Section: Application Domains

Multivariate decompositions

Multivariate decompositions are an important tool to model complex data such as brain activation images: for instance, one might be interested in extracting an atlas of brain regions from a given dataset, such as regions depicting similar activities during a protocol, across multiple protocols, or even in the absence of protocol (during resting-state). These data can often be factorized into spatial-temporal components, and thus can be estimated through regularized Principal Components Analysis (PCA) algorithms, which share some common steps with regularized regression.

Let $X$ be a neuroimaging dataset written as an $(n_{subj},n_{voxels})$ matrix, after proper centering; the model reads

where $D$ represents a set of $n_{comp}$ spatial maps, hence a matrix of shape $(n_{comp},n_{voxels})$, and $A$ the associated subject-wise loadings. While traditional PCA and independent components analysis are limited to reconstruct components $D$ within the space spanned by the column of $X$, it seems desirable to add some constraints on the rows of $D$, that represent spatial maps, such as sparsity, and/or smoothness, as it makes the interpretation of these maps clearer in the context of neuroimaging.

This yields the following estimation problem:

where $\left(A_i\right),i\in \{1..n_f\}$ represents the columns of $A$. $\Psi$ can be chosen such as in Eq. (2 ) in order to enforce smoothness and/or sparsity constraints.

The problem is not jointly convex in all the variables but each penalization given in Eq (2 ) yields a convex problem on $D$ for $A$ fixed, and conversely. This readily suggests an alternate optimization scheme, where $D$ and $A$ are estimated in turn, until convergence to a local optimum of the criterion. As in PCA, the extracted components can be ranked according to the amount of fitted variance. Importantly, also, estimated PCA models can be interpreted as a probabilistic model of the data, assuming a high-dimensional Gaussian distribution (probabilistic PCA).