Asymmetric Kernel Training via Recursive Matrix Inversion
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Solution Overview
Problem
Training asymmetric kernels for Determinantal Point Processes is computationally intensive, requiring O(N^3) time and significant resources, making it inefficient for large datasets.
Innovation Solution
Calculating the inverse matrix of the sum of the asymmetric kernel and identity matrix in a recursive manner to reduce computational time and resources, allowing for the training of asymmetric kernels using O((K+M)N^2) time, where K and M are smaller than N.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If asymmetric kernel matrix is used in DPP, then probability distribution quality is improved, but training time and computational resources increase significantly
Solution Approach 1:
The patent segments the inverse matrix calculation into iterative updates. Instead of computing the full inverse matrix at once (O(N^3)), it updates the inverse matrix incrementally using the formula: (K + λI)^-1 = (K^-1 - K^-1 * (λK^-1 + I)^-1 * K^-1) / λ, where K is the asymmetric kernel matrix and λ is a regularization parameter. This segmentation reduces computational complexity to O(N^2) per iteration.
Solution Approach 2:
The patent performs preliminary factorization of the asymmetric kernel matrix K into L * L^T (Cholesky decomposition or similar factorization) before the main training process. This preliminary action allows subsequent inverse matrix calculations to operate on the factored form, significantly reducing computational burden during iterative training updates.
2Measurement precision
If asymmetric kernel matrix is used in DPP, then probability distribution quality is improved, but computational resources increase significantly
Solution Approach 1:
The patent segments the computational workload by dividing the inverse matrix calculation into manageable iterative updates. Each update operates on smaller matrix operations rather than the full N×N matrix, reducing peak memory usage and computational resource requirements from O(N^3) to O(N^2) per iteration.
Solution Approach 2:
The patent computes only the necessary portions of the inverse matrix needed for the current training iteration rather than the full inverse matrix. This partial computation approach reduces computational resources by calculating only (K + λI)^-1 * b vectors needed for gradient updates, avoiding unnecessary full matrix inversion operations.
Data Source
AI summary
A method for a determinantal Point Process-based prediction includes obtaining, using a hardware processor, a training data set stored on one or more computer readable storage mediums operably coupled to the hardware processor, training an asymmetric kernel of a Determinantal Point Process (DPP) from a training data set by calculating an inverse matrix of a sum of the asymmetric kernel and an identity matrix in a recursive manner to reduce time and computational resources utilized, and determining a prediction model by training the asymmetric kernel as at least part of a prediction model to make a prediction.