I recently asked a question on Physics SE regarding the validity of using the saddle point technique when the saddle point is not only degenerate, but forms a continuous subspace of the parameter space. It has not yet been answered. and I think it worth asking here to get a more relevant answer.
I am specifically concerned with the estimation of the ground state energy for a statistical mechanical system, in which one has a form
$$U = \lim_{\beta \rightarrow \infty} \frac{\displaystyle \int d\boldsymbol{\theta}\,H(\theta)e^{-\beta H(\boldsymbol{\theta})}}{Z}$$
where $Z = \displaystyle\int d\boldsymbol{\theta}\,e^{-\beta H(\boldsymbol{\theta})}$.
Now the usual saddle point technique, would expand the Hamiltonian about a single point in the parameter space, and in the limit the lowest (constant) term in the expansion would be the only one remaining.
What would one formally do if the extremum of the Hamiltonian is in fact an extremal subspace?
I am sure there is a way fo going about this, for instance using a dimensional reduction of this subspace and then expanding only perpendicular to this subspace, but then I wouldn't know what has become of the integral.
Essentially I am interested in the formal treatment of a system with a subspace extremum, on which I wish to use the saddle point technique.
EDIT: to motivate why this is important; in some cases in physics we have a system that depends on a number of sets of variables. Say two sets, $\mathcal{D}$ and $\boldsymbol{\theta}$. Because the sets of variable vary on different time scales, the relevant partition function is
$$Z(\mathcal{D}) = \displaystyle\int d\boldsymbol{\theta}\,e^{-\beta H(\boldsymbol{\theta}, \mathcal{D})}$$
An infinitesimal change in the set of (continuous) parameters $\mathcal{D}$ may make the difference between $H(\boldsymbol{\theta}, \mathcal{D})$ having a single point in the $\theta$ parameter space a sa saddle point, or a continous subspace. If the saddle point method cannot be generalised to a continuous subspace, then it is obviously wrong to write about a problem in all generality, using the saddle point method.
A second example: in statistics when we are trying to find the maximum likelihood estimator. I have seen many approaches where the authors look to statistical physics techniques, and turn the problem of finding and MLE to minimised in the ground state energy of a statistical mechanical system. Whether the MLE exists or not depends on the data $\mathcal{D}$ (note the connection to the symbols used above). In particular, whether the data is completely or quasi separated (in which case the MLE does not exist) or not. Clearly, a continuous change in a single data point can make the difference!
In this paper on overfitting in the regularised cox model, for instance, the authors use this 'finding the ground state energy of a nt technique' method for minimising the Kullbeck Leibler divergence and thus finding an overfitting parameter. I am not sure at all thatthis is valid, if there is no solution to the steepest descent!