Gaussian Process Kernel Matrix Inversion via Dominant Eigenvalue
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Solution Overview
Problem
The practical use of behavior models for non-deterministic technical systems is hindered by computationally elaborate operations with matrices, making it difficult to operate these systems effectively.
Innovation Solution
A computer-implemented method that involves presetting data points for a Gaussian process, determining a positive semi-definite kernel matrix, and calculating its inverse using an estimated dominant eigenvalue and 1-Lipschitz mapping, allowing for parallelized calculations and improved computing speed.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If behavior models of non-deterministic technical systems are developed based on observations, then the model accuracy is improved, but computationally elaborate operations with matrices are necessary making the practical use difficult
Solution Approach 1:
The patent transforms the computational problem by changing parameters from direct matrix operations to eigenvalue-based operations. By computing the dominant eigenvalue and corresponding eigenvector of the kernel matrix, the system reduces complex matrix inversions to simpler operations that scale better, maintaining model accuracy while reducing computational burden.
Solution Approach 2:
The patent extracts only the most critical component from the full kernel matrix - specifically the dominant eigenvalue and its corresponding eigenvector. This extraction allows the system to work with a simplified representation that captures the essential behavior needed for predictions, eliminating the need for computationally intensive operations with the complete matrix.
2Measurement precision
If matrix operations are performed for Gaussian process predictions, then accurate predictions are obtained, but the computing speed decreases
Solution Approach 1:
The patent changes the computational parameters from full matrix operations to eigenvalue-based operations. By using the dominant eigenvalue decomposition, the system maintains prediction accuracy while significantly improving computing speed, as eigenvalue computation is less computationally intensive than full matrix inversion and multiplication.
Solution Approach 2:
The patent extracts only the dominant eigenvalue and eigenvector from the kernel matrix, which contain the most significant information for making predictions. This extraction enables faster computation while preserving the essential predictive capability of the full matrix operations.
3Measurement precision
If more data points are acquired during operation to improve prediction accuracy, then the model quality improves, but the computational load increases
Solution Approach 1:
The patent extracts only the dominant eigenvalue and eigenvector from the kernel matrix constructed from observed data points. This extraction allows the system to incorporate multiple data points for improved accuracy while keeping computational energy consumption manageable by working with a simplified representation rather than the full data matrix.
Data Source
AI summary
Apparatus and computer-implemented method, including: presetting data points which include pairs of mutually assigned input and output of a Gaussian process; determining a positive semi-definite kernel matrix from inputs predetermined by the data points; determining an inverse of the kernel matrix depending on an estimation for an inverse of a 1-Lipschitz mapping of the kernel matrix; presetting an input for the Gaussian process; determining a prediction for an expected value of the Gaussian process, and/or a prediction for a variance of the Gaussian process; determining a probable output variable of a sensor and/or a control variable for a machine depending on at least one of the predictions.