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Register a new userConditional Selective Inference for Robust Regression and Outlier Detection using Piecewise-Linear Homotopy Continuation
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In practical data analysis under noisy environment, it is common to first use robust methods to identify outliers, and then to conduct further analysis after removing the outliers. In this paper, we consider statistical inference of the model estimated after outliers are removed, which can be interpreted as a selective inference (SI) problem. To use conditional SI framework, it is necessary to characterize the events of how the robust method identifies outliers. Unfortunately, the existing methods cannot be directly used here because they are applicable to the case where the selection events can be represented by linear/quadratic constraints. In this paper, we propose a conditional SI method for popular robust regressions by using homotopy method. We show that the proposed conditional SI method is applicable to a wide class of robust regression and outlier detection methods and has good empirical performance on both synthetic data and real data experiments.
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In support vector machine (SVM) applications with unreliable data that contains a portion of outliers, non-robustness of SVMs often causes considerable performance deterioration. Although many approaches for improving the robustness of SVMs have been studied, two major challenges remain in robust SVM learning. First, robust learning algorithms are essentially formulated as non-convex optimization problems. It is thus important to develop a non-convex optimization method for robust SVM that can find a good local optimal solution. The second practical issue is how one can tune the hyperparameter that controls the balance between robustness and efficiency. Unfortunately, due to the non-convexity, robust SVM solutions with slightly different hyper-parameter values can be significantly different, which makes model selection highly unstable. In this paper, we address these two issues simultaneously by introducing a novel homotopy approach to non-convex robust SVM learning. Our basic idea is to introduce parametrized formulations of robust SVM which bridge the standard SVM and fully robust SVM via the parameter that represents the influence of outliers. We characterize the necessary and sufficient conditions of the local optimal solutions of robust SVM, and develop an algorithm that can trace a path of local optimal solutions when the influence of outliers is gradually decreased. An advantage of our homotopy approach is that it can be interpreted as simulated annealing, a common approach for finding a good local optimal solution in non-convex optimization problems. In addition, our homotopy method allows stable and efficient model selection based on the path of local optimal solutions. Empirical performances of the proposed approach are demonstrated through intensive numerical experiments both on robust classification and regression problems.
Conditional selective inference (SI) has been actively studied as a new statistical inference framework for data-driven hypotheses. The basic idea of conditional SI is to make inferences conditional on the selection event characterized by a set of linear and/or quadratic inequalities. Conditional SI has been mainly studied in the context of feature selection such as stepwise feature selection (SFS). The main limitation of the existing conditional SI methods is the loss of power due to over-conditioning, which is required for computational tractability. In this study, we develop a more powerful and general conditional SI method for SFS using the homotopy method which enables us to overcome this limitation. The homotopy-based SI is especially effective for more complicated feature selection algorithms. As an example, we develop a conditional SI method for forward-backward SFS with AIC-based stopping criteria and show that it is not adversely affected by the increased complexity of the algorithm. We conduct several experiments to demonstrate the effectiveness and efficiency of the proposed method.
Outliers are ubiquitous in modern data sets. Distance-based techniques are a popular non-parametric approach to outlier detection as they require no prior assumptions on the data generating distribution and are simple to implement. Scaling these techniques to massive data sets without sacrificing accuracy is a challenging task. We propose a novel algorithm based on the intuition that outliers have a significant influence on the quality of divergence-based clustering solutions. We propose sensitivity - the worst-case impact of a data point on the clustering objective - as a measure of outlierness. We then prove that influence, a (non-trivial) upper-bound on the sensitivity, can be computed by a simple linear time algorithm. To scale beyond a single machine, we propose a communication efficient distributed algorithm. In an extensive experimental evaluation, we demonstrate the effectiveness and establish the statistical significance of the proposed approach. In particular, it outperforms the most popular distance-based approaches while being several orders of magnitude faster.
Conditional selective inference (SI) has been studied intensively as a new statistical inference framework for data-driven hypotheses. The basic concept of conditional SI is to make the inference conditional on the selection event, which enables an exact and valid statistical inference to be conducted even when the hypothesis is selected based on the data. Conditional SI has mainly been studied in the context of model selection, such as vanilla lasso or generalized lasso. The main limitation of existing approaches is the low statistical power owing to over-conditioning, which is required for computational tractability. In this study, we propose a more powerful and general conditional SI method for a class of problems that can be converted into quadratic parametric programming, which includes generalized lasso. The key concept is to compute the continuum path of the optimal solution in the direction of the selected test statistic and to identify the subset of the data space that corresponds to the model selection event by following the solution path. The proposed parametric programming-based method not only avoids the aforementioned major drawback of over-conditioning, but also improves the performance and practicality of SI in various respects. We conducted several experiments to demonstrate the effectiveness and efficiency of our proposed method.
We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions $f$ that may be represented as the difference $phi_1 - phi_2$ for a choice of convex functions $phi_1, phi_2$. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the $ell_infty$ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.
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