In supervised learning, the learning algorithm is provided outcome data in advance, in the form of a pre-labeled set of instances. It is from this set that the algorithm is expected to learn what to do when it encounters future, previously unseen instances. Classification is a form of supervised learning. As an example, take the biological taxonomic hierarchy. Organisms are grouped into successfully more specific ranks of domain, kingdom, phylum, etc.

In part one, we showed how the machine learning process is like the scientific thinking process, and in part two, we introduced a benchmark task and showed how to get your machine learning system up and running with a simple nearest neighbors model. Time to crank it up! In this post, we continue the scientific thinking process by extending our simple starting model into increasingly powerful models. We'll show how to implement a particularly useful kind of model in Tensorflow. Shortcutting this process is a common failure mode--ye've been warned! But familiarity with common patterns of errors, diagnoses, and solutions allow experts to zip through the iterations. From the perspective of someone who is learning, it might look like the expert is skipping steps and going straight to the complicated stuff.

The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adaptive normal distribution (LAND) is a generalization of the normal distribution to the "manifold" setting, where data is assumed to lie near a potentially low-dimensional manifold embedded in $\mathbb{R}^D$. The LAND is parametric, depending only on a mean and a covariance, and is the maximum entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, and provide the corresponding EM algorithm. We demonstrate the efficiency of the LAND to fit non-trivial probability distributions over both synthetic data, and EEG measurements of human sleep.

Mixture models and topic models generate each observation from a single cluster, but standard variational posteriors for each observation assign positive probability to all possible clusters. This requires dense storage and runtime costs that scale with the total number of clusters, even though typically only a few clusters have significant posterior mass for any data point. We propose a constrained family of sparse variational distributions that allow at most $L$ non-zero entries, where the tunable threshold $L$ trades off speed for accuracy. Previous sparse approximations have used hard assignments ($L=1$), but we find that moderate values of $L>1$ provide superior performance. Our approach easily integrates with stochastic or incremental optimization algorithms to scale to millions of examples. Experiments training mixture models of image patches and topic models for news articles show that our approach produces better-quality models in far less time than baseline methods.

We consider the task of predicting lexical entailment using distributional vectors. We perform a novel qualitative analysis of one existing model which was previously shown to only measure the prototypicality of word pairs. We find that the model strongly learns to identify hypernyms using Hearst patterns, which are well known to be predictive of lexical relations. We present a novel model which exploits this behavior as a method of feature extraction in an iterative procedure similar to Principal Component Analysis. Our model combines the extracted features with the strengths of other proposed models in the literature, and matches or outperforms prior work on multiple data sets.

In this post, we take a tour of the most popular machine learning algorithms. It is useful to tour the main algorithms in the field to get a feeling of what methods are available. It is useful to tour the main algorithms in the field to get a feeling of what methods are available. There are so many algorithms available and it can feel overwhelming when algorithm names are thrown around and you are expected to just know what they are and where they fit. I want to give you two ways to think about and categorize the algorithms you may come across in the field.

This paper presents a speed-optimized and cache-friendly implementation for multivariate classification called FastBDT. FastBDT is one order of magnitude faster during the fitting-phase and application-phase, in comparison with popular implementations in software frameworks like TMVA, scikit-learn and XGBoost. The concepts used to optimize the execution time and performance studies are discussed in detail in this paper. The key ideas include: An equal-frequency binning on the input data, which allows replacing expensive floating-point with integer operations, while at the same time increasing the quality of the classification; a cache-friendly linear access pattern to the input data, in contrast to usual implementations, which exhibit a random access pattern.

I have a little secret: I don't like the terminology, notation, and style of writing in statistics. I find it unnecessarily complicated. This shows up when trying to read about Markov Chain Monte Carlo methods. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of Biostatistics. Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Successive random selections form a Markov chain, the stationary distribution of which is the target distribution.

We propose an extension to Hawkes processes by treating the levels of self-excitation as a stochastic differential equation. Our new point process allows better approximation in application domains where events and intensities accelerate each other with correlated levels of contagion. We generalize a recent algorithm for simulating draws from Hawkes processes whose levels of excitation are stochastic processes, and propose a hybrid Markov chain Monte Carlo approach for model fitting. Our sampling procedure scales linearly with the number of required events and does not require stationarity of the point process. A modular inference procedure consisting of a combination between Gibbs and Metropolis Hastings steps is put forward. We recover expectation maximization as a special case. Our general approach is illustrated for contagion following geometric Brownian motion and exponential Langevin dynamics.

Independent component analysis (ICA) is a method for recovering statistically independent signals from observations of unknown linear combinations of the sources. Some of the most accurate ICA decomposition methods require searching for the inverse transformation which minimizes different approximations of the Mutual Information, a measure of statistical independence of random vectors. Two such approximations are the Kernel Generalized Variance or the Kernel Canonical Correlation which has been shown to reach the highest performance of ICA methods. However, the computational effort necessary just for computing these measures is cubic in the sample size. Hence, optimizing them becomes even more computationally demanding, in terms of both space and time. Here, we propose a couple of alternative novel measures based on randomized features of the samples - the Randomized Generalized Variance and the Randomized Canonical Correlation. The computational complexity of calculating the proposed alternatives is linear in the sample size and provide a controllable approximation of their Kernel-based non-random versions. We also show that optimization of the proposed statistical properties yields a comparable separation error at an order of magnitude faster compared to Kernel-based measures.