Riemannian Gradient Flow |5 Nov. 2019|Previously we looked at , - and its resulting flow - as the result from mapping to the sphere. However, we saw that for this flow convergences to , with such that for we have . Hence, it is interesting to look at the flow from an optimization point of view: .
A fair question would be, is our flow not simply Riemannian gradient ascent for this problem? As common with these kind of problems, such a hypothesis is something you just feel. Now, using the tools from (CU1994, p.311) we can compute the gradient of (on the sphere) via , where is a vector field normal to , e.g., . From there we obtain To make our life a bit easier, use instead of the map . Moreover, set . Then it follows that Of course, to make the computation cleaner, we changed to , but the relation between and is beautiful. Somehow, mapping trajectories of , for some , to the sphere corresponds to (Riemannian) gradient ascent applied to the problem .
Now, we formalize the previous analysis a bit a show how fast we convergence. Assume that the eigenvectors are ordered such that eigenvector corresponds to the largest eigenvalue of . Then, the solution to is given by Let be expressed in eigenvector coordinates, with all (normalized). Moreover, assume all eigenvalues are distinct. Then, to measure if is near , we compute , which is if and only if is parallel to . To simplify the analysis a bit, we look at , for some perturbation , this yields Next, take the the (natural) logarithm on both sides: This log-sum-exp terms are hard to deal with, but we can apply the so-called ‘‘log-sum-exp trick’’: In our case, we set and obtain We clearly observe that for the LHS approaches from below, which means that from above, like intended. Of course, we also observe that the mentioned method is not completely general, we already assume distinct eigenvalues, but there is more. We do also not convergence when , which is however a set of measure on the sphere . More interestingly, we see that the convergence rate is largely dictated by the ‘‘spectral gap/eigengap’’ . Specifically, to have a particular projection error , such that , we need Comparing this to the resulting flow from , , we see that we have the same flow, but with . This is interesting, since and have the same eigenvectors, yet a different (scaled) spectrum. With respect to the convergence rate, we have to compare and for any with (the additional is not so interesting). It is obvious what we will happen, the crux is, is larger or smaller than ? Can we immediately extend this to a Newton-type algorithm? Well, this fails (globally) since we work in instead of purely with . To be concrete, , we never have degrees of freedom. Of course, these observations turned out to be far from new, see for example (AMS2008, sec. 4.6). (AMS2008) P.A. Absil, R. Mahony and R. Sepulchre: ‘‘Optimization Algorithms on Matrix Manifolds’’, 2008 Princeton University Press. |