When choosing a suitable technique for regression and classification with multivariate predictor variables, one is often faced with a tradeoff between interpretability and high predictive accuracy. To give a classical example, classification and regression trees are easy to understand and interpret. Tree ensembles like Random Forests provide usually more accurate predictions. Yet tree ensembles are also more difficult to analyze than single trees and are often criticized, perhaps unfairly, as `black box' predictors. Node harvest is trying to reconcile the two aims of interpretability and predictive accuracy by combining positive aspects of trees and tree ensembles. Results are very sparse and interpretable and predictive accuracy is extremely competitive, especially for low signal-to-noise data. The procedure is simple: an initial set of a few thousand nodes is generated randomly. If a new observation falls into just a single node, its prediction is the mean response of all training observation within this node, identical to a tree-like prediction. A new observation falls typically into several nodes and its prediction is then the weighted average of the mean responses across all these nodes. The only role of node harvest is to `pick' the right nodes from the initial large ensemble of nodes by choosing node weights, which amounts in the proposed algorithm to a quadratic programming problem with linear inequality constraints. The solution is sparse in the sense that only very few nodes are selected with a nonzero weight. This sparsity is not explicitly enforced. Maybe surprisingly, it is not necessary to select a tuning parameter for optimal predictive accuracy. Node harvest can handle mixed data and missing values and is shown to be simple to interpret and competitive in predictive accuracy on a variety of data sets.

We propose in this work a new family of kernels for variable-length time series. Our work builds upon the vector autoregressive (VAR) model for multivariate stochastic processes: given a multivariate time series x, we consider the likelihood function p_{\theta}(x) of different parameters \theta in the VAR model as features to describe x. To compare two time series x and x', we form the product of their features p_{\theta}(x) p_{\theta}(x') which is integrated out w.r.t \theta using a matrix normal-inverse Wishart prior. Among other properties, this kernel can be easily computed when the dimension d of the time series is much larger than the lengths of the considered time series x and x'. It can also be generalized to time series taking values in arbitrary state spaces, as long as the state space itself is endowed with a kernel \kappa. In that case, the kernel between x and x' is a a function of the Gram matrices produced by \kappa on observations and subsequences of observations enumerated in x and x'. We describe a computationally efficient implementation of this generalization that uses low-rank matrix factorization techniques. These kernels are compared to other known kernels using a set of benchmark classification tasks carried out with support vector machines.

We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a proximal minimization algorithm. We theoretically show under some conditions that DAL converges super-linearly in a non-asymptotic and global sense. Due to a special modelling of sparse estimation problems in the context of machine learning, the assumptions we make are milder and more natural than those made in conventional analysis of augmented Lagrangian algorithms. In addition, the new interpretation enables us to generalize DAL to wide varieties of sparse estimation problems. We experimentally confirm our analysis in a large scale $\ell_1$-regularized logistic regression problem and extensively compare the efficiency of DAL algorithm to previously proposed algorithms on both synthetic and benchmark datasets.

For a density f on R^d, a high-density cluster is any connected component of {x: f(x) >= c}, for some c > 0. The set of all high-density clusters form a hierarchy called the cluster tree of f. We present a procedure for estimating the cluster tree given samples from f. We give finite-sample convergence rates for our algorithm, as well as lower bounds on the sample complexity of this estimation problem.

Network models are widely used to capture interactions among component of complex systems, such as social and biological. To understand their behavior, it is often necessary to analyze functionally related components of the system, corresponding to subsystems. Therefore, the analysis of subnetworks may provide additional insight into the behavior of the system, not evident from individual components. We propose a novel approach for incorporating available network information into the analysis of arbitrary subnetworks. The proposed method offers an efficient dimension reduction strategy using Laplacian eigenmaps with Neumann boundary conditions, and provides a flexible inference framework for analysis of subnetworks, based on a group-penalized principal component regression model on graphs. Asymptotic properties of the proposed inference method, as well as the choice of the tuning parameter for control of the false positive rate are discussed in high dimensional settings. The performance of the proposed methodology is illustrated using simulated and real data examples from biology.

Motivated by an application to unsupervised part-of-speech tagging, we present an algorithm for the Euclidean embedding of large sets of categorical data based on co-occurrence statistics. We use the CODE model of Globerson et al. but constrain the embedding to lie on a high-dimensional unit sphere. This constraint allows for efficient optimization, even in the case of large datasets and high embedding dimensionality. Using k-means clustering of the embedded data, our approach efficiently produces state-of-the-art results. We analyze the reasons why the sphere constraint is beneficial in this application, and conjecture that these reasons might apply quite generally to other large-scale tasks.

We propose a new variational EM algorithm for fitting factor analysis models with mixed continuous and categorical observations. The algorithm is based on a simple quadratic bound to the log-sum-exp function. In the special case of fully observed binary data, the bound we propose is significantly faster than previous variational methods. We show that EM is significantly more robust in the presence of missing data compared to treating the latent factors as parameters, which is the approach used by exponential family PCA and other related matrix-factorization methods. A further benefit of the variational approach is that it can easily be extended to the case of mixtures of factor analyzers, as we show. We present results on synthetic and real data sets demonstrating several desirable properties of our proposed method.

The problem of controlling the margin of a classifier is studied. A detailed analytical study is presented on how properties of the classification risk, such as its optimal link and minimum risk functions, are related to the shape of the loss, and its margin enforcing properties. It is shown that for a class of risks, denoted canonical risks, asymptotic Bayes consistency is compatible with simple analytical relationships between these functions. These enable a precise characterization of the loss for a popular class of link functions. It is shown that, when the risk is in canonical form and the link is inverse sigmoidal, the margin properties of the loss are determined by a single parameter. Novel families of Bayes consistent loss functions, of variable margin, are derived. These families are then used to design boosting style algorithms with explicit control of the classification margin. The new algorithms generalize well established approaches, such as LogitBoost. Experimental results show that the proposed variable margin losses outperform the fixed margin counterparts used by existing algorithms. Finally, it is shown that best performance can be achieved by cross-validating the margin parameter.

We consider the problem of learning a coefficient vector x0 from noisy linear observation y=Ax0+w. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator. In this case, a popular approach consists in solving an l1-penalized least squares problem known as the LASSO or BPDN. For sequences of matrices A of increasing dimensions, with iid gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic risk of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Through simulations on real data matrices (gene expression data and hospital medical records) we observe that these results can be relevant in a broad array of practical applications.

We consider the problem of learning a local metric to enhance the performance of nearest neighbor classification. Conventional metric learning methods attempt to separate data distributions in a purely discriminative manner; here we show how to take advantage of information from parametric generative models. We focus on the bias in the information-theoretic error arising from finite sampling effects, and find an appropriate local metric that maximally reduces the bias based upon knowledge from generative models. As a byproduct, the asymptotic theoretical analysis in this work relates metric learning with dimensionality reduction, which was not understood from previous discriminative approaches. Empirical experiments show that this learned local metric enhances the discriminative nearest neighbor performance on various datasets using simple class conditional generative models.