arXiv subjects

Reza Gheissari

Publications and source records attributed to Reza Gheissari.

At least 37 records · Page 2Linked to original sources

arXiv2024

Metastability cascades and prewetting in the SOS model

We study Glauber dynamics for the low temperature $(2+1)$D Solid-On-Solid model on a box of side-length $n$ with a floor at height $0$ (inducing entropic repulsion) and a competing bulk external field $λ$ pointing down (the prewetting problem). In 1996, Cesi and Martinelli showed that if the inverse-temperature $β$ is large enough, then along a decreasing sequence of critical points $(λ_c^{(k)})_{k=0}^{K_β}$ the dynamics is torpid: its inverse spectral gap is $O(1)$ when $λ\in (λ_c^{(k+1)},λ_c^{(k)})$ whereas it is $\exp[Θ(n)]$ at each $λ_c^{(k)}$ for each $k\leq K_β$, due to a coexistence of rigid phases at heights $k+1$ and $k$. Our focus is understanding (a) the onset of metastability as $λ_n\uparrowλ_c^{(k)}$; and (b) the effect of an unbounded number of layers, as we remove the restriction $k\le K_β$, and even allow for $λ_n\to 0$ towards the $λ= 0$ case which has $O(\log n)$ layers and was studied by Caputo et al. (2014). We show that for any $k$, possibly growing with $n$, the inverse gap is $\exp[\tildeΘ(1/|λ_n-λ_c^{(k)}|)]$ as $λ\uparrow λ_c^{(k)}$ up to distance $n^{-1+o(1)}$ from this critical point, due to a metastable layer at height $k$ on the way to forming the desired layer at height $k+1$. By taking $λ_n = n^{-α}$ (corresponding to $k_n\asymp \log n$), this also interpolates down to the behavior of the dynamics when $λ=0$. We complement this by extending the fast mixing to all $λ$ uniformly bounded away from $(λ_c^{(k)})_{k=0}^\infty$. Together, these results provide a sharp understanding of the predicted infinite sequence of dynamical phase transitions governed by the layering phenomenon.

math.PR↗

arXiv2024

Critical wetting in the (2+1)D Solid-On-Solid model

In this note, we study the low temperature $(2+1)$D SOS interface above a hard floor with critical pinning potential $λ_w= \log (\frac{1}{1-e^{-4β}})$. At $λ<λ_w$ entropic repulsion causes the surface to delocalize and be rigid at height $\frac1{4β}\log n+O(1)$; at $λ>λ_w$ it is localized at some $O(1)$ height. We show that at $λ=λ_w$, there is delocalization, with rigidity now at height $\lfloor \frac1{6β}\log n+\frac13\rfloor$, confirming a conjecture of Lacoin.

math.PR↗

arXiv2024

Entropic repulsion of 3D Ising interfaces conditioned to stay above a floor

We study the interface of the Ising model in a box of side-length $n$ in $\mathbb Z^3$ at low temperature $1/β$ under Dobrushin's boundary conditions, conditioned to stay in a half-space above height $h$ (a hard floor). Without this conditioning, Dobrushin showed in 1972 that typically most of the interface is flat at height $0$. With the floor, for small $h$, the model is expected to exhibit {\it entropic repulsion}, where the typical height of the interface lifts off of $0$. Detailed understanding of the SOS model -- a more tractable height function approximation of 3D Ising -- due to Caputo et al., suggests that there is a single integer value $-h_n^* \sim -c\log n$ of the floor height, delineating the transition between rigidity at height $0$ and entropic repulsion. We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for the typical Ising interface above a hard floor at $h$, all but an $ε(β)$-fraction of the sites are propelled to be above height $0$ if $h < h_n^*-1$, whereas all but an $ε(β)$-fraction of the sites remain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that the typical height of the unconditional maximum is $(2c_\star + o(1))\log n$; this confirms scaling predictions from the SOS approximation.

math.PR↗

arXiv2023

Spatial mixing and the random-cluster dynamics on lattices

An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the Glauber dynamics for the Ising and Potts models. Glauber dynamics for the random-cluster model does not naturally fit into this spin systems framework because its transition rules are not local. In this paper, we present various implications between weak spatial mixing, strong spatial mixing, and the newer notion of spatial mixing within a phase, and mixing time bounds for the random-cluster dynamics in finite subsets of $\mathbb Z^d$ for general $d\ge 2$. These imply a host of new results, including optimal $O(N\log N)$ mixing for the random cluster dynamics on torii and boxes on $N$ vertices in $\mathbb Z^d$ at all high temperatures and at sufficiently low temperatures, and for large values of $q$ quasi-polynomial (or quasi-linear when $d=2$) mixing time bounds from random phase initializations on torii at the critical point (where by contrast the mixing time from worst-case initializations is exponentially large). In the same parameter regimes, these results translate to fast sampling algorithms for the Potts model on $\mathbb Z^d$ for general $d$.

math.PR↗

arXiv2023

High-dimensional limit theorems for SGD: Effective dynamics and critical scaling

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step-size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.

