Online variants of the Expectation Maximization (EM) algorithm have recently been proposed to perform parameter inference with large data sets or data streams, in independent latent models and in hidden Markov models. Nevertheless, the convergence properties of these algorithms remain an open problem at least in the hidden Markov case. This contribution deals with a new online EM algorithm which updates the parameter at some deterministic times. Some convergence results have been derived even in general latent models such as hidden Markov models. These properties rely on the assumption that some intermediate quantities are available in closed form or can be approximated by Monte Carlo methods when the Monte Carlo error vanishes rapidly enough. In this paper, we propose an algorithm which approximates these quantities using Sequential Monte Carlo methods. The convergence of this algorithm and of an averaged version is established and their performance is illustrated through Monte Carlo experiments.
This is a supplementary material to the paper "Online Expectation Maximization based algorithms for inference in hidden Markov models". It contains further technical derivations and additional simulation results.
The Expectation Maximization (EM) algorithm is a versatile tool for model parameter estimation in latent data models. When processing large data sets or data stream however, EM becomes intractable since it requires the whole data set to be available at each iteration of the algorithm. In this contribution, a new generic online EM algorithm for model parameter inference in general Hidden Markov Model is proposed. This new algorithm updates the parameter estimate after a block of observations is processed (online). The convergence of this new algorithm is established, and the rate of convergence…
Expectation-Maximization algorithm to estimate the forcing parameter of a nonlinear McKean-Vlasov diffusion
Instituto de Estadística, Universidad de Valparaíso（瓦尔帕莱索大学统计研究所）；Universidad Adventista de Chile（智利基督复临安息日会大学）；Departamento de Ingeniería Matemática, Universidad de Concepción（康塞普西翁大学数学工程系）；Cimfav - Ingemat, Universidad de Valparaíso（瓦尔帕莱索大学CIMFAV-Ingemat）
We present a family of expectation-maximization (EM) algorithms for binary and negative-binomial logistic regression, drawing a sharp connection with the variational-Bayes algorithm of Jaakkola and Jordan (2000). Indeed, our results allow a version of this variational-Bayes approach to be re-interpreted as a true EM algorithm. We study several interesting features of the algorithm, and of this previously unrecognized connection with variational Bayes. We also generalize the approach to sparsity-promoting priors, and to an online method whose convergence properties are easily established. This …
Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data which is particularly the case of medical images. We propose to use an optimized Monte Carlo Markov Chain method into a stochastic Expectation Maximization algorithm in order to e…
This paper deals with parameter estimation when the data are randomly right censored. The maximum likelihood estimates from censored samples are obtained by using the expectation-maximization (EM) and Monte Carlo EM (MCEM) algorithms. We introduce the concept of the EM and MCEM algorithms and develop parameter estimation methods for a variety of distributions such as normal, Laplace and Rayleigh distributions. These proposed methods are illustrated with three examples.
Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaus…
We propose the misclassified Ising Model; a framework for analyzing dependent binary data where the binary state is susceptible to error. We extend the theoretical results of the model selection method presented in Ravikumar et. al. (2010) to show that the method will still correctly identify edges in the underlying graphical model under suitable misclassification settings. With knowledge of the misclassification process, an expectation maximization algorithm is developed that accounts for misclassification during model selection. We illustrate the increase of performance of the proposed expec…
University of California, Berkeley（加州大学伯克利分校）；Google Research（谷歌研究院）
We show that a large class of Estimation of Distribution Algorithms, including, but not limited to, Covariance Matrix Adaption, can be written as a Monte Carlo Expectation-Maximization algorithm, and as exact EM in the limit of infinite samples. Because EM sits on a rigorous statistical foundation and has been thoroughly analyzed, this connection provides a new coherent framework with which to reason about EDAs.
