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Tensorized Feature Extraction Technique for Multimodality Preserving Manifold Visualization

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Abstract

The intrinsic multimodal structures are often encountered in real life cases. Also, reducing the dimensionality of data without losing the local and multimodal information is important in multivariate data analysis. This paper proposes a locality preserving technique, which we refer to as Multimodality Preserving Tensorized Feature Extraction (MPTFE). MPTFE works in tensor space extracting the features directly from the matrix patterns of images. Thus the local topological information of pixels in the images can be clearly preserved. In extracting the informative features, MPTFE aims at finding a large margin representation of data and keeping the intrinsic characteristics of data in addition to separating inter-class points. The orthogonal projection matrix of MPTFE is efficiently obtained by eigen-decomposition, thus the similarity based on Euclidean distance can be effectively preserved. The validity and feasibility of the proposed algorithm are verified through conducting extensive simulations, including data visualization and clustering, on three benchmark real life problems. The delivered visualization results show that our MPTFE algorithm is able to capture the intrinsic characteristic of the given data and exhibits large margins between multiple objects and cluster. Clustering evaluation results show that MPTFE delivers comparable accuracy to some widely used neighborhood preserving techniques.

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Acknowledgements

The authors would like to express our sincere thanks to the anonymous reviewers’ comments and suggestions which have made the paper a higher standard.

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Authors and Affiliations

  1. College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Si-yuan Wu

  2. Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Zhao Zhang

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  1. Si-yuan Wu
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  2. Zhao Zhang
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Correspondence to Zhao Zhang.

Appendix: Computational Analysis of U y and V y

Appendix: Computational Analysis of U y and V y

Let W (XY) and Q (XY) be two diagonal matrices whose entries are column and row sums of the matrix A (XY) respectively, that is \(W_{ii}^{( XY )} = \sum_{j} A_{i,j}^{( XY )}\) and \(Q_{jj}^{( XY)} = \sum_{i} A_{i,j}^{( XY )}\). Since ∥M∥2=tr(MM T), we can obtain

(38)

where

Similarly

(39)

where

$$L_{V}^{ \leftarrow} = \sum_{i} D_{ii}^{( YY)}Y_{i}V_{y}V_{y}^{\mathrm{T}}Y_{i}^{\mathrm{T}},\quad S_{V}^{ \leftarrow} =\sum_{i,j} A_{i,j}^{( YY)}Y_{i}V_{y}V_{y}^{\mathrm{T}}Y_{j}^{\mathrm{T}}$$

and D (YY) is a diagonal matrix whose elements are column (or row) sums of A (YY), i.e. \(D_{ii}^{( YY )} = \sum_{j} A_{i,j}^{( YY )}\). Similarly, because ∥M∥2=tr(M T M), we also have

(40)

where

Similarly

(41)

where

Thus the projection matrices U y and V y can be obtained by solving the following two eigenvector problems in (25) and (26).

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Wu, Sy., Zhang, Z. Tensorized Feature Extraction Technique for Multimodality Preserving Manifold Visualization. J Math Imaging Vis 44, 295–314 (2012). https://doi.org/10.1007/s10851-012-0327-1

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