Rolling Riemannian Manifolds to Solve the Multi-class Classification Problem

Abstract

In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds [31,27]. A popular framework, valid over any Riemannian manifold, was proposed in [31] for binary classification. Once moving from binary to multi-class classification this paradigm is not valid anymore, due to the spread of multiple positive classes on the manifold [27]. It is then natural to ask whether the multi-class paradigm could be extended to operate on a large class of Riemannian manifolds. We propose a mathematically well-founded classification paradigm that allows to extend the work in [31] to multi-class models, taking into account the structure of the space. The idea is to project all the data from the manifold onto an affine tangent space at a particular point. To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21,16]. The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed.

Image Set-Based Collaborative Representation for Face Recognition

“…Many non-parametric set modeling methods have also been proposed, including subspace [10], [1], [15], manifold [16], [17], [4], [11], [18], affine hull [5], [7], convex hull [5], and covariance matrix based ones [18], [19], [20]. The method in [10] employs canonical correlation to measure the similarity between two sets.…”

“…In [18], an image set is represented by a covariance matrix and a Riemannian kernel function is defined to measure the similarity between two image sets by a mapping from the Riemannian manifold to a Euclidean space. With the kernel function between two image sets, traditional discriminant learning methods, e.g., linear discriminative analysis [26], partial least squares [27], kernel machines, can be used for image set classification [19], [20]. The disadvantages of covariance matrix based methods include the computational complexity of eigen-decomposition of symmetric positive-definite (SPD) matrices and the curse of dimensionality with limited number of training sets.…”

Geometry of the rolling ellipsoid

“…Illustration of this algorithms has been done for the ellipsoid equipped with a non-Euclidean metric in [16] and for other Riemannian manifolds in [10]. The same idea has been proposed in [5] to solve multiclass classification problems in the context of pattern recognition.…”

Efficient clustering on Riemannian manifolds: A kernelised random projection approach

“…Unfortunately, the kernel-based approaches cannot scale easily, as the Gram matrix computation is O(n 2 ) where n is the number of data points. Also, it is often quite challenging to kernelise the existing algorithms that do not have known kernelised versions [22]. Furthermore, Nikhil et al demonstrate that clustering data in the RKHS may lead to unexpected results since the clusters obtained in the…”