## Section: Research Program

### 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 $\mathrm{\pi \x9d\x90\x97}$ be a neuroimaging dataset written as an $({n}_{subj},{n}_{voxels})$ matrix, after proper centering; the model reads

where $\mathrm{\pi \x9d\x90\x83}$ represents a set of ${n}_{comp}$ spatial maps, hence a matrix of shape $({n}_{comp},{n}_{voxels})$, and $\mathrm{\pi \x9d\x90\x80}$ the associated subject-wise loadings. While traditional PCA and independent components analysis are limited to reconstruct components $\mathrm{\pi \x9d\x90\x83}$ within the space spanned by the column of $\mathrm{\pi \x9d\x90\x97}$, it seems desirable to add some constraints on the rows of $\mathrm{\pi \x9d\x90\x83}$, 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:

${\text{min}}_{\mathrm{\pi \x9d\x90\x83},\mathrm{\pi \x9d\x90\x80}}{\beta \x88\u20af\mathrm{\pi \x9d\x90\x97}-\mathrm{\pi \x9d\x90\x80\pi \x9d\x90\x83}\beta \x88\u20af}^{2}+\mathrm{\Xi \xa8}\left(\mathrm{\pi \x9d\x90\x83}\right)\phantom{\rule{4.pt}{0ex}}\text{s.t.}\phantom{\rule{4.pt}{0ex}}\beta \x88\u20af{\mathrm{\pi \x9d\x90\x80}}_{i}\beta \x88\u20af=1\phantom{\rule{0.277778em}{0ex}}\beta \x88\x80i\beta \x88\x88\{1..{n}_{f}\},$ | (6) |

where $\left({\mathrm{\pi \x9d\x90\x80}}_{i}\right),\phantom{\rule{0.277778em}{0ex}}i\beta \x88\x88\{1..{n}_{f}\}$ represents the columns of $\mathrm{\pi \x9d\x90\x80}$. $\mathrm{\Xi \xa8}$ 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 $\mathrm{\pi \x9d\x90\x83}$ for $\mathrm{\pi \x9d\x90\x80}$ fixed, and conversely. This readily suggests an alternate optimization scheme, where $\mathrm{\pi \x9d\x90\x83}$ and $\mathrm{\pi \x9d\x90\x80}$ 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).