Positive Curvature and Hamiltonian Monte Carlo
The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. We show that positive curvature can be used to prove theoretical concentration results for HMC Markov chains.
This paper seem to have 3 main components:1. The authors make quite a nice general setup for relating Ollivier's notion of curvature to some ideas in differential geometry.2. The authors derive a straightforward application of Ollivier/Joulin's concentration result.3. The authors do some careful computations getting very nice analytic answers for an important simple example.The first seems quite important for bringing this to a general audience; the second is trivial but nice to write down; the third seems very important for showing that their approach really works (and proving that HMC `works well' for at least one example).Since HMC is still not well understood even for simple examples, I welcome this theoretical contribution. That said, there is one technical issues which vexes me; I am not even sure if the proposed methodology is sound because of this. The issue is that the authors estimate the minimum of curvatures from simulations. I am extremely unclear on how they are taking the "empirical infimum" of curvatures to get a lower bound on curvature and then apply bounds of Ollivier and Joulin. The bounds that they cite require the actual infimum, and it isn't at all clear to me that this empirical thing makes any sense. This sort of thing can certainly be resolved, but I don't know of any trivial way to do it, and the authors seem to gloss over the entire issue as far as I can see. They don't give even any intuition on why this must work. Minor quibbles:1. Remark 2.7 and Remark 3.3: In both of these remarks, the authors discuss calculating the curvature of the whole Markov chain by running the chain for a while and taking the sample infimum. This has a whole host of obvious problems. To mention a few: A. calculating an infimum via Monte Carlo is always dangerous in the absence of regularity assumptions, B. Calculating ANY measurement of mixing by running MCMC is even more dangerous, even with regularity assumptions, since the relevant chain may simply not have mixed. Of course, one can be lucky and everything can work out, but this seems like it careless to write down such a thing without any justification. Perhaps I am missing something here...2. They say near the bottom of pp. 5 that $\kappa \approx \frac{ Inf Sec}{6}$. In Lemma 4.1, the sectional curvatures are random variables. How does one go from the distribution of a random variable to an infimum in this setting? How does this relate to the simulation at the end of the paper?3. A few typos are scattered through out the paper; these must be corrected e.g., Equality 2.2: Sec(u,v) should be Sec_{x}(u,v)