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.

Recent biological experimental findings have shown that synaptic plasticity depends on the relative timing of the pre- and postsynaptic spikes. This determines whether long-term potentiation (LTP) or long-term depression (LTD) is induced. This synaptic plasticity has been called temporally asymmetric Hebbian plasticity (TAH). Many authors have numerically demonstrated that neural networks are capable of storing spatiotemporal patterns. However, the mathematical mechanism of the storage of spatiotemporal patterns is still unknown, and the effect of LTD is particularly unknown. In this article, we employ a simple neural network model and show that interference between LTP and LTD disappears in a sparse coding scheme. On the other hand, the covariance learning rule is known to be indispensable for the storage of sparse patterns. We also show that TAH has the same qualitative effect as the covariance rule when spatiotemporal patterns are embedded in the network.

Recent biological experimental (cid:12)ndings have shown that the synap- tic plasticity depends on the relative timing of the pre- and post- synaptic spikes which determines whether Long Term Potentiation (LTP) occurs or Long Term Depression (LTD) does. The synaptic plasticity has been called \Temporally Asymmetric Hebbian plas- ticity (TAH)". Many authors have numerically shown that spatio- temporal patterns can be stored in neural networks. However, the mathematical mechanism for storage of the spatio-temporal pat- terns is still unknown, especially the e(cid:11)ects of LTD. In this paper, we employ a simple neural network model and show that inter- ference of LTP and LTD disappears in a sparse coding scheme. On the other hand, it is known that the covariance learning is in- dispensable for storing sparse patterns. We also show that TAH qualitatively has the same e(cid:11)ect as the covariance learning when spatio-temporal patterns are embedded in the network.