Training Gaussian Mixture Models at Scale via Coresets

How can we train a statistical mixture model on a massive data set? In this work we show how to construct coresets for mixtures of Gaussians. A coreset is a weighted subset of the data, which guarantees that models fitting the coreset also provide a good fit for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size polynomial in dimension and the number of mixture components, while being independent of the data set size. Hence, one can harness computationally intensive algorithms to compute a good approximation on a significantly smaller data set. More importantly, such coresets can be efficiently constructed both in distributed and streaming settings and do not impose restrictions on the data generating process. Our results rely on a novel reduction of statistical estimation to problems in computational geometry and new combinatorial complexity results for mixtures of Gaussians. Empirical evaluation on several real-world datasets suggests that our coreset-based approach enables significant reduction in training-time with negligible approximation error

AutoBayes: A System for Generating Data Analysis Programs from Statistical Models

Data analysis is an important scientific task which is required whenever information needs to be extracted from raw data. Statistical approaches to data analysis, which use methods from probability theory and numerical analysis, are well-founded but difficult to implement: the development of a statistical data analysis program for any given application is time-consuming and requires substantial knowledge and experience in several areas. In this paper, we describe AutoBayes, a program synthesis system for the generation of data analysis programs from statistical models. A statistical model specifies the properties for each problem variable (i.e., observation or parameter) and its dependencies in the form of a probability distribution. It is a fully declarative problem description, similar in spirit to a set of differential equations. From such a model, AutoBayes generates optimized and fully commented C/C++ code which can be linked dynamically into the Matlab and Octave environments. Code is produced by a schema-guided deductive synthesis process. A schema consists of a code template and applicability constraints which are checked against the model during synthesis using theorem proving technology. AutoBayes augments schema-guided synthesis by symbolic-algebraic computation and can thus derive closed-form solutions for many problems. It is well-suited for tasks like estimating best-fitting model parameters for the given data. Here, we describe AutoBayes's system architecture, in particular the schema-guided synthesis kernel. Its capabilities are illustrated by a number of advanced textbook examples and benchmarks

Sequence to Sequence Mixture Model for Diverse Machine Translation

Sequence to sequence (SEQ2SEQ) models often lack diversity in their generated translations. This can be attributed to the limitation of SEQ2SEQ models in capturing lexical and syntactic variations in a parallel corpus resulting from different styles, genres, topics, or ambiguity of the translation process. In this paper, we develop a novel sequence to sequence mixture (S2SMIX) model that improves both translation diversity and quality by adopting a committee of specialized translation models rather than a single translation model. Each mixture component selects its own training dataset via optimization of the marginal loglikelihood, which leads to a soft clustering of the parallel corpus. Experiments on four language pairs demonstrate the superiority of our mixture model compared to a SEQ2SEQ baseline with standard or diversity-boosted beam search. Our mixture model uses negligible additional parameters and incurs no extra computation cost during decoding.Comment: 11 pages, 5 figures, accepted to CoNLL201

Statistical inference with anchored Bayesian mixture of regressions models: A case study analysis of allometric data

We present a case study in which we use a mixture of regressions model to improve on an ill-fitting simple linear regression model relating log brain mass to log body mass for 100 placental mammalian species. The slope of this regression model is of particular scientific interest because it corresponds to a constant that governs a hypothesized allometric power law relating brain mass to body mass. A specific line of investigation is to determine whether the regression parameters vary across subgroups of related species. We model these data using an anchored Bayesian mixture of regressions model, which modifies the standard Bayesian Gaussian mixture by pre-assigning small subsets of observations to given mixture components with probability one. These observations (called anchor points) break the relabeling invariance typical of exchangeable model specifications (the so-called label-switching problem). A careful choice of which observations to pre-classify to which mixture components is key to the specification of a well-fitting anchor model. In the article we compare three strategies for the selection of anchor points. The first assumes that the underlying mixture of regressions model holds and assigns anchor points to different components to maximize the information about their labeling. The second makes no assumption about the relationship between x and y and instead identifies anchor points using a bivariate Gaussian mixture model. The third strategy begins with the assumption that there is only one mixture regression component and identifies anchor points that are representative of a clustering structure based on case-deletion importance sampling weights. We compare the performance of the three strategies on the allometric data set and use auxiliary taxonomic information about the species to evaluate the model-based classifications estimated from these models

EMMIXcskew: an R Package for the Fitting of a Mixture of Canonical Fundamental Skew t-Distributions

This paper presents an R package EMMIXcskew for the fitting of the canonical fundamental skew t-distribution (CFUST) and finite mixtures of this distribution (FM-CFUST) via maximum likelihood (ML). The CFUST distribution provides a flexible family of models to handle non-normal data, with parameters for capturing skewness and heavy-tails in the data. It formally encompasses the normal, t, and skew-normal distributions as special and/or limiting cases. A few other versions of the skew t-distributions are also nested within the CFUST distribution. In this paper, an Expectation-Maximization (EM) algorithm is described for computing the ML estimates of the parameters of the FM-CFUST model, and different strategies for initializing the algorithm are discussed and illustrated. The methodology is implemented in the EMMIXcskew package, and examples are presented using two real datasets. The EMMIXcskew package contains functions to fit the FM-CFUST model, including procedures for generating different initial values. Additional features include random sample generation and contour visualization in 2D and 3D