We propose a novel accelerated exact k-means algorithm, which performs better than the current state-of-the-art low-dimensional algorithm in 18 of 22 experiments, running up to 3 times faster. We also propose a general improvement of existing state-of-the-art accelerated exact k-means algorithms through better estimates of the distance bounds used to reduce the number of distance calculations, and get a speedup in 36 of 44 experiments, up to 1.8 times faster. We have conducted experiments with our own implementations of existing methods to ensure homogeneous evaluation of performance, and we show that our implementations perform as well or better than existing available implementations. Finally, we propose simplified variants of standard approaches and show that they are faster than their fully-fledged counterparts in 59 of 62 experiments.

Consider the problem of sparse clustering, where it is assumed that only a subset of the features are useful for clustering purposes. In the framework of the COSA method of Friedman and Meulman, subsequently improved in the form of the Sparse K-means method of Witten and Tibshirani, a natural and simpler hill-climbing approach is introduced. The new method is shown to be competitive with these two methods and others.

In my last post, I showed how to find similar Game of Thrones episodes based on the characters that appear in different episodes. This allowed us to find similar episodes on an episode by episode basis, but I was curious whether there were groups of similar episodes that we could identify. A clustering algorithm groups similar documents together, where similarity is based on calculating a'distance' between documents. Documents separated by a small distance would be in the same cluster, whereas if there's a large distance between episodes then they'd probably be in different clusters. The KMeans algorithm clusters data by trying to separate samples in n groups of equal variance, minimizing a criterion known as the inertia or within-cluster sum-of-squares.

We derive generalization error bounds for traditional time-series forecasting models. Our results hold for many standard forecasting tools including autoregressive models, moving average models, and, more generally, linear state-space models. These non-asymptotic bounds need only weak assumptions on the data-generating process, yet allow forecasters to select among competing models and to guarantee, with high probability, that their chosen model will perform well. We motivate our techniques with and apply them to standard economic and financial forecasting tools---a GARCH model for predicting equity volatility and a dynamic stochastic general equilibrium model (DSGE), the standard tool in macroeconomic forecasting. We demonstrate in particular how our techniques can aid forecasters and policy makers in choosing models which behave well under uncertainty and mis-specification.

Sorry if this is too trivial, but let me start at the "very beginning:" Linear regression. The goal of (ordinary least-squares) linear regression is to find the optimal weights that -- when linearly combined with the inputs -- result in a model that minimizes the vertical offsets between the target and explanatory variables, but let's not get distracted by model fitting, which is a different topic;). So, in linear regression, we compute a linear combination of weights and inputs (let's call this function the "net input function"). Next, let's consider logistic regression. Here, we put the net input z through a non-linear "activation function" -- the logistic sigmoid function where.

An implementation of support vector machines (SVM) is available in the JMSL Numerical Library as of release 7.3. The data mining functionality in JMSL continues to expand with the latest release, including areas such as decision trees and bootstrap aggregation. The documentation of the IMSL Libraries is detailed and robust, but the algorithm discussion and examples can only cover a finite set of use cases. This series of blog posts walks through some additional examples with a focus on classification, starting with the textbook examples part of most SVM resources. Notes and key points are highlighted throughout to provide a complementary resource for users new to SVM or new to using the JMSL Library.

Songbirds provide a model system that neuroscientists use to understand how the brain learns and controls speech and similar skills. Much like infants learning to speak from their parents, songbirds learn their song from a tutor and practice it millions of times before reaching maturity. Also like humans, songbirds have evolved special brain regions for learning and producing their vocalizations. These newly-evolved brain regions in songbirds, known as the song system, are found within broader brain areas shared by birds and humans across evolution. So by studying how the song system works, we can learn about our own brains.

The focus of this talk is scalable machine learning using the H2O R and Python packages. H2O is an open source distributed machine learning platform designed for big data, with the added benefit that it's easy to use on a laptop (in addition to a multi-node Hadoop or Spark cluster). The core machine learning algorithms of H2O are implemented in high-performance Java; however, fully featured APIs are available in R, Python, Scala, REST/JSON and also through a web interface. Since H2O's algorithm implementations are distributed, this allows the software to scale to very large datasets that may not fit into RAM on a single machine. H2O currently features distributed implementations of generalized linear models, gradient boosting machines, random forest, deep neural nets, dimensionality reduction methods (PCA, GLRM), clustering algorithms (K-means), and anomaly detection methods, among others.

Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood estimator and standard Bayesian procedures do not admit the root-$n$ parametric rate of convergence. We propose a general framework for the identification of singularity structures of the parameter space of finite mixtures, and study the impacts of the singularity levels on minimax lower bounds and rates of convergence for the maximum likelihood estimator over a compact parameter space. Our study makes explicit the deep links between model singularities, parameter estimation convergence rates and minimax lower bounds, and the algebraic geometry of the parameter space for mixtures of continuous distributions. The theory is applied to establish concrete convergence rates of parameter estimation for finite mixture of skewnormal distributions. This rich and increasingly popular mixture model is shown to exhibit a remarkably complex range of asymptotic behaviors which have not been hitherto reported in the literature.

Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on how many projections are needed to accurately preserve the geometry of these manifolds, given their intrinsic dimensionality, volume and curvature. However, such works employ definitions of volume and curvature that are inherently difficult to compute. Therefore such theory cannot be easily tested against numerical simulations to understand the tightness of the proven bounds. We instead study typical distortions arising in random projections of an ensemble of smooth Gaussian random manifolds. We find explicitly computable, approximate theoretical bounds on the number of projections required to accurately preserve the geometry of these manifolds. Our bounds, while approximate, can only be violated with a probability that is exponentially small in the ambient dimension, and therefore they hold with high probability in cases of practical interest. Moreover, unlike previous work, we test our theoretical bounds against numerical experiments on the actual geometric distortions that typically occur for random projections of random smooth manifolds. We find our bounds are tighter than previous results by several orders of magnitude.