You'll find the rest of the chapter in the full version of the IPython Cookbook, by Cyrille Rossant, Packt Publishing, 2014.

The variational autoencoder (VAE) is a generative model with continuous latent variables where a pair of probabilistic encoder (bottom-up) and decoder (top-down) is jointly learned by stochastic gradient variational Bayes. We first elaborate Gaussian VAE, approximating the local covariance matrix of the decoder as an outer product of the principal direction at a position determined by a sample drawn from Gaussian distribution. We show that this model, referred to as VAE-ROC, better captures the data manifold, compared to the standard Gaussian VAE where independent multivariate Gaussian was used to model the decoder. Then we extend the VAE-ROC to handle mixed categorical and continuous data. To this end, we employ Gaussian copula to model the local dependency in mixed categorical and continuous data, leading to {\em Gaussian copula variational autoencoder} (GCVAE). As in VAE-ROC, we use the rank-one approximation for the covariance in the Gaussian copula, to capture the local dependency structure in the mixed data. Experiments on various datasets demonstrate the useful behaviour of VAE-ROC and GCVAE, compared to the standard VAE.

Factor analysis aims to determine latent factors, or traits, which summarize a given data set. Inter-battery factor analysis extends this notion to multiple views of the data. In this paper we show how a nonlinear, nonparametric version of these models can be recovered through the Gaussian process latent variable model. This gives us a flexible formalism for multi-view learning where the latent variables can be used both for exploratory purposes and for learning representations that enable efficient inference for ambiguous estimation tasks. Learning is performed in a Bayesian manner through the formulation of a variational compression scheme which gives a rigorous lower bound on the log likelihood. Our Bayesian framework provides strong regularization during training, allowing the structure of the latent space to be determined efficiently and automatically. We demonstrate this by producing the first (to our knowledge) published results of learning from dozens of views, even when data is scarce.

Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric variational family, which adapts its shape to match complex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity. We prove a universal approximation theorem for the VGP, demonstrating its representative power for learning any model. For inference we present a variational objective inspired by auto-encoders and perform black box inference over a wide class of models. The VGP achieves new state-of-the-art results for unsupervised learning, inferring models such as the deep latent Gaussian model and the recently proposed DRAW.

Abstract--A denoising algorithm seeks to remove noise, errors, or perturbations from a signal. Extensive research has been devoted to this arena over the last several decades, and as a result, todays denoisers can effectively remove large amounts of additive white Gaussian noise. A compressed sensing (CS) reconstruction algorithm seeks to recover a structured signal acquired using a small number of randomized measurements. Typical CS reconstruction algorithms can be cast as iteratively estimating a signal from a perturbed observation. This paper answers a natural question: How can one effectively employ a generic denoiser in a CS reconstruction algorithm? In response, we develop an extension of the approximate message passing (AMP) framework, called Denoising-based AMP (DAMP), that can integrate a wide class of denoisers within its iterations. We demonstrate that, when used with a high performance denoiser for natural images, DAMP offers state-of-the-art CS recovery performance while operating tens of times faster than competing methods. We explain the exceptional performance of DAMP by analyzing some of its theoretical features. A key element in DAMP is the use of an appropriate Onsager correction term in its iterations, which coerces the signal perturbation at each iteration to be very close to the white Gaussian noise that denoisers are typically designed to remove. The fundamental challenge faced by a compressed sensing (CS) reconstruction algorithm is to reconstruct a highdimensional signal from a small number of measurements. In a single pixel camera, Φ might be a sequence of 1s and 0s representing the modulation of a micromirror array [3]. " Ψu with sparse u, where Ψ represents the inverse transform matrix. C. Metzler and R. Baraniuk are with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77023 USA (email: chris.metzler@rice.edu and richb@rice.edu). A. Maleki is with the Department of Statistics, Columbia University, New York, NY 10023 USA (email: arian@stat.columbia.edu). The work of C. Metzler supported by the NSF GRF Program and the DoD NDSEG Program. The work of A. Maleki was supported by the grant NSF CCF-1420328. However, when dealing with large signals, such as images, these convex programs are extremely computationally demanding. Therefore, lower cost iterative algorithms were developed; including matching pursuit [6], orthogonal matching pursuit [7], iterative hard-thresholding [8], compressive sampling matching pursuit [9], approximate message passing [10], and iterative soft-thresholding [11]-[16], to name just a few. See [17], [18] for a complete set of references. Here, δ " m{n is a measure of the under-determinacy of the problem, x y denotes the average of a vector, and The role of this term is illustrated in Figure 1. A QQplot is a visual inspection tool for checking the Gaussianity of the data. In a QQplot, deviation from a straight line is an evidence of non-Gaussianity.

At Geneia, a health care technology and consulting company, they use big data along with machine learning to help health care organizations deliver better patient care at a lower cost. "Geneia brings data in from a lot of different sources," said Lavoie. We've built Theon, a unified platform to integrate the data and allow us to apply machine learning techniques." Using machine learning, Geneia can match and determine missing values, as well as perform principal component analysis and look at patterns in the data – clusters that help them see trends and causality. "Machine learning allows us to see patterns in the data that we couldn't see before.

In these situations, it is not always necessary to think about samples and populations, or to think about a model that expresses a scientific idea. It doesn't make sense to me, because if I were to build a regression model I would still need to think about my samples and population. I don't understand why I should just plug my sample data into R and hope for the best without any idea my sample is about. The sentence doesn't add anything, it's confusing and technically incorrect. Instead the goal is to simply find an equation or algorithm that makes reasonably correct predictons sounds doggy to me.

Time-varying mixture densities occur in many scenarios, for example, the distributions of keywords that appear in publications may evolve from year to year, video frame features associated with multiple targets may evolve in a sequence. Any models that realistically cater to this phenomenon must exhibit two important properties: the underlying mixture densities must have an unknown number of mixtures, and there must be some "smoothness" constraints in place for the adjacent mixture densities. The traditional Hierarchical Dirichlet Process (HDP) may be suited to the first property, but certainly not the second. This is due to how each random measure in the lower hierarchies is sampled independent of each other and hence does not facilitate any temporal correlations. To overcome such shortcomings, we proposed a new Smoothed Hierarchical Dirichlet Process (sHDP). The key novelty of this model is that we place a temporal constraint amongst the nearby discrete measures $\{G_j\}$ in the form of symmetric Kullback-Leibler (KL) Divergence with a fixed bound $B$. Although the constraint we place only involves a single scalar value, it nonetheless allows for flexibility in the corresponding successive measures. Remarkably, it also led us to infer the model within the stick-breaking process where the traditional Beta distribution used in stick-breaking is now replaced by a new constraint calculated from $B$. We present the inference algorithm and elaborate on its solutions. Our experiment using NIPS keywords has shown the desirable effect of the model.

In this post you will discover the k-Nearest Neighbors (KNN) algorithm for classification and regression. After reading this post you will know. This post was written for developers and assumes no background in statistics or mathematics. The focus is on how the algorithm works and how to use it for predictive modeling problems. If you have any questions, leave a comment and I will do my best to answer.

However, if the argument to exp(wT x) is large enough to cause overflow, wouldn't that also be the case for standard binary logistic regression as well, since negative-log-likelihood in that case contains the sigmoid function, which also has exp(wT x)? However, I don't think log-sum-exp can be applied to binary logistic regression, right?