Fast Combinatorial Algorithm for Linearly Constrained Least Squares
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Solution Overview
Problem
Current algorithms for solving linearly constrained least squares problems, particularly in large-scale multivariate image analysis, are inefficient due to excessive computational requirements and lack of scalability, especially when dealing with multiple right-hand-side vectors and non-negativity constraints.
Innovation Solution
A fast combinatorial algorithm that rearranges calculations to minimize arithmetic operations by precomputing constant parts of the pseudoinverse, partitioning column indices, and solving triangular sets of linear equations, allowing for efficient handling of equality and inequality constraints.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If standard NNLS algorithm is applied to multiple RHS problems, then mathematically rigorous least squares solutions are obtained, but computational speed becomes unacceptably slow
Solution Approach 1:
The patent precomputes the pseudoinverse of matrix A and stores it for reuse across multiple RHS problems. By performing the computationally intensive pseudoinverse calculation once before processing multiple right-hand side vectors, the algorithm avoids repeating this operation for each RHS, thereby maintaining mathematical rigor while dramatically improving computational efficiency for multiple RHS problems
Solution Approach 2:
The patent creates a universal solution framework where the pseudoinverse of A serves as a reusable computational object that can be applied to any number of RHS vectors. This universal approach allows the same precomputed pseudoinverse to efficiently solve multiple different least squares problems with the same coefficient matrix A but different RHS vectors, eliminating redundant calculations
2Measurement precision
If full pseudoinverse calculation is performed for each observation vector, then accurate solutions are obtained, but computational burden becomes excessive
Solution Approach 1:
The patent extracts the computationally intensive part of the pseudoinverse calculation that remains constant across multiple RHS problems. By separating the pseudoinverse computation into a one-time preliminary step and a reusable component, the algorithm eliminates redundant calculations for each RHS vector while preserving the accuracy of the least squares solutions
Solution Approach 2:
The patent performs the expensive pseudoinverse calculation in advance before processing multiple RHS vectors. This preliminary computation stores the essential mathematical structure that can be reused across all subsequent RHS problems, reducing the computational burden from O(n³) per RHS to O(n²) per RHS where n is the number of variables
3Reliability
If rigorous NNLS methods are used to ensure true least squares solutions, then mathematical correctness is maintained, but algorithm complexity increases
Solution Approach 1:
The patent precomputes the pseudoinverse matrix which encapsulates all the complex mathematical operations needed for rigorous least squares solutions. By moving the complexity into a one-time preliminary calculation, the subsequent processing of multiple RHS vectors becomes simpler and more straightforward, maintaining mathematical correctness while reducing implementation complexity for the main algorithm loop
Data Source
AI summary
A fast combinatorial algorithm can significantly reduce the computational burden when solving general equality and inequality constrained least squares problems with large numbers of observation vectors. The combinatorial algorithm provides a mathematically rigorous solution and operates at great speed by reorganizing the calculations to take advantage of the combinatorial nature of the problems to be solved. The combinatorial algorithm exploits the structure that exists in large-scale problems in order to minimize the number of arithmetic operations required to obtain a solution.