Multilinear Discriminant Analysis Via Invariant Theory
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Solution Overview
Problem
Current machine learning techniques, such as Linear Discriminant Analysis (LDA), are not well-suited for classifying higher order data sets like tensors due to the loss of spatial locality and information during vectorization, leading to reduced accuracy and efficiency in analysis and classification.
Innovation Solution
The method employs invariant theory, specifically using an optimization algorithm like Kempf-Ness theory, to compute a change of coordinates for each mode of a multilinear data set, transforming it into a relocated data set that preserves the tensor rank and allows for accurate classification based on distances between coordinates.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If vectorization is applied to transform tensor data into vectors for classification, then the data can be processed by classical methods like LDA, but spatial locality and important information are lost
Solution Approach 1:
The patent applies multilinear transformation to map tensor data from its original high-dimensional space into a new coordinate system while preserving the tensor structure. This dimensional transformation allows the data to maintain its spatial relationships and structural information while being suitable for classification, avoiding the information loss associated with vectorization.
Solution Approach 2:
The patent changes the coordinate system parameters through multilinear transformation, defining new basis vectors and transformation matrices that preserve the essential structural parameters of the tensor data. This parameter transformation enables the data to be processed by classical methods without losing spatial information, as the transformation is designed to preserve key structural invariants.
2Loss of information
If tensor data is kept in its original multilinear structure for analysis, then spatial locality and structural information are preserved, but classical classification methods like LDA cannot be directly applied
Solution Approach 1:
The patent introduces multilinear transformation as an intermediary step between the original tensor data and classical classification methods. This intermediary transformation maps the tensor data into a transformed coordinate system where the data retains its structural information while becoming compatible with classical classification algorithms, thus bridging the gap between structured data and traditional methods.
3Productivity
If data is transformed using traditional methods, then classification can be performed, but the speed and accuracy of analysis for higher order data sets are reduced
Solution Approach 1:
The patent performs preliminary multilinear transformation of the tensor data before classification, pre-processing the data in a way that preserves its structural information and optimizes it for subsequent classification. This preliminary action of transforming to a suitable coordinate system enables both faster processing and higher accuracy by maintaining the essential features of the data throughout the analysis pipeline.
Data Source
AI summary
Methods and systems for identifying and classifying multilinear data sets into a plurality of classes using invariant theory are disclosed herein. An example method includes receiving an input data set; computing a change of coordinates for each mode of the plurality of modes for the input data set using an invariant theory optimization algorithm by (i) constructing a chosen group and (ii) determining a group element in the chosen group; transforming the input data set into a relocated data set by applying each change of coordinates for each respective mode of the plurality of modes for the input data set by multiplying the subset of the input data set for each mode by the at least one matrix corresponding to each respective mode; and classifying, based on distances between coordinates in the relocated data set, the input data set into the plurality of classes.