A \(k\)-points-based distance for robust geometric inference (Q2203630)

Let \(P\) be a Borel probability measure on \(\mathbb{R}^d\), \(x\in\mathbb{R}^d\), and let \(h\in[0,1]\) be a mass parameter. The \textit{distance} of \(x\) \textit{to the measure} \(P\), \(\mathrm{d}_{P,h}(x)\), is defined by the equation \[ \mathrm{d}_{P,h}^2(x)= P_{x,h}\|x-\cdot\|^2, \] where \(P_{x,h}\) is the probability distribution defined as the restriction of the distribution \(P\) to the ball centered at \(x\) with \(P\)-mass \(h\), and \(P_{x,h}\|x-\cdot\|^2\) denotes the expectation of the function \(\|x-\cdot\|^2\) with respect to distribution \(P_{x,h}\). The main contribution of the paper is the construction of \textit{\(k\)-power distance to measure} for a distribution \(P\) and a mass parameter \(h\): \[ \mathrm{d}_{P,h,k}(x)=\sqrt{\min_{i\in\{1,\ldots,k\}}\|x-\tau_i\|+\omega^2_{P,h}(\tau_i)}, \] where \(\tau_1,\tau_2,\ldots, \tau_k\) are specially selected elements in \(\mathbb{R}^d\), and \[ \omega^2_{P,h}(\tau)=\sup_{x\in\mathbb{R}^d}\mathrm{d}_{P,h}^2(x)-\|x-\tau\|^2 \] for an arbitrary \(\tau\in\mathbb{R}^d\). The authors of the paper prove that this \(k\)-points power distance is robust to noise and is a provably good approximation of the distance to measure. The algorithm is provided to compute \textit{\(k\)-power distance to measure}. The presented numerical experiments illustrate the good behavior of the presented \(k\)-points approximation in a noisy topological inference framework.