Inspired from quantum Monte Carlo, by using unbiased estimators all the time and sampling discrete and continuous variables at the same time using Metropolis algorithm, we present a novel, fast, and accurate high performance Monte Carlo Parametric Expectation Maximization (MCPEM) algorithm. We named it Randomized Parametric Expectation Maximization (RPEM). In particular, we compared RPEM with Monolix's SAEM and Certara's QRPEM for a realistic two-compartment Voriconazole model with ordinary differential equations (ODEs) and using simulated data. We show that RPEM is 3 to 4 times faster than SA…
We present a noise-injected version of the Expectation-Maximization (EM) algorithm: the Noisy Expectation Maximization (NEM) algorithm. The NEM algorithm uses noise to speed up the convergence of the EM algorithm. The NEM theorem shows that injected noise speeds up the average convergence of the EM algorithm to a local maximum of the likelihood surface if a positivity condition holds. The generalized form of the noisy expectation-maximization (NEM) algorithm allow for arbitrary modes of noise injection including adding and multiplying noise to the data. We demonstrate these noise benefits on E…
A Maximum Likelihood recursive state estimator is derived for non-linear and non-Gaussian state-space models. The estimator combines a particle filter to generate the conditional density and the Expectation Maximization algorithm to compute the maximum likelihood state estimate iteratively. Algorithms for maximum likelihood state filtering, prediction and smoothing are presented. The convergence properties of these algorithms, which are inherited from the Expectation Maximization algorithm, are proven and examined in two examples. It is shown that, with randomized reinitialization, which is fe…
Maximum likelihood estimation (MLE) is one of the most important methods in machine learning, and the expectation-maximization (EM) algorithm is often used to obtain maximum likelihood estimates. However, EM heavily depends on initial configurations and fails to find the global optimum. On the other hand, in the field of physics, quantum annealing (QA) was proposed as a novel optimization approach. Motivated by QA, we propose a quantum annealing extension of EM, which we call the deterministic quantum annealing expectation-maximization (DQAEM) algorithm. We also discuss its advantage in terms …
The University of Tokyo（东京大学）；University of Innsbruck（因斯布鲁克大学）
Clustering algorithms are a cornerstone of machine learning applications. Recently, a quantum algorithm for clustering based on the k-means algorithm has been proposed by Kerenidis, Landman, Luongo and Prakash. Based on their work, we propose a quantum expectation-maximization (EM) algorithm for Gaussian mixture models (GMMs). The robustness and quantum speedup of the algorithm is demonstrated. We also show numerically the advantage of GMM over k-means for non-trivial cluster data.
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）；Centre de Mathématiques Appliquées（应用数学中心）；Ecole Polytechnique（巴黎综合理工学院）；HSE University（高等经济大学）；The Chinese University of Hong Kong（香港中文大学）
The Expectation Maximization (EM) algorithm is of key importance for inference in latent variable models including mixture of regressors and experts, missing observations. This paper introduces a novel EM algorithm, called \texttt{SPIDER-EM}, for inference from a training set of size $n$, $n \gg 1$. At the core of our algorithm is an estimator of the full conditional expectation in the {\sf E}-step, adapted from the stochastic path-integrated differential estimator ({\tt SPIDER}) technique. We derive finite-time complexity bounds for smooth non-convex likelihood: we show that for convergence t…
Probabilistic approach to Boolean matrix factorization can provide solutions robustagainst noise and missing values with linear computational complexity. However,the assumption about latent factors can be problematic in real world applications.This study proposed a new probabilistic algorithm free of assumptions of latentfactors, while retaining the advantages of previous algorithms. Real data experimentshowed that our algorithm was favourably compared with current state-of-the-artprobabilistic algorithms.
We study a class of weakly identifiable location-scale mixture models for which the maximum likelihood estimates based on $n$ i.i.d. samples are known to have lower accuracy than the classical $n^{- \frac{1}{2}}$ error. We investigate whether the Expectation-Maximization (EM) algorithm also converges slowly for these models. We provide a rigorous characterization of EM for fitting a weakly identifiable Gaussian mixture in a univariate setting where we prove that the EM algorithm converges in order $n^{\frac{3}{4}}$ steps and returns estimates that are at a Euclidean distance of order ${ n^{- \…
In this paper, we firstly give a brief introduction of expectation maximization (EM) algorithm, and then discuss the initial value sensitivity of expectation maximization algorithm. Subsequently, we give a short proof of EM's convergence. Then, we implement experiments with the expectation maximization algorithm (We implement all the experiments on Gaussion mixture model (GMM)). Our experiment with expectation maximization is performed in the following three cases: initialize randomly; initialize with result of K-means; initialize with result of K-medoids. The experiment result shows that expe…
