Order-preserving factor analysis (OPFA)

With the advent of high-throughput data collection techniques, low-dimensional matrix factorizations have become an essential tool for pre-processing, interpreting or compressing high-dimensional data. They are widely used in a variety of signal processing domains including electrocardiogram [1], image [2], or sound [3] processing. These methods can take advantage of a large range of a priori knowledge on the form of the factors, enforcing it through constraints on sparsity or patterns in the factors. However, these methods do not work well when there are unknown misalignments between subjects in the population, e.g., unknown subject-specific time shifts. In such cases, one cannot apply standard patterning constraints without first aligning the data; a difficult task. An alternative approach, explored in this paper, is to impose a factorization constraint that is invariant to factor misalignments but preserves the relative ordering of the factors over the population. This order-preserving factor analysis is accomplished using a penalized least squares formulation using shift-invariant yet order-preserving model selection (group lasso) penalties on the factorization. As a byproduct the factorization produces estimates of the factor ordering and the order-preserving time shifts. In traditional matrix factorization, the data is modeled as a linear combination of a number of factors.