We investigate the relationship between three fundamental problems in machine learning: binary classification, bipartite ranking, and binary class probability estimation (CPE). It is known that a good binary CPE model can be used to obtain a good binary classification model (by thresholding at 0.5), and also to obtain a good bipartite ranking model (by using the CPE model directly as a ranking model); it is also known that a binary classification model does not necessarily yield a CPE model. However, not much is known about other directions. Formally, these relationships involve regret transfer bounds. In this paper, we introduce the notion of weak regret transfer bounds, where the mapping needed to transform a model from one problem to another depends on the underlying probability distribution (and in practice, must be estimated from data). We then show that, in this weaker sense, a good bipartite ranking model can be used to construct a good classification model (by thresholding at a suitable point), and more surprisingly, also to construct a good binary CPE model (by calibrating the scores of the ranking model).

We consider the problem of accurately estimating a high-dimensional sparse vector using a small number of linear measurements that are contaminated by noise. It is well known that standard computationally tractable sparse recovery algorithms, such as the Lasso, OMP, and their various extensions, perform poorly when the measurement matrix contains highly correlated columns. We develop a simple greedy algorithm, called SWAP, that iteratively swaps variables until a desired loss function cannot be decreased any further. SWAP is surprisingly effective in handling measurement matrices with high correlations. We prove that SWAP can be easily used as a wrapper around standard sparse recovery algorithms for improved performance. We theoretically quantify the statistical guarantees of SWAP and complement our analysis with numerical results on synthetic and real data.

We propose a semiparametric procedure for estimating high dimensional sparse inverse covariance matrix. Our method, named ALICE, is applicable to the elliptical family. Computationally, we develop an efficient dual inexact iterative projection (${\rm D_2}$P) algorithm based on the alternating direction method of multipliers (ADMM). Theoretically, we prove that the ALICE estimator achieves the parametric rate of convergence in both parameter estimation and model selection. Moreover, ALICE calibrates regularizations when estimating each column of the inverse covariance matrix. So it not only is asymptotically tuning free, but also achieves an improved finite sample performance. We present numerical simulations to support our theory, and a real data example to illustrate the effectiveness of the proposed estimator.

We propose a fast algorithm for ridge regression when the number of features is much larger than the number of observations ($p \gg n$). The standard way to solve ridge regression in this setting works in the dual space and gives a running time of $O(n^2p)$. Our algorithm (SRHT-DRR) runs in time $O(np\log(n))$ and works by preconditioning the design matrix by a Randomized Walsh-Hadamard Transform with a subsequent subsampling of features. We provide risk bounds for our SRHT-DRR algorithm in the fixed design setting and show experimental results on synthetic and real datasets.

Biological tissue is often composed of cells with similar morphologies replicated throughout large volumes and many biological applications rely on the accurate identification of these cells and their locations from image data. Here we develop a generative model that captures the regularities present in images composed of repeating elements of a few different types. Formally, the model can be described as convolutional sparse block coding. For inference we use a variant of convolutional matching pursuit adapted to block-based representations. We extend the K-SVD learning algorithm to subspaces by retaining several principal vectors from the SVD decomposition instead of just one. Good models with little cross-talk between subspaces can be obtained by learning the blocks incrementally. We perform extensive experiments on simulated images and the inference algorithm consistently recovers a large proportion of the cells with a small number of false positives. We fit the convolutional model to noisy GCaMP6 two-photon images of spiking neurons and to Nissl-stained slices of cortical tissue and show that it recovers cell body locations without supervision. The flexibility of the block-based representation is reflected in the variability of the recovered cell shapes.

Measuring similarity is crucial to many learning tasks. It is also a richer and broader notion than what most metric learning algorithms can model. For example, similarity can arise from the process of aggregating the decisions of multiple latent components, where each latent component compares data in its own way by focusing on a different subset of features. In this paper, we propose Similarity Component Analysis (SCA), a probabilistic graphical model that discovers those latent components from data. In SCA, a latent component generates a local similarity value, computed with its own metric, independently of other components. The final similarity measure is then obtained by combining the local similarity values with a (noisy-)OR gate. We derive an EM-based algorithm for fitting the model parameters with similarity-annotated data from pairwise comparisons. We validate the SCA model on synthetic datasets where SCA discovers the ground-truth about the latent components. We also apply SCA to a multiway classification task and a link prediction task. For both tasks, SCA attains significantly better prediction accuracies than competing methods. Moreover, we show how SCA can be instrumental in exploratory analysis of data, where we gain insights about the data by examining patterns hidden in its latent components' local similarity values.

Conditional random fields, which model the distribution of a multivariate response conditioned on a set of covariates using undirected graphs, are widely used in a variety of multivariate prediction applications. Popular instances of this class of models such as categorical-discrete CRFs, Ising CRFs, and conditional Gaussian based CRFs, are not however best suited to the varied types of response variables in many applications, including count-valued responses. We thus introduce a “novel subclass of CRFs”, derived by imposing node-wise conditional distributions of response variables conditioned on the rest of the responses and the covariates as arising from univariate exponential families. This allows us to derive novel multivariate CRFs given any univariate exponential distribution, including the Poisson, negative binomial, and exponential distributions. Also in particular, it addresses the common CRF problem of specifying feature'' functions determining the interactions between response variables and covariates. We develop a class of tractable penalized $M$-estimators to learn these CRF distributions from data, as well as a unified sparsistency analysis for this general class of CRFs showing exact structure recovery can be achieved with high probability."

This paper presents an approach to multilabel classification (MLC) with a large number of labels. Our approach is a reduction to binary classification in which label sets are represented by low dimensional binary vectors. This representation follows the principle of Bloom filters, a space-efficient data structure originally designed for approximate membership testing. We show that a naive application of Bloom filters in MLC is not robust to individual binary classifiers' errors. We then present an approach that exploits a specific feature of real-world datasets when the number of labels is large: many labels (almost) never appear together. Our approch is provably robust, has sublinear training and inference complexity with respect to the number of labels, and compares favorably to state-of-the-art algorithms on two large scale multilabel datasets.

The detection of anomalous activity in graphs is a statistical problem that arises in many applications, such as network surveillance, disease outbreak detection, and activity monitoring in social networks. Beyond its wide applicability, graph structured anomaly detection serves as a case study in the difficulty of balancing computational complexity with statistical power. In this work, we develop from first principles the generalized likelihood ratio test for determining if there is a well connected region of activation over the vertices in the graph in Gaussian noise. Because this test is computationally infeasible, we provide a relaxation, called the Lov\'asz extended scan statistic (LESS) that uses submodularity to approximate the intractable generalized likelihood ratio. We demonstrate a connection between LESS and maximum a-posteriori inference in Markov random fields, which provides us with a poly-time algorithm for LESS. Using electrical network theory, we are able to control type 1 error for LESS and prove conditions under which LESS is risk consistent. Finally, we consider specific graph models, the torus, $k$-nearest neighbor graphs, and $\epsilon$-random graphs. We show that on these graphs our results provide near-optimal performance by matching our results to known lower bounds.

We use the notion of local Rademacher complexity to design new algorithms for learning kernels. Our algorithms thereby benefit from the sharper learning bounds based on that notion which, under certain general conditions, guarantee a faster convergence rate. We devise two new learning kernel algorithms: one based on a convex optimization problem for which we give an efficient solution using existing learning kernel techniques, and another one that can be formulated as a DC-programming problem for which we describe a solution in detail. We also report the results of experiments with both algorithms in both binary and multi-class classification tasks.