Regression-aware decompositions

Linear least-squares regression with a “design” matrix A approximates a given matrix B via minimization of the spectral- or Frobenius-norm discrepancy ||AX-B|| over every conformingly sized matrix X. Another popular approximation is low-rank approximation via principal component analysis (PCA) – which is essentially singular value decomposition (SVD) – or interpolative decomposition (ID). Classically, PCA/SVD and ID operate solely with the matrix B being approximated, not supervised by any auxiliary matrix A. However, linear least-squares regression models can inform the ID, yielding regression-aware ID. As a bonus, this provides an interpretation as regression-aware PCA for a kind of canonical correlation analysis between A and B. The regression-aware decompositions effectively enable supervision to inform classical dimensionality reduction, which classically has been totally unsupervised. The regression-aware decompositions reveal the structure inherent in B that is relevant to regression against A.