Sparsest piecewise-linear regression of one-dimensional data

We study the problem of one-dimensional regression data points with total-variation (TV) regularization (in sense measures) on second derivative, which is known to promote piecewise-linear solutions few knots. While there are efficient algorithms for determining such adaptive splines, difficulty TV that solution generally non-unique, an aspect often ignored in practice. In this paper, we present a systematic analysis results complete description set clear distinction between cases where unique and those, much more frequent, it not. For latter scenario, identify sparsest solutions, i.e., those minimum number knots, derive formula compute knots based solely points. To achieve this, first consider exact interpolation leads easier theoretical analysis. Next, relax requirement setting, penalized optimization strictly convex data-fidelity cost function. show underlying can be reformulated as constrained problem, thus all our previous still apply. Based analysis, propose simple fast two-step algorithm, agnostic uniqueness, reach problem.