Multi-dimensional Sparse Super-resolution
For researchers in imaging and signal processing, this work fills a gap in the theoretical understanding of multi-dimensional super-resolution, though it is incremental as it builds on existing BLASSO theory.
This paper extends the theoretical analysis of sparse super-resolution via the BLASSO method to arbitrary dimensions, focusing on support recovery. It establishes a connection between dual solutions and Hermite polynomial interpolation, and provides a sharp analysis of how signal-to-noise ratio must scale with spike separation for a pair of spikes in 2-D.
This paper studies sparse super-resolution in arbitrary dimensions. More precisely, it develops a theoretical analysis of support recovery for the so-called BLASSO method, which is an off-the-grid generalisation of l1 regularization (also known as the LASSO). While super-resolution is of paramount importance in overcoming the limitations of many imaging devices, its theoretical analysis is still lacking beyond the 1-dimensional (1-D) case. The reason is that in the 2-dimensional (2-D) case and beyond, the relative position of the spikes enters the picture, and different geometrical configurations lead to different stability properties. Our first main contribution is a connection, in the limit where the spikes cluster at a given point, between solutions of the dual of the BLASSO problem and Hermite polynomial interpolation ideals. Polynomial bases for these ideals, introduced by De Boor, can be computed by Gaussian elimination, and lead to an algorithmic description of limiting solutions to the dual problem. With this construction at hand, our second main contribution is a detailed analysis of the support stability and super-resolution effect in the case of a pair of spikes. This includes in particular a sharp analysis of how the signal-to-noise ratio should scale with respect to the separation distance between the spikes. Lastly, numerical simulations on different classes of kernels show the applicability of this theory and highlight the richness of super-resolution in 2-D.