|
|
|
| e-Pub |
Previous | 
Home
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 be a neuroimaging dataset written as an matrix, after proper centering; the model reads
where represents a set of spatial maps, hence a matrix of shape , and the associated subject-wise loadings. While traditional PCA and independent components analysis are limited to reconstruct components within the space spanned by the column of , it seems desirable to add some constraints on the rows of , 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 represents the columns of . 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 for fixed, and conversely. This readily suggests an alternate optimization scheme, where and 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).