  This paper presents a new approach to selecting knots at the same time as
estimating the B-spline regression model. Such simultaneous selection of knots
and model is not trivial, but our strategy can make it possible by employing a
nonconvex regularization on the least square method that is usually applied.
More specifically, motivated by the constraint that directly designates (the
upper bound of) the number of knots to be used, we present an (unconstrained)
regularized least square reformulation, which is later shown to be equivalent
to the motivating cardinality-constrained formulation. The obtained formulation
is further modified so that we can employ a proximal gradient-type algorithm,
known as GIST, for a class of non-smooth non-convex optimization problems. We
show that under a mild technical assumption, the algorithm is shown to reach a
local minimum of the problem. Since it is shown that any local minimum of the
problem satisfies the cardinality constraint, the proposed algorithm can be
used to obtain a spline regression model that depends only on a designated
number of knots at most. Numerical experiments demonstrate how our approach
performs on synthetic and real data sets.
