  The subspace selection problem seeks a subspace that maximizes an objective
function under some constraint. This problem includes several important machine
learning problems such as the principal component analysis and sparse
dictionary selection problem. Often, these problems can be solved by greedy
algorithms. Here, we are interested in why these problems can be solved by
greedy algorithms, and what classes of objective functions and constraints
admit this property. To answer this question, we formulate the problems as
optimization problems on lattices. Then, we introduce a new class of functions,
directional DR-submodular functions, to characterize the approximability of
problems. We see that the principal component analysis, sparse dictionary
selection problem, and these generalizations have directional
DR-submodularities. We show that, under several constraints, the directional
DR-submodular function maximization problem can be solved efficiently with
provable approximation factors.
