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Learning binary latent variable models: A tensor eigenpair approach

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arXiv 2018年

作者： Jaffe, Ariel Weiss, Roi Carmi, Shai Kluger, Yuval Nadler, Boaz Dept. of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot7610001 Israel Braun School of Public Health and Community Medicine Hebrew University of Jerusalem Jerusalem9112102 Israel Interdepartmental Program in Computational Biology and Bioinformatics Yale University New HavenCT06511 United States Dept. of Pathology Yale Cancer Center Yale University School of Medicine New HavenCT06520 United States Program of Applied Mathematics Yale University New HavenCT06511 United States

Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and third order moment tensor of the observed data. We prove that under mild non-degeneracy conditions, our method consistently estimates the model parameters at the optimal parametric rate. Our tensor-based method generalizes previous orthogonal tensor decomposition approaches, where the hidden units were assumed to be either statistically independent or mutually exclusive. We illustrate the consistency of our method on simulated data and demonstrate its usefulness in learning a common model for population mixtures in genetics. Copyright © 2018, The Authors. All rights reserved.

关键词： Tensors

Extraordinary Optical and Transport Properties of Disordered Stealthy Hyperuniform Two-Phase Media

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arXiv 2024年

作者： Kim, Jaeuk Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Materials Institute PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered stealthy hyperuniform two-phase media are a special subset of hyperuniform structures with novel physical properties due to their hybrid crystal-liquid nature. We have previously shown that the rapidly converging strong-contrast expansion of a linear fractional form of the effective dynamic dielectric constant Εe(k1,ω) [Phys. Rev. X 11, 296 021002 (2021)] leads to accurate approximations for both hyperuniform and nonhyperuniform two-phase composite media when truncated at the two-point level for distinctly different types of microstructural symmetries in three dimensions. In this paper, we further elucidate the extraordinary optical and transport properties of disordered stealthy hyperuniform media. Among other results, we provide detailed proofs that stealthy hyperuniform layered and transversely isotropic media are perfectly transparent (i.e., no Anderson localization, in principle) within finite wavenumber intervals through the third-order terms. Remarkably, these results imply that there can be no Anderson localization within the predicted perfect transparency interval in stealthy hyperuniform layered and transversely isotropic media in practice because the localization length (associated with only possibly negligibly small higher-order contributions) should be very large compared to any practically large sample size. We further contrast and compare the extraordinary physical properties of stealthy hyperuniform two-phase layered, transversely isotropic media, and fully 3D isotropic media to other model nonstealthy microstructures, including their attenuation characteristics, as measured by the imaginary part of Εe(k1,ω), and transport properties, as measured by the time-dependent diffusion spreadability S(t). We demonstrate that there are cross-property relations between them, namely, we quantify how the imaginary parts of Εe(k1,ω) and the spreadability at long times are positively correlated as the structures span from nonhyperuniform, nonstealthy hy

关键词： Transport properties

Rate-achievability strategies for two-hop interference flows

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Rate-achievability strategies for two-hop interference flows

Annual Allerton Conference on Communication, Control, and Computing

作者： P.S. Chandrashekhar Thejaswi Amir Bennatan Junshan Zhang Robert Calderbank Douglas Cochran Department of Electrical Engineering Arizona Stare University Tempe 85287 USA Program in Applied and Computational Mathematics (PACM) at Princeton University Princeton NJ USA Department of Electrical Engineering Arizona State University Tempe AZ USA Department of Electrical Engineering Arizona Stare University Tempe AZ USA

We consider a basic model for two-hop transmissions of two information flows which interfere with each other. In this model, two sources simultaneously transmit to two relays (in the first hop), which then simultaneously transmit to two destinations (in the second hop). While the transmission during the first hop is essentially the transmission over a classical interference channel, the transmission in the second hop enjoys an interesting advantage. Specifically, as a byproduct of the Han-Kobayashi transmission scheme applied to the first hop, each of the relays (in the second hop) has access to some of the data that is intended to the other destination, in addition to its own data. As recently observed by Simeone et al., this opens the door to cooperation between the relays. In this paper, we observe that the cooperation can take the form of distributed MIMO broadcast, thus greatly enhancing its effectiveness at high SNR. However, since each relay is only aware of part of the data beyond its own, full cooperation is not possible. We propose several approaches that combine MIMO broadcast strategies (including ldquodirty paperrdquo) with standard non-cooperative strategies for the interference channel. Numerical results are provided, which indicate that our approaches provide substantial benefits at high SNR.

