Tensor Decomposition With Sparse Core for Data Completion
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Solution Overview
Problem
Existing methods for processing multi-dimensional data, such as color images and time-dependent user ratings, face challenges when data is represented in more than two dimensions, as Singular Value Decomposition (SVD) and its generalizations like Higher-Order SVD and Canonical Polyadic Decomposition are not directly applicable and lack consistency, preventing effective completion and compression of such data.
Innovation Solution
A multi-dimensional data manipulation engine decomposes tensor representations into a sparse core and unitary matrix multipliers, allowing for efficient completion or compression of data by truncating these components, thereby reconstructing missing elements or reducing data complexity, using constrained Lp norm optimization and algorithms like Augmented Lagrange Method.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If SVD methods are used for matrix data, then completion and compression can be achieved, but these methods are not directly applicable to multi-dimensional data
Solution Approach 1:
The patent extends the two-dimensional SVD framework to multi-dimensional tensors by introducing n-mode multiplication and tensor decomposition. This allows the same mathematical principles to be applied across any number of dimensions while maintaining consistency with the original SVD approach for matrices.
2Adaptability or versatility
If known generalizations like HOSVD and CPD are used, then multi-dimensional data can be processed, but they are not strictly consistent with SVD and prevent generalization of common algorithms
Solution Approach 1:
The patent creates a universal decomposition framework that works for both two-dimensional matrices and multi-dimensional tensors. By defining n-mode multiplication and tensor decomposition in a unified manner, common algorithms can be generalized across different dimensions without requiring dimension-specific implementations.
3Productivity
If the sparse core is truncated for compression, then data complexity is reduced, but some information is lost
Solution Approach 1:
The patent optimizes the threshold parameter for truncating the sparse core, balancing compression efficiency with information preservation. By carefully selecting the truncation threshold, the method achieves high compression ratios while retaining the most important structural information in the tensor.
Data Source
AI summary
Original data that represents a real-world object or activity and organized along three or more dimensions is received. The original data is represented as a product of several multipliers including a sparse core, such that the sparse core has fewer non-zero values than a tensor representation of the original data, and one or more unitary matrix multipliers. Modified data is generated based on the original data using the multipliers. This includes compressing, or reconstructing missing elements in, the tensor representation of the original data, such that the modified data provides a description of the real-world object or activity that is less complete or more complete, respectively, relative to the original data.