Manifold Transform for High-Dimensional Data Modeling
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Solution Overview
Problem
Existing solutions struggle to effectively model high-dimensional data sets in a low-dimensional space while providing probabilistic modeling that is computationally tractable, as current methods like GANs and VAEs fail to learn complex high-dimensional spaces and neglect underlying probabilities, and normalizing flows maintain dimensionality without effectively characterizing low-dimensional manifolds.
Innovation Solution
A computer model employing a manifold transform using conformal flows to map high-dimensional data to a low-dimensional space, followed by a density transformation to a base probability distribution, allowing for effective density estimation and tractable inversion between spaces, enabling the representation of high-dimensional data as a manifold in a low-dimensional space.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If normalizing flows are used to provide probabilistic information, then probability distribution is maintained, but data dimensionality cannot be reduced to low-dimensional manifold space
Solution Approach 1:
The patent applies dimensionality change by introducing a low-dimensional manifold space that captures the essential structure of high-dimensional data. The model transforms data from high-dimensional space to low-dimensional manifold coordinates, enabling probabilistic modeling in reduced dimensions while maintaining reliability through the manifold's geometric structure.
Solution Approach 2:
The patent introduces a manifold as an intermediary between high-dimensional data and low-dimensional probabilistic representation. This manifold acts as a mediator that preserves probabilistic information while enabling dimensionality reduction, solving the contradiction between maintaining reliability and reducing complexity.
2Productivity
If GAN models are used to generate data in high-dimensional space, then data generation capability is improved, but underlying probability distribution is not modeled
Solution Approach 1:
The patent changes the parameterization approach by representing data in terms of manifold coordinates rather than raw high-dimensional coordinates. This parameter change enables the model to capture the underlying probability distribution on the manifold while maintaining effective data generation capability through the transformed coordinate system.
3Loss of information
If high-dimensional data is modeled directly without manifold transformation, then complete data range representation is maintained, but computational tractability decreases
Solution Approach 1:
The patent transforms the problem from high-dimensional space to low-dimensional manifold space, changing the dimensionality to achieve computational tractability. The manifold transformation preserves the essential data range representation while reducing computational complexity by operating in the lower-dimensional manifold coordinates.
Data Source
AI summary
A computer models a high-dimensional data with a low-dimensional manifold in conjunction with a low-dimensional base probability density. A first transform (a manifold transform) may be used to transform the high-dimensional data to a low-dimensional manifold, and a second transform (a density transform) may be used to transform the low-dimensional manifold to a low-dimensional probability distribution. To enable the model to tractably learn the manifold transformation from the high-dimensional to low-dimensional spaces, the manifold transformation includes conformal flows, which simplify the probabilistic volume transform and enables tractable learning of the transform. This may also allow the manifold transform to be jointly learned with density transform.