effective population size trajectories that avoids discretization of the likelihood integrals. In simulations, conditioning on the bound reduces the median sum of squared errors in two of three settings, with less favourable results in the most rapidly varying setting. We illustrate the method using severe acute respiratory syndrome coronavirus 2 sequence data from Washington State.

stat.ME↗

arXiv2026

Spectral clustering in the Gaussian mixture block model

Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex $i$ with a latent feature vector $u_i \in \mathbb{R}^d$ sampled from a mixture of Gaussians, and we add edge $(i,j)$ if and only if the feature vectors are sufficiently similar, in that $\langle u_i,u_j \rangle \ge τ$ for a pre-specified threshold $τ$. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features -- for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component). In this paper we initiate the study of clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors $d\to \infty$ as the size of the network $n \to \infty$. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures, and begin to sketch out the information-computation landscape for clustering and embedding in these models.

stat.ML↗

arXiv2026

Stable algorithms cannot reliably find isolated perceptron solutions

We study the binary perceptron, a random constraint satisfaction problem that asks to find a Boolean vector in the intersection of independently chosen random halfspaces. A striking feature of this model is that at every positive constraint density, it is expected that a $1-o_N(1)$ fraction of solutions are \emph{strongly isolated}, i.e. separated from all others by Hamming distance $Ω(N)$. At the same time, efficient algorithms are known to find solutions at certain positive constraint densities. This raises a natural question: can any isolated solution be algorithmically visible? We answer this in the negative: no algorithm whose output is stable under a tiny Gaussian resampling of the disorder can \emph{reliably} locate isolated solutions. We show that any stable algorithm has success probability at most $\frac{3\sqrt{17}-9}{4}+o_N(1)\leq 0.84233$. Furthermore, every stable algorithm that finds a solution with probability $1-o_N(1)$ finds an isolated solution with probability $o_N(1)$. The class of stable algorithms we consider includes degree-$D$ polynomials up to $D\leq o(N/\log N)$; under the low-degree heuristic \cite{hopkins2018statistical}, this suggests that locating strongly isolated solutions requires running time $\exp(\widetildeΘ(N))$. Our proof does not use the overlap gap property. Instead, we show via Pitt's correlation inequality that after a random perturbation of the disorder, the number of solutions located close to a pre-existing isolated solution cannot concentrate at $1$.

cs.CC↗

arXiv2026

Detection of local geometry in random graphs: information-theoretic and computational limits

We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all remaining edges follow the Erdős--Rényi model $\mathcal{G}(n, p)$. The random geometric graph is generated by thresholding inner products of latent vectors on $\mathbb{S}^{d-1}$, with each edge having marginal probability equal to $p$. This implies that $\mathcal{G}(n, p, d, k)$ and $\mathcal{G}(n, p)$ are indistinguishable at the level of the marginals, and the signal lies entirely in the edge dependencies induced by the local geometry. We investigate both the information-theoretic and computational limits of detection. On the information-theoretic side, our upper bounds follow from three tests based on signed triangle counts: a global test, a scan test, and a constrained scan test; our lower bounds follow from two complementary methods: truncated second moment via Wishart--GOE comparison, and tensorization of KL divergence. These results together settle the detection threshold at $d = \widetildeΘ(k^2 \vee k^6/n^3)$ for fixed $p$, and extend the state-of-the-art bounds from the full model (i.e., $k = n$) for vanishing $p$. On the computational side, we identify a computational--statistical gap and provide evidence via the low-degree polynomial framework, as well as the suboptimality of signed cycle counts of length $\ell \geq 4$.

math.ST↗

arXiv2025

Some easy optimization problems have the overlap-gap property

We show that the shortest $s$-$t$ path problem has the overlap-gap property in (i) sparse $\mathbf{G}(n,p)$ graphs and (ii) complete graphs with i.i.d. Exponential edge weights. Furthermore, we demonstrate that in sparse $\mathbf{G}(n,p)$ graphs, shortest path is solved by $O(\log n)$-degree polynomial estimators, and a uniform approximate shortest path can be sampled in polynomial time. This constitutes the first example in which the overlap-gap property is not predictive of algorithmic intractability for a (non-algebraic) average-case optimization problem.

cs.CC↗

arXiv2025

Discrepancy Algorithms for the Binary Perceptron

The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-κ$. We analyze the performance of the canonical discrepancy minimization algorithms of Lovett-Meka and Rothvoss/Eldan-Singh for the asymmetric binary perceptron problem. We obtain new algorithmic results in the $κ= 0$ case and in the large-$|κ|$ case. In the $κ\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.

cs.DS↗

arXiv2024

Synthetic continued pretraining

Pretraining on large-scale, unstructured internet text enables language models to acquire a significant amount of world knowledge. However, this knowledge acquisition is data-inefficient--to learn a given fact, models must be trained on hundreds to thousands of diverse representations of it. This poses a challenge when adapting a pretrained model to a small corpus of domain-specific documents, where each fact may appear rarely or only once. We propose to bridge this gap with synthetic continued pretraining: using the small domain-specific corpus to synthesize a large corpus more amenable to learning, and then performing continued pretraining on the synthesized corpus. We instantiate this proposal with EntiGraph, a synthetic data augmentation algorithm that extracts salient entities from the source documents and then generates diverse text by drawing connections between the sampled entities. Synthetic continued pretraining with EntiGraph enables a language model to answer questions and follow generic instructions related to the source documents without access to them. If, instead, the source documents are available at inference time, we show that the knowledge acquired through our approach compounds with retrieval-augmented generation. To better understand these results, we build a simple mathematical model of EntiGraph, and show how synthetic data augmentation can "rearrange" knowledge to enable more data-efficient learning.

cs.LG↗

arXiv2022

Mean Field Behavior during the Big Bang Regime for Coalescing Random Walks

In this paper we consider coalescing random walks on a general connected graph $G=(V,E)$. We set up a unified framework to study the leading order of the decay rate of $P_t$, the expectation of the fraction of occupied sites at time $t$, particularly for the `Big Bang' regime where $t\ll t_{\text{coal}}:=\mathbb{E}[\inf\{s:\text{There is only one particle at time }s\}]$. Our results show that $P_t$ satisfies certain mean field behavior, if the graphs satisfy certain transience-like conditions. We apply this framework to two families of graphs: (1) graphs given by the configuration model with a degree distribution supported in $[3,\bar d]$ for some $\bar d\geq 3$, and (2) finite and infinite vertex-transitive graphs. In the first case, we show that for $1 \ll t \ll |V|$, $P_t$ decays in the order of $t^{-1}$, and $(tP_t)^{-1}$ is approximately the probability that two particles starting from the root of the corresponding unimodular Galton-Watson tree never collide after one of them leaves the root, which is also roughly $|V|/(2t_{\text{meet}})$, where $t_{\text{meet}}$ is the mean meeting time of two walkers. By taking the local weak limit, for the unimodular Galton-Watson tree we prove the convergence of $tP_t$ as $t\to\infty$. For the second family of graphs, if we take a sequence of finite graphs $G_n=(V_n, E_n)$, such that $t_{\text{meet}}=O(|V_n|)$ and the inverse of the spectral gap $t_{\text{rel}}$ is $o(|V_n|)$, then for $t_{\text{rel}}\ll t\ll t_{\text{coal}}$, $(tP_t)^{-1}$ is approximately the probability that two random walks never meet before time $t$, and also $|V|/(2t_{\text{meet}})$. In addition, we define a natural uniform transience condition, and show that it implies the above for all $1\ll t\ll t_{\text{coal}}$. Such estimates of $tP_t$ are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.

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