Sub-Cauchy Sampling: Escaping the Dark Side of the Moon was recently posted on arXiv by Sebastiano Grazzi (Warwick), Sifan Liu, Gareth O. Roberts (Warwick), and Jun Yang. With an hommage to Pink Floyd’s 1973 album both Gareth and I listened to at the time. (This was for sure my first Pink Floyd album!)
This highly original work is a sequel to the stereographic projection paper by Yang, Latuszýnski and Roberts (which was itself vaguely connected to our unpublished origami sampler). As in the stereographic projection method, the Euclidean space supporting the target is turned into a spherical cap of a hyper-sphere, referred to as the complement of the dark side of the Moon (or its bright side), and defined with respect to an observer ο who was at the north pole in the original method. The proposed MCMC algorithm, the Sub-Cauchy Projection Sampler (SCS), is a random-walk-type Metropolis algorithm on the bright side and it gets its name from being uniformly ergodic for sub-Cauchy targets. An explanation for this massive achievement is that points at infinity in the Euclidean space are now mapped to the (d − 1)-dimensional boundary of the dark side rather than at the north pole of the hypersphere. Meaning that the push-forward density may remain bounded. (The random walk on the bright side involves projections for proposals ending on the dark side, while keeping the target intact.) There are several calibration parameters to the algorithm that can be tuned by variational arguments (and the goal of getting near a uniform distribution over the bright side), since optimal acceptance rates no longer apply.
