Anisotropic diffusion on sub-manifolds with application to Earth structure classification

Kushnir, Dan; Haddad, Abdolhosein; Coifman, Ronald R.

doi:10.1016/j.acha.2011.06.002

Cited by 52 publications

(55 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the methodology of Singer and Coifman (5). and Kushnir et al (19),W R converges to a diffusion operator that reveals the low-dimensional manifold and the eigenvectors give a parameterization of the underlying processes. Specifically, in case of independent factors, the eigenvectors recover d proxies for the controlling factors.…”

Section: Intrinsic Modelingmentioning

confidence: 99%

Empirical intrinsic geometry for nonlinear modeling and time series filtering

Talmon

Coifman

2013

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

Self Cite

107

View full text Add to dashboard Cite

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...

show abstract

Section: Intrinsic Modelingmentioning

confidence: 99%

Empirical intrinsic geometry for nonlinear modeling and time series filtering

Talmon

Coifman

2013

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

Self Cite

107

View full text Add to dashboard Cite

show abstract

“…In other words, when there are no observation-specific variables, i.e., = η = 0, the metric we build based on local applications of CCA is a modified Mahalanobis distance, which was presented and analyzed in [18], [19], [34] for the purpose of recovering the intrinsic representation from nonlinear observation data.…”

Section: Methodsmentioning

confidence: 99%

“…The more efficient algorithm is presented in Algorithm 2, where the CCA matrices A (x i ) are constructed only for a subset of L ≤ N points x i ∈ X L ⊆ X (without the need to directly address the middle point), i.e., A (x i ) is computed only L times. This modification does not affect the algorithm; Theorem 3.2 presented in [34] states that the entries of matrix M calculated in Step 3 in Algorithm 2 are approximations of the entries of the matrix M calculated in Step 2 in Algorithm 1. For more details, see [34].…”

Section: Algorithm 2 Diffusion Maps Of Two Datasets Without Middle Pomentioning

confidence: 99%

“…This modification does not affect the algorithm; Theorem 3.2 presented in [34] states that the entries of matrix M calculated in Step 3 in Algorithm 2 are approximations of the entries of the matrix M calculated in Step 2 in Algorithm 1. For more details, see [34]. The modification gives rise to two benefits.…”

Section: Algorithm 2 Diffusion Maps Of Two Datasets Without Middle Pomentioning

confidence: 99%

“…In order to relaxe the above assumption and to reduce the computational complexity, we present an algorithm based on [34]. The more efficient algorithm is presented in Algorithm 2, where the CCA matrices A (x i ) are constructed only for a subset of L ≤ N points x i ∈ X L ⊆ X (without the need to directly address the middle point), i.e., A (x i ) is computed only L times.…”

Section: Algorithm 2 Diffusion Maps Of Two Datasets Without Middle Pomentioning

confidence: 99%

See 2 more Smart Citations

Local Canonical Correlation Analysis for Nonlinear Common Variables Discovery

Yair

Talmon

2017

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

Abstract-In this paper, we address the problem of hidden common variables discovery from multimodal data sets of nonlinear high-dimensional observations. We present a metric based on local applications of canonical correlation analysis (CCA) and incorporate it in a kernel-based manifold learning technique. We show that this metric discovers the hidden common variables underlying the multimodal observations by estimating the Euclidean distance between them. Our approach can be viewed both as an extension of CCA to a nonlinear setting as well as an extension of manifold learning to multiple data sets. Experimental results show that our method indeed discovers the common variables underlying high-dimensional nonlinear observations without assuming prior rigid model assumptions.

show abstract