PCA is basically a decomposition of data into a low rank Euclidean space (where the principal axis accounts for the bulk of the variation in the data and each s

PCA is basically a decomposition of data into a low rank Euclidean space (where the principal axis accounts for the bulk of the variation in the data and each successive axis is orthogonal and accounts for the remaining variation). Notably though, most problems do not lie in a Euclidean vector space — they often exist on a lower dimensional manifold embedded into such a space. More recent work considers the problem of performing low rank decompositions if you assume constraints on the factors, i.e., general priors. The work on this is quite recent and may be of interest to someone. You can essentially perform PCA with constraints using a message passing algorithm (originally derived from physics applications): https://arxiv.org/abs/1701.00858 https://arxiv.org/abs/1701.00858