arXiv subjects

Nike Sun

Publications and source records attributed to Nike Sun.

At least 19 recordsLinked to original sources

arXiv2026

Algorithmic threshold for high-dimensional projection pursuit I: general theory

We study a null model of high-dimensional projection pursuit: we are given $M$ points sampled i.i.d. from a standard gaussian in $N$ dimensions, where $M,N\to\infty$ with $M/N\to\alpha\in(0,\infty)$. Our goal is to characterize the possible empirical distributions of these points' projections along a data-dependent direction $x$, which ranges over either the sphere $S_N=\sqrt{N}\mathbb{S}^{N-1}$ or cube $\Sigma_N=\{-1,+1\}^N$. We consider this problem in an algorithmic setting, where $x$ must be the output of an algorithm with dimension-free Lipschitz dependence on the input; this class of algorithms includes general gradient-based methods such as Langevin dynamics and approximate message passing (AMP). Our main result exactly characterizes the set of empirical distributions attainable by this class in terms of a one-dimensional stochastic control problem. As a consequence of our main result, we obtain exact algorithmic thresholds for optimizing the Hamiltonian of a spherical or Ising perceptron model with general bounded continuous activation. For the spherical problem, independent work of Montanari and Zhou (2024) characterized the empirical distributions attainable by a related two-stage AMP algorithm, also in terms of stochastic control. Our proof of hardness builds on the branching overlap gap property introduced in earlier work by the first two authors. Our main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold. Notably, our methods apply even though the non-algorithmic problem of characterizing all feasible projections remains a major outstanding challenge. For the matching algorithmic result, we construct a new incremental AMP algorithm that acts on a Brownian-bridge revelation of the gaussian disorder and simulates the same family of controlled SDEs.

math.PR↗

arXiv2026

Resolution of the Detection Threshold Conjecture for Random Geometric Graphs in the $d>n$ Regime

A random geometric graph (RGG) is generated by first sampling latent points $x_1,\ldots,x_n$ independently and uniformly from the unit sphere in $\mathbb{R}^d$, and then connecting each pair $(i,j)$ if $\langle x_i,x_j\rangle$ exceeds some threshold $\tau$. We study the sharp detection threshold -- the largest dimension at which the RGG can be statistically distinguished from the Erd\H{o}s--R\'enyi graph with the same edge density $p$. This threshold is conjectured to be $d \asymp (nh(p))^3$, where $h(p)=p \log \frac{1}{p} + (1-p) \log \frac{1}{1-p}$ is the binary entropy function. Previous works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with $p=\Theta(1/n)$. In this paper, we prove that detection is impossible when $d\gg (nh(p))^3$ and $d\ge (1+\epsilon) n$ for any constant $\epsilon>0$, thereby resolving the conjecture in the regime $p\gtrsim n^{-2/3}/\log n$ and improving upon the state of the art in the regime $1/n \ll p \ll n^{-2/3}/\log n$. The key to our proof is a sharp analysis of the posterior distribution of the latent points given the observed graph, obtained through an information-theoretic comparison argument combined with strong log-concavity.

math.PR↗

arXiv2023

Sharp thresholds in inference of planted subgraphs

A major question in the study of the Erd\H{o}s--R\'enyi random graph is to understand the probability that it contains a given subgraph. This study originated in classical work of Erd\H{o}s and R\'enyi (1960). More recent work studies this question both in building a general theory of sharp versus coarse transitions (Friedgut and Bourgain 1999; Hatami, 2012) and in results on the location of the transition (Kahn and Kalai, 2007; Talagrand, 2010; Frankston, Kahn, Narayanan, Park, 2019; Park and Pham, 2022). In inference problems, one often studies the optimal accuracy of inference as a function of the amount of noise. In a variety of sparse recovery problems, an ``all-or-nothing (AoN) phenomenon'' has been observed: Informally, as the amount of noise is gradually increased, at some critical threshold the inference problem undergoes a sharp jump from near-perfect recovery to near-zero accuracy (Gamarnik and Zadik, 2017; Reeves, Xu, Zadik, 2021). We can regard AoN as the natural inference analogue of the sharp threshold phenomenon in random graphs. In contrast with the general theory developed for sharp thresholds of random graph properties, the AoN phenomenon has only been studied so far in specific inference settings. In this paper we study the general problem of inferring a graph $H=H_n$ planted in an Erd\H{o}s--R\'enyi random graph, thus naturally connecting the two lines of research mentioned above. We show that questions of AoN are closely connected to first moment thresholds, and to a generalization of the so-called Kahn--Kalai expectation threshold that scans over subgraphs of $H$ of edge density at least $q$. In a variety of settings we characterize AoN, by showing that AoN occurs if and only if this ``generalized expectation threshold'' is roughly constant in $q$. Our proofs combine techniques from random graph theory and Bayesian inference.

math.ST↗

arXiv2022

Locality of critical percolation on expanding graph sequences

We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probability for the emergence of an infinite component on $G$ coincide with the critical probability for the emergence of a linear-sized component on $G_n$? In this short article we give a positive answer provided the graphs $G_n$ satisfy an expansion condition, and the limiting graph $G$ has finite expected root degree. The main result of Benjamini, Nachmias, and Peres (2011), where this question was first formulated, showed the result assuming the $G_n$ satisfy a uniform degree bound and uniform expansion condition, and converge to a deterministic limit $G$. Later work of Sarkar (2021) extended the result to allow for a random limit $G$, but still required a uniform degree bound and uniform expansion for $G_n$. Our result replaces the degree bound on $G_n$ with the (milder) requirement that $G$ must have finite expected root degree. Our proof is a modification of the previous results, using a pruning procedure and the second moment method to control unbounded degrees.

math.PR↗

arXiv2022

A second moment proof of the spread lemma

This note concerns a well-known result which we term the ``spread lemma,'' which establishes the existence (with high probability) of a desired structure in a random set. The spread lemma was central to two recent celebrated results: (a) the improved bounds of Alweiss, Lovett, Wu, and Zhang (2019) on the Erd\H{o}s-Rado sunflower conjecture; and (b) the proof of the fractional Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park (2019). While the lemma was first proved (and later refined) by delicate counting arguments, alternative proofs have also been given, via Shannon's noiseless coding theorem (Rao, 2019), and also via manipulations of Shannon entropy bounds (Tao, 2020). In this note we present a new proof of the spread lemma, that takes advantage of an explicit recasting of the proof in the language of Bayesian statistical inference. We show that from this viewpoint the proof proceeds in a straightforward and principled probabilistic manner, leading to a truncated second moment calculation which concludes the proof. The proof can also be viewed as a demonstration of the ``planting trick'' introduced by Achlioptas and Coga-Oghlan (2008) in the study of random constraint satisfaction problems.

math.CO↗

arXiv2022

On the Second Kahn--Kalai Conjecture

For any given graph $H$, we are interested in $p_\mathrm{crit}(H)$, the minimal $p$ such that the Erd\H{o}s-R\'enyi graph $G(n,p)$ contains a copy of $H$ with probability at least $1/2$. Kahn and Kalai (2007) conjectured that $p_\mathrm{crit}(H)$ is given up to a logarithmic factor by a simpler "subgraph expectation threshold" $p_\mathrm{E}(H)$, which is the minimal $p$ such that for every subgraph $H'\subseteq H$, the Erd\H{o}s-R\'enyi graph $G(n,p)$ contains \emph{in expectation} at least $1/2$ copies of $H'$. It is trivial that $p_\mathrm{E}(H) \le p_\mathrm{crit}(H)$, and the so-called "second Kahn-Kalai conjecture" states that $p_\mathrm{crit}(H) \lesssim p_\mathrm{E}(H) \log e(H)$ where $e(H)$ is the number of edges in $H$. In this article, we present a natural modification $p_\mathrm{E, new}(H)$ of the Kahn--Kalai subgraph expectation threshold, which we show is sandwiched between $p_\mathrm{E}(H)$ and $p_\mathrm{crit}(H)$. The new definition $p_\mathrm{E, new}(H)$ is based on the simple observation that if $G(n,p)$ contains a copy of $H$ and $H$ contains \emph{many} copies of $H'$, then $G(n,p)$ must also contain \emph{many} copies of $H'$. We then show that $p_\mathrm{crit}(H) \lesssim p_\mathrm{E, new}(H) \log e(H)$, thus proving a modification of the second Kahn--Kalai conjecture. The bound follows by a direct application of the set-theoretic "spread" property, which led to recent breakthroughs in the sunflower conjecture by Alweiss, Lovett, Wu and Zhang and the first fractional Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park.

