Sparsest Continuous Piecewise-Linear Representation of Data

We study the problem of interpolating one-dimensional data with total variation regularization on the second derivative, which is known to promote piecewise-linear solutions with few knots. In a first scenario, we consider the problem of exact interpolation. We thoroughly describe the form of the solutions of the underlying constrained optimization problem, including the sparsest piecewise-linear solutions, i.e., with the minimum number of knots. Next, we relax the exact interpolation requirement, and consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. We propose a simple and fast two-step algorithm to reach a sparsest solution of this constrained problem.