Manifold learning techniques for dynamical systems and time series have shown their utility for a broad spectrum of applications in recent years. While these methods are effective at learning a low-dimensional representation, they are often insufficient for visualizing the global and local structure of the data. In this paper, we present DIG (Dynamical Information Geometry), a visualization method for multivariate time series data that extracts an information geometry from a diffusion framework. Specifically, we implement a novel group of distances in the context of diffusion operators, which may be useful to reveal structure in the data that may not be accessible by the commonly used diffusion distances. Finally, we present a case study applying our visualization tool to EEG data to visualize sleep stages.
 We present a new technique called “t-SNE” that visualizes high-dimensional data by giving each datapoint a location in a two or three-dimensional map. The technique is a variation of Stochastic Neighbor Embedding (Hinton and Roweis, 2002) that is much easier to optimize, and produces significantly better visualizations by reducing the tendency to crowd points together in the center of the map. t-SNE is better than existing techniques at creating a single map that reveals structure at many different scales. This is particularly important for high-dimensional data that lie on several different, but related, low-dimensional manifolds, such as images of objects from multiple classes seen from multiple viewpoints. For visualizing the structure of very large datasets, we show how t-SNE can use random walks on neighborhood graphs to allow the implicit structure of all of the data to influence the way in which a subset of the data is displayed. We illustrate the performance of t-SNE on a wide variety of datasets and compare it with many other non-parametric visualization techniques, including Sammon mapping, Isomap, and Locally Linear Embedding. The visualizations produced by t-SNE are significantly better than those produced by the other techniques on almost all of the datasets.  Many areas of science depend on exploratory data analysis and visualization. The need to analyze large amounts of multivariate data raises the fundamental problem of dimensionality reduction: how to discover compact representations of high-dimensional data. Here, we introduce locally linear embedding (LLE), an unsupervised learning algorithm that computes low-dimensional, neighborhood-preserving embeddings of high-dimensional inputs. Unlike clustering methods for local dimensionality reduction, LLE maps its inputs into a single global coordinate system of lower dimensionality, and its optimizations do not involve local minima. By exploiting the local symmetries of linear reconstructions, LLE is able to learn the global structure of nonlinear manifolds, such as those generated by images of faces or documents of text. How do we judge similarity? Our mental representations of the world are formed by processing large numbers of sensory in  Scientists working with large volumes of high-dimensional data, such as global climate patterns, stellar spectra, or human gene distributions, regularly confront the problem of dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. The human brain confronts the same problem in everyday perception, extracting from its high-dimensional sensory inputs—30,000 auditory nerve fibers or 106 optic nerve fibers—a manageably small number of perceptually relevant features. Here we describe an approach to solving dimensionality reduction problems that uses easily measured local metric information to learn the underlying global geometry of a data set. Unlike classical techniques such as principal component analysis (PCA) and multidimensional scaling (MDS), our approach is capable of discovering the nonlinear degrees of freedom that underlie complex natural observations, such as human handwriting or images of a face under different viewing conditions. In contrast to previous algorithms for nonlinear dimensionality reduction, ours efficiently computes a globally optimal solution, and, for an important class of data manifolds, is guaranteed to converge asymptotically to the true structure. A benchmarking analysis on single-cell RNA-seq and mass cytometry data reveals the best-performing technique for dimensionality reduction. I. INTRODUCTION AND FOUNDATIONS. 1. Introduction and Foundations. II. VECTOR SPACES AND LINEAR ALGEBRA. 2. Signal Spaces. 3. Representation and Approximation in Vector Spaces. 4. Linear Operators and Matrix Inverses. 5. Some Important Matrix Factorizations. 6. Eigenvalues and Eigenvectors. 7. The Singular Value Decomposition. 8. Some Special Matrices and Their Applications. 9. Kronecker Products and the Vec Operator. III. DETECTION, ESTIMATION, AND OPTIMAL FILTERING. 10. Introduction to Detection and Estimation, and Mathematical Notation. 11. Detection Theory. 12. Estimation Theory. 13. The Kalman Filter. IV. ITERATIVE AND RECURSIVE METHODS IN SIGNAL PROCESSING. 14. Basic Concepts and Methods of Iterative Algorithms. 15. Iteration by Composition of Mappings. 16. Other Iterative Algorithms. 17. The EM Algorithm in Signal Processing. V. METHODS OF OPTIMIZATION. 18. Theory of Constrained Optimization. 19. Shortest-Path Algorithms and Dynamic Programming. 20. Linear Programming. APPENDIXES. A. Basic Concepts and Definitions. B. Completing the Square. C. Basic Matrix Concepts. D. Random Processes. E. Derivatives and Gradients. F. Conditional Expectations of Multinomial and Poisson r.v.s.