Patch-to-tensor embedding

Salhov, Moshe; Wolf, Guy; Averbuch, Amir

doi:10.1016/j.acha.2011.11.003

Cited by 7 publications

(15 citation statements)

References 38 publications

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“…Thus, the matrix O xy represents a linear-projection between these tangent spaces, and, in some sense, the similarity between them. As in [10], it is referred to as a tangent similarity matrix.…”

Section: B Diffusion Mapsmentioning

confidence: 99%

“…These linear projection operators express some important properties of the manifold structure, e.g., curvatures between patches and differences in orientation. More details on the properties of this super-kernel are given in [10], [13]. It is convenient to use the vectors o i x and o j y to apply a double-indexing scheme by using the notation…”

Section: B Diffusion Mapsmentioning

confidence: 99%

“…In other words, let G be the super-kernel that is constructed on the dataset M according to [10]. We aim to find two matrices E ∈ R ηsd×nd andG ∈ R ηd×ηd such that…”

Section: A the Approximation Of The Spectral Decomposition Of Super-mentioning

confidence: 99%

“…Recently, DM was extended in several different ways to handle the orientation in local tangent spaces [10]- [13]. The relation between two patches is described by a matrix instead of a scalar value.…”

Section: Introductionmentioning

confidence: 99%

“…We describe the necessary condition for updating a non-scalar dictionary for achieving a bound on the approximation error. Although the proposed method is applicable to many such kernels, we focus on the linear-projection super-kernel construction described in [10]. The super-kernel construction there analyzes patches in the manifold instead of analyzing single data points on the manifold.…”

Section: Introductionmentioning

confidence: 99%

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Dictionary Construction for Patch-to-Tensor Embedding

Salhov

Wolf

Bermanis

et al. 2012

Advances in Intelligent Data Analysis XI

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The incorporation of a matrix relations, which encompass multidimensional similarities between local neighborhoods of data points in the underlying manifold of a data, improves the utilization of kernel based data analysis methodologies. However, the utilization of multidimensional similarities results in a larger kernel and hence the computational complexity of the corresponding spectral decomposition increases dramatically. In this paper, we propose an efficient approximation to a spectral decomposition of a multidimensional similarity based kernel. Furthermore, we propose a dictionary construction that approximates the oversized kernel in this case and its associated embedding. The performance of the proposed dictionary construction is demonstrated on an example of a super-kernel that utilizes the Diffusion Maps methodology together with linear-projection operators between tangent spaces in the manifold. I. INTRODUCTIONRecent methods for advanced massive high dimensional data analysis utilize a manifold structure on which data points are assumed to lie. This manifold is immersed (or submersed) in an ambient space that is defined by observable parameters. Kernel methods such as k-PCA and Diffusion Maps (DM) [4] have provided good results in analyzing such massive high dimensional data. The defined kernel can be thought of as an adjacency matrix of a graph whose vertices are the data points in the dataset. The analysis of the eigenvalues and the corresponding eigenvectors of this matrix reveals many properties and connections in the graph. These methods are based on the spectral decomposition of a kernel that was designed to incorporate a scalar similarity measure between data points. The resulting embedding of the data points into an Euclidean space preserves the qualities represented by the designed kernel. This approach extends the core of the classical Multi-Dimensional Scaling (MDS) method [6], [9] by considering non-linear relations instead of just a linear one in its original Gram matrix.Recently, DM was extended in several different ways to handle the orientation in local tangent spaces [10]- [13]. The relation between two patches is described by a matrix instead of a scalar value. The resulting kernel captures enriched similarities between local structures in the underlying manifold. These enriched similarities can be used to analyze local areas around data points instead of analyzing their specific locations. For example, this analysis can be beneficial in image processing (analyzing regions instead of individual pixels) and when the data points are perturbed so that their surrounding area is more important than their specific position. Since the constructions of these similarities are based on local tangent spaces, they provide methods to manipulate tangential vector fields (e.g., perform out-ofsample extensions). These manipulations are beneficial when the analyzed data consists of directional information in addition to positional information on the manifold. For example, the goal in [2] is ...

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Section: B Diffusion Mapsmentioning

confidence: 99%

Section: B Diffusion Mapsmentioning

confidence: 99%

“…In other words, let G be the super-kernel that is constructed on the dataset M according to [10]. We aim to find two matrices E ∈ R ηsd×nd andG ∈ R ηd×ηd such that…”

Section: A the Approximation Of The Spectral Decomposition Of Super-mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dictionary Construction for Patch-to-Tensor Embedding

Salhov

Wolf

Bermanis

et al. 2012

Advances in Intelligent Data Analysis XI

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Patch-Based Data Analysis Using Linear-Projection Diffusion

Salhov

Wolf

Averbuch

et al. 2012

Advances in Intelligent Data Analysis XI

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Learning from patches by efficient spectral decomposition of a structured kernel

et al. 2015

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We present a kernel based method that learns from a small neighborhoods (patches) of multidimensional data points. This method is based on spectral decomposition of a large structured kernel accompanied by an out-of-sample extension method. In many cases, the performance of a spectral based learning mechanism is limited due to the use of a distance metric among the multidimensional data points in the kernel construction. Recently, different distance metrics have been proposed that are based on a spectral decomposition of an appropriate kernel prior to the application of learning mechanisms. The diffusion distance metric is a typical example where a distance metric is computed by incorporating the relation of a single measurement to the entire input dataset. A method, which is called patch-to-tensor embedding (PTE), generalizes the diffusion distance metric that incorporates matrix similarity relations into the kernel construction that replaces its scalar entries with matrices. The use of multidimensional similarities in PTE based spectral decomposition results in a bigger kernel that significantly increases its computational complexity. In this paper, we propose an efficient dictionary construction that approximates the oversized PTE kernel and its associated spectral decomposition. It is supplemented by providing an out-of-sample extension for vector fields. Furthermore, the approximation error is analyzed and the advantages of the proposed dictionary construction are demonstrated on several image processing tasks.

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Patch-to-tensor embedding

Cited by 7 publications

References 38 publications

Dictionary Construction for Patch-to-Tensor Embedding

Dictionary Construction for Patch-to-Tensor Embedding

Patch-Based Data Analysis Using Linear-Projection Diffusion

Learning from patches by efficient spectral decomposition of a structured kernel

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