Sound Field Estimation Using Manifold Learning From Partial Observations

The patent transforms the optimization problem from the original parameter space to a tangent space of a manifold, adding a geometric dimension to the problem. By representing filter coefficients as points on a manifold and performing optimization in the tangent space, the system achieves faster convergence while maintaining tractable computational complexity. This dimensional transformation allows the use of Riemannian optimization techniques that exploit the geometric structure of the parameter space.

The patent changes the parameterization of the filter coefficients by constraining them to lie on a manifold (e.g., orthogonal matrices or unitary matrices). This parameter change transforms the unconstrained optimization problem into a constrained one on a manifold, enabling the use of geometric optimization methods that converge faster while maintaining the structural properties of the filter coefficients.

The patent applies partial action by using only the necessary subset of measurements rather than requiring complete microphone array data. By formulating the problem in terms of manifold learning, the system can estimate sound fields from limited or partial observations, achieving acceptable accuracy without the complexity of deploying and processing data from large microphone arrays.

The patent creates a simplified representation (copy) of the sound field by learning the underlying manifold structure from available measurements. Instead of directly processing all raw microphone data, the system learns a low-dimensional manifold representation that captures the essential acoustic characteristics, enabling accurate sound field estimation with reduced measurement requirements.

The patent performs preliminary action by pre-learning the manifold structure and tangent space transformations during an offline training phase. This preliminary learning enables the system to perform fast online optimization by simply applying the pre-computed geometric transformations, significantly reducing the computational resources required for real-time operation while maintaining high responsiveness.

The patent substitutes traditional iterative optimization methods with geometric optimization on manifolds, replacing computationally intensive mechanical-like iterative adjustments with elegant geometric projections and retractions. This substitution reduces computational complexity and enables real-time operation with fewer resources by exploiting the inherent geometric structure of the optimization problem.