Hessian Matrix Computation Using Sparse-Dense Column Segmentation
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Solution Overview
Problem
Existing algorithms for logistic regression, such as Newton's method and Iteratively Reweighted Least Squares (IRLS), face significant computational bottlenecks in calculating the Hessian matrix, particularly when dealing with large and dense input matrices, which can lead to increased processing time and iterations.
Innovation Solution
The approach involves determining the density of input matrix columns, sparsifying the matrix by selecting maximum values, and using sparse matrix operations to efficiently compute the Hessian matrix, thereby reducing unnecessary operations and speeding up the computation process.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If dense matrix operations are used for Hessian computation, then computation accuracy is maintained, but processing time increases significantly
Solution Approach 1:
The patent segments the input matrix into dense columns and sparse columns based on a density threshold. Different storage formats and computation methods are applied to each segment: dense matrix operations for dense columns and sparse matrix operations for sparse columns. This segmentation allows the system to maintain accuracy for dense portions while achieving significant speedup for sparse portions during Hessian computation.
Solution Approach 2:
The patent applies local quality by using different data representations and computational approaches for different parts of the matrix. Specifically, dense matrix formats and operations are used for dense columns where accuracy is critical, while sparse matrix formats and operations are used for sparse columns where computational efficiency is more important. This localized adaptation optimizes the trade-off between accuracy and speed for each region of the matrix.
2Loss of time
If sparse matrix operations are used for Hessian computation, then processing time is reduced, but computation accuracy may deteriorate
Solution Approach 1:
The patent segments the input matrix into dense columns and sparse columns based on a density threshold. Different storage formats and computation methods are applied to each segment: dense matrix operations for dense columns and sparse matrix operations for sparse columns. This segmentation allows the system to maintain accuracy for dense portions while achieving significant speedup for sparse portions during Hessian computation.
Solution Approach 2:
The patent applies local quality by using different data representations and computational approaches for different parts of the matrix. Specifically, dense matrix formats and operations are used for dense columns where accuracy is critical, while sparse matrix formats and operations are used for sparse columns where computational efficiency is more important. This localized adaptation optimizes the trade-off between accuracy and speed for each region of the matrix.
3Reliability
If full Hessian matrix computation is performed, then algorithm convergence is achieved, but computational resources are wasted on redundant operations
Solution Approach 1:
The patent extracts and identifies sparse columns from the input matrix and applies sparse matrix operations specifically to these columns during Hessian computation. By taking out the sparse columns and handling them differently from dense columns, the system eliminates redundant operations on zero or near-zero elements while maintaining the necessary computational accuracy for algorithm convergence.
Solution Approach 2:
The patent changes the computational parameters (storage format and operation type) based on the density parameter of each column. By dynamically selecting between dense and sparse matrix operations based on column density thresholds, the system adapts its computational approach to minimize redundant operations while ensuring sufficient accuracy for convergence.
Data Source
AI summary
According to one technique, a modeling computer computes a Hessian matrix by determining whether an input matrix contains more than a threshold number of dense columns. If so, the modeling computer computes a sparsified version of the input matrix and uses the sparsified matrix to compute the Hessian. Otherwise, the modeling computer identifies which columns are dense and which columns are sparse. The modeling computer then partitions the input matrix by column density and uses sparse matrix format to store the sparse columns and dense matrix format to store the dense columns. The modeling computer then computes component parts which combine to form the Hessian, wherein component parts that rely on dense columns are computed using dense matrix multiplication and component parts that rely on sparse columns are computed using sparse matrix multiplication.