TASK DEFINITION: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
PROBLEM: Metric learning has been shown to be highly effective to improve the performance of nearest neighbor classification. In this paper, we address the problem of metric learning for symmetric positive definite (SPD) matrices such as covariance matrices, which arise in many real-world applications. Naively using standard Mahalanobis metric learning methods under the Euclidean geometry for SPD matrices is not appropriate, because the difference of SPD matrices can be a non-SPD matrix and thus the obtained solution can be uninterpretable. To cope with this problem, we propose to use a properly parameterized LogEuclidean distance and optimize the metric with respect to kernel-target alignment, which is a supervised criterion for kernel learning. Then the resulting non-trivial optimization problem is solved by utilizing the Riemannian geometry. Finally, we experimentally demonstrate the usefulness of our LogEuclidean metric learning algorithm on real-world classification tasks for EEG signals and texture patches.

SOLUTION: Supervised LogEuclidean Metric Learning for Symmetric Positive Definite Matrices

PROBLEM: Surely we want solid foundations. What kind of castle can we build on sand? What is the point of devoting effort to balconies and minarets, if the foundation may be so weak as to allow the structure to collapse of its own weight? We want our foundations set on bedrock, designed to last for generations. Who would want an architect who cannot certify the soundness of the foundations of his buildings?

SOLUTION: Why Do We Need Foundations for Modelling Uncertainties?

PROBLEM: Given a teacher that holds a function f : X R from some class of functions C. The teacher can receive from the learner an element d in the domain X (a query) and returns the value of the function in d, f(d) R. The learner goal is to find f with a minimum number of queries, optimal time complexity, and optimal resources. In this survey, we present some of the results known from the literature, different techniques used, some new problems, and open problems. ar X iv :1 70 6. 03 93 5v 1 [ cs .L G ] 1 3 Ju n 20 17

SOLUTION:
Exact Learning from an Honest Teacher That Answers Membership Queries