math.CO↗

arXiv2022

Sharp threshold sequence and universality for Ising perceptron models

We study a family of Ising perceptron models with $\{0,1\}$-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins--Xu (2021) and Abbe--Li--Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).

math.PR↗

arXiv2021

Gardner formula for Ising perceptron models at small densities

We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth--Mezard 1989, see also Gardner--Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero.

math.PR↗

arXiv2019

Breaking of 1RSB in random MAX-NAE-SAT

For several models of random constraint satisfaction problems, it was conjectured by physicists and later proved that a sharp satisfiability transition occurs. For random $k$-SAT and related models it happens at clause density $α$ around $2^k$. Just below the threshold, further results suggest that the solution space has a "1RSB" structure of a large bounded number of near-orthogonal clusters inside the space of variable assignments $\{0,1\}^N$. In the unsatisfiable regime, it is natural to consider max-satisfiability: violating the least number of constraints. For a simplified variant, the strong refutation problem, there is strong evidence that an algorithmic transition occurs around $α= N^{k/2-1}$. For $α$ bounded in $N$, a very precise estimate of the max-sat value was obtained by Achlioptas, Naor, and Peres (2007), but it is not sharp enough to indicate the nature of the energy landscape. Later work (Sen, 2016; Panchenko, 2016) shows that for $α$ very large (roughly, above $64^k$) the max-sat value approaches the mean-field (complete graph) limit: this is conjectured to have an "FRSB" structure where near-optimal configurations form clusters within clusters, in an ultrametric hierarchy of infinite depth inside $\{0,1\}^N$. A stronger form of FRSB was shown in several recent works to have algorithmic implications (again, in complete graphs). Consequently we find it of interest to understand how the model transitions from 1RSB near the satisfiability threshold, to (conjecturally) FRSB for large $α$. In this paper we show that in the random regular $k$-NAE-SAT model, the 1RSB description breaks down already above $α\asymp 4^k/k^3$. This is proved by an explicit perturbation in the 2RSB parameter space, inspired by the "bug proliferation" mechanism proposed by physicists (Montanari and Ricci-Tersenghi, 2003; Krzakala, Pagnani, and Weigt, 2004).

math.PR↗

arXiv2018

Capacity lower bound for the Ising perceptron

We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube $\{-1,+1\}^N$ intersected by $M$ random half-spaces. The perceptron's capacity is $α_N \equiv M_N/N$ for the largest integer $M_N$ such that the intersection in nonempty. It is conjectured by Krauth and Mézard (1989) that the (random) ratio $α_N$ converges in probability to an explicit constant $α_\star \doteq 0.83$. Kim and Roche (1998) proved the existence of a positive constant $γ$ such that $γ\le α_N \le 1-γ$ with high probability; see also Talagrand (1999). In this paper we show that the Krauth--Mézard conjecture $α_\star$ is a lower bound with positive probability, under the condition that an explicit univariate function $S(λ)$ is maximized at $λ=0$. Our proof is an application of the second moment method to a certain slice of perceptron configurations, as selected by the so-called TAP (Thouless, Anderson, and Palmer, 1977) or AMP (approximate message passing) iteration, whose scaling limit has been characterized by Bayati and Montanari (2011) and Bolthausen (2012). For verifying the condition on $S(λ)$ we outline one approach, which is implemented in the current version using (nonrigorous) numerical integration packages. In a future version of this paper we intend to complete the verification by implementing a rigorous numerical method.

