This is the fundamental unit of PosDefManifold. It contains functions
for manipulating points in the Riemannian manifold of
Symmetric Positive Definite (SPD) or Hermitian Positive Definite (HPD) matrices. In Julia those are Hermitian matrices, see typecasting matrices.
The functions are divided in six categories:
| Category | Output |
|---|---|
| 1. Geodesic equations | interpolation, extrapolation, weighted mean of two matrices, ... |
| 2. Distances | length of geodesics |
| 3. Graphs and Laplacians | inter-distance matrices, spectral embedding, eigenmaps, ... |
| 4. Means | mid-points of geodesics, Fréchet means of several points, midrange,... |
| 5. Tangent Space operations | maps from the manifold to the tangent space and viceversa, parallel transport,... |
| 6. Procrustes problems | data matching, transfer learning (domain adaptation), ... |
⋅
| Function | Description |
|---|---|
geodesic |
Geodesic equations (weighted mean of two positive definite matrices) for any metric |
⋅
geodesic
| Function | Description |
|---|---|
distanceSqr, distance² |
Squared distance between positive definite matrices |
distance |
Distance between positive definite matrices |
⋅
distanceSqr
distance
| Function | Description |
|---|---|
distanceSqrMat, distance²Mat |
Lower triangular matrix of all squared inter-distances |
distanceMat |
Lower triangular matrix of all inter-distances |
laplacian |
Laplacian of a squared inter-distances matrix |
laplacianEigenMaps, laplacianEM |
Eigen maps (eigenvectors) of a Laplacian |
spectralEmbedding, spEmb |
Spectral Embedding (the above functions run in series) |
⋅
distanceSqrMat
distanceMat
laplacian
laplacianEigenMaps
spectralEmbedding
| Function | Description |
|---|---|
mean |
Weighted Fréchet mean (wFm) of a scalar or matrix set using any metric |
means |
As above for several sets at once |
generalizedMean |
Generalized wFm of a matrix set |
geometricMean, gMean |
wFm of a matrix set minimizing the dispersion according to the Fisher metric (iterative) |
geometricpMean, gpMean |
robust wFm of a matrix set minimizing the p-dispersion according to the Fisher metric (iterative) |
logdet0Mean, ld0Mean |
wFm of a matrix set according to the logdet0 metric (iterative) |
wasMean |
wFm of a matrix set according to the Wasserstein metric (iterative) |
powerMean |
Power wFm of a matrix set (iterative) |
inductiveMean, indMean |
Recursive Fréchet mean of a matrix set (constructive) |
midrange |
Geometric midrange of two matrices |
⋅
mean
means
generalizedMean
geometricMean
geometricpMean
logdet0Mean
wasMean
powerMean
inductiveMean
midrange
| Function | Description |
|---|---|
logMap |
Logarithmic map (from manifold to tangent space) |
expMap |
Exponential map (from tangent space to manifold) |
vecP |
vectorization of matrices in the tangent space |
matP |
matrization of matrices in the tangent space (inverse of vecp) |
parallelTransport, pt |
Parallel transport of tangent vectors and matrices |
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logMap
expMap
vecP
matP
parallelTransport
| Function | Description |
|---|---|
procrustes |
Solution to the Procrustes problem in the manifold of positive definite matrices |
⋅
procrustes