Modeling natural and artificial systems has a key role in various applications, and has long been a task that drew enormous efforts. In this work, instead of exploring predefined models, we aim at implicitly identifying the system degrees of freedom. This approach circumvents the dependency of a specific predefined model for a specific task or system, and enables a generic data-driven method to characterize a system based solely on its output observations. We claim that each system can be viewed as a black box controlled by several independent parameters. Moreover, we assume that the perceptual characterization of the system output is determined by these independent parameters. Consequently, by recovering the independent controlling parameters, we find in fact a generic modeling for the system. In this work, we propose a supervised algorithm to recover the controlling parameters of natural and artificial linear systems. The proposed algorithm relies on nonlinear independent component analysis using diffusion kernels and spectral analysis. Employment of the proposed algorithm on both synthetic and real examples has shown accurate recovery of parameters.

Many natural and artificial high-dimensional time series are often controlled by a set of lower-dimensional independent factors. In this paper anisotropic diffusion is combined with local dynamical models to provide intrinsic global modeling that reveals these factors. The obtained model is shown to be invariant to the measuring equipment and can be efficiently extended. These two properties are paramount for sequential processing and provide a foundation for probabilistic analysis. The widely applicable approach is demonstrated on nonlinear tracking problems based on both simulated and recorded data.

Patch-based de-noising algorithms and patch manifold smoothing have emerged as efficient de-noising methods. This paper provides a new insight on these methods, such as the Non Local Means or the image graph de-noising, by showing its use for filtering a selected pattern.

In 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 lowdimensional 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.