math.PR↗

arXiv2016

Spectral algorithms for tensor completion

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the SOS approach does not scale well to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.

cs.DS↗

arXiv2016

The number of solutions for random regular NAE-SAT

Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain "survey propagation model" associated to this problem. We believe that these methods may be applicable in a wide class of related problems.

math.PR↗

arXiv2015

Shotgun assembly of random regular graphs

Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider this question for random regular graphs of fixed degree $d\ge3$. A result of Bollob\'as (1982) implies efficient recovery at $R = (1 + \epsilon) \frac12 \log_{d-1}n$ with high probability -- moreover, this recovery algorithm uses only a summary of the distances in each neighborhood. We show that using the full neighborhood structure gives a sharper bound \[ R = \frac{\log n + \log\log n}{2\log(d-1)} + O(1)\,, \] which we prove is tight up to the $O(1)$ term. One consequence of our proof is that if $G,H$ are independent graphs where $G$ follows the random regular law, then with high probability the graphs are non-isomorphic; furthermore, this can be efficiently certified by testing the $R$-neighborhood list of $H$ against the $R$-neighborhood of a single adversarially chosen vertex of $G$.

math.PR↗

arXiv2015

On the asymptotics of dimers on tori

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show that the dimer partition function on the quotient $\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 + \text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$, $f_0$ is the free energy density in the bulk, and $\text{fsc}$ is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for $\mathbb{L}$ non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.

math-ph↗

arXiv2015

Supercritical minimum mean-weight cycles

We study the weight and length of the minimum mean-weight cycle in the stochastic mean-field distance model, i.e., in the complete graph on $n$ vertices with edges weighted by independent exponential random variables. Mathieu and Wilson showed that the minimum mean-weight cycle exhibits one of two distinct behaviors, according to whether its mean weight is smaller or larger than $1/(ne)$; and that both scenarios occur with positive probability in the limit $n\to\infty$. If the mean weight is $< 1/(ne)$, the length is of constant order. If the mean weight is $> 1/(ne)$, it is concentrated just above $1/(n e)$, and the length diverges with $n$. The analysis of Mathieu--Wilson gives a detailed characterization of the subcritical regime, including the (non-degenerate) limiting distributions of the weight and length, but leaves open the supercritical behavior. We determine the asymptotics for the supercritical regime, showing that with high probability, the minimum mean weight is $(n e)^{-1}[1 + π^2/(2 \log^2 n) + O((\log n)^{-3})]$, and the cycle achieving this minimum has length on the order of $(\log n)^3$.

math.PR↗

arXiv2014

Proof of the satisfiability conjecture for large k

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_*(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha<\alpha_*$, and unsatisfiable for $\alpha>\alpha_*$. We show that the threshold $\alpha_*(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the 1-RSB universality class.

math.PR↗

arXiv2014

The Hausdorff dimension of the CLE gasket

The conformal loop ensemble $\mathrm{CLE}_κ$ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter $κ$ varies between $8/3$ and $8$; $\mathrm{CLE}_{8/3}$ is empty while $\mathrm {CLE}_8$ is a single space-filling loop. In this work, we study the geometry of the $\mathrm{CLE}$ gasket, the set of points not surrounded by any loop of the $\mathrm{CLE}$. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by $2-(8-κ)(3κ-8)/(32κ)$ when $4<κ<8$. Together with the work of Schramm-Sheffield-Wilson [Comm. Math. Phys. 288 (2009) 43-53] giving the upper bound for all $κ$ and the work of Nacu-Werner [J. Lond. Math. Soc. (2) 83 (2011) 789-809] giving the matching lower bound for $κ\le4$, this completes the determination of the $\mathrm{CLE}_κ$ gasket dimension for all values of $κ$ for which it is defined. The dimension agrees with the prediction of Duplantier-Saleur [Phys. Rev. Lett. 63 (1989) 2536-2537] for the FK gasket.

math.PR↗

arXiv2013

Factor models on locally tree-like graphs

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $ϕ$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $ϕ$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $ϕ$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $ϕ$. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.

math.PR↗