Many real world tasks such as reasoning and physical interaction require identification and manipulation of conceptual entities. A first step towards solving these tasks is the automated discovery of distributed symbol-like representations. In this paper, we explicitly formalize this problem as inference in a spatial mixture model where each component is parametrized by a neural network. Based on the Expectation Maximization framework we then derive a differentiable clustering method that simultaneously learns how to group and represent individual entities. We evaluate our method on the (seque…
Expectation maximization (EM) is a technique for estimating maximum-likelihood parameters of a latent variable model given observed data by alternating between taking expectations of sufficient statistics, and maximizing the expected log likelihood. For situations where sufficient statistics are intractable, stochastic approximation EM (SAEM) is often used, which uses Monte Carlo techniques to approximate the expected log likelihood. Two common implementations of SAEM, Batch EM (BEM) and online EM (OEM), are parameterized by a "learning rate", and their efficiency depend strongly on this param…
CNRS（法国国家科学研究中心）；Ecole Polytechnique（巴黎综合理工学院）；The Chinese University of Hong Kong（香港中文大学）
The Expectation Maximization (EM) algorithm is a key reference for inference in latent variable models; unfortunately, its computational cost is prohibitive in the large scale learning setting. In this paper, we propose an extension of the Stochastic Path-Integrated Differential EstimatoR EM (SPIDER-EM) and derive complexity bounds for this novel algorithm, designed to solve smooth nonconvex finite-sum optimization problems. We show that it reaches the same state of the art complexity bounds as SPIDER-EM; and provide conditions for a linear rate of convergence. Numerical results support our fi…
State University of New York at Buffalo（纽约州立大学布法罗分校）
In this paper, we study the problem of estimating latent variable models with arbitrarily corrupted samples in high dimensional space ({\em i.e.,} $d\gg n$) where the underlying parameter is assumed to be sparse. Specifically, we propose a method called Trimmed (Gradient) Expectation Maximization which adds a trimming gradients step and a hard thresholding step to the Expectation step (E-step) and the Maximization step (M-step), respectively. We show that under some mild assumptions and with an appropriate initialization, the algorithm is corruption-proofing and converges to the (near) optimal…
VU University Amsterdam（阿姆斯特丹自由大学）；University of Leiden（莱顿大学）
We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in ${\mathbb R}^H$ with arbitrary dimension $H$. Each IFS defines a probability distribution on ${\mathbb R}^H$, so that the fractal inverse problem can be cast as a problem of parameter estimation. We show that the algorithm reconstructs well-known fractals from data, with the model converging to high precision parameters. We…
In this paper we develop an Expectation Maximization(EM) algorithm to estimate the parameter of a Yule-Simon distribution. The Yule-Simon distribution exhibits the "rich get richer" effect whereby an 80-20 type of rule tends to dominate. These distributions are ubiquitous in industrial settings. The EM algorithm presented provides both frequentist and Bayesian estimates of the $λ$ parameter. By placing the estimation method within the EM framework we are able to derive Standard errors of the resulting estimate. Additionally, we prove convergence of the Yule-Simon EM algorithm and study the rat…
Error propagation formulae are derived for the expectation-maximization iterative unfolding algorithm regularized by a smoothing step. The effective number of parameters in the fit to the observed data is defined for unfolding procedures. Based upon this definition, the Akaike information criterion is proposed as a principle for choosing the smoothing parameters in an automatic, data-dependent manner. The performance and the frequentist coverage of the resulting method are investigated using simulated samples. A number of issues of general relevance to all unfolding techniques are discussed, i…
Dramatic increases in the size and dimensionality of many recent data sets make crucial the need for sophisticated methods that can exploit inherent structure and handle missing values. In this article we derive an expectation-maximization (EM) algorithm for the matrix normal distribution, a distribution well-suited for naturally structured data such as spatio-temporal data. We review previously established maximum likelihood matrix normal estimates, and then consider the situation involving missing data. We apply our EM method in a simulation study exploring errors across different dimensions…
We propose a unified framework that extends the inference methods for classical hidden Markov models to continuous settings, where both the hidden states and observations occur in continuous time. Two different settings are analyzed: hidden jump process with a finite state space, and hidden diffusion process with a continuous state space. For each setting, we first estimate the hidden states given the observations and model parameters, showing that the posterior distribution of the hidden states can be described by differential equations in continuous time. We then consider the estimation of u…
We study Sinkhorn EM (sEM), a variant of the expectation maximization (EM) algorithm for mixtures based on entropic optimal transport. sEM differs from the classic EM algorithm in the way responsibilities are computed during the expectation step: rather than assign data points to clusters independently, sEM uses optimal transport to compute responsibilities by incorporating prior information about mixing weights. Like EM, sEM has a natural interpretation as a coordinate ascent procedure, which iteratively constructs and optimizes a lower bound on the log-likelihood. However, we show theoretica…
