Geometry is resolved once when a model is fitted and retained for prediction.
For SPD matrices, "intrinsic" means "affine_invariant" and
"extrinsic" means "log_euclidean". On the sphere the corresponding
names are "round" and "chordal". Euclidean aliases coincide.
Landmark geometry currently identifies configurations under the full orthogonal
group, including reflections. A capability records an available operation;
its validation status must also be considered before interpreting results.
Value
riem.geometry returns a serializable geometry specification.
riem.capabilities returns the operation registry as a data frame.
Examples
x <- wrap.spd(list(diag(2), 2 * diag(2)))
riem.geometry(x, "log_euclidean")
#> Geometry: log_euclidean on spd
#> Representation: 2 x 2 ; intrinsic dimension: 3
#> Validation: core_contract
riem.capabilities()
#> manifold_id backend geometry_id distance mean
#> 1 spd intrinsic affine_invariant TRUE TRUE
#> 2 sphere intrinsic round TRUE TRUE
#> 3 euclidean intrinsic euclidean TRUE TRUE
#> 4 landmark intrinsic shape_orthogonal TRUE TRUE
#> 5 grassmann intrinsic principal_angles TRUE TRUE
#> 6 stiefel intrinsic stiefel_intrinsic_unavailable FALSE FALSE
#> 7 rotation intrinsic rotation_frobenius TRUE TRUE
#> 8 multinomial intrinsic fisher_rao TRUE TRUE
#> 9 spdk intrinsic factor_procrustes TRUE TRUE
#> 10 correlation intrinsic correlation_quotient_unavailable FALSE FALSE
#> 11 spd extrinsic log_euclidean TRUE TRUE
#> 12 sphere extrinsic chordal TRUE TRUE
#> 13 euclidean extrinsic euclidean TRUE TRUE
#> 14 landmark extrinsic procrustes_chordal TRUE FALSE
#> 15 grassmann extrinsic projector_chordal TRUE TRUE
#> 16 stiefel extrinsic frame_chordal TRUE TRUE
#> 17 rotation extrinsic rotation_chordal TRUE TRUE
#> 18 multinomial extrinsic sqrt_chordal TRUE TRUE
#> 19 spdk extrinsic factor_extrinsic_unavailable FALSE FALSE
#> 20 correlation extrinsic correlation_extrinsic_unavailable FALSE FALSE
#> median tangent validation_status median_estimand
#> 1 TRUE TRUE core_contract frechet_median
#> 2 TRUE TRUE core_contract frechet_median
#> 3 TRUE TRUE core_contract frechet_median
#> 4 TRUE TRUE audit_pending frechet_median
#> 5 TRUE FALSE audit_pending frechet_median
#> 6 FALSE FALSE restricted_inconsistent_geometry frechet_median
#> 7 TRUE FALSE audit_pending frechet_median
#> 8 TRUE FALSE audit_pending frechet_median
#> 9 TRUE FALSE audit_pending frechet_median
#> 10 FALSE FALSE restricted_inconsistent_geometry frechet_median
#> 11 TRUE TRUE core_contract frechet_median
#> 12 TRUE FALSE core_contract projected_ambient_median
#> 13 TRUE TRUE core_contract frechet_median
#> 14 FALSE FALSE audit_pending projected_ambient_median
#> 15 TRUE FALSE audit_pending projected_ambient_median
#> 16 TRUE FALSE audit_pending projected_ambient_median
#> 17 TRUE FALSE audit_pending projected_ambient_median
#> 18 TRUE FALSE audit_pending projected_ambient_median
#> 19 FALSE FALSE restricted_inconsistent_geometry projected_ambient_median
#> 20 FALSE FALSE restricted_inconsistent_geometry projected_ambient_median