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作者：Caragea, Andrei; Petersen, Philipp; Voigtlaender, Felix
作者单位：University of Vienna
摘要：We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization us-ing networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input d...
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作者：Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan
作者单位：University of Wisconsin System; University of Wisconsin Madison; Beijing Institute of Technology; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
摘要：In this paper we study the perturbation theory of ?42 model on the whole plane via stochastic quantization. We use integration by parts formula (i.e., Dyson-Schwinger equations) to generate the perturbative expansion for the k-point correlation functions, and prove bounds on the remainder of the trun-cated expansion using PDE estimates; this in particular proves that the ex-pansion is asymptotic. Furthermore, we derive short distance behaviors of the 2-point function and the connected 4-point ...
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作者：Genovese, Giuseppe
作者单位：University of Zurich
摘要：We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming inde-pendent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for case.
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作者：Mueller, Carl; Neuman, Eyal
作者单位：University of Rochester; Imperial College London
摘要：We study elastic manifolds with self-repelling terms and estimate their effective radius. This class of manifolds is modelled by a self-repelling vector-valued Gaussian free field with Neumann boundary conditions over the domain [-N,N](d)boolean AND Z(d), that takes values in R-d. Our main result states that in two dimensions (d=2), the effective radius R-N of the manifold is approximately N. This verifies the conjecture of Kantor, Kardar and Nelson (Phys. Rev. Lett. 58 (1987) 1289-1292) up to...
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作者：Anagnostakis, Alexis; Lejay, Antoine; Villemonais, Denis
作者单位：Universite de Lorraine; Inria; Centre National de la Recherche Scientifique (CNRS)
摘要：We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to (1/4) perpendicular to (1/p) in terms of the maximum cell size of the grid, for any p-Wasserstein distance. We also show that it is possible to achieve any rate strictly inf...
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作者：Croydon, David A.; Fukushima, Ryoki; Junk, Stefan
作者单位：Kyoto University; University of Tsukuba; Tohoku University
摘要：We derive an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time befor...
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作者：Henning, Florian; Kuelske, Christof
作者单位：Ruhr University Bochum
摘要：We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invari-ant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation inv...
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作者：Lacker, Daniel; Le Flem, Luc
作者单位：Columbia University
摘要：This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points, as n & RARR; & INFIN;, of closed -loop approximate equilibria from the corresponding n-player games. This ex-tends to the common noise setting a recent result of the first author, while also simplifying a key step in the proof and allowing unbounded coefficients and non-i.i.d. initial conditions. Co...
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作者：Damron, Michael; Hanson, Jack; Lam, Wai -Kit
作者单位：University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY); National Taiwan University
摘要：We consider first-passage percolation (FPP) on the triangular lattice with vertex weights (t(v)) whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zeroweight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T (0, partial derivative B(n)) the first-passage time from 0 to {x : parallel to x parallel to(infinity) = n}, we show existence of a time constant and find its ex...
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作者：Yang, Jun; Rosenthal, Jeffrey S.
作者单位：University of Oxford; University of Toronto
摘要：This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization ap-proach, which establishes generalized drift conditions defined in subsets of the state space. The subsets are called the large sets, and are chosen to rule out some bad states which have poor drift property when the dimension of the state space gets large. Using the large sets together with a f...
