Fast Singular Value Decomposition for Large Matrix Analysis
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Solution Overview
Problem
Traditional singular value decomposition (SVD) methods are computationally intensive and inefficient when handling large volumes of data, leading to significant computing time bottlenecks in computer analysis systems, particularly when dealing with huge matrices, which limits their practical applications.
Innovation Solution
A fast singular value decomposition method that involves inputting a first matrix, computing an average vector, performing overlapped random number sub-grouping, computing variance matrices, reorganizing space coordinates, adjusting column vectors using QR decomposition, and combining results to form a new matrix, thereby optimizing the SVD process and enhancing computing speed.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional SVD method is used to decompose large matrices, then decomposition accuracy is maintained, but computing time increases significantly
Solution Approach 1:
The patent divides the large matrix decomposition task into multiple sub-groupings. The matrix is segmented into smaller blocks that can be processed independently or in parallel, reducing the overall computational burden while maintaining decomposition accuracy through coordinated processing of these segments.
Solution Approach 2:
The patent performs preliminary actions by precomputing certain matrix properties, performing initial decompositions on smaller sub-matrices, and preparing data structures in advance. This preliminary processing reduces the complexity of the main decomposition task and accelerates the overall computing process.
2Quantity of substance
If traditional SVD method processes large volume of data, then comprehensive analysis is achieved, but computing speed decreases
Solution Approach 1:
The patent segments the large data matrix into multiple smaller sub-matrices or blocks. These segmented data blocks can be processed in parallel or sequentially with reduced computational overhead, thereby improving computing speed while still analyzing the complete data volume through aggregation of results.
Solution Approach 2:
The patent applies partial action by performing decomposition on representative sub-samples or sub-groups of the data first, then extending results to the full dataset. This approach provides sufficient analytical insight without processing every single data point in full detail, thus improving computing speed.
3Reliability
If traditional SVD decomposes huge matrices, then complete factorization is obtained, but computing complexity increases
Solution Approach 1:
The patent segments the huge matrix decomposition into manageable sub-tasks. By dividing the matrix into smaller blocks and processing them separately, the computational complexity of each individual task is reduced, while the complete factorization is reconstructed by combining the results from all segments.
Solution Approach 2:
The patent performs preliminary computations on matrix properties, dimensions, and structures before the main decomposition process. This preliminary analysis simplifies the subsequent factorization steps and reduces the overall computing complexity while ensuring complete and accurate factorization results.
Data Source
AI summary
The present invention uses a computer analysis system of a fast singular value decomposition to overcome the bottleneck of a traditional singular value decomposition that takes much computing time for decomposing a huge number of objects, and the invention can also process a matrix in any form without being limited to symmetric matrixes only. The decomposition and subgroup concept of the fast singular value decomposition works together with the decomposition of a variance matrix and the adjustment of an average vector of a column vector are used for optimizing the singular value decomposition to improve the overall computing speed of the computer analysis system.