Non-Euclidean Conditional Expectation and Filtering

TL;DR

This paper introduces a non-Euclidean generalization of conditional expectation for manifold-valued data, providing a computationally feasible approach to filtering and numerical forecasting that respects the geometric structure.

Contribution

It develops a novel non-Euclidean conditional expectation framework and derives practical filtering equations for manifold-valued signals, enabling more accurate numerical forecasts.

Findings

Provides a tractable formulation for non-Euclidean conditional expectation.

Derives filtering equations that incorporate geometric structure.

Demonstrates improved numerical forecasts of portfolios.

Abstract

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.