Regularization and Kernelization of the Maximin Correlation Approach

Robust classification becomes challenging when each class consists of multiple subclasses. Examples include multi-font optical character recognition and automated protein function prediction. In correlation-based nearest-neighbor classification, the maximin correlation approach (MCA) provides the worst-case optimal solution by minimizing the maximum misclassification risk through an iterative procedure. Despite the optimality, the original MCA has drawbacks that have limited its wide applicability in practice. That is, the MCA tends to be sensitive to outliers, cannot effectively handle nonlinearities in datasets, and suffers from having high computational complexity. To address these limitations, we propose an improved solution, named regularized maximin correlation approach (R-MCA). We first reformulate MCA as a quadratically constrained linear programming (QCLP) problem, incorporate regularization by introducing slack variables in the primal problem of the QCLP, and derive the corresponding Lagrangian dual. The dual formulation enables us to apply the kernel trick to R-MCA so that it can better handle nonlinearities. Our experimental results demonstrate that the regularization and kernelization make the proposed R-MCA more robust and accurate for various classification tasks than the original MCA. Furthermore, when the data size or dimensionality grows, R-MCA runs substantially faster by solving either the primal or dual (whichever has a smaller variable dimension) of the QCLP.Comment: Submitted to IEEE Acces

Maximin projection learning for optimal treatment decision with heterogeneous individualized treatment effects

A salient feature of data from clinical trials and medical studies is inhomogeneity. Patients not only differ in baseline characteristics, but also in the way that they respond to treatment. Optimal individualized treatment regimes are developed to select effective treatments based on patient's heterogeneity. However, the optimal treatment regime might also vary for patients across different subgroups. We mainly consider patients’ heterogeneity caused by groupwise individualized treatment effects assuming the same marginal treatment effects for all groups. We propose a new maximin projection learning method for estimating a single treatment decision rule that works reliably for a group of future patients from a possibly new subpopulation. Based on estimated optimal treatment regimes for all subgroups, the proposed maximin treatment regime is obtained by solving a quadratically constrained linear programming problem, which can be efficiently computed by interior point methods. Consistency and asymptotic normality of the estimator are established. Numerical examples show the reliability of the methodology proposed

Learning Probabilistic Graphical Models for Image Segmentation

Probabilistic graphical models provide a powerful framework for representing image structures. Based on this concept, many inference and learning algorithms have been developed. However, both algorithm classes are NP-hard combinatorial problems in the general case. As a consequence, relaxation methods were developed to approximate the original problems but with the benefit of being computationally efficient. In this work we consider the learning problem on binary graphical models and their relaxations. Two novel methods for determining the model parameters in discrete energy functions from training data are proposed. Learning the model parameters overcomes the problem of heuristically determining them. Motivated by common learning methods which aim at minimizing the training error measured by a loss function we develop a new learning method similar in fashion to structured SVM. However, computationally more efficient. We term this method as linearized approach (LA) as it is restricted to linearly dependent potentials. The linearity of LA is crucial to come up with a tight convex relaxation, which allows to use off-the-shelf inference solvers to approach subproblems which emerge from solving the overall problem. However, this type of learning methods almost never yield optimal solutions or perfect performance on the training data set. So what happens if the learned graphical model on the training data would lead to exact ground segmentation? Will this give a benefit when predicting? Motivated by the idea of inverse optimization, we take advantage of inverse linear programming to develop a learning approach, referred to as inverse linear programming approach (invLPA). It further refines the graphical models trained, using the previously introduced methods and is capable to perfectly predict ground truth on training data. The empirical results from implementing invLPA give answers to our questions posed before. LA is able to learn both unary and pairwise potentials jointly while with invLPA this is not possible due to the representation we use. On the other hand, invLPA does not rely on a certain form for the potentials and thus is flexible in the choice of the fitting method. Although the corrected potentials with invLPA always result in ground truth segmentation of the training data, invLPA is able to find corrections on the foreground segments only. Due to the relaxed problem formulation this does not affect the final segmentation result. Moreover, as long as we initialize invLPA with model parameters of a learning method performing sufficiently well, this drawback of invLPA does not significantly affect the final prediction result. The performance of the proposed learning methods is evaluated on both synthetic and real world datasets. We demonstrate that LA is competitive in comparison to other parameter learning methods using loss functions based on Maximum a Posteriori Marginal (MPM) and Maximum Likelihood Estimation (MLE). Moreover, we illustrate the benefits of learning with inverse linear programming. In a further experiment we demonstrate the versatility of our learning methods by applying LA to learning motion segmentation in video sequences and comparing it to state-of-the-art segmentation algorithms

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum