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In this paper, we present a method for time series analysis based on empirical intrinsic geometry (EIG). EIG enables one to reveal the low-dimensional parametric manifold as well as to infer the underlying dynamics of high-dimensional time series. By incorporating concepts of information geometry, this method extends existing geometric analysis tools to support stochastic settings and parametrizes the geometry of empirical distributions. However, the statistical models are not required as priors; hence, EIG may be applied to a wide range of real signals without existing definitive models. We show that the inferred model is noise-resilient and invariant under different observation and instrumental modalities. In addition, we show that it can be extended efficiently to newly acquired measurements in a sequential manner. These two advantages enable us to revisit the Bayesian approach and incorporate empirical dynamics and intrinsic geometry into a nonlinear filtering framework. We show applications to nonlinear and non-Gaussian tracking problems as well as to acoustic signal localization.anisotropic diffusion | manifold learning | signal processing I n recent years, there has been much progress in the development of new methods for parameterizing and embedding highdimensional data in a low-dimensional space (1-4). The method proposed by Singer and Coifman (5) is of particular interest because the data are assumed to be inaccessible and can be observed only via unknown nonlinear functions. By integrating local principal components with diffusion maps, the approach of Singer and Coifman (5) provides modeling of the underlying parametric manifold, whereas classical manifold learning methods provide a parameterization of the observable manifold. However, geometric methods usually suffer from two significant disadvantages. First, in many natural systems, the mapping of the low-dimensional data into a subset of high-dimensional observations is stochastic. In addition, a random measurement noise usually corrupts the observations. Thus, the geometry of the observations may not convey the appropriate information on the underlying parametric manifold. Second, these methods provide modeling to a given dataset and do not model a stream of new incoming measurements well.In this paper, we propose a framework for sequential processing of time series based on empirical intrinsic geometric models. We adopt the nonlinear filtering formalism and propose a method consisting of two steps: novel data-driven modeling of stochastic data, which is resilient to noise as well as invariant to the measurement modality, and Bayesian filtering of new incoming measurements based on the learned model. High-dimensional time series often exhibit highly redundant representations and can be compactly represented by a dynamical process on a lowdimensional manifold. Thus, in the first stage, we reveal the lowdimensional manifold and infer the underlying process using anisotropic diffusion. The proposed approach relies on the observation that the local s...

Highlights d Populations of layer 2-3 pyramidal neurons in M1 report motor performance outcome d Success and failure activity is late, prolonged, and dissociated from kinematics and reward d At trial start, layer 5 pyramidal tract activity is affected by previous outcome d Post-movement activity in M1 is required for motor performance and learning

Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of "repeated eigendirections," which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.

The problem of domain adaptation has become central in many applications from a broad range of fields. Recently, it was proposed to use Optimal Transport (OT) to solve it. In this paper, we model the difference between the two domains by a diffeomorphism and use the polar factorization theorem to claim that OT is indeed optimal for domain adaptation in a well-defined sense, up to a volume preserving map. We then focus on the manifold of Symmetric and Positive-Definite (SPD) matrices, whose structure provided a useful context in recent applications. We demonstrate the polar factorization theorem on this manifold. Due to the uniqueness of the weighted Riemannian mean, and by exploiting existing regularized OT algorithms, we formulate a simple algorithm that maps the source domain to the target domain. We test our algorithm on two Brain-Computer Interface (BCI) data sets and observe state of the art performance.Preprint. Under review.