Efficient regularized isotonic regression with application to gene--gene interaction search

Isotonic regression is a nonparametric approach for fitting monotonic models to data that has been widely studied from both theoretical and practical perspectives. However, this approach encounters computational and statistical overfitting issues in higher dimensions. To address both concerns, we present an algorithm, which we term Isotonic Recursive Partitioning (IRP), for isotonic regression based on recursively partitioning the covariate space through solution of progressively smaller "best cut" subproblems. This creates a regularized sequence of isotonic models of increasing model complexity that converges to the global isotonic regression solution. The models along the sequence are often more accurate than the unregularized isotonic regression model because of the complexity control they offer. We quantify this complexity control through estimation of degrees of freedom along the path. Success of the regularized models in prediction and IRPs favorable computational properties are demonstrated through a series of simulated and real data experiments. We discuss application of IRP to the problem of searching for gene--gene interactions and epistasis, and demonstrate it on data from genome-wide association studies of three common diseases.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS504 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Signal Decomposition Using Masked Proximal Operators

We consider the well-studied problem of decomposing a vector time series signal into components with different characteristics, such as smooth, periodic, nonnegative, or sparse. We describe a simple and general framework in which the components are defined by loss functions (which include constraints), and the signal decomposition is carried out by minimizing the sum of losses of the components (subject to the constraints). When each loss function is the negative log-likelihood of a density for the signal component, this framework coincides with maximum a posteriori probability (MAP) estimation; but it also includes many other interesting cases. Summarizing and clarifying prior results, we give two distributed optimization methods for computing the decomposition, which find the optimal decomposition when the component class loss functions are convex, and are good heuristics when they are not. Both methods require only the masked proximal operator of each of the component loss functions, a generalization of the well-known proximal operator that handles missing entries in its argument. Both methods are distributed, i.e., handle each component separately. We derive tractable methods for evaluating the masked proximal operators of some loss functions that, to our knowledge, have not appeared in the literature.Comment: The manuscript has 61 pages, 22 figures and 2 tables. Also hosted at https://web.stanford.edu/~boyd/papers/sig_decomp_mprox.html. For code, see https://github.com/cvxgrp/signal-decompositio

Complex event detection using semantic saliency and nearly-isotonic SVM

Copyright © 2015 by the author(s). We aim to detect complex events in long Internet videos that may last for hours. A major challenge in this setting is that only a few shots in a long video are relevant to the event of interest while others are irrelevant or even misleading. Instead of indifferently pooling the shots, we first define a novel notion of semantic saliency that assesses the relevance of each shot with the event of interest. We then prioritize the shots according to their saliency scores since shots that are semantically more salient are expected to contribute more to the final event detector. Next, we propose a new isotonic regularizer that is able to exploit the semantic ordering information. The resulting nearly-isotonic SVM classifier exhibits higher discriminative power. Computationally, we develop an efficient implementation using the proximal gradient algorithm, and we prove new, closed-form proximal steps. We conduct extensive experiments on three real-world video datasets and confirm the effectiveness of the proposed approach

Primal-Dual Active-Set Methods for Convex Quadratic Optimization with Applications

Primal-dual active-set (PDAS) methods are developed for solving quadratic optimization problems (QPs). Such problems arise in their own right in optimal control and statistics–two applications of interest considered in this dissertation–and as subproblems when solving nonlinear optimization problems. PDAS methods are promising as they possess the same favorable properties as other active-set methods, such as their ability to be warm-started and to obtain highly accurate solutions by explicitly identifying sets of constraints that are active at an optimal solution. However, unlike traditional active-set methods, PDAS methods have convergence guarantees despite making rapid changes in active-set estimates, making them well suited for solving large-scale problems.Two PDAS variants are proposed for efficiently solving generally-constrained convex QPs. Both variants ensure global convergence of the iterates by enforcing montonicity in a measure of progress. Besides identifying an estimate set estimate, a novel uncertain set is introduced into the framework in order to house indices of variables that have been identified as being susceptible to cycling. The introduction of the uncertainty set guarantees convergence of the algorithm, and with techniques proposed to keep the set from expanding quickly, the practical performance of the algorithm is shown to be very efficient. Another PDAS variant is proposed for solving certain convex QPs that commonly arise when discretizing optimal control problems. The proposed framework allows inexactness in the subproblem solutions, which can significantly reduce computational cost in large-scale settings. By controlling the level inexactness either by exploiting knowledge of an upper bound of a matrix inverse or by dynamic estimation of such a value, the method achieves convergence guarantees and is shown to outperform a method that employs exact solutions computed by direct factorization techniques.Finally, the application of PDAS techniques for applications in statistics, variants are proposed for solving isotonic regression (IR) and trend filtering (TR) problems. It is shown that PDAS can solve an IR problem with n data points with only O(n) arithmetic operations. Moreover, the method is shown to outperform the state-of-the-art method for solving IR problems, especially when warm-starting is considered. Enhancements to themethod are proposed for solving general TF problems, and numerical results are presented to show that PDAS methods are viable for a broad class of such problems