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Wei-Kuo Chen

Publications and source records attributed to Wei-Kuo Chen.

At least 19 recordsLinked to original sources
arXiv2026

Complexity of the p-spin Hamiltonian with a Non-Rotationally Invariant Potential

We investigate the complexity of the Hamiltonian in the pure $p$-spin spin glass model accompanied with a polynomial-type potential on $\mathbb{R}^N$. In this Hamiltonian, the Gaussian field is anisotropic, and the potential lacks rotational invariance. Our main result derives the logarithmic limit for the expected number of critical points in terms of a variational formula. As a consequence, by identifying the critical location of the phase transition from our representation, we provide an upper bound for the ground state energy of the model.

math.PR↗
arXiv2025

Joint parameter estimations for spin glasses

Spin glass models with quadratic-type Hamiltonians are disordered statistical physics systems with competing ferromagnetic and anti-ferromagnetic spin interactions. The corresponding Gibbs measures belong to the exponential family parametrized by (inverse) temperature $β>0$ and external field $h\in\mathbb{R}$. Given a sample from these Gibbs measures, a statistically fundamental question is to infer the temperature and external field parameters. In 2007, Chatterjee (Ann. Statist. 35 (2007), no.5, 1931-1946) first proved that in the absence of external field $h=0$, the maximum pseudolikelihood estimator for $β$ is $\sqrt{N}$-consistent under some mild assumptions on the disorder matrices. It was left open whether the same method can be used to estimate the temperature and external field simultaneously. In this paper, under some easily verifiable conditions, we prove that the bivariate maximum pseudolikelihood estimator is indeed jointly $\sqrt{N}$-consistent for the temperature and external field parameters. The examples cover the classical Sherrington-Kirkpatrick model and its diluted variants.

math.PR↗
arXiv2025

On the Fitness Landscape in the $NK$ Model

The $NK$ model, introduced by Kauffman, Levin, and Weinberger, is a random field used to describe the fitness landscape of certain species with $N$ genetic loci, each interacting with $K$ others. The model has wide applications in understanding evolutionary and natural selection as it captures ruggedness feature of the fitness landscape. Earlier literature has been focused on the case $K$ being a fixed positive integer and used tools from Ergodic and Markov theory. In this paper, by viewing it as a statistical physics object, we investigate the $NK$ model in the regime $K/N\toα\in(0,1]$ via the spin glass methodologies. Our main result identifies the exact limits for the free energy at any temperature and the maximum fitness. Moreover, we show that the $NK$ model exhibits a multiple-peak structure, namely, the number of near-fittest genomes that are asymptotically orthogonal to each other is exponentially large. Based on establishing the overlap gap properties, we obtain quantitative descriptions for the geometry of the fitness landscape and deduce that, in particular, near-fittest evolutionary paths become impossible as the fitness levels of the genomes approach the global maximum for any $α\in (0,1]$. Nevertheless, we also show that by choosing $α$ sufficiently small, an evolutionary path maintained at a given fitness level can be constructed with high probability.

math.PR↗
arXiv2025

Disorder Chaos in Short-Range, Diluted, and Lévy Spin Glasses

In a recent breakthrough [arXiv:2301.04112], Chatterjee proved site disorder chaos in the Edwards-Anderson (EA) short-range spin glass model utilizing the Hermite spectral method. In this paper, we demonstrate the further usefulness of this Hermite spectral approach by extending the validity of site disorder chaos in three related spin glass models. The first, called the mixed even $p$-spin short-range model, is a generalization of the EA model where the underlying graph is a deterministic bounded degree hypergraph consisting of hyperedges with even number of vertices. The second model is the diluted mixed $p$-spin model, which is allowed to have hyperedges with both odd and even number of vertices. For both models, our results hold under general symmetric disorder distributions. The main novelty of our argument is played by an elementary algebraic equation for the Fourier-Hermite series coefficients for the two-spin correlation functions. It allows us to deduce necessary geometric conditions to determine the contributing coefficients in the overlap function, which in spirit is the same as the crucial Lemma 1 in [arXiv:2301.04112]. Finally, we also establish disorder chaos in the Lévy model with stable index $α\in (1, 2)$.

