Iterative Singular Value Decomposition Parallel Processing
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Solution Overview
Problem
Current methods for singular value decomposition (SVD) are computationally intensive and require significant resources, making them inefficient for large matrices and real-time applications, especially when performed on devices with limited processing power, and they do not leverage modern multi-processor architectures for parallel processing.
Innovation Solution
A regular iterative technique for estimating singular values of a matrix is introduced, which is more computationally efficient and allows for parallelization across multiple processors, reducing computational time by performing iteration operations in a pipelined fashion.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Cholesky decomposition is used to compute SVD, then the decomposition can be performed using standard algorithms, but the computational complexity increases significantly for large matrices
Solution Approach 1:
The patent segments the SVD computation into two distinct phases: (1) an iterative phase that computes only the top k singular values and vectors for the most significant components, and (2) a Cholesky decomposition phase that completes the full SVD. This segmentation allows the computationally intensive part to be approximated efficiently while maintaining overall accuracy, reducing the complexity burden on large matrices.
Solution Approach 2:
The patent applies partial action by computing only the necessary portion of the SVD (top k singular values) using iterative methods, rather than computing all singular values through full Cholesky decomposition. This partial computation approach significantly reduces computational complexity while providing sufficient accuracy for most practical applications where only the most significant singular values are needed.
2Ease of manufacture
If sequential Cholesky decomposition is used, then the algorithm is straightforward to implement, but the processing speed is limited due to inability to parallelize
Solution Approach 1:
The patent segments the computation into iterative phases (which can be parallelized across multiple processors to compute different singular values simultaneously) and a final Cholesky phase. This segmentation enables parallel processing in the iterative section, dramatically improving processing speed while maintaining implementation feasibility through the structured two-phase approach.
Solution Approach 2:
The patent performs preliminary computation of the top k singular values and vectors using iterative methods before applying Cholesky decomposition. This preliminary action prepares the data in a form that reduces the remaining computational burden and enables more efficient parallel processing in subsequent stages, thereby improving overall processing speed.
3Reliability
If full SVD is computed on large matrices, then complete decomposition is achieved, but significant computational resources and power are consumed
Solution Approach 1:
The patent computes only the top k singular values and vectors using iterative methods, which provides sufficient accuracy for most applications without the need to compute all singular values. This partial computation dramatically reduces computational resource consumption and energy usage while maintaining the reliability needed for practical big data applications.
Solution Approach 2:
The patent extracts and computes only the most significant singular values and vectors (the top k components) that contain the majority of the useful information, rather than computing the complete SVD. This extraction approach reduces computational resource consumption and energy usage while preserving the essential features needed for data analysis tasks.
4Productivity
If iterative methods are used to estimate singular values, then computational efficiency improves, but multiple iteration cycles are required which may increase time complexity
Solution Approach 1:
The patent performs a limited number of iteration cycles (typically 3-10 iterations) to compute the top k singular values, which provides sufficient accuracy without requiring excessive iteration time. This partial iteration approach achieves computational efficiency by stopping the iterative process early when adequate convergence is reached, balancing computational efficiency with acceptable iteration time.
Data Source
AI summary
Matrix processing includes: accessing an original matrix; iteratively determining a plurality of estimated singular vectors of the original matrix, a plurality of estimated singular values of the original matrix, or both, using a plurality of iteration cycles; wherein at least some of the plurality of iteration cycles are performed in parallel on a plurality of processors; and outputting the plurality of estimated singular vectors of the original matrix, the plurality of estimated singular values of the original matrix, or both.