Neural Network Eigen Decomposition for Matrix Efficiency
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Solution Overview
Problem
Conventional algorithms for eigen value/vector decomposition, such as Jacobi iterations, are computationally inefficient and complex, especially for high-dimensional matrices, and require iterative solutions that may not converge without assumptions.
Innovation Solution
A deep neural network is employed to predict dominant eigen information by applying convolutional and pooling layers to the input covariance matrix, which is more efficient and estimates eigen values/vectors with comparable accuracy to Jacobi iterations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional algorithms like Jacobi iterations are used for eigen value/vector decomposition, then accuracy can be maintained, but computational efficiency deteriorates and complexity increases
Solution Approach 1:
The patent replaces conventional iterative numerical algorithms (mechanical computational processes) with a neural network model that performs eigen decomposition through learned transformations. The neural network substitutes the iterative Jacobi algorithm with a direct computational approach using convolutional and pooling layers, eliminating the need for repeated iterations while maintaining decomposition accuracy.
Solution Approach 2:
The patent transforms the input covariance matrix through specific neural network operations (convolution with kernel size equal to matrix dimension, followed by pooling) that change the computational parameters from iterative updates to single-pass transformations. This parameter change in the computational approach achieves both speed improvement and accuracy preservation.
2Measurement precision
If conventional iterative algorithms are used for high-dimensional matrices, then decomposition can be achieved, but computational complexity increases significantly
Solution Approach 1:
The patent segments the eigen decomposition task into distinct neural network processing stages: convolutional layers for feature extraction, pooling layers for dimensionality reduction, and output layers for eigen value/vector computation. This segmentation transforms a monolithic iterative algorithm into modular operations that reduce overall computational complexity while handling high-dimensional matrices effectively.
Solution Approach 2:
The patent introduces a new computational dimension by using neural network layers with specific kernel sizes matching the matrix dimension. The convolution operation with kernel size equal to the matrix dimension creates a transformation that operates in a different computational space, reducing the complexity from O(n^3) iterative operations to a more efficient single-pass neural network computation.
3Reliability
If iterative solutions are used for eigen decomposition, then convergence can be achieved under certain assumptions, but the method becomes less reliable without those assumptions
Solution Approach 1:
The neural network model is trained to automatically handle the convergence issue by learning the eigen decomposition mapping directly from input matrices to output eigen values and vectors. The model serves itself by incorporating convergence guarantees into its training process, eliminating the need for external assumptions about matrix properties or iterative convergence conditions that plague traditional algorithms.
Data Source
AI summary
A method performs eigen decomposition with an artificial deep neural network. The deep neural network receives an input covariance matrix. The deep neural network has a number of convolutional layers and also a number of pooling layers. The deep neural network predicts dominant eigen information of the input covariance matrix, after applying the convolutional layers and the pooling layers to the input covariance matrix. The input covariance matrix may be a real-valued covariance matrix or a complex-valued covariance matrix having a concatenated pair of matrices, including a first matrix of real components and a second matrix of imaginary components. The dominant eigen information may be absolute values of a pair of dominant eigen values and sign information of the pair of dominant eigen values, and/or absolute values of a pair of dominant eigen vectors and sign information of the pair of dominant eigen vectors.