MainModule

This is the main unit containing the PosDefManifoldML module.

dependencies

types

The following types are used.

MLmodel

This type is created (a struct in Julia) to hold a ML model. The abstract type for all machine learning models is

abstract type MLmodel end

The abstract type for all machine learning models acting on the positive definite (PD) manifold (for example, see MDM) is

abstract type PDmodel<:MLmodel end

The abstract type for all machine learning models acting on the tangent space (for example, see ENLR) is

abstract type TSmodel<:MLmodel end

Conditioner

Conditioners (also called pre-conditioners) are data transformations for the manifold of positive-definite matrices. A sequence of conditioners forms a pipeline (see Pipeline). Their role of a transforming pipeline is to improve the classification performance and/or to reduce the computational complexity.

A type is created (a struct in Julia) to specify a conditioner. The abstract type for all conditioners. is

abstract type Conditioner end # all SPD matrix transformations

These here below are the types for the conditioners currently implemented:

    abstract type Tikhonov      <: Conditioner end # Tikhonov regularization
    abstract type Recentering   <: Conditioner end # Recentering with or w/o dim reduction
    abstract type Compressing   <: Conditioner end # Compressing (global scaling)
    abstract type Equalizing    <: Conditioner end # Equalizing (individual scaling)
    abstract type Shrinking     <: Conditioner end # Geodesic Shrinking

IntVector

In all concerned functions, class labels are given as a vector of integers, of type

IntVector=Vector{Int}

Class labels must be natural numbers in $[1,...,z]$, where $z$ is the number of classes.

Tips & Tricks

working with metrics

In order to work with metrics for the manifold of positive definite matrices, make sure you install the PosDefManifold package.

the ℍVector type

Check this documentation on typecasting matrices.

notation & nomenclature

Throughout the code of this package the following notation is followed:

• scalars and vectors are denoted using lower-case letters, e.g., y,
• matrices using upper case letters, e.g., X
• sets (vectors) of matrices using bold upper-case letters, e.g., 𝐗.

The following nomenclature is used consistently:

• 𝐏Tr: a training set of positive definite matrices
• 𝐏Te: a testing set of positive definite matrices
• 𝐏, 𝐐: a generic set of positive definite matrices.
• w: a weights vector of non-negative real numbers
• yTr: a training set class labels vector of positive integer numbers (1, 2,...)
• yTe: a testing set class labels vector of positive integer numbers
• y: a generic class labels vector of positive integer numbers.
• z: number of classes of a ML model
• k: number of matrices in a set

In the examples, bold upper-case letters are replaced by upper case letters in order to allow reading in the REPL.

acronyms

• MDM: minimum distance to mean
• ENLR: Elastic-Net Logistic Regression
• SVM: Support-Vector Machine
• cv: cross-validation