stat.ML↗

arXiv2023

Sampling from Potts on random graphs of unbounded degree via random-cluster dynamics

We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establish that for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal $Θ(n\log n)$ steps on $n$-vertex random graphs having a prescribed degree sequence with bounded average branching $γ$ throughout the full high-temperature uniqueness regime $p<p_u(q,γ)$. The family of random graph models we consider includes the Erdős--Rényi random graph $G(n,γ/n)$, and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erdős--Rényi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our $Θ(n \log n)$ bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.

math.PR↗

arXiv2023

Cutoff for the Glauber dynamics of the lattice free field

The Gaussian Free Field (GFF) is a canonical random surface in probability theory generalizing Brownian motion to higher dimensions. In two dimensions, it is critical in several senses, and is expected to be the universal scaling limit of a host of random surface models in statistical physics. It also arises naturally as the stationary solution to the stochastic heat equation with additive noise. Focusing on the dynamical aspects of the corresponding universality class, we study the mixing time, i.e., the rate of convergence to stationarity, for the canonical prelimiting object, namely the discrete Gaussian free field (DGFF), evolving along the (heat-bath) Glauber dynamics. While there have been significant breakthroughs made in the study of cutoff for Glauber dynamics of random curves, analogous sharp mixing bounds for random surface evolutions have remained elusive. In this direction, we establish that on a box of side-length $n$ in $\mathbb Z^2$, when started out of equilibrium, the Glauber dynamics for the DGFF exhibit cutoff at time $\frac{2}{π^2}n^2 \log n$.

math.PR↗

arXiv2021

Approximate domain Markov property for rigid Ising interfaces

Consider the Ising model on a centered box of side length $n$ in $\mathbb Z^d$ with $\mp$-boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions $d\ge 3$, at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has $O(1)$ height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-$h$ level curve of the interface, bounding a set $S \subset \mathbb Z^{d-1}$, along with the entire interface outside the cylinder $S\times \mathbb Z$: what does the interface in $S\times \mathbb Z$ look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on $S$ only depend on the heights on $S^c$ through the heights on $\partial S$. The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside $S\times \mathbb Z$. Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin's result to this setting, showing the interface in $S\times \mathbb Z$ is rigid about height $h$, with exponential tails on its height oscillations. Then we show that the typical tall pillars in $S\times \mathbb Z$ are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and Gumbel tail bounds on the maximum oscillations in $S\times \mathbb Z$ about height $h$, showing that these only depend on the conditioning through the cardinality of $S$.

math.PR↗

arXiv2021

Online stochastic gradient descent on non-convex losses from high-dimensional inference

Stochastic gradient descent (SGD) is a popular algorithm for optimization problems arising in high-dimensional inference tasks. Here one produces an estimator of an unknown parameter from independent samples of data by iteratively optimizing a loss function. This loss function is random and often non-convex. We study the performance of the simplest version of SGD, namely online SGD, from a random start in the setting where the parameter space is high-dimensional. We develop nearly sharp thresholds for the number of samples needed for consistent estimation as one varies the dimension. Our thresholds depend only on an intrinsic property of the population loss which we call the information exponent. In particular, our results do not assume uniform control on the loss itself, such as convexity or uniform derivative bounds. The thresholds we obtain are polynomial in the dimension and the precise exponent depends explicitly on the information exponent. As a consequence of our results, we find that except for the simplest tasks, almost all of the data is used simply in the initial search phase to obtain non-trivial correlation with the ground truth. Upon attaining non-trivial correlation, the descent is rapid and exhibits law of large numbers type behavior. We illustrate our approach by applying it to a wide set of inference tasks such as phase retrieval, and parameter estimation for generalized linear models, online PCA, and spiked tensor models, as well as to supervised learning for single-layer networks with general activation functions.

stat.ML↗

arXiv2021

Random-cluster dynamics on random regular graphs in tree uniqueness

We establish rapid mixing of the random-cluster Glauber dynamics on random $Δ$-regular graphs for all $q\ge 1$ and $p 2$ the Glauber dynamics on random $Δ$-regular graphs undergoes an exponential slowdown at $p_u(q,Δ)$. More precisely, we show that for every $q\ge 1$, $Δ\ge 3$, and $p<p_u(q,Δ)$, with probability $1-o(1)$ over the choice of a random $Δ$-regular graph on $n$ vertices, the Glauber dynamics for the random-cluster model has $Θ(n \log n)$ mixing time. As a corollary, we deduce fast mixing of the Swendsen--Wang dynamics for the Potts model on random $Δ$-regular graphs for every $q\ge 2$, in the tree uniqueness region. Our proof relies on a sharp bound on the "shattering time", i.e., the number of steps required to break up any configuration into $O(\log n)$ sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.

