32 research outputs found
On the Spectral Gap of Spherical Spin Glass Dynamics
We consider the time to equilibrium for the Langevin dynamics of the
spherical -spin glass model of system size . We show that the log-Sobolev
constant and spectral gap are order in at sufficiently high temperature
whereas the spectral gap decays exponentially in at sufficiently low
temperatures. These verify the existence of a dynamical high temperature phase
and a dynamical glass phase at the level of the spectral gap. Key to these
results are the understanding of the extremal process and restricted free
energy of Subag--Zeitouni and Subag.Comment: 21 page
Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions
The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region beta<beta_c(q) of the q-state Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model.
On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 Lambda_n. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p 1 and p != p_c(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^2 log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures
Algorithmic thresholds for tensor PCA
We study the algorithmic thresholds for principal component analysis of
Gaussian -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.Comment: 34 pages. The manuscript has been updated to add a proof of what was
Conjecture 1 in the first versio
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
for general . These imply a host of new results,
including optimal mixing for the random cluster dynamics on torii
and boxes on vertices in at all high temperatures and at
sufficiently low temperatures, and for large values of quasi-polynomial (or
quasi-linear when ) 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 for general .Comment: 34 page
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 and a cluster weight . We establish
that for every , the random-cluster Glauber dynamics mixes in optimal
steps on -vertex random graphs having a prescribed degree
sequence with bounded average branching throughout the full
high-temperature uniqueness regime .
The family of random graph models we consider include the Erd\H{o}s--R\'enyi
random graph , and so we provide the first polynomial-time
sampling algorithm for the ferromagnetic Potts model on the Erd\H{o}s--R\'enyi
random graphs that works for all in the full uniqueness regime. We
accompany our results with mixing time lower bounds (exponential in the maximum
degree) for the Potts Glauber dynamics, in the same settings where our
bounds for the random-cluster Glauber dynamics apply. This
reveals a significant computational advantage of random-cluster based
algorithms for sampling from the Potts Gibbs distribution at high temperatures
in the presence of high-degree vertices.Comment: 45 pages, 3 figure