Embedding statistical models into symmetric positive-definite matrix cones

Determine conditions and explicit constructions for embedding exponential families or parametric statistical models of order m into higher-dimensional symmetric positive-definite matrix cones P(m′), with m′>m, equipped with a scaled trace metric.

Background

Embedding Fisher–Rao manifolds into SPD cones can yield tractable distances and bounds, as exemplified by Calvo–Oller embeddings for elliptical families. A general embedding theory for broader classes of models could standardize lower bounds and geometric tools by leveraging well-understood SPD geometry.