We consider the problem of multivariate regression in a setting where the relevant predictors could be shared among different responses. We propose an algorithm which decomposes the coefficient matrix into the product of a long matrix and a wide matrix, with an elastic net penalty on the former and an $\ell_1$ penalty on the latter. The first matrix linearly transforms the predictors to a set of latent factors, and the second one regresses the responses on these factors. Our algorithm simultaneously performs dimension reduction and coefficient estimation and automatically estimates the number of latent factors from the data. Our formulation results in a non-convex optimization problem, which despite its flexibility to impose effective low-dimensional structure, is difficult, or even impossible, to solve exactly in a reasonable time. We specify an optimization algorithm based on alternating minimization with three different sets of updates to solve this non-convex problem and provide theoretical results on its convergence and optimality. Finally, we demonstrate the effectiveness of our algorithm via experiments on simulated and real data.
We review sufficient dimension reduction (SDR) estimators with multivariate response in this paper. A wide range of SDR methods are characterized as inverse regression SDR estimators or forward regression SDR estimators. The inverse regression family include pooled marginal estimators, projective resampling estimators, and distance-based estimators. Ordinary least squares, partial least squares, and semiparametric SDR estimators, on the other hand, are discussed as estimators from the forward regression family.
Tree-based ensembles such as the Random Forest are modern classics among statistical learning methods. In particular, they are used for predicting univariate responses. In case of multiple outputs the question arises whether we separately fit univariate models or directly follow a multivariate approach. For the latter, several possibilities exist that are, e.g. based on modified splitting or stopping rules for multi-output regression. In this work we compare these methods in extensive simulations to help in answering the primary question when to use multivariate ensemble techniques.
We develop Distributionally Robust Optimization (DRO) formulations for Multivariate Linear Regression (MLR) and Multiclass Logistic Regression (MLG) when both the covariates and responses/labels may be contaminated by outliers. The DRO framework uses a probabilistic ambiguity set defined as a ball of distributions that are close to the empirical distribution of the training set in the sense of the Wasserstein metric. We relax the DRO formulation into a regularized learning problem whose regularizer is a norm of the coefficient matrix. We establish out-of-sample performance guarantees for the s…
Multivariate Regression with Gross Errors on Manifold-valued Data
Bioinformatics Institute, A*STAR（新加坡科技研究局生物信息学研究所）；School of Computing, National University of Singapore（新加坡国立大学计算机学院）；Singapore Institute for Neurotechnology, National University of Singapore（新加坡国立大学新加坡神经技术研究所）
Calibrated Multivariate Regression with Localized PIT Mappings
National University of Singapore（新加坡国立大学）；University of Brasília（巴西利亚大学）；University of New South Wales（新南威尔士大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
A multiple objective space-time forecasting approach is presented involving cyclical curve log-regression, and multivariate time series spatial residual correlation analysis. Specifically, the mean quadratic loss function is minimized in the framework of trigonometric regression. While, in our subsequent spatial residual correlation analysis, maximization of the likelihood allows us to compute the posterior mode in a Bayesian multivariate time series soft-data framework. The presented approach is applied to the analysis of COVID-19 mortality in the first wave affecting the Spanish Communities,…
Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate density functions (modal regression). Rates of convergence are established and, in many cases, provide superior rates to current standard estimators such as those based on kernels, including kernel density estimators and kernel regression functions. Numerical illustrations are presented.
Multivariate time series (MTS) regression tasks are common in many real-world data mining applications including finance, cybersecurity, energy, healthcare, prognostics, and many others. Due to the tremendous success of deep learning (DL) algorithms in various domains including image recognition and computer vision, researchers started adopting these techniques for solving MTS data mining problems, many of which are targeted for safety-critical and cost-critical applications. Unfortunately, DL algorithms are known for their susceptibility to adversarial examples which also makes the DL regress…
STOR-i Centre for Doctoral Training（STOR-i博士培训中心）；Lancaster University（兰卡斯特大学）；Stanford University（斯坦福大学）
Many single-target regression problems require estimates of uncertainty along with the point predictions. Probabilistic regression algorithms are well-suited for these tasks. However, the options are much more limited when the prediction target is multivariate and a joint measure of uncertainty is required. For example, in predicting a 2D velocity vector a joint uncertainty would quantify the probability of any vector in the plane, which would be more expressive than two separate uncertainties on the x- and y- components. To enable joint probabilistic regression, we propose a Natural Gradient …
German Aerospace Center (DLR)（德国航空航天中心）；Institute for Simulation Intelligence（模拟智能研究所）
There is significant need for principled uncertainty reasoning in machine learning systems as they are increasingly deployed in safety-critical domains. A new approach with uncertainty-aware neural networks (NNs), based on learning evidential distributions for aleatoric and epistemic uncertainties, shows promise over traditional deterministic methods and typical Bayesian NNs, yet several important gaps in the theory and implementation of these networks remain. We discuss three issues with a proposed solution to extract aleatoric and epistemic uncertainties from regression-based neural networks…
Random Forest (Breiman, 2001) is a successful and widely used regression and classification algorithm. Part of its appeal and reason for its versatility is its (implicit) construction of a kernel-type weighting function on training data, which can also be used for targets other than the original mean estimation. We propose a novel forest construction for multivariate responses based on their joint conditional distribution, independent of the estimation target and the data model. It uses a new splitting criterion based on the MMD distributional metric, which is suitable for detecting heterogene…
National Technical University of Athens（雅典国家技术大学）；University of Pennsylvania（宾夕法尼亚大学）
In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial time for any $\ell_p$ approximation error. Based on these results, we propose a novel method for piecewise-linear fitting of convex multivariate functions, with optimality guarantees for the model parameters and an approximately minimum number of affine regions.
In this paper, we propose methods for functional predictor selection and the estimation of smooth functional coefficients simultaneously in a scalar-on-function regression problem under high-dimensional multivariate functional data setting. In particular, we develop two methods for functional group-sparse regression under a generic Hilbert space of infinite dimension. We show the convergence of algorithms and the consistency of the estimation and the selection (oracle property) under infinite-dimensional Hilbert spaces. Simulation studies show the effectiveness of the methods in both the selec…
Multivariate Regression with Grossly Corrupted Observations: A Robust Approach and its Applications
Bioinformatics Institute, A*STAR（新加坡科技研究局生物信息研究所）；Institute of Psychology, Chinese Academy of Sciences（中国科学院心理研究所）；School of Computing, National University of Singapore（新加坡国立大学计算学院）
This work aims to give non-asymptotic results for estimating the first principal component of a multivariate random process. We first define the covariance function and the covariance operator in the multivariate case. We then define a projection operator. This operator can be seen as a reconstruction step from the raw data in the functional data analysis context. Next, we show that the eigenelements can be expressed as the solution to an optimization problem, and we introduce the LASSO variant of this optimization problem and the associated plugin estimator. Finally, we assess the estimator's…
