Transformation-invariant representation and NMF

Non-negative matrix factorization (NMF) is a method for the decomposition of multivariate data into strictly positive activations and basis vectors. Here, instead of using unstructured data vectors, we assume that something is known in advance about the type of transformations that either the input data or the basis vectors may undergo. This would be the case e.g. if we assume input vectors that are translationally shifted versions of each other, but it applies to any other transformations as well. The key idea is that we factorize the data into activations and basis vectors modulo the transformations. We show that this can be done by extending NMF in a natural way. The gained factorization thus provides a transformation-invariant and compact encoding that is optimal for the given transformation constraints.