Cramer-Rao bound and Ricci flow II


The paper I presented about this matter (see here) has been accepted by EuRad 2009 Conference. This will result in a publication in IEEE Proceedings. IEEE is the most important engineering society. I cannot made public this paper until it will appear in the proceedings. After this date you can read it at IEEE Xplore where you can find another paper of mine about scattering of electromagnetic waves by a rough surface (see here). As you can see, the way publishing is operated by engineers is quite different from that of physicists.

Anyhow, the idea is quite simple and use the fact that for a two-dimensional Riemann manifold one has always a conformal metric. Then, Fischer information matrix can be expressed in a diagonal form with new estimators that are always optimal with respect to Cramer-Rao bound. So, due to the fact that there exists a vast set of probability distributions with two parameters, the application areas of this result are huge. In my paper I make the case of sea clutter for radar applications but what I prove is a theorem in statistics and you can realize by yourself the importance.

The n-parameter case can also be made but here there are two more demanding requests: the existence of a conformal metric and the existence of a potential for a vector field that satisfies Liouville equation. These cannot always be satisfied and so the two-dimensional case appears a rather lucky one.

Liouville theory in the infrared limit


Today I want to report a quite interesting result that I have discussed in the comments of a preceding post of mine (see here):  2d general relativity has no confinement as a quantum field theory. 2d general relativity can be written down as


being \Lambda a cosmological constant. This equation is the same as the Liouville equation

\partial_t^2\phi-\partial_x^2\phi+\Lambda e^{b\phi}=0

and all the problem is to find the scalar function \phi. As you know this equation can be solved exactly. About quantum field theory for 2d gravity there is really a large body of literature due the importance of this equation. I just point out to you this paper but there is much more about.

So, if you want to study this equation in the infrared limit, you have just to take the cosmological constant going to infinity. Then, to solve this problem we have to use strong perturbation theory (or a gradient expansion) giving at the leading order the equation for the propagator

\partial_t^2G+\Lambda e^{bG}=\delta(t)

and this equation can be solved exactly:


being \epsilon and \phi two arbitrary constants that may depend on the spatial coordinate. This Green function solves for the propagator after we have rescaled time by\sqrt{\epsilon}\tanh\phi and the \Lambda constant as \Lambda/\epsilon\tanh^2\phi. What can we learn from it? We see that this is not a periodic function and so it cannot be expressed through a Fourier series. This implies that the quantum spectrum is not discrete and so the theory has no bound states in the infrared limit of an increasingly large cosmological constant. This is a substantial difference with respect to a quartic scalar field theory that has a discrete spectrum in the same limit producing confinement.

As shocking as this result may seem, it can be straightforwardly extended to general relativity. We know that the solution, in the gradient expansion of the Einstein equations, is the Kasner solution that is not periodic at all. The situation is made more complicate by BKL scenario. In this case we have a sequence of oscillatory epochs making an overall chaotic scenario. So, we cannot find a class of periodic solutions to build an infrared quantum field theory that in this way seems to have no bound state again in a regime of strong nonlinearities (strong gravitational fields). I should say that a more detailed analysis would be helpful here opening the possibility to have an infrared formulation of QFT for Einstein equations.

Cramer-Rao bound and Ricci flow


Two dimensional Ricci flow is really easy to manage. In this case the equation takes a very simple form and a wealth of results can be extracted. As you know from my preceding posts, I have been able to prove in a rigorous way that in this case the Ricci flow arises from Brownian motion (see here). So, the equation for  Einstein manifolds in this case takes the very simple form, R=\Lambda being \Lambda a constant, that is also the equation for a Ricci soliton. This equation is rather well-knwon to physicists as is the equation of 2d Einstein gravity. This equation is nothing else than Liouville equation

\Delta_2\phi+\Lambda e^{\phi}=0

that admits an exact solution notwithstanding being non-linear. There is an unexpected application of all this machinery of Riemann geometry to the case of statistics. Statistics has a wide body of application fields as radar tracking, digital communications and so on. Then, any new result about can be translated into a wealthy number of applications.

The problem one meets in this case is that of parameter estimation of a given probability distribution. For a sample of measured data the question is to determine the best probability distribution with respect to the spread of the data themselves with a proper choice of the parameters. A known result in this area is the so called Cramer-Rao bound. This inequality gives limit for the optimality of the chosen estimators of the data entering into the distribution. The result I have found is that, for a probability distribution with two parameters, an infinite class of optimal estimators exists that are all efficient. These estimators are given by the solution of Liouville equation! The result can be extended to the n-dimensional case granted the existence of isothermal coordinates that are the conformal ones.

This result arises from the deep link between differential geometry and statistics that was put forward by Calayampudi Radhakrishna Rao. My personal interest in this matter was arisen working in radar tracking but one can think on a large number of other areas. I should say, as a final consideration, that the work of Hamilton and Perelman can have a deep impact in a large body of our knowledge. We are just at the beginning.

Ricci solitons in two dimensions


Today I have read recent changes to DispersiveWiki. This is a beautiful site about differential equations that is maintained at University of Toronto by Jim Colliander and has notable contributors as the Fields medallist Terence Tao. Terry introduced a new page about Liouville’s equation as he got involved with it in a way you can read here. Physicists working on quantum gravity has been aware of this equation since eighties as it is the equation of two-dimensional quantum gravity and comes out quite naturally in string theory. A beautiful paper about quantum field theory of Liouville equation is due to Roman Jackiw and one of his collaborators Eric D’Hoker (see here). But what people could have overlooked is that Liouville’s equation is the equation of the Ricci soliton in two dimensions. The reason is that in this case a set of isothermal coordinates can always be found and the metric is always conformal, that is


being g_0 the Euclidean metric. The Ricci tensor takes here a quite simple form


being \epsilon_{ik}=diag(1,1) . Then the Ricci flow is

\frac{\partial\phi}{\partial t}=e^{-2\phi}(\partial^2_x+\partial^2_y)\phi

and finally for the Ricci soliton one has

(\partial^2_x+\partial^2_y)\phi = H e^{2\phi}

being H a constant. After a simple rescaling we are left with the Euclidean Liouville’s equation

(\partial^2_x+\partial^2_y)u = \Lambda e^{u}.

Turning back to the Jackiw and D’Hoker paper, we can see that a 2D gravity theory emerges naturally as the equilibrium (Ricci soliton) solution of a Fokker-Planck (Ricci flow) equation. This scenario seems a beautiful starting point to build an understanding of quantum gravity. I am still thinking about a lot and I will put all this on a paper one day.

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