关键词： Relays Interference channels Decoding MIMO Broadcasting Context Wireless networks mathematics Electronic mail

Predicting permeability via statistical learning on higher-order microstructural information

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arXiv 2020年

作者： Röding, Magnus Ma, Zheng Torquato, Salvatore RISE Research Institutes of Sweden Göteborg41276 Sweden Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Quantitative structure-property relationships are crucial for the understanding and prediction of the physical properties of complex materials. For fluid flow in porous materials, characterizing the geometry of the pore microstructure facilitates prediction of permeability, a key property that has been extensively studied in material science, geophysics and chemical engineering. In this work, we study the predictability of different structural descriptors via both linear regressions and neural networks. A large data set of 30,000 virtual, porous microstructures of different types, including both granular and continuous solid phases, is created for this end. We compute permeabilities of these structures using the lattice Boltzmann method, and characterize the pore space geometry using one-point correlation functions (porosity, specific surface), two-point surface-surface, surface-void, and void-void correlation functions, as well as the geodesic tortuosity as an implicit descriptor. Then, we study the prediction of the permeability using different combinations of these descriptors. We obtain significant improvements of performance when compared to a Kozeny-Carman regression with only lowest-order descriptors (porosity and specific surface). We find that combining all three two-point correlation functions and tortuosity provides the best prediction of permeability, with the void-void correlation function being the most informative individual descriptor. Moreover, the combination of porosity, specific surface, and geodesic tortuosity provides very good predictive performance. This shows that higher-order correlation functions are extremely useful for forming a general model for predicting physical properties of complex materials. Additionally, our results suggest that artificial neural networks are superior to the more conventional regression methods for establishing quantitative structure-property relationships. We make the data and code used publicly available to fac

关键词： Forecasting

Frequency-Resolved Optical Gating Recovery via Smoothing Gradient

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arXiv 2019年

作者： Pinilla, Samuel Bendory, Tamir Eldar, Yonina C. Arguello, Henry Department of Electrical Engineering Universidad Industrial de Santander BucaramangaSantander680002 Colombia Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Weizmann Institute of Science Rehovot7610001 Israel Department of Computer Science Universidad Industrial de Santander BucaramangaSantander680002 Colombia

Frequency-resolved optical gating (FROG) is a popular technique for complete characterization of ultrashort laser pulses. The acquired data in FROG, called FROG trace, is the Fourier magnitude of the product of the unknown pulse with a time-shifted version of itself, for several different shifts. To estimate the pulse from the FROG trace, we propose an algorithm that minimizes a smoothed non-convex least-squares objective function. The method consists of two steps. First, we approximate the pulse by an iterative spectral algorithm. Then, the attained initialization is refined based upon a sequence of block stochastic gradient iterations. The algorithm is theoretically simple, numerically scalable, and easy-to-implement. Empirically, our approach outperforms the state-of-the-art when the FROG trace is incomplete, that is, when only few shifts are recorded. Simulations also suggest that the proposed algorithm exhibits similar computational cost compared to a state-of-the-art technique for both complete and incomplete data. In addition, we prove that in the vicinity of the true solution, the algorithm converges to a critical point. A Matlab implementation is publicly available at https://***/samuelpinilla/FROG. Copyright © 2019, The Authors. All rights reserved.