math.PR↗

arXiv2021

Local and global geometry of the 2D Ising interface in critical pre-wetting

Consider the Ising model at low-temperatures and positive external field $λ$ on an $N\times N$ box with Dobrushin boundary conditions that are plus on the north, east, and west boundaries and minus on the south boundary. If $λ= 0$, the interface separating the plus and minus phases is diffusive, having $O(\sqrt N)$ height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are $O(1)$ and the interface is only partially wetted, being pinned to its southern boundary. We study the critical pre-wetting regime of $λ_N \downarrow 0$, where the height fluctuations are expected to scale as $λ^{ -1/3}$ and the rescaled interface is predicted to converge to the Ferrari--Spohn diffusion. Velenik (2004) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts. In this paper, we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical pre-wetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving $e^{ - Θ(x^{3/2})}$ upper tails reminiscent of the Tracy--Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines, and the poly-logarithmic pre-factor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface, with novel local resampling and coupling schemes.

math.PR↗

arXiv2021

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $μ$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

math.PR↗

arXiv2020

Tightness and tails of the maximum in 3D Ising interfaces

Consider the 3D Ising model on a box of side length $n$ with minus boundary conditions above the $xy$-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height $M_n$ of this interface: for every $β$ large, $M_n/ \log n\to c_β$ in probability as $n\to\infty$. Here we show that the laws of the centered maxima $(M_n - \mathbb{E}[M_n])_{n\geq 1}$ are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to $O(1)$ precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.

math.PR↗

arXiv2020

Maximum and shape of interfaces in 3D Ising crystals

Dobrushin (1972) showed that the interface of a 3D Ising model with minus boundary conditions above the $xy$-plane and plus below is rigid (has $O(1)$-fluctuations) at every sufficiently low temperature. Since then, basic features of this interface -- such as the asymptotics of its maximum -- were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D Solid-On-Solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side-length $n$ with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for $M_n$, its maximum: if the inverse-temperature $β$ is large enough, then $M_n / \log n \to 2/α_β$ as $n\to\infty$, in probability, where $α_β$ is given by a large deviation rate in infinite volume. We further show that, on the large deviation event that the interface connects the origin to height $h$, it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in $h$) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number $T$ of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in $T$. These results generalize to every dimension $d\geq 3$.

math.PR↗

arXiv2019

Local minima in disordered mean-field ferromagnets

We consider the complexity of random ferromagnetic landscapes on the hypercube $\{\pm 1\}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $\mathcal G(N,p)$. Previous results had shown that, with high probability as $N\to\infty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as $N\to\infty$. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.

cond-mat.dis-nn↗

arXiv2019

Bounding flows for spherical spin glass dynamics

We introduce a new approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, we obtain an approximate phase diagram for the evolution of the energy $H$ and its gradient under Langevin dynamics for spherical $p$-spin models. We then derive several consequences of this phase diagram. For example, at any temperature, uniformly over all starting points, the process must reach and remain in an absorbing region of large negative values of $H$ and large (in norm) gradients in order 1 time. Furthermore, if the process starts in a neighborhood of a critical point of $H$ with negative energy, then both the gradient and energy must increase macroscopically under this evolution, even if this critical point is a saddle with index of order $N$. As a key technical tool, we estimate Sobolev norms of spin glass Hamiltonians, which are of independent interest.

math.PR↗

arXiv2019

Nature vs. Nurture: Dynamical Evolution in Disordered Ising Ferromagnets

We study the predictability of zero-temperature Glauber dynamics in various models of disordered ferromagnets. This is analyzed using two independent dynamical realizations with the same random initialization (called twins). We derive, theoretically and numerically, trajectories for the evolution of the normalized magnetization and twin overlap as the system size tends to infinity. The systems we treat include mean-field ferromagnets with light-tailed and heavy-tailed coupling distributions, as well as highly-disordered models with a variety of other geometries. In the mean-field setting with light-tailed couplings, the disorder averages out and the limiting trajectories of the magnetization and twin overlap match those of the homogenous Curie--Weiss model. On the other hand, when the coupling distribution has heavy tails, or the geometry changes, the effect of the disorder persists in the thermodynamic limit. Nonetheless, qualitatively all such random ferromagnets share a similar time evolution for their twin overlap, wherein the two twins initially decorrelate, before either partially or fully converging back together due to the ferromagnetic drift.

cond-mat.stat-mech↗

arXiv2019

Algorithmic thresholds for tensor PCA

We study the algorithmic thresholds for principal component analysis of Gaussian $k$-tensors with a planted rank-one spike, via Langevin dynamics and gradient descent. In order to efficiently recover the spike from natural initializations, the signal to noise ratio must diverge in the dimension. Our proof shows that the mechanism for the success/failure of recovery is the strength of the "curvature" of the spike on the maximum entropy region of the initial data. To demonstrate this, we study the dynamics on a generalized family of high-dimensional landscapes with planted signals, containing the spiked tensor models as specific instances. We identify thresholds of signal-to-noise ratios above which order 1 time recovery succeeds; in the case of the spiked tensor model these match the thresholds conjectured for algorithms such as Approximate Message Passing. Below these thresholds, where the curvature of the signal on the maximal entropy region is weak, we show that recovery from certain natural initializations takes at least stretched exponential time. Our approach combines global regularity estimates for spin glasses with point-wise estimates, to study the recovery problem by a perturbative approach.

math.PR↗