Discovery of factors in matrices with grades
This work addresses factor analysis for ordinal data in domains like image analysis or product features, but it appears incremental as it builds on existing lattice structures and algorithms.
The paper tackles the problem of decomposing matrices with ordinal data into a product of two matrices with a small number of factors, using a greedy approximation algorithm based on geometric insights from formal concepts. The result includes an experimental evaluation, but no concrete numbers are provided in the abstract.
We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a product has a given feature, or a person performs well in a test. We assume that the grades form a bounded scale equipped with certain aggregation operators and conforms to the structure of a complete residuated lattice. We present a greedy approximation algorithm for the problem of decomposition of such matrix in a product of two matrices with grades under the restriction that the number of factors be small. Our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions. These factors correspond to formal concepts of the input data and allow an easy interpretation of the decomposition. We present illustrative examples and experimental evaluation.