Section: Research Program

Multivariate decompositions

Multivariate decompositions provide a way 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 exhibiting similar activity 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 $(n_{subjects},n_{voxels})$ matrix, after proper centering; the model reads

where $\left({𝐀}_i\right),i\in \{1..n_{features}\}$ 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).

Ultimately, the main limitations to these algorithms is the cost due to the memory requirements: holding datasets with large dimension and large number of samples (as in recent neuroimaging cohorts) leads to inefficient computation. To solve this issue, online methods are particularly attractive [24].