Parametrization of Linear Systems Using Diffusion Kernels

Talmon, Ronen; Kushnir, Dan; Coifman, Ronald R.; Cohen, Israel; Gannot, Sharon

doi:10.1109/tsp.2011.2177973

Cited by 45 publications

(34 citation statements)

References 36 publications

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“…The use of DM is well suited for this task as a nonlinear dimensionality reduction method. This method can help to model the relation between intrinsic latent parameters of musical pieces [28], such as Key, Timbre, Genre, Harmonic measures and more.…”

Section: Dimensionality Reductionmentioning

confidence: 99%

Musical key extraction using diffusion maps

2015

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Section: Dimensionality Reductionmentioning

confidence: 99%

Musical key extraction using diffusion maps

2015

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“…We revisit the experiment presented by Talmon et al (23,24) and test the ability of the EIG method to recover the location of a sound source (25) from a single microphone in a room. Following is a brief description of the experiment setup.…”

Section: Acoustic Localizationmentioning

confidence: 99%

Empirical intrinsic geometry for nonlinear modeling and time series filtering

Talmon

Coifman

2013

Proc. Natl. Acad. Sci. U.S.A.

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107

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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...

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“…As proposed in [29,34,35], a nonsymmetric pairwise metric between any new measurement z t and a reference measurement z s is defined, similarly to the Mahalanobis distance (10), by…”

Section: Sequential Processingmentioning

confidence: 99%

Intrinsic modeling of stochastic dynamical systems using empirical geometry

Talmon

Coifman

2015

Applied and Computational Harmonic Analysis

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In a broad range of natural and real-world dynamical systems, measured signals are controlled by underlying processes or drivers. As a result, these signals exhibit highly redundant representations, while their temporal evolution can often be compactly described by dynamical processes on a low-dimensional manifold. In this paper, we propose a graph-based method for revealing the low-dimensional manifold and inferring the processes. This method provides intrinsic models for measured signals, which are noise resilient and invariant under different random measurements and instrumental modalities. Such intrinsic models may enable mathematical calibration of complex measurements and build an empirical geometry driven by the observations, which is especially suitable for applications without a priori knowledge of models and solutions. We exploit the temporal dynamics and natural small perturbations of the signals to explore the local tangent spaces of the low-dimensional manifold of empirical probability densities. This information is used to define an intrinsic Riemannian metric, which in turn gives rise to the construction of a graph that represents the desired low-dimensional manifold. Such a construction is equivalent to an inverse problem, which is formulated as a nonlinear differential equation and is solved empirically through eigenvectors of an appropriate Laplace operator. We examine our method on two nonlinear filtering applications: a nonlinear and non-Gaussian tracking problem as well as 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, in an entirely data-driven nonparametric way.

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Parametrization of Linear Systems Using Diffusion Kernels

Cited by 45 publications

References 36 publications

Musical key extraction using diffusion maps

Musical key extraction using diffusion maps

Empirical intrinsic geometry for nonlinear modeling and time series filtering

Intrinsic modeling of stochastic dynamical systems using empirical geometry

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