In statistics/machine learning the individual samples often come in the form of 2-D arrays, eg, a set of population counts of different species (one axis) at different points in time (second axis). Standard regression collapses these arrays into vectors and thus loses the structure in the regression process. Bilinear regression attempts to use the array structure by using the samples as matrices.

Details

Basic model

The bilinear predictor function takes the form

(1)$f(X) = tr(U^T X V) + b = \sum_{i=1:m} u^T_i X v_i + b$

Noe that, as is particularly apparent in the $(u_i,v_i)$ form, there is a freedom to move a multiplicative factor between the $u_i$ columns and the matching $v_i$ forms.

When performing regularised fitting the score function is