Recovery of Noisy Points on Band-limited Surfaces: Kernel Methods Re-explained
This work provides a theoretical foundation for discrete-continuous domain processing of signals on graphs, which is incremental but addresses a specific challenge in high-dimensional data analysis.
The paper tackles the problem of recovering points on a band-limited surface from noisy measurements in high-dimensional space by showing that exponential maps satisfy annihilation relations, enabling perfect recovery under certain sampling conditions using nuclear norm minimization and an iterative reweighted algorithm with the kernel trick.
We introduce a continuous domain framework for the recovery of points on a surface in high dimensional space, represented as the zero-level set of a bandlimited function. We show that the exponential maps of the points on the surface satisfy annihilation relations, implying that they lie in a finite dimensional subspace. The subspace properties are used to derive sampling conditions, which will guarantee the perfect recovery of the surface from finite number of points. We rely on nuclear norm minimization to exploit the low-rank structure of the maps to recover the points from noisy measurements. Since the direct estimation of the surface is computationally prohibitive in very high dimensions, we propose an iterative reweighted algorithm using the "kernel trick". The iterative algorithm reveals deep links to Laplacian based algorithms widely used in graph signal processing; the theory and the sampling conditions can serve as a basis for discrete-continuous domain processing of signals on a graph.