SEMF: Supervised Expectation-Maximization Framework for Predicting Intervals
HEC, University of Lausanne, Lausanne, Switzerland（洛桑大学HEC学院，洛桑，瑞士）；HEC, University of Lausanne, Expertise Center for Climate Extremes (ECCE), Lausanne, Switzerland（洛桑大学HEC学院，洛桑，瑞士）
We propose a graphical model framework for goal-conditioned RL, with an EM algorithm that operates on the lower bound of the RL objective. The E-step provides a natural interpretation of how 'learning in hindsight' techniques, such as HER, to handle extremely sparse goal-conditioned rewards. The M-step reduces policy optimization to supervised learning updates, which greatly stabilizes end-to-end training on high-dimensional inputs such as images. We show that the combined algorithm, hEM significantly outperforms model-free baselines on a wide range of goal-conditioned benchmarks with sparse r…
The correlated binomial (CB) distribution was proposed by Luceño (Computational Statistics $\&$ Data Analysis, 20, 1995, 511-520) as an alternative to the binomial distribution for the analysis of the data in the presence of correlations among events. Due to the complexity of the mixture likelihood of the model, it may be impossible to derive analytical expressions of the maximum likelihood estimators (MLEs) of the unknown parameters. To overcome this difficulty, we develop an expectation-maximization algorithm for computing the MLEs of the CB parameters. Numerical results from simulation stud…
This paper is concerned with the image reconstruction problem when the measured data are solar hard X-ray modulation profiles obtained from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI)} instrument. Our goal is to demonstrate that a statistical iterative method classically applied to the image deconvolution problem is very effective when utilized for the analysis of count modulation profiles in solar hard X-ray imaging based on Rotating Modulation Collimators. The algorithm described in this paper solves the maximum likelihood problem iteratively and encoding a positivity c…
The performance of ensemble-based data assimilation techniques that estimate the state of a dynamical system from partial observations depends crucially on the prescribed uncertainty of the model dynamics and of the observations. These are not usually known and have to be inferred. Many approaches have been proposed to tackle this problem, including fully Bayesian, likelihood maximization and innovation-based techniques. This work focuses on maximization of the likelihood function via the expectation-maximization (EM) algorithm to infer the model error covariance combined with ensemble Kalman …
Nansen Environmental and Remote Sensing Center（南森环境与遥感中心）；Sorbonne University（索邦大学）；University of Reading（雷丁大学）；Utrecht University（乌得勒支大学）
The reconstruction from observations of high-dimensional chaotic dynamics such as geophysical flows is hampered by (i) the partial and noisy observations that can realistically be obtained, (ii) the need to learn from long time series of data, and (iii) the unstable nature of the dynamics. To achieve such inference from the observations over long time series, it has been suggested to combine data assimilation and machine learning in several ways. We show how to unify these approaches from a Bayesian perspective using expectation-maximization and coordinate descents. In doing so, the model, the…
Training deep generative models with maximum likelihood remains a challenge. The typical workaround is to use variational inference (VI) and maximize a lower bound to the log marginal likelihood of the data. Variational auto-encoders (VAEs) adopt this approach. They further amortize the cost of inference by using a recognition network to parameterize the variational family. Amortized VI scales approximate posterior inference in deep generative models to large datasets. However it introduces an amortization gap and leads to approximate posteriors of reduced expressivity due to the problem known…
University of Amsterdam（阿姆斯特丹大学）；UC, San Diego（加州大学圣地亚哥分校）
The iterative nature of the expectation maximization (EM) algorithm presents a challenge for privacy-preserving estimation, as each iteration increases the amount of noise needed. We propose a practical private EM algorithm that overcomes this challenge using two innovations: (1) a novel moment perturbation formulation for differentially private EM (DP-EM), and (2) the use of two recently developed composition methods to bound the privacy "cost" of multiple EM iterations: the moments accountant (MA) and zero-mean concentrated differential privacy (zCDP). Both MA and zCDP bound the moment gener…
The present work proposes hybridization of Expectation-Maximization (EM) and K-Means techniques as an attempt to speed-up the clustering process. Though both K-Means and EM techniques look into different areas, K-means can be viewed as an approximate way to obtain maximum likelihood estimates for the means. Along with the proposed algorithm for hybridization, the present work also experiments with the Standard EM algorithm. Six different datasets are used for the experiments of which three are synthetic datasets. Clustering fitness and Sum of Squared Errors (SSE) are computed for measuring the…
The Expectation Maximization (EM) algorithm is widely used as an iterative modification to maximum likelihood estimation when the data is incomplete. We focus on a semi-supervised case to learn the model from labeled and unlabeled samples. Existing work in the semi-supervised case has focused mainly on performance rather than convergence guarantee, however we focus on the contribution of the labeled samples to the convergence rate. The analysis clearly demonstrates how the labeled samples improve the convergence rate for the exponential family mixture model. In this case, we assume that the po…
Model error covariances play a central role in the performance of data assimilation methods applied to nonlinear state-space models. However, these covariances are largely unknown in most of the applications. A misspecification of the model error covariance has a strong impact on the computation of the posterior probability density function, leading to unreliable estimations and even to a total failure of the assimilation procedure. In this work, we propose the combination of the Expectation-Maximization algorithm (EM) with an efficient particle filter to estimate the model error covariance, u…
Spatial generalized linear mixed models (SGLMMs) are popular and flexible models for non-Gaussian spatial data. They are useful for spatial interpolations as well as for fitting regression models that account for spatial dependence, and are commonly used in many disciplines such as epidemiology, atmospheric science, and sociology. Inference for SGLMMs is typically carried out under the Bayesian framework at least in part because computational issues make maximum likelihood estimation challenging, especially when high-dimensional spatial data are involved. Here we provide a computationally effi…
The direction of arrival (DOA) estimation of sound sources has been a popular signal processing research topic due to its widespread applications. Using spherical microphone arrays (SMA), DOA estimation can be applied in the spherical harmonic (SH) domain without any spatial ambiguity. However, the environment reverberation and noise can degrade the estimation performance. In this paper, we propose a new expectation maximization (EM) algorithm for deterministic maximum likelihood (ML) DOA estimation of L sound sources in the presence of spatially nonuniform noise in the SH domain. Furthermore …
Positive-definite matrix-variate data is becoming popular in computer vision. The computer vision data descriptors in the form of Region Covariance Descriptors (RCD) are positive definite matrices, which extract the key features of the images. The RCDs are extensively used in image set classification. Some classification methods treating RCDs as Wishart distributed random matrices are being proposed. However, the majority of the current methods preclude the potential correlation among the RCDs caused by the so-called non-voxel information (e.g., subjects' ages and nose widths, etc). Modeling c…
Fast convergence of a Federated Expectation-Maximization Algorithm
Institute of Mathematics, Ecole Polytechnique Federale de Lausanne（洛桑联邦理工学院数学研究所）；Department of Operations Research and Financial Engineering（运筹学与金融工程系）；Princeton University（普林斯顿大学）
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University of Houston（休斯顿大学）；Rutgers University（罗格斯大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）
(Gradient) Expectation Maximization (EM) is a widely used algorithm for estimating the maximum likelihood of mixture models or incomplete data problems. A major challenge facing this popular technique is how to effectively preserve the privacy of sensitive data. Previous research on this problem has already lead to the discovery of some Differentially Private (DP) algorithms for (Gradient) EM. However, unlike in the non-private case, existing techniques are not yet able to provide finite sample statistical guarantees. To address this issue, we propose in this paper the first DP version of (Gra…
The expectation-maximization (EM) algorithm is an iterative computational method to calculate the maximum likelihood estimators (MLEs) from the sample data. It converts a complicated one-time calculation for the MLE of the incomplete data to a series of relatively simple calculations for the MLEs of the complete data. When the MLE is available, we naturally want the Fisher information matrix (FIM) of unknown parameters. The FIM is, in fact, a good measure of the amount of information a sample of data provides and can be used to determine the lower bound of the variance and the asymptotic varia…
Domain adaptation (DA) is the task of classifying an unlabeled dataset (target) using a labeled dataset (source) from a related domain. The majority of successful DA methods try to directly match the distributions of the source and target data by transforming the feature space. Despite their success, state of the art methods based on this approach are either involved or unable to directly scale to data with many features. This article shows that domain adaptation can be successfully performed by using a very simple randomized expectation maximization (EM) method. We consider two instances of t…
Qualcomm AI Research（高通AI研究院）；Qualcomm Technologies, Inc.（高通技术公司）；University of Amsterdam（阿姆斯特丹大学）
Federated learning describes the distributed training of models across multiple clients while keeping the data private on-device. In this work, we view the server-orchestrated federated learning process as a hierarchical latent variable model where the server provides the parameters of a prior distribution over the client-specific model parameters. We show that with simple Gaussian priors and a hard version of the well known Expectation-Maximization (EM) algorithm, learning in such a model corresponds to FedAvg, the most popular algorithm for the federated learning setting. This perspective on…
