A few weeks ago Lydia Dishman wrote a fantastic piece characterizing the top tech companies as defined by the "quality" of their talent using the Paysa CompanyRank algorithm and how these companies change in rank over time. CompanyRank is an Expectation-Maximization algorithm applied over space and time to quantifying the network-wide flux of tech workers to/fro/retained at each technical company at each month in time…a distant cousin to Google's PageRank algorithm. Another example of emergent structure learned from perceived independent chaos of a large, sparsely-connected complex system. Figure 1 depicts the Paysa CompanyRank time-series of Uber, Facebook, Google and Zynga over time. The top slot, e.g., most talent dense company gets rank of #1 and decays down to an arbitrary low rank (high number)…think 1 is better than 2 is better than 3 and so forth.

The Hidden Markov Model or HMM is all about learning sequences. A lot of the data that would be very useful for us to model is in sequences. Stock prices are sequences of prices. Language is a sequence of words. Credit scoring involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you're going to default.

We consider the setting where a collection of time series, modeled as random processes, evolve in a causal manner, and one is interested in learning the graph governing the relationships of these processes. A special case of wide interest and applicability is the setting where the noise is Gaussian and relationships are Markov and linear. We study this setting with two additional features: firstly, each random process has a hidden (latent) state, which we use to model the internal memory possessed by the variables (similar to hidden Markov models). Secondly, each variable can depend on its latent memory state through a random lag (rather than a fixed lag), thus modeling memory recall with differing lags at distinct times. Under this setting, we develop an estimator and prove that under a genericity assumption, the parameters of the model can be learned consistently. We also propose a practical adaption of this estimator, which demonstrates significant performance gains in both synthetic and real-world datasets.

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaussians. This is achieved by (i) studying the sequence of parameters from idealized execution of EM in the infinite sample limit, and fully characterizing the limit points of the sequence in terms of the initial parameters; and then (ii) based on this convergence analysis, establishing statistical consistency (or lack thereof) for the actual sequence of parameters produced by EM.

We present a unified approach for learning the parameters of Sum-Product networks (SPNs). We prove that any complete and decomposable SPN is equivalent to a mixture of trees where each tree corresponds to a product of univariate distributions. Based on the mixture model perspective, we characterize the objective function when learning SPNs based on the maximum likelihood estimation (MLE) principle and show that the optimization problem can be formulated as a signomial program. We construct two parameter learning algorithms for SPNs by using sequential monomial approximations (SMA) and the concave-convex procedure (CCCP), respectively. The two proposed methods naturally admit multiplicative updates, hence effectively avoiding the projection operation. With the help of the unified framework, we also show that, in the case of SPNs, CCCP leads to the same algorithm as Expectation Maximization (EM) despite the fact that they are different in general.

Three years earlier, Apple had been the first major tech company to integrate a smart assistant into its operating system. Siri was the company's adaptation of a standalone app it had purchased, along with the team that created it, in 2010. Initial reviews were ecstatic, but over the next few months and years, users became impatient with its shortcomings. All too often, it erroneously interpreted commands. So Apple moved Siri voice recognition to a neural-net based system for US users on that late July day (it went worldwide on August 15, 2014.)

We introduce a new criterion to determine the order of an autoregressive model fitted to time series data. It has the benefits of the two well-known model selection techniques, the Akaike information criterion and the Bayesian information criterion. When the data is generated from a finite order autoregression, the Bayesian information criterion is known to be consistent, and so is the new criterion. When the true order is infinity or suitably high with respect to the sample size, the Akaike information criterion is known to be efficient in the sense that its prediction performance is asymptotically equivalent to the best offered by the candidate models; in this case, the new criterion behaves in a similar manner. Different from the two classical criteria, the proposed criterion adaptively achieves either consistency or efficiency depending on the underlying true model. In practice where the observed time series is given without any prior information about the model specification, the proposed order selection criterion is more flexible and robust compared with classical approaches. Numerical results are presented demonstrating the adaptivity of the proposed technique when applied to various datasets.

It always amazes me how I can hear a statement uttered in the space of a few seconds about some aspect of machine learning that then takes me countless hours to understand. I first heard about Gaussian Processes on an episode of the Talking Machines podcast and thought it sounded like a really neat idea. I promptly procured myself a copy of the classic text on the subject, Gaussian Processes for Machine Learning by Rasmussen and Williams, but my tenuous grasp on the Bayesian approach to machine learning meant I got stumped pretty quickly. That's when I began the journey I described in my last post, From both sides now: the math of linear regression. Gaussian Processes (GPs) are the natural next step in that journey as they provide an alternative approach to regression problems.

To construct flexible nonlinear predictive distributions, the paper introduces a family of softplus function based regression models that convolve, stack, or combine both operations by convolving countably infinite stacked gamma distributions, whose scales depend on the covariates. Generalizing logistic regression that uses a single hyperplane to partition the covariate space into two halves, softplus regressions employ multiple hyperplanes to construct a confined space, related to a single convex polytope defined by the intersection of multiple half-spaces or a union of multiple convex polytopes, to separate one class from the other. The gamma process is introduced to support the convolution of countably infinite (stacked) covariate-dependent gamma distributions. For Bayesian inference, Gibbs sampling derived via novel data augmentation and marginalization techniques is used to deconvolve and/or demix the highly complex nonlinear predictive distribution. Example results demonstrate that softplus regressions provide flexible nonlinear decision boundaries, achieving classification accuracies comparable to that of kernel support vector machine while requiring significant less computation for out-of-sample prediction.

In Part I of this series, we introduced the theory and intuition behind the VAE, an exciting development in machine learning for combined generative modeling and inference--"machines that imagine and reason." To recap: VAEs put a probabilistic spin on the basic autoencoder paradigm--treating their inputs, hidden representations, and reconstructed outputs as probabilistic random variables within a directed graphical model. With this Bayesian perspective, the encoder becomes a variational inference network, mapping observed inputs to (approximate) posterior distributions over latent space, and the decoder becomes a generative network, capable of mapping arbitrary latent coordinates back to distributions over the original data space. The beauty of this setup is that we can take a principled Bayesian approach toward building systems with a rich internal "mental model" of the observed world, all by training a single, cleverly-designed deep neural network. These benefits derive from an enriched understanding of data as merely the tip of the iceberg--the observed result of an underlying causative probabilistic process.