Manifold-valued data: Regression and Fitting on PDFs

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The paper addresses the problem of spline interpolations on $\mathcal{P}$, the space of probability density functions when only a few observations $p_i \in \mathcal{P}$ are available. Given a finite set of $n+1$ distinct time instants $t_i$ and corresponding data points $p_i \in \mathcal{P}$, we consider the general problem of estimating a spline as a special regularized function $\gamma$ on $\mathcal{P}$ with $\gamma(t_i)=p_i$. In particular, we focus on estimating missing data using smooth temporal splines to overcome the discrete nature of observations. In addition to generalizing splines on $\mathcal{P}$ with minimal squared-norm of the acceleration, we give numerical schemes for solving $C^1$ and $C^2$ splines from data points $p_i \in \mathcal{P}$. The two solutions are then shown to be computationally efficient, geometrically simpler, extensible, and can be transposed to other spaces and applications.