Non-Euclidean Conditional Expectation and Filtering
This work addresses filtering challenges for manifold-valued data, such as in finance, by providing a novel computational approach, though it appears incremental as it builds on existing conditional expectation concepts.
The paper tackles the problem of filtering for manifold-valued signals by introducing a non-Euclidean generalization of conditional expectation, which is characterized as a minimizer of expected intrinsic squared-distance. This leads to computationally tractable filtering equations that are applied to achieve accurate numerical forecasts of efficient portfolios, incorporating their geometric structure into estimates.
A non-Euclidean generalization of conditional expectation is introduced and characterized as the minimizer of expected intrinsic squared-distance from a manifold-valued target. The computational tractable formulation expresses the non-convex optimization problem as transformations of Euclidean conditional expectation. This gives computationally tractable filtering equations for the dynamics of the intrinsic conditional expectation of a manifold-valued signal and is used to obtain accurate numerical forecasts of efficient portfolios by incorporating their geometric structure into the estimates.