Empirical Intrinsic Modeling of Signals and Information Geometry

many natural and real-world applications, the measured signals are controlled by underlying processes or drivers. As a result, these signals exhibit highly redundant representations and their temporal evolution can be compactly described by a dynamical process on a low-dimensional manifold. In this paper, we propose a graph-based method for revealing the low-dimensional manifold and inferring the underlying process. This method provides intrinsic modeling for signals using empirical information geometry. We construct an intrinsic representation of the underlying parametric manifold from noisy measurements based on local density estimates. This construction is shown to be equivalent to an inverse problem, which is formulated as a nonlinear differential equation and is solved empirically through eigenvectors of an appropriate Laplace operator. The learned intrinsic nonlinear model exhibits two important properties. We show that it is invariant under different observation and instrumental modalities and is noise resilient. In addition, the learned model can be efficiently extended to newly acquired measurements in a sequential manner. We examine our method on two nonlinear filtering applications: a nonlinear and non-Gaussian tracking problem and a non-stationary hidden Markov chain scheme. The experimental results demonstrate the power of our theory by extracting the underlying processes, which were measured through different nonlinear instrumental conditions.