Texture separation via a reference set

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

doi:10.1016/j.acha.2013.05.002

Cited by 23 publications

(33 citation statements)

References 6 publications

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“…3 Specifically, when the underlying process consists of independent coordinates, the leading d eigenvectors (except the trivial), which are a local canonical/intrinsic coordinate system for the manifold [33], recover d proxies for the underlying process coordinates up to a monotonic scaling [29]. In other words, they are empirical solutions to the inverse problem described by the differential equation in (12). In addition, the eigenvectors are independent in case the manifold is flat [11].…”

Section: Laplace Operatormentioning

confidence: 99%

“…It implies that two measurements are similar if they "see" the reference measurements in the same way. Furthermore, it is shown in [29] and [12] that the elements of the extended kernel are proportional to a Gaussian defined similarly to (22) with the corresponding Mahalanobis distances between pairs of new measurements.…”

Section: Sequential Processingmentioning

confidence: 99%

“…Specifically, the eigenvectors provide an embedding (or a parametrization) of the underlying processes on the intrinsic manifold. We remark that the construction of the graph is implemented using a reference set of measurements [12,13]. This allows for the parameterization of a training signal in advance, and in turn, for the extension of the parameterization to newly acquired signal samples in a sequential manner.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Laplace Operatormentioning

confidence: 99%

Section: Sequential Processingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intrinsic modeling of stochastic dynamical systems using empirical geometry

Talmon

Coifman

2015

Applied and Computational Harmonic Analysis

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“…Technically, this kernel form has been considered in several related works such as [30,14,21,19,15,27] and references therein. These works consider the relations between the analyzed dataset and a reference set, which typically is significantly smaller than the dataset.…”

Section: Definition 31 (Measure-based Gaussian Correlation Kernel)mentioning

confidence: 99%

“…Another approach, which is presented in [4,15], is to analyze the data by considering their relations to a given reference set, which can either be part of the input data, or designed for specific applications. Therefore, instead of representing pairwise similarities within the data, the kernel in these cases takes an asymmetric form consisting of relations between data points and reference points.…”

Section: Introductionmentioning

confidence: 99%

Diffusion-based kernel methods on Euclidean metric measure spaces

Bermanis

Wolf

Averbuch

2016

Applied and Computational Harmonic Analysis

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Bi-stochastic kernels via asymmetric affinity functions

Coifman

Hirn

2013

Applied and Computational Harmonic Analysis

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In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function {\alpha}. The affinity function {\alpha} measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or through solving an optimization problem, the construction presented here is quite simple. Furthermore, it can be viewed through the lens of out of sample extensions, making it useful for massive data sets.Comment: 5 pages. v2: Expanded upon the first paragraph of subsection 2.1. v3: Minor changes and edits. v4: Edited comments and added DO

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Texture separation via a reference set

Cited by 23 publications

References 6 publications

Intrinsic modeling of stochastic dynamical systems using empirical geometry

Intrinsic modeling of stochastic dynamical systems using empirical geometry

Diffusion-based kernel methods on Euclidean metric measure spaces

Bi-stochastic kernels via asymmetric affinity functions

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