math.PR↗
arXiv2024

On the Replica Symmetric Solution in General Diluted Spin Glasses

We present a unifying approach to studying the replica symmetric solution in general diluted spin glass models on random $p$-uniform hypergraphs with sparsity parameter $α$. Our result shows that there exist two key regimes in which the model exhibits replica symmetry and the free energy can be explicitly represented as the evaluation of an energy functional at the unique fixed point of a recursive distributional equation. One is called the high temperature regime, where the temperature and the sparsity parameter are essentially inversely proportional to each other; the other is the subcritical regime defined as $αp (p-1)\leq 1$. In particular, the fact that the second regime is independent of the temperature parameter further allows us to deduce an analogous representation of the ground state energy in the subcritical regime. Along the way, we revisit several well-known formulas and also derive new ones for the free and ground state energies in the constraint satisfaction problem, Potts model, XY model, and continuous hardcore model.

math.PR↗
arXiv2024

A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses

For any convex function $F$ of $n$-dimensional Gaussian vector $g$ with $\mathbb{E} e^{λF(g)}<\infty$ for any $λ>0$, we show that $λ^{-1}\ln \mathbb{E} e^{λF(g)}$ is convex in $λ\in\mathbb{R}$. Based on this convexity, we draw three major consequences. The first recovers a version of the Paouris-Valettas lower deviation inequality for Gaussian convex functions with an improved exponent. The second establishes a quantitative bound for the Dotsenko-Franz-Mézard conjecture arising from the study of the Sherrington-Kirkpatrick (SK) mean-field spin glass model, which states that in the absence of external field, the annealed free energy of negative replica is asymptotically equal to the free energy. The last further establishes the differentiability for this annealed free energy with respect to the negative replica variable at any temperature and external field.

math.PR↗
arXiv2024

Some Rigorous Results on the Lévy Spin Glass Model

We study the Lévy spin glass model, a fully connected model on $N$ vertices with heavy-tailed interactions governed by a power law distribution of order $0<α<2.$ Our investigation is divided into three cases $0<α<1$, $α=1$, and $1<α<2.$ When $1<α<2,$ we identify a high temperature regime, in which the limit and fluctuation of the free energy are explicitly obtained and the site and bond overlaps are shown to exhibit concentration, interestingly, while the former is concentrated around zero, the latter obeys a positivity behavior. At any temperature, we further establish the existence of the limiting free energy and derive a variational formula analogous to Panchenko's framework in the setting of the Poissonian Viana-Bray model. For $α=1$, the free energy scales super-linearly and converges to a constant proportional to $β$ in probability at any temperature. In the case of $0<α<1$, the scaling for the free energy is again super-linear, however, it converges weakly to the sum of a Poisson Point Process at any temperature. Additionally, we show that the Gibbs measure puts most of its mass on the configurations that align with signs of the polynomially many heaviest edge weights.

math.PR↗
arXiv2024

On the TAP equations via the cavity approach in the generic mixed $p$-spin models

In 1977, Thouless, Anderson, and Palmer (TAP) derived a system of consistent equations in terms of the effective magnetization in order to study the free energy in the Sherrington-Kirkpatrick (SK) spin glass model. The solutions to their equations were predicted to contain vital information about the landscapes in the SK Hamiltonian and the TAP free energy and moreover have direct connections to Parisi's replica ansatz. In this work, we aim to investigate the validity of the TAP equations in the generic mixed $p$-spin model. By utilizing the ultrametricity of the overlaps, we show that the TAP equations are asymptotically satisfied by the conditional local magnetizations on the asymptotic pure states.

math.PR↗
arXiv2023

Free energy of a diluted spin glass model with quadratic Hamiltonian

We study a diluted mean-field spin glass model with a quadratic Hamiltonian. Our main result establishes the limiting free energy in terms of an integral of a family of random variables that are the weak limits of the quenched variances of the spins in the system with varying edge connectivity. The key ingredient in our argument is played by the identification of these random variables as the unique solution to a recursive distributional equation. Our results in particular provide the first example of the diluted Shcherbina-Tirozzi model, whose limiting free energy can be derived at any inverse temperature and external field.

math.PR↗
arXiv2023

Universality of Superconcentration in the Sherrington-Kirkpatrick Model

We study the universality of superconcentration for the free energy in the Sherrington-Kirkpatrick (SK) model. In arXiv:0907.3381, Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log{N}$, in contrast to the $O(N)$ bound obtained from the Gaussian-Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing function $f$ that arises in the representation of the disorder as $f(g)$ for $g$ standard normal. Under an additional regularity assumption on $f$, we further show that the variance is of order at most $N/\log{N}$.

