Variational inference using the reparameterization trick has enabled large-scale approximate Bayesian inference in complex probabilistic models, leveraging stochastic optimization to sidestep intractable expectations. The reparameterization trick is applicable when we can simulate a random variable by applying a differentiable deterministic function on an auxiliary random variable whose distribution is fixed. For many distributions of interest (such as the gamma or Dirichlet), simulation of random variables relies on acceptance-rejection sampling. The discontinuity introduced by the accept-reject step means that standard reparameterization tricks are not applicable. We propose a new method that lets us leverage reparameterization gradients even when variables are outputs of a acceptance-rejection sampling algorithm. Our approach enables reparameterization on a larger class of variational distributions. In several studies of real and synthetic data, we show that the variance of the estimator of the gradient is significantly lower than other state-of-the-art methods. This leads to faster convergence of stochastic gradient variational inference.

High signal to noise ratio (SNR) consistency of model selection criteria in linear regression models has attracted a lot of attention recently. However, most of the existing literature on high SNR consistency deals with model order selection. Further, the limited literature available on the high SNR consistency of subset selection procedures (SSPs) is applicable to linear regression with full rank measurement matrices only. Hence, the performance of SSPs used in underdetermined linear models (a.k.a compressive sensing (CS) algorithms) at high SNR is largely unknown. This paper fills this gap by deriving necessary and sufficient conditions for the high SNR consistency of popular CS algorithms like $l_0$-minimization, basis pursuit de-noising or LASSO, orthogonal matching pursuit and Dantzig selector. Necessary conditions analytically establish the high SNR inconsistency of CS algorithms when used with the tuning parameters discussed in literature. Novel tuning parameters with SNR adaptations are developed using the sufficient conditions and the choice of SNR adaptations are discussed analytically using convergence rate analysis. CS algorithms with the proposed tuning parameters are numerically shown to be high SNR consistent and outperform existing tuning parameters in the moderate to high SNR regime.

Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. To this end, explicit feature maps of kernels for vectorial data have been extensively studied. As many real-world data is structured, various kernels for complex data like graphs have been proposed. Indeed, many of them directly compute feature maps. However, the kernel trick is employed when the number of features is very large or the individual vertices of graphs are annotated by real-valued attributes. Can we still compute explicit feature maps efficiently under these circumstances? Triggered by this question, we investigate how general convolution kernels are composed from base kernels and construct corresponding feature maps. We apply our results to widely used graph kernels and analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive feature maps for random walk and subgraph matching kernels and apply them to real-world graphs with discrete labels. Thereby, our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively. Moreover, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and Graph Invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time.

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the joint embedding method identifies a linear subspace spanned by rank one symmetric matrices and projects adjacency matrices of graphs into this subspace. The projection coefficients can be treated as features of the graphs. We also propose a random graph model which generalizes classical random graph model and can be used to model multiple graphs. We show through theory and numerical experiments that under the model, the joint embedding method produces estimates of parameters with small errors. Via simulation experiments, we demonstrate that the joint embedding method produces features which lead to state of the art performance in classifying graphs. Applying the joint embedding method to human brain graphs, we find it extract interpretable features that can be used to predict individual composite creativity index.

A framework is presented for unsupervised learning of representations based on infomax principle for large-scale neural populations. We use an asymptotic approximation to the Shannon's mutual information for a large neural population to demonstrate that a good initial approximation to the global information-theoretic optimum can be obtained by a hierarchical infomax method. Starting from the initial solution, an efficient algorithm based on gradient descent of the final objective function is proposed to learn representations from the input datasets, and the method works for complete, overcomplete, and undercomplete bases. As confirmed by numerical experiments, our method is robust and highly efficient for extracting salient features from input datasets. Compared with the main existing methods, our algorithm has a distinct advantage in both the training speed and the robustness of unsupervised representation learning. Furthermore, the proposed method is easily extended to the supervised or unsupervised model for training deep structure networks.

This post originally appeared on the Yhat blog. Yhat is a Brooklyn based company whose goal is to make data science applicable for developers, data scientists, and businesses alike. Yhat provides a software platform for deploying and managing predictive algorithms as REST APIs, while eliminating the painful engineering obstacles associated with production environments like testing, versioning, scaling and security. In this post, I'll detail how you can use K-Means clustering to help with some of the exploratory aspects of customer segmentation. I'll be walking through the example using Yhat's own Python IDE, Rodeo, which you can download for Windows, Mac or Linux here.

I had the opportunity to start using xgboost machine learning algorithm, it is fast and shows good results. Here I will be using multiclass prediction with the iris dataset from scikit-learn. In order to work with the data, I need to install various scientific libraries for python. The best way I have found is to use Anaconda. It simply installs all the libs and helps to install new ones.

All the regression theory developed by statisticians over the last 200 years (related to the general linear model) is useless. Regression can be performed as accurately without statistical models, including the computation of confidence intervals (for estimates, predicted values or regression parameters). The non-statistical approach is also more robust than theory described in all statistics textbooks and taught in all statistical courses. It does not require Map-Reduce when data is really big, nor any matrix inversion, maximum likelihood estimation, or mathematical optimization (Newton algorithm). It is indeed incredibly simple, robust, easy to interpret, and easy to code (no statistical libraries required).

Indian Buffet Process based models are an elegant way for discovering underlying features within a data set, but inference in such models can be slow. Inferring underlying features using Markov chain Monte Carlo either relies on an uncollapsed representation, which leads to poor mixing, or on a collapsed representation, which leads to a quadratic increase in computational complexity. Existing attempts at distributing inference have introduced additional approximation within the inference procedure. In this paper we present a novel algorithm to perform asymptotically exact parallel Markov chain Monte Carlo inference for Indian Buffet Process models. We take advantage of the fact that the features are conditionally independent under the beta-Bernoulli process. Because of this conditional independence, we can partition the features into two parts: one part containing only the finitely many instantiated features and the other part containing the infinite tail of uninstantiated features. For the finite partition, parallel inference is simple given the instantiation of features. But for the infinite tail, performing uncollapsed MCMC leads to poor mixing and hence we collapse out the features. The resulting hybrid sampler, while being parallel, produces samples asymptotically from the true posterior.

Although neural networks are routinely and successfully trained in practice using simple gradient-based methods, most existing theoretical results are negative, showing that learning such networks is difficult, in a worst-case sense over all data distributions. In this paper, we take a more nuanced view, and consider whether specific assumptions on the "niceness" of the input distribution, or "niceness" of the target function (e.g. in terms of smoothness, non-degeneracy, incoherence, random choice of parameters etc.), are sufficient to guarantee learnability using gradient-based methods. We provide evidence that neither class of assumptions alone is sufficient: On the one hand, for any member of a class of "nice" target functions, there are difficult input distributions. On the other hand, we identify a family of simple target functions, which are difficult to learn even if the input distribution is "nice". To prove our results, we develop some tools which may be of independent interest, such as extending Fourier-based hardness techniques developed in the context of statistical queries \cite{blum1994weakly}, from the Boolean cube to Euclidean space and to more general classes of functions.