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Tensor SOM and tensor GTM: Nonlinear tensor analysis by topographic mappings.
Tohru Iwasaki1, Tetsuo Furukawa1
1Department of Human Intelligence Systems, Kyushu Institute of Technology, Japan.
Summary
We introduce nonlinear tensor analysis methods, Tensor Self-Organizing Map (TSOM) and Tensor Generative Topographic Mapping (TGTM), for analyzing complex multimodal relational data. These techniques offer novel ways to organize and visualize high-dimensional tensorial datasets.
Area of Science:
- Data Science
- Machine Learning
- Multivariate Statistics
Background:
- Analyzing high-dimensional and multimodal relational data presents significant challenges.
- Existing methods often struggle to capture the complex interdependencies within tensorial structures.
- The need for advanced visualization and analysis tools for such data is critical.
Purpose of the Study:
- To propose novel nonlinear tensor analysis methods: Tensor Self-Organizing Map (TSOM) and Tensor Generative Topographic Mapping (TGTM).
- To provide tools for effective analysis and visualization of tensorial data, particularly multimodal relational data.
- To extend existing topographic mapping techniques to handle the complexities of tensorial data structures.
Main Methods:
- Developed Tensor Self-Organizing Map (TSOM) as an extension of Self-Organizing Maps for tensorial data.
- Formulated Tensor Generative Topographic Mapping (TGTM) as a probabilistic generative model providing theoretical grounding for TSOM.
- Implemented algorithms for organizing n-mode topographic maps and exploring tensorial data spaces.
Main Results:
- TSOM and TGTM enable simultaneous organization of multiple topographic maps for n-mode relational data.
- These methods facilitate interactive visualization of relationships between different modes within the tensorial data.
- Demonstrated the utility of TSOM and its variations through applications on real-world relational datasets.
Conclusions:
- TSOM and TGTM are powerful nonlinear methods for analyzing and visualizing tensorial data.
- These techniques offer significant advantages for exploring complex multimodal relational datasets.
- The proposed methods provide a comprehensive framework for understanding and manipulating tensorial data structures.
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