Fantope Regularization in Metric Learning

Abstract

This paper introduces a regularization method to explicitly control the rank of a learned symmetric positive semidefinite distance matrix in distance metric learning. To this end, we propose to incorporate in the objective function a linear regularization term that minimizes the k smallest eigenvalues of the distance matrix. It is equivalent to minimizing the trace of the product of the distance matrix with a matrix in the convex hull of rank-k projection matrices, called a Fantope. Based on this new regularization method, we derive an optimization scheme to efficiently learn the distance matrix. We demonstrate the effectiveness of the method on synthetic and challenging real datasets of face verification and image classification with relative attributes, on which our method outperforms state-of-the-art metric learning algorithms.

Distance Metric Learning by Optimization on the Stiefel Manifold

“…In [23], the authors add a regularization function to the convex objective that biases the solution such that similar points in a small neighborhood lie on a low-dimensional manifold. Law et al [12] also use a regularization function that elegantly incorporates explicit control over the rank of the learned Mahalanobis matrix in the objective function. An ADMM based algorithm is proposed to learn the low-rank metric.…”

“…Without a regularizer in the objective function, the technique requires an early stopping criterion, which is difficult to tune for each dataset [12]. Other methods [7,9,11] take a projection free approach and use special regularization functions like the log det Bregman divergence, which implicitly maintains the positive semidefiniteness constraint and the rank of the learned Mahalanobis matrix.…”

“…The notion of similarity and dissimilarity is based on the semantics of the application. Many popular techniques [7,9,11,12] set up the metric learning problem in a constrained optimization framework. The imposed constraints capture the intuition that same class point pairs have small distances, while sample points from different classes have a large distance.…”

“…Techniques that rely on projection on to S n + like [12,20,22], usually require an eigendecomposition or SVD in each iteration resulting in an additional cost of O(n 3 ). Projection free approaches like [7,9,11] use special regularization functions leading to updates that guarantee positive semidefiniteness.…”

Absolute geo-localization thanks to Hidden Markov Model and exemplar-based metric learning

“…It is to be noted that other regularization schemes could also be used, e.g. based on the Frobenius, nuclear norm [20] or methods giving an explicit control of the matrix rank as [15].…”