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This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure operator. On one hand, by linking this operator with other optimality measures, we…

The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of nonpositive curvature. We…

This paper is concerned with an important issue in finite mixture modelling, the selection of the number of mixing components. We propose a new penalized likelihood method for model selection of finite multivariate Gaussian mixture models.…

We analyze the classical EM algorithm for parameter estimation in the symmetric two-component Gaussian mixtures in $d$ dimensions. We show that, even in the absence of any separation between components, provided that the sample size…

Problems of probabilistic inference and decision making under uncertainty commonly involve continuous random variables. Often these are discretized to a few points, to simplify assessments and computations. An alternative approximation is…

We consider approximate maximum likelihood parameter estimation in nonlinear state-space models. We discuss both direct optimization of the likelihood and expectation--maximization (EM). For EM, we also give closed-form expressions for the…

The Mixture Transition Distribution (MTD) model was introduced by Raftery to face the need for parsimony in the modeling of high-order Markov chains in discrete time. The particularity of this model comes from the fact that the effect of…

The likelihood function of a finite mixture model is a non-convex function with multiple local maxima and commonly used iterative algorithms such as EM will converge to different solutions depending on initial conditions. In this paper we…

Expectation-Maximization (EM) is a prominent approach for parameter estimation of hidden (aka latent) variable models. Given the full batch of data, EM forms an upper-bound of the negative log-likelihood of the model at each iteration and…

Initialisation of the EM algorithm in model-based clustering is often crucial. Various starting points in the parameter space often lead to different local maxima of the likelihood function and, so to different clustering partitions. Among…

Since introduced by Martinet and Rockafellar, the proximal point algorithm was generalized in many fruitful directions. More recently, in 2002, Pennanen studied the proximal point algorithm without monotonicity. A year later, Iusem and…

Mixtures of $r$ independent distributions for two discrete random variables can be represented by matrices of nonnegative rank $r$. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry…

We introduce a new approach to a linear-circular regression problem that relates multiple linear predictors to a circular response. We follow a modeling approach of a wrapped normal distribution that describes angular variables and angular…

Zhu and Melnykov (2018) develop a model to fit mixture models when the components are derived from the Manly transformation. Their EM algorithm utilizes Nelder-Mead optimization in the M-step to update the skew parameter,…

Semi- and non-parametric mixture of regressions are a very useful flexible class of mixture of regressions in which some or all of the parameters are non-parametric functions of the covariates. These models are, however, based on the…

In this paper we address the problem of building a class of robust factorization algorithms that solve for the shape and motion parameters with both affine (weak perspective) and perspective camera models. We introduce a Gaussian/uniform…

Due to the highly non-convex nature of large-scale robust parameter estimation, avoiding poor local minima is challenging in real-world applications where input data is contaminated by a large or unknown fraction of outliers. In this paper,…

The Expectation-Maximization (EM) algorithm is one of the most popular methods used to solve the problem of parametric distribution-based clustering in unsupervised learning. In this paper, we propose to analyze a generalized EM (GEM)…

We propose a new constrained EM algorithm that is applicable to general constrained estimation problems. The proposed method is based on a novel framework, the `dual-homotopy framework,' which combines deterministic annealing EM with a…

Expectation Maximization (EM) is the standard method to learn Gaussian mixtures. Yet its classic, centralized form is often infeasible, due to privacy concerns and computational and communication bottlenecks. Prior work dealt with data…