Dirichlet process mixtures are flexible non-parametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are typically not consistent for the number of clusters if the concentration parameter is held fixed and data come from a finite mixture. Here we show that consistency for the number of clusters can be achieved if the concentration parameter is adapted in a fully Bayesian way, as commonly done in practice. Our results are derived for data coming from a class of finite mixtures, with mild assumptions on the prior for the concentration parameter and for a variety of choices of likelihood kernels for the mixture.

Abstract: The application of the lasso is espoused in high-dimensional settings where only a small number of the regression coefficients are believed to be nonzero. Moreover, statistical properties of high-dimensional lasso estimators are often proved under the assumption that the correlation between the predictors is bounded. In this vein, coordinatewise methods, the most common means of computing the lasso solution, work well in the presence of low to moderate multicollinearity. Motivated by these limitations, we propose the novel "Deterministic Bayesian Lasso" algorithm for computing the lasso solution. This algorithm is developed by considering a limiting version of the Bayesian lasso.

An essential part of a regression analysis is to understand if we can use a linear model or not for solving our ML problem. There are many ways to do this, and, generally, we have to use multiple ways to understand if our data are really linear distributed. In this article, we will see two different graphical methods for analyzing the residuals in a regression problem: but those are just two methods useful for understanding if our data are linearly distributed. You can use just one of these methods, or even both, but you will need the help of other metrics to better validate your hypothesis (the model to be used is linear): we'll see other methods in future articles. But first of all…what are the residuals in a regression problem?

In the era of big data, there is an increasing demand for new methods for analyzing and forecasting 2-dimensional data. The current research aims to accomplish these goals through the combination of time-series modeling and multilinear algebraic systems. We expand previous autoregressive techniques to forecast multilinear data, aptly named the L-Transform Tensor autoregressive (L-TAR for short). Tensor decompositions and multilinear tensor products have allowed for this approach to be a feasible method of forecasting. We achieve statistical independence between the columns of the observations through invertible discrete linear transforms, enabling a divide and conquer approach. We present an experimental validation of the proposed methods on datasets containing image collections, video sequences, sea surface temperature measurements, stock prices, and networks.

High dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves, results in empirically observed "stickiness" and poor theoretical mixing properties -- lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the mixing time decreases with dimension.

This work presents strategies to learn an Energy-Based Model (EBM) according to the desired length of its MCMC sampling trajectories. MCMC trajectories of different lengths correspond to models with different purposes. Our experiments cover three different trajectory magnitudes and learning outcomes: 1) shortrun sampling for image generation; 2) midrun sampling for classifier-agnostic adversarial defense; and 3) longrun sampling for principled modeling of image probability densities. To achieve these outcomes, we introduce three novel methods of MCMC initialization for negative samples used in Maximum Likelihood (ML) learning. With standard network architectures and an unaltered ML objective, our MCMC initialization methods alone enable significant performance gains across the three applications that we investigate. Our results include state-of-the-art FID scores for unnormalized image densities on the CIFAR-10 and ImageNet datasets; state-of-the-art adversarial defense on CIFAR-10 among purification methods and the first EBM defense on ImageNet; and scalable techniques for learning valid probability densities. Code for this project can be found at https://github.com/point0bar1/ebm-life-cycle.

This book constitutes the refereed post-conference proceedings of the First International Conference on Pan-African Intelligence and Smart Systems, PAAISS 2021, which was held in Windhoek, Namibia, in September 2021. The 17 revised full papers presented were carefully selected from 41 submissions. The theme of PAAISS 2021 was "Advancing AI research in Africa" and the papers are arranged according to subject areas: Deep Learning; Classification and Pattern Recognition; Neural Networks and Support Vector Machines; Smart Systems.

The unsupervised K- Nearest Neighbour (KNN) algorithm is perhaps the most straightforward machine learning algorithm. However, a simple algorithm does not mean that analyzing the results is equally simple. As per my research, there are not many documented approaches to analyzing the results of the KNN algorithm. In this article, I will show you how to analyze and understand the results of the unsupervised KNN algorithm. I will be using a dataset on cars.

When starting out with Data Science, there is so much to learn it can become quite overwhelming. This guide will help aspiring data scientists and machine learning engineers gain better knowledge and experience. I will list different types of machine learning algorithms, which can be used with both Python and R. Linear Regression is the simplest Machine learning algorithm that branches off from Supervised Learning. It is primarily used to solve regression problems and make predictions on continuous dependent variables with the knowledge from independent variables. The goal of Linear Regression is to find the line of best fit, which can help predict the output for continuous dependent variables.

The binomial deviance and the SVM hinge loss functions are two of the most widely used loss functions in machine learning. While there are many similarities between them, they also have their own strengths when dealing with different types of data. In this work, we introduce a new exponential family based on a convex relaxation of the hinge loss function using softness and class-separation parameters. This new family, denoted Soft-SVM, allows us to prescribe a generalized linear model that effectively bridges between logistic regression and SVM classification. This new model is interpretable and avoids data separability issues, attaining good fitting and predictive performance by automatically adjusting for data label separability via the softness parameter. These results are confirmed empirically through simulations and case studies as we compare regularized logistic, SVM, and Soft-SVM regressions and conclude that the proposed model performs well in terms of both classification and prediction errors.