gCCA

Generalized Canonical Correlation Analysis (gCCA) is a mutiple approximate joint diagonalization prodedure generalizing the canonical correlation analysis (CCA) to the situation $m>2$ (number of datasets), as for CCA with $k=1$ (one observation). As the CCA is an MCA carried out on whitened data, so the gCCA is a gMCA carried out on whitened data.

Let ${X_1,...,X_m}$ be a set of $m$ data matrices of dimension $n⋅t$, where $n$ is the number of variables and $t$ the number of samples, both common to all datasets. From these data matrices let us estimate

$C_{ij}=\frac{1}{t}X_iX_j^H$, for all $i,j∈[1...m]$, $\hspace{1cm}$ [gcca.1]

i.e., all covariance ($i=j$) and cross-covariance ($i≠j$) matrices.

The gMCA seeks $m$ matrices $F_1,...,F_m$ diagonalizing as much as possible all products

$F_i^H C_{ij} F_j$, for all $i≠j∈[1...m]$. $\hspace{1cm}$ [gcca.2]

under costraint

$F_i^H C_{ii} F_i=I$, for all $i∈[1...m]$. $\hspace{1cm}$ [gcca.3]

permutation for gCCA

Given constraint [gcca.3], the scaling of approximate diagonalizers $F_1,...,F_m$ are fixed, however there is still a sign and permutation ambiguity (see scale and permutation). Diagonalizations.jl attempts to solve them by finding signed permutation matrices for $F_1,...,F_m$ so as to make all diagonal elements of [gcca.2] positive and sorted in descending order.

Let

$λ=[λ_1...λ_n]$$\hspace{1cm}$ [gcca.4]

be the diagonal elements of

$\frac{1}{m^2-m}\sum_{i≠j=1}^m(F_i^H C_{ij} F_j)$$\hspace{1cm}$ [gcca.5]

and $σ_{TOT}=\sum_{i=1}^nλ_i$ be the total correlation.

We denote $\widetilde{F}_i=[f_{i1} \ldots f_{ip}]$ the matrix holding the first $p<n$ column vectors of $F_i$, where $p$ is the subspace dimension. The explained variance is given by

$σ_p=\frac{\sum_{i=1}^pλ_i}{σ_{TOT}}$$\hspace{1cm}$ [gcca.6]

and the accumulated regularized eigenvalues (arev) by

$σ_j=\sum_{i=1}^j{σ_i}$, for $j=[1 \ldots n]$, $\hspace{1cm}$ [gcca.7]

where $σ_i$ is given by Eq. [gcca.6].

For setting the subspace dimension $p$ manually, set the eVar optional keyword argument of the gCCA constructors either to an integer or to a real number, this latter establishing $p$ in conjunction with argument eVarMeth using the arev vector (see subspace dimension). By default, eVar is set to 0.999.

Solution

There is no closed-form solution to the AJD problem in general. See Algorithms.

Note that solving algorithms constraining the solution to the general linear group, like NoJoB, do not suit gCCA as they do not ensure constraint [gcca.2].

Constructors

One constructor is available (see here below). The constructed LinearFilter object holding the gCCA will have fields:

.F: vector of matrices $\widetilde{F}_1,...,\widetilde{F}_m$ with columns holding the first $p$ eigenvectors in $F_1,...,F_m$, or just $F_1,...,F_m$ if $p=n$

.iF: the vector of the left-inverses of the matrices in .F

.D: the leading $p⋅p$ block of $Λ$, i.e., the elements [gcca.4] associated to the matrices in .F in diagonal form.

.eVar: the explained variance [gcca.6] for the chosen value of $p$.

.ev: the vector $λ$ [gcca.4].

.arev: the accumulated regularized eigenvalues, defined by [gcca.7]