scholarly journals Sharp Guarantees and Optimal Performance for Inference in Binary and Gaussian-Mixture Models

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 178
Author(s):  
Hossein Taheri ◽  
Ramtin Pedarsani ◽  
Christos Thrampoulidis
Keyword(s):  
Mixture Models ◽  
Linear Models ◽  
Gaussian Mixture Models ◽  
Gaussian Mixture ◽  
Loss Functions ◽  
Risk Minimization ◽  
Small Problem ◽  
Empirical Risk ◽  
High Dimensional Inference ◽  
Convex Loss

We study convex empirical risk minimization for high-dimensional inference in binary linear classification under both discriminative binary linear models, as well as generative Gaussian-mixture models. Our first result sharply predicts the statistical performance of such estimators in the proportional asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit to prove bounds on the best achievable performance. Notably, we show that the proposed bounds are tight for popular binary models (such as signed and logistic) and for the Gaussian-mixture model by constructing appropriate loss functions that achieve it. Our numerical simulations suggest that the theory is accurate even for relatively small problem dimensions and that it enjoys a certain universality property.

scholarly journals Differentially Private Empirical Risk Minimization with Smooth Non-Convex Loss Functions: A Non-Stationary View

2019 ◽  
Vol 33 ◽  
pp. 1182-1189
Author(s):  
Di Wang ◽  
Jinhui Xu
Keyword(s):  
Dimensional Space ◽  
Upper Bounds ◽  
Loss Functions ◽  
Log P ◽  
Empirical Risk Minimization ◽  
Risk Minimization ◽  
Projected Gradient ◽  
Empirical Risk ◽  
Convex Loss ◽  
Real World Datasets

In this paper, we study the Differentially Private Empirical Risk Minimization (DP-ERM) problem with non-convex loss functions and give several upper bounds for the utility in different settings. We first consider the problem in low-dimensional space. For DP-ERM with non-smooth regularizer, we generalize an existing work by measuring the utility using ℓ2 norm of the projected gradient. Also, we extend the error bound measurement, for the first time, from empirical risk to population risk by using the expected ℓ2 norm of the gradient. We then investigate the problem in high dimensional space, and show that by measuring the utility with Frank-Wolfe gap, it is possible to bound the utility by the Gaussian Width of the constraint set, instead of the dimensionality p of the underlying space. We further demonstrate that the advantages of this result can be achieved by the measure of ℓ2 norm of the projected gradient. A somewhat surprising discovery is that although the two kinds of measurements are quite different, their induced utility upper bounds are asymptotically the same under some assumptions. We also show that the utility of some special non-convex loss functions can be reduced to a level (i.e., depending only on log p) similar to that of convex loss functions. Finally, we test our proposed algorithms on both synthetic and real world datasets and the experimental results confirm our theoretical analysis.

A robust non-rigid point set registration method based on inhomogeneous Gaussian mixture models

2017 ◽  
Vol 34 (10) ◽  
pp. 1399-1414 ◽  
Author(s):  
Wanxia Deng ◽  
Huanxin Zou ◽  
Fang Guo ◽  
Lin Lei ◽  
Shilin Zhou ◽  
...  
Keyword(s):  
Mixture Models ◽  
Gaussian Mixture Models ◽  
Gaussian Mixture ◽  
Registration Method ◽  
Point Set Registration ◽  
Point Set

scholarly journals Data Assimilation with Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part I: Theory and Scheme

2013 ◽  
Vol 141 (6) ◽  
pp. 1737-1760 ◽  
Author(s):  
Thomas Sondergaard ◽  
Pierre F. J. Lermusiaux
Keyword(s):  
Data Assimilation ◽  
Mixture Models ◽  
Gaussian Mixture Models ◽  
Expectation Maximization Algorithm ◽  
Planck Equation ◽  
Information Criterion ◽  
Gaussian Mixture ◽  
Field Equations ◽  
Dynamical Equations ◽  
Prior Probabilities

Abstract This work introduces and derives an efficient, data-driven assimilation scheme, focused on a time-dependent stochastic subspace that respects nonlinear dynamics and captures non-Gaussian statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical applications, but that also rigorously utilizes the governing dynamical equations with information theory and learning theory for efficient Bayesian data assimilation. Building on the foundations of classical filters, the underlying theory and algorithmic implementation of the new filter are developed and derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive stochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively approximating the Fokker–Planck equation. At assimilation times, the DO realizations are fit to semiparametric Gaussian Mixture Models (GMMs) using the Expectation-Maximization algorithm and the Bayesian Information Criterion. Bayes’s law is then efficiently carried out analytically within the evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example. Variations of the GMM-DO filter are also provided along with comparisons with related schemes.

Sign in / Sign up

Export Citation Format

Share Document