High-Dimensional Data Analysis via Reduced Base Sampling
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Solution Overview
Problem
Current methods for estimating uncertainty in inverse problems, particularly in high-dimensional spaces, are inefficient due to high computational costs and limitations in handling large parameterizations and costly forward evaluations.
Innovation Solution
The development of automated and semi-automated systems that reduce the dimensionality of the model parameter space using orthogonal transformations and other model reduction techniques, allowing for efficient sampling within a lower-dimensional reduced base to generate output model parameter sets compatible with observed data.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Bayesian network-based frameworks are used to estimate uncertainty, then uncertainty estimation is achieved, but computational costs become very high
Solution Approach 1:
The patent transforms the high-dimensional model parameter space into a lower-dimensional reduced base space through orthogonal transformations. This dimensionality reduction allows uncertainty estimation to be performed in a compressed space, dramatically reducing computational costs while maintaining estimation accuracy. The reduced base captures the essential variability of the system with fewer parameters.
Solution Approach 2:
The patent changes the parameter representation from the original high-dimensional model parameters to a transformed coordinate system in the reduced base. By expressing model parameters as linear combinations of basis vectors in the reduced space, the system achieves the same uncertainty estimation functionality with significantly fewer parameters and lower computational burden.
2Reliability
If sampling is performed in high-dimensional model space, then comprehensive uncertainty coverage is achieved, but sampling efficiency deteriorates
Solution Approach 1:
The patent performs sampling in the lower-dimensional reduced base space rather than the original high-dimensional model space. The orthogonal transformation preserves the essential uncertainty information while reducing the sampling space dimensionality, thereby improving sampling efficiency without sacrificing uncertainty coverage.
Solution Approach 2:
The patent creates a transformed copy of the model parameter space through orthogonal basis vectors. This copied reduced space maintains the statistical properties and uncertainty relationships of the original space but with reduced dimensionality, enabling efficient sampling that can be mapped back to the original parameter space.
3Measurement precision
If the number of model parameters is increased for better parameterization, then model accuracy improves, but computational complexity increases
Solution Approach 1:
The patent transforms the parameter representation from individual high-dimensional parameters to a compact set of coefficients in the reduced base. This parameter transformation maintains the model's ability to represent complex systems while reducing the number of parameters that need to be estimated and sampled, thereby lowering computational complexity.
Solution Approach 2:
The orthogonal basis vectors in the reduced base serve multiple functions: they span the essential model variability, provide a compressed parameter representation, and enable efficient sampling. This multi-functionality allows the reduced base to handle complex high-dimensional problems with fewer resources.
Data Source
AI summary
Described herein is a framework for analyzing data in high-dimensional space. In accordance with one implementation, observed data and at least one input model parameter set is received. The input model parameter set serves as a solution candidate of a predefined problem (e.g., inverse or optimization problem) and is related to the observed data via a model. To provide enhanced computational efficiency, a reduced base with lower dimensionality is determined based on the input model parameter set. The reduced base is associated with a set of coefficients, which represents the coordinates of any model parameter set in the reduced base. Sampling is performed within the reduced base to generate an output model parameter set in the reduced base. The output model parameter set is compatible with the input model parameter set and fits the observed data, via the model, within a predetermined threshold.