math.PR↗
arXiv2021

Performance of Bayesian linear regression in a model with mismatch

In this paper we analyze, for a model of linear regression with gaussian covariates, the performance of a Bayesian estimator given by the mean of a log-concave posterior distribution with gaussian prior, in the high-dimensional limit where the number of samples and the covariates' dimension are large and proportional. Although the high-dimensional analysis of Bayesian estimators has been previously studied for Bayesian-optimal linear regression where the correct posterior is used for inference, much less is known when there is a mismatch. Here we consider a model in which the responses are corrupted by gaussian noise and are known to be generated as linear combinations of the covariates, but the distributions of the ground-truth regression coefficients and of the noise are unknown. This regression task can be rephrased as a statistical mechanics model known as the Gardner spin glass, an analogy which we exploit. Using a leave-one-out approach we characterize the mean-square error for the regression coefficients. We also derive the log-normalizing constant of the posterior. Similar models have been studied by Shcherbina and Tirozzi and by Talagrand, but our arguments are much more straightforward. An interesting consequence of our analysis is that in the quadratic loss case, the performance of the Bayesian estimator is independent of a global "temperature" hyperparameter and matches the ridge estimator: sampling and optimizing are equally good.

math.PR↗
arXiv2021

The SK model is infinite step replica symmetry breaking at zero temperature

We prove that the Parisi measure of the mixed p-spin model at zero temperature has infinitely many points in its support. This establishes Parisi's prediction that the functional order parameter of the Sherrington-Kirkpatrick model is not a step function at zero temperature. As a consequence, we show that the number of levels of broken replica symmetry in the Parisi formula of the free energy diverges as the temperature goes to zero.

math.PR↗
arXiv2021

On convergence of the cavity and Bolthausen's TAP iterations to the local magnetization

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work [Comm. Math. Phys., 325(1):333-366, 2014], Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen's iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen's scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated.

math-ph↗
arXiv2020

On $\ell_p$-Gaussian-Grothendieck problem

For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell_p$-Gaussian-Grothendieck problem defined as \begin{align*}\max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}.\end{align*} The case $p=2$ corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble; when $p=\infty$, the maximum value is essentially the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty.$ For the former, we compute the limit of the $\ell_p$-Gaussian-Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order.

math.PR↗
arXiv2020

Phase transition in random tensors with multiple independent spikes

Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian $p$-tensor for $p\geq 3$ and signal in the form of a symmetric low-rank random tensor. The latter is defined as a linear combination of $k$ independent symmetric rank-one random tensors, referred to as spikes, with weights referred to as signal-to-noise ratios (SNRs). The entries of the vectors that determine the spikes are i.i.d. sampled from general probability distributions supported on bounded subsets of $\mathbb{R}$. This work focuses on the problem of detecting the presence of these spikes, and establishes the phase transition of this detection problem for any fixed $k \geq 1$. In particular, it shows that for a set of relatively low SNRs it is impossible to distinguish between the spiked and non-spiked Gaussian tensors. Furthermore, in the interior of the complement of this set, where at least one of the $k$ SNRs is relatively high, these two tensors are distinguishable by the likelihood ratio test. In addition, when the total number of low-rank components, $k$, of the $p$-tensor of size $N$ grows in the order $o(N^{(p-2)/4})$ as $N$ tends to infinity, the problem exhibits an analogous phase transition. This theory for spike detection is also shown to imply that recovery of the spikes by the minimum mean square error exhibits the same phase transition. The main methods used in this work arise from the study of mean field spin glass models, where the phase transition thresholds are identified as the critical inverse temperatures distinguishing the high and low-temperature regimes of the free energies. In particular, our result formulates the first full characterization of the high temperature regime for vector-valued spin glass models with independent coordinates.

math.PR↗
arXiv2020

The generalized TAP free energy

We consider the mixed $p$-spin mean-field spin glass model with Ising spins and investigate its free energy in the spirit of the TAP approach, named after Thouless, Anderson, and Palmer. More precisely, we define and compute the generalized TAP correction, and establish the corresponding generalized TAP representation for the free energy. In connection with physicists' replica theory, we introduce the notion of generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameters match the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, our result shows that the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards-Anderson parameter.

math.PR↗
arXiv2020

Universality of Approximate Message Passing Algorithms

We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an $n\times n$ random symmetric matrix $A$. We establish universality in noise for this AMP in the $n$-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.

math.PR↗