Error Metrics for Learning Reliable Manifolds from Streaming Data

Spectral dimensionality reduction is frequently used to identify low-dimensional structure in high-dimensional data. However, learning manifolds, especially from the streaming data, is computationally and memory expensive. In this paper, we argue that a stable manifold can be learned using only a fraction of the stream, and the remaining stream can be mapped to the manifold in a significantly less costly manner. Identifying the transition point at which the manifold is stable is the key step. We present error metrics that allow us to identify the transition point for a given stream by quantitatively assessing the quality of a manifold learned using Isomap. We further propose an efficient mapping algorithm, called S-Isomap, that can be used to map new samples onto the stable manifold. We describe experiments on a variety of data sets that show that the proposed approach is computationally efficient without sacrificing accuracy

Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging

Various metrics for comparing diffusion tensors have been recently proposed in the literature. We consider a broad family of metrics which is indexed by a single power parameter. A likelihood-based procedure is developed for choosing the most appropriate metric from the family for a given dataset at hand. The approach is analogous to using the Box-Cox transformation that is frequently investigated in regression analysis. The methodology is illustrated with a simulation study and an application to a real dataset of diffusion tensor images of canine hearts

Learning Generative Models across Incomparable Spaces

Generative Adversarial Networks have shown remarkable success in learning a distribution that faithfully recovers a reference distribution in its entirety. However, in some cases, we may want to only learn some aspects (e.g., cluster or manifold structure), while modifying others (e.g., style, orientation or dimension). In this work, we propose an approach to learn generative models across such incomparable spaces, and demonstrate how to steer the learned distribution towards target properties. A key component of our model is the Gromov-Wasserstein distance, a notion of discrepancy that compares distributions relationally rather than absolutely. While this framework subsumes current generative models in identically reproducing distributions, its inherent flexibility allows application to tasks in manifold learning, relational learning and cross-domain learning.Comment: International Conference on Machine Learning (ICML