## 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 $\mathbf{X}$ be a neuroimaging dataset written as an $({n}_{subj},{n}_{voxels})$ matrix, after proper centering; the model reads

where $\mathbf{D}$ represents a set of ${n}_{comp}$ spatial maps, hence a matrix of shape $({n}_{comp},{n}_{voxels})$, and $\mathbf{A}$ the associated subject-wise loadings. While traditional PCA and independent components analysis are limited to reconstruct components $\mathbf{D}$ within the space spanned by the column of $\mathbf{X}$, it seems desirable to add some constraints on the rows of $\mathbf{D}$, 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:

${\text{min}}_{\mathbf{D},\mathbf{A}}{\parallel \mathbf{X}-\mathrm{\mathbf{A}\mathbf{D}}\parallel}^{2}+\Psi \left(\mathbf{D}\right)\phantom{\rule{4.pt}{0ex}}\text{s.t.}\phantom{\rule{4.pt}{0ex}}\parallel {\mathbf{A}}_{i}\parallel =1\phantom{\rule{0.277778em}{0ex}}\forall i\in \{1..{n}_{f}\},$ | (6) |

where $\left({\mathbf{A}}_{i}\right),\phantom{\rule{0.277778em}{0ex}}i\in \{1..{n}_{f}\}$ represents the columns of $\mathbf{A}$. $\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 $\mathbf{D}$ for $\mathbf{A}$ fixed, and conversely. This readily suggests an alternate optimization scheme, where $\mathbf{D}$ and $\mathbf{A}$ 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).