We introduce a notion called entropic independence that is an entropic analog of spectral notions of highdimensional expansion. Informally, entropic independence of a background distribution $\mu$ on $k$sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponentialsized state spaces.
In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified logSobolev inequalities, for downup random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified logSobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: $\mu$ is entropically independent exactly when a transformed version of the generating polynomial of $\mu$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence.
We apply our results to obtain (1) tight modified logSobolev inequalities and mixing times for multistep downup walks on fractionally logconcave distributions, (2) the tight mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$, and (3) nearlylinear time $\widetilde O_{\delta}(n)$ samplers for the hardcore and Ising models on $n$node graphs that have $\delta$relative gap to the treeuniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree $\Delta$ of the graph, and is therefore optimal even for highdegree graphs, and in fact, is sublinear in the size of the graph for highdegree graphs.
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Sampling Approximately LowRank Ising Models: MCMC meets Variational Methods
We consider Ising models on the hypercube with a general interaction matrix 𝐽, and give a polynomial time sampling algorithm when all but 𝑂(1) eigenvalues of 𝐽 lie in an interval of length one, a situation which occurs in many models of interest. This was previously known for the Glauber dynamics when \emph{all} eigenvalues fit in an interval of length one; however, a single outlier can force the Glauber dynamics to mix torpidly. Our general result implies the first polynomial time sampling algorithms for lowrank Ising models such as Hopfield networks with a fixed number of patterns and Bayesian clustering models with lowdimensional contexts, and greatly improves the polynomial time sampling regime for the antiferromagnetic/ferromagnetic Ising model with inconsistent field on expander graphs. It also improves on previous approximation algorithm results based on the naive meanfield approximation in variational methods and statistical physics. Our approach is based on a new fusion of ideas from the MCMC and variational inference worlds. As part of our algorithm, we define a new nonconvex variational problem which allows us to sample from an exponential reweighting of a distribution by a negative definite quadratic form, and show how to make this procedure provably efficient using stochastic gradient descent. On top of this, we construct a new simulated tempering chain (on an extended state space arising from the HubbardStratonovich transform) which overcomes the obstacle posed by large positive eigenvalues, and combine it with the SGDbased sampler to solve the full problem.
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 Award ID(s):
 1704417
 NSFPAR ID:
 10354700
 Date Published:
 Journal Name:
 Proceedings of Thirty Fifth Conference on Learning Theory, PMLR
 Volume:
 178
 Page Range / eLocation ID:
 49454988
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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null (Ed.)The primary objectives of International Ocean Discovery Program (IODP) Expedition 367/368 to the northern South China Sea (SCS) margin were to (1) examine its history of continental breakup and (2) compare it with other nonvolcanic or magmapoor rifted margins with the broader goal of testing models for continental breakup. A secondary objective was to further our understanding of the paleoceanographic and environmental development of the SCS and southeast Asia during the Cenozoic. Four primary sites were selected for the overall program: one in the outer margin high (OMH) and three seaward of the OMH on distinct, marginparallel basement ridges. These three ridges are informally labeled A, B, and C and are located in the continent–ocean transition (COT) zone ranging from the OMH to the interpreted steadystate oceanic crust (Ridge C) of the SCS. 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In addition, Expedition 368 added two more sites on the OMH (Sites U1504 and U1505). Expedition 367/368 completed operations at six of the seven sites (U1499–U1502, U1504, and U1505). Site U1503, however, was not completed beyond casing without coring to 990 m because of mechanical problems with the drilling equipment that prevented the expedition, after 25 May 2017, from operating with a drill string longer than 3400 m. New alternate Site U1504, proposed during Expedition 367, met this condition. Original Site U1505 also met the operational constraints of the 3400 m drill string (total) and was an alternate site for the alreadydrilled Site U1501. At Site U1499, we cored to 1081.8 m in 22.1 days with 52% recovery and then logged downhole data from 655 to 1020 m. In 31 days at Site U1500, we penetrated to 1529 m, cored a total of 1012.8 m with 37% recovery, and collected log data from 842 to 1133 m. At Site U1501, we cored to 697.1 m in 9.4 days with 78.5% recovery. 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Coring at Site U1504 on the OMH, located ~45 km east of Site U1501, recovered mostly foliated, greenschist facies metamorphic rocks below late Eocene(?) carbonate rocks (partly reef debris) and early Miocene to Pleistocene sediments. At Site U1505, we cored to 480.15 m through Pleistocene to late Oligocene mainly carbonaceous ooze followed at depth by early Oligocene siliciclastic sediments. Efforts were made at every drill site to correlate the core with the seismic data and seismic stratigraphic unconformities interpreted in the Eocene to Plio–Pleistocene sedimentary sequence prior to drilling. The predrilling interpretation of ages of these unconformities was in general confirmed by drilling results, although some nontrivial corrections can be expected from detailed postexpedition work on integrating seismic stratigraphic interpretations with detailed bio and lithostratigraphy. 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