Metric Learning on Manifolds
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated than in Euclidean space, and thus many techniques for processing and analysis taken for granted in Euclidean space are difficult on manifolds. A priori, it is not obvious how we may generalize such methods to manifolds. We consider specifically the problem of distance metric learning, and present a framework that solves it on a large class of manifolds, such that similar data are located in closer proximity with respect to the manifold distance function. In particular, we extend the existing metric learning algorithms, and derive the corresponding sample complexity rates for the case of manifolds. Additionally, we demonstrate an improvement of performance in $k$-means clustering and $k$-nearest neighbor classification on real-world complex networks using our methods.
Code (0)
등록된 구현이 없습니다.
Tasks
ClusteringMetric LearningSimilar Papers 제목 키워드 기반
Quantifying Manifolds: Do the manifolds learned by Generative Adversarial Networks converge to the real data manifold
This paper presents our experiments to quantify the manifolds learned by ML models (in our experiment, we use a GAN model) as they train. We compare the manifolds learned at each epoch to the real manifolds representing …
Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds
This paper presents two new metrics on the Symmetric Positive Definite (SPD) manifold via the Cholesky manifold, i.e., the space of lower triangular matrices with positive diagonal elements. We first unveil that the exis…
Investigating Image Manifolds of 3D Objects: Learning, Shape Analysis, and Comparisons
Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object cl…
Lie PCA: Density estimation for symmetric manifolds
We introduce an extension to local principal component analysis for learning symmetric manifolds. In particular, we use a spectral method to approximate the Lie algebra corresponding to the symmetry group of the underlyi…
Density EstimationCurvatures of Stiefel manifolds with deformation metrics
We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{\"u}per, Markina and Silva Leite, which includes the well-known embedded and canonical metrics on Stiefel manifolds as…