PosDefManifold Documentation
Requirements
Julia version ≥ 1.3
Installation
Execute the following command in Julia's REPL:
]add PosDefManifold
To obtain the latest development version execute instead
]add PosDefManifold#master
About the Author
Marco Congedo is a Research Director of CNRS (Centre National de la Recherche Scientifique), working in Grenoble, France.
Overview
Riemannian geometry studies smooth manifolds, multi-dimensional curved spaces with peculiar geometries endowed with non-Euclidean metrics. In these spaces Riemannian geometry allows the definition of angles, geodesics (shortest path between two points), distances between points, centers of mass of several points, etc.
In this package we are concerned with the manifold P of positive definite matrices, either symmetric positive definite or Hermitian positive definite.
In several fields of research such as computer vision and brain-computer interface, treating data in the P manifold has allowed the introduction of machine learning approaches with remarkable characteristics, such as simplicity of use, excellent classification accuracy, as demonstrated by the winning score obtained in six international data classification competitions, and the ability to operate transfer learning (Congedo et al., 2017)🎓).
For a formal introduction to the P manifold the reader is referred to the monography written by Bhatia (2007)🎓.
For an introduction to Riemannian geometry and an overview of mathematical tools implemented in this package, see Intro to Riemannian Geometry in this documentation.
For starting using this package, browse the code units listed here below and execute the many code examples you will find therein. The core functions are contained in unit riemannianGeometry.jl.
Code units
PosDefManifold includes six code units (.jl files):
Unit | Description |
---|---|
MainModule (PosDefManifold.jl) | Main module, constants, types, aliases, tips & tricks |
riemannianGeometry.jl | The fundamental unit collecting all functions acting on the P manifold |
linearAlgebra.jl | Collection of linear algebra routines |
statistics.jl | Collection of statistics routines |
signalProcessing.jl | Collection of signal processing routines |
test.jl | Unit performing all tests |
Contents
- PosDefManifold Documentation
- Intro to Riemannian Geometry
- MainModule (PosDefManifold.jl)
- riemannianGeometry.jl
- linearAlgebra.jl
- statistics.jl
- signalProcessing.jl
- test.jl
Index
LinearAlgebra.tr
PosDefManifold.DiagOfProd
PosDefManifold.choInv
PosDefManifold.choInv!
PosDefManifold.choL
PosDefManifold.colNorm
PosDefManifold.colProd
PosDefManifold.congruence
PosDefManifold.det1
PosDefManifold.dim
PosDefManifold.distance
PosDefManifold.distanceMat
PosDefManifold.distanceSqr
PosDefManifold.distanceSqrMat
PosDefManifold.evd
PosDefManifold.expMap
PosDefManifold.fDiag
PosDefManifold.fVec
PosDefManifold.fidelity
PosDefManifold.frf
PosDefManifold.generalizedMean
PosDefManifold.geodesic
PosDefManifold.geometricMean
PosDefManifold.geometricpMean
PosDefManifold.gram
PosDefManifold.inductiveMean
PosDefManifold.invfrf
PosDefManifold.invsqrt
PosDefManifold.isSquare
PosDefManifold.ispos
PosDefManifold.laplacian
PosDefManifold.laplacianEigenMaps
PosDefManifold.logMap
PosDefManifold.logdet0Mean
PosDefManifold.matP
PosDefManifold.means
PosDefManifold.mgs
PosDefManifold.midrange
PosDefManifold.nearestOrthogonal
PosDefManifold.nearestPosDef
PosDefManifold.normalizeCol!
PosDefManifold.parallelTransport
PosDefManifold.pow
PosDefManifold.powerIterations
PosDefManifold.powerMean
PosDefManifold.procrustes
PosDefManifold.quadraticForm
PosDefManifold.randChi²
PosDefManifold.randEigvals
PosDefManifold.randEigvalsMat
PosDefManifold.randPosDefMat
PosDefManifold.randUnitaryMat
PosDefManifold.regularize!
PosDefManifold.remove
PosDefManifold.softmax
PosDefManifold.spectralEmbedding
PosDefManifold.spectralFunctions
PosDefManifold.sqr
PosDefManifold.sumOfSqr
PosDefManifold.sumOfSqrDiag
PosDefManifold.sumOfSqrTril
PosDefManifold.tr1
PosDefManifold.trade
PosDefManifold.typeofMatrix
PosDefManifold.typeofVector
PosDefManifold.vecP
PosDefManifold.wasMean
Statistics.mean
Statistics.std