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.
DM has been extended to dynamical systems previously @cite_30 @cite_6 @cite_0 @cite_33 . In particular, Talmon and Coifman @cite_6 @cite_0 introduced an approach called empirical intrinsic geometry (EIG) that builds a diffusion geometry using a noise resilient distance. The resulting embedding learned from the geometry is thus noise-free and captures the true structure of the underlying process. However, EIG and other extensions of DM to dynamical systems are still not optimized for visualization as the learned structure of the data is encoded in higher dimensions. In this work, we introduce a new visualization method DIG that is well-suited for visualizing high-dimensional dynamical processes by preserving an information distance between the diffusion probabilities constructed from a noise resilient distance. This results in a visualization method that represents the true structure of the underlying dynamical process.
@cite_30 @cite_0 @cite_33 @cite_6
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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. This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset @math , associated with the Euclidean metric, with points in the cube @math and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in @math . Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in @math , depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving @math norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. We study the flow of information and the evolution of internal representations during deep neural network (DNN) training, aiming to demystify the compression aspect of the information bottleneck theory. The theory suggests that DNN training comprises a rapid fitting phase followed by a slower compression phase, in which the mutual information @math between the input @math and internal representations @math decreases. Several papers observe compression of estimated mutual information on different DNN models, but the true @math over these networks is provably either constant (discrete @math ) or infinite (continuous @math ). This work explains the discrepancy between theory and experiments, and clarifies what was actually measured by these past works. To this end, we introduce an auxiliary (noisy) DNN framework for which @math is a meaningful quantity that depends on the network's parameters. This noisy framework is shown to be a good proxy for the original (deterministic) DNN both in terms of performance and the learned representations. We then develop a rigorous estimator for @math in noisy DNNs and observe compression in various models. By relating @math in the noisy DNN to an information-theoretic communication problem, we show that compression is driven by the progressive clustering of hidden representations of inputs from the same class. Several methods to directly monitor clustering of hidden representations, both in noisy and deterministic DNNs, are used to show that meaningful clusters form in the @math space. Finally, we return to the estimator of @math employed in past works, and demonstrate that while it fails to capture the true (vacuous) mutual information, it does serve as a measure for clustering. This clarifies the past observations of compression and isolates the geometric clustering of hidden representations as the true phenomenon of interest. We present a new method for inferring dense data to model correspondences, focusing on the application of human pose estimation from depth images. Recent work proposed the use of regression forests to quickly predict correspondences between depth pixels and points on a 3D human mesh model. That work, however, used a proxy forest training objective based on the classification of depth pixels to body parts. In contrast, we introduce Metric Space Information Gain (MSIG), a new decision forest training objective designed to directly minimize the entropy of distributions in a metric space. When applied to a model surface, viewed as a metric space defined by geodesic distances, MSIG aims to minimize image-to-model correspondence uncertainty. A naive implementation of MSIG would scale quadratically with the number of training examples. As this is intractable for large datasets, we propose a method to compute MSIG in linear time. Our method is a principled generalization of the proxy classification objective, and does not require an extrinsic isometric embedding of the model surface in Euclidean space. Our experiments demonstrate that this leads to correspondences that are considerably more accurate than state of the art, using far fewer training images.