关键词： Ultrashort pulses

Structure learning from time series with false discovery control

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arXiv 2018年

作者： Pegueroles, Bernat Guillen Vinzamuri, Bhanukiran Shanmugam, Karthikeyan Hedden, Steve Moyer, Jonathan D. Varshney, Kush R. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States IBM Research Yorktown HeightsNY10598 United States Frederick S. Pardee Center for International Futures Josef Korbel School of International Studies University of Denver DenverCO80210 United States

We consider the Granger causal structure learning problem from time series data. Granger causal algorithms predict a 'Granger causal effect' between two variables by testing if prediction error of one decreases significantly in the absence of the other variable among the predictor covariates. Almost all existing Granger causal algorithms condition on a large number of variables (all but two variables) to test for effects between a pair of variables. We propose a new structure learning algorithm called MMPC-p inspired by the well known MMHC algorithm for non-time series data. We show that under some assumptions, the algorithm provides false discovery rate control. The algorithm is sound and complete when given access to perfect directed information testing oracles. We also outline a novel tester for the linear Gaussian case. We show through our extensive experiments that the MMPC-p algorithm scales to larger problems and has improved statistical power compared to existing state of the art for large sparse graphs. We also apply our algorithm on a global development dataset and validate our findings with subject matter experts. Copyright © 2018, The Authors. All rights reserved.

关键词： Time series

Feature selection using stochastic gates

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arXiv 2018年

作者： Yamada, Yutaro Lindenbaum, Ofir Negahban, Sahand Kluger, Yuval Statistics Department Yale University New HavenCT United States Applied Mathematics Program Yale University New HavenCT United States Computational Biology and Bioinformatics Yale University New HavenCT United States Department of Pathology Yale University New HavenCT United States

Feature selection problems have been extensively studied for linear estimation, for instance, Lasso, but less emphasis has been placed on feature selection for non-linear functions. In this study, we propose a method for feature selection in high-dimensional non-linear function estimation problems. The new procedure is based on minimizing the `0 norm of the vector of indicator variables that represent if a feature is selected or not. Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via gradient descent. This general framework simultaneously minimizes a loss function while selecting relevant features. Furthermore, we provide an information-theoretic justification of incorporating Bernoulli distribution into our approach and demonstrate the potential of the approach on synthetic and real-life applications. Copyright © 2018, The Authors. All rights reserved.

关键词： Feature extraction

Coarse-grained descriptions of dynamics for networks with both intrinsic and structural heterogeneities

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arXiv 2017年

作者： Bertalan, Tom Wu, Yan Laing, Carlo William Gear, C. Kevrekidis, Ioannis G. Department of Chemical and Biological Engineering Princeton University PrincetonNJ United States Department of Bioengineering University of California San DiegoCA United States Institute of Natural and Mathematical Sciences Massey University Auckland New Zealand Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Finding accurate reduced descriptions for large, complex, dynamically evolving networks is a crucial enabler to their simulation, analysis, and, ultimately, design. Here we propose and illustrate a systematic and powerful approach to obtaining good collective coarse-grained observables-variables successfully summarizing the detailed state of such networks. Finding such variables can naturally lead to successful reduced dynamic models for the networks. The main premise enabling our approach is the assumption that the behavior of a node in the network depends (after a short initial transient) on the node identity: a set of descriptors that quantify the node properties, whether intrinsic (e.g. parameters in the node evolution equations) or structural (imparted to the node by its connectivity in the particular network structure). The approach creates a natural link with modeling and "computational enabling technology" developed in the context of Uncertainty Quantification. In our case, however, we will not focus on ensembles of different realizations of a problem, each with parameters randomly selected from a distribution. We will instead study many coupled heterogeneous units, each characterized by randomly assigned (heterogeneous) parameter value(s). One could then coin the term Heterogeneity Quantification for this approach, which we illustrate through a model dynamic network consisting of coupled oscillators with one intrinsic heterogeneity (oscillator individual frequency) and one structural heterogeneity (oscillator degree in the undirected network). The computational implementation of the approach, its shortcomings and possible extensions are also discussed. Copyright © 2017, The Authors. All rights reserved.

关键词： Data mining

Minuscule corrections to near-surface solar internal rotation using mode-coupling

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arXiv 2023年

作者： Das, Srijan Bharati Kashyap, Samarth G. Oktay, Deniz Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Computer Science Princeton University PrincetonNJ United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow — differential rotation — is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency-splittings do not consider the effect of mode-coupling. This approximation worsens for high-angular-degree modes, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ` ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a non-linear inversion using an eigenvalue solver. Differences in inverted profiles for frequency splitting measurements from MDI and HMI are compared and discussed. We find that corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields. © 2023, CC BY.

关键词： Rotation