riemannianGeometry.jl
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), ... |
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Geodesic equations
| Function | Description |
|---|---|
geodesic | Geodesic equations (weighted mean of two positive definite matrices) for any metric |
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Missing docstring for geodesic. Check Documenter's build log for details.
Distances
| Function | Description |
|---|---|
distanceSqr, distance² | Squared distance between positive definite matrices |
distance | Distance between positive definite matrices |
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Missing docstring for distanceSqr. Check Documenter's build log for details.
Missing docstring for distance. Check Documenter's build log for details.
Graphs and Laplacians
| 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) |
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Missing docstring for distanceSqrMat. Check Documenter's build log for details.
Missing docstring for distanceMat. Check Documenter's build log for details.
Missing docstring for laplacian. Check Documenter's build log for details.
Missing docstring for laplacianEigenMaps. Check Documenter's build log for details.
Missing docstring for spectralEmbedding. Check Documenter's build log for details.
Means
| 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 |
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Missing docstring for mean. Check Documenter's build log for details.
Missing docstring for means. Check Documenter's build log for details.
Missing docstring for generalizedMean. Check Documenter's build log for details.
Missing docstring for geometricMean. Check Documenter's build log for details.
Missing docstring for geometricpMean. Check Documenter's build log for details.
Missing docstring for logdet0Mean. Check Documenter's build log for details.
Missing docstring for wasMean. Check Documenter's build log for details.
Missing docstring for powerMean. Check Documenter's build log for details.
Missing docstring for inductiveMean. Check Documenter's build log for details.
Missing docstring for midrange. Check Documenter's build log for details.
Tangent Space operations
| 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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Missing docstring for logMap. Check Documenter's build log for details.
Missing docstring for expMap. Check Documenter's build log for details.
Missing docstring for vecP. Check Documenter's build log for details.
Missing docstring for matP. Check Documenter's build log for details.
Missing docstring for parallelTransport. Check Documenter's build log for details.
Procrustes problems
| Function | Description |
|---|---|
procrustes | Solution to the Procrustes problem in the manifold of positive definite matrices |
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Missing docstring for procrustes. Check Documenter's build log for details.