Exact Biobjective k-Sparse Nonnegative Least Squares

The
$k$
-sparse nonnegative least squares (NNLS) problem is a variant of the standard least squares problem, where the solution is constrained to be nonnegative and to have at most
$k$
nonzero entries. Several methods exist to tackle this NP-hard problem, including fast but approximate heuristics, and exact methods based on brute-force or branch-and-bound algorithms. Although intuitive, the
$k$
-sparse constraint is sometimes limited; the parameter
$k$
can be hard to tune, especially in the case of NNLS with multiple right-hand sides (MNNLS) where the relevant
$k$
could differ between columns. In this work, we propose a novel biobjective formulation of the
$k$
-sparse nonnegative least squares problem. We present an extension of Arborescent, a branch-and-bound algorithm for exact
$k$
-sparse NNLS, that computes the whole Pareto front (that is, the set of optimal solutions for all values of
$k$
) instead of only the
$k$
-sparse solution, for virtually the same computing cost. We also present a method for MNNLS that enforces a matrix-wise sparsity constraint, by first computing the Pareto front for each column and then selecting one solution per column to build a globally optimal solution matrix. We show the advantages of the proposed approach for the unmixing of hyperspectral images.