Local Distance Correlation Embedding for Time-Series Analysis on Riemannian Manifolds

This paper proposes a time-series data embedding technique that preserves curvature and orientation, with a focus on visualizing temporal manifold-valued data. Manifold-valued data provide pair-wise local distances on which the proposed method is built. First, we introduce a simpler form of our method, conformal folding embedding (CFE), as an interpretable straightforward algorithm to perfectly preserve the angles between adjacent velocity vectors, maintaining local geometric structure while forgoing global structure. Then we introduce the general formulation, dubbed local distance correlation embedding (LDCE), that maximizes distance correlation between input manifoldvalued data and the embedded ones while preserving both global distance structure and local geometric structure. Although the two algorithms are different in formulation, we show their theoretical connection by proving that CFE is a special case of LDCE. We empirically showcase the effectiveness of LDCE in preserving curvature/orientation by visualizing simulated data. The method is also applied to analyze the temporal information encoded at a population level in the inferior temporal cortex of monkeys.