arXiv subjects

Shuangping Li

Publications and source records attributed to Shuangping Li.

13 recordsLinked to original sources

arXiv2026

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

The low-degree heuristic has become a widely used framework for predicting computational thresholds in average-case planted-versus-null problems. However, a recent sequence of counterexamples shows that low-degree indistinguishability does not, in general, rule out efficient noise-tolerant distinguishers; see Buhai et al. (2025) and Mao (2026). Motivated by these developments, Hsieh et al. (2026) initiated the study of rigorous consequences of the low-degree heuristic. In this work, we continue this program for planted-graph problems. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $\Gamma_n$ 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 $\operatorname{tw}(\Gamma_n)=o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. Here $\operatorname{tw}(\Gamma_n)$ denotes the treewidth of $\Gamma_n$, a measure of how efficiently the graph can be decomposed into tree-like pieces. In the critical and subcritical regimes $0<c\leq 1$, the same conclusion holds whenever $D_n=\omega(\log n)$, without any treewidth assumption. Our proof has two main ingredients. First, we uncover a correspondence between the subgraph-count and automorphism factors in the Fourier expansion and counts of isomorphism triples. Second, we cut the decomposition tree into subtrees, breaking each large Fourier support into low-degree pieces that meet at only a few interface vertices, and use noise to absorb the cost of reassembling them. At and below criticality, the low-degree assumption rules out short cycles, while noise destroys the remaining long cycles.

cs.CC↗

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

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 $\Omega(N)$. 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 $\exp(\widetilde{\Theta}(N))$. 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 Erd\H{o}s--R\'enyi 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 $d = \widetilde{\Theta}(k^2 \vee k^6/n^3)$ 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↗

arXiv2024

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↗

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↗

arXiv2024

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 $-\kappa$. 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 $\kappa = 0$ case and in the large-$|\kappa|$ case. In the $\kappa\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.

cs.DS↗

arXiv2023

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 $\langle u_i,u_j \rangle \ge \tau$ for a pre-specified threshold $\tau$. 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↗

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.

math.PR↗

arXiv2021

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognormal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

math.PR↗

arXiv2021

Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster

It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu '21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.

math.PR↗

arXiv2020

Learning Sparse Graphons and the Generalized Kesten-Stigum Threshold

The problem of learning graphons has attracted considerable attention across several scientific communities, with significant progress over the recent years in sparser regimes. Yet, the current techniques still require diverging degrees in order to succeed with efficient algorithms in the challenging cases where the local structure of the graph is homogeneous. This paper provides an efficient algorithm to learn graphons in the constant expected degree regime. The algorithm is shown to succeed in estimating the rank-$k$ projection of a graphon in the $L_2$ metric if the top $k$ eigenvalues of the graphon satisfy a generalized Kesten-Stigum condition.

math.ST↗

arXiv2019

A Trace Theorem For Sobolev Spaces On The Sierpinski Gasket

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the L2 domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.

math.CA↗