Am I understanding the relevance of the curse of dimensionality to this correctly: Generally, our goal is to find a simple pattern in some high-dimensional data. However, due to the high dimensionality there are exponentially many possible data points and, practically, we can only observe a very small fraction of that, so curse is that we are often left with an immense list of candidates for the true pattern. All we can do is to limit this list of candidates with certain heuristic priors, for example that the true pattern is a smooth, compact manifold (that worked well e.g. for relativity and machine learning, but for example quantum mechanics looks more like that the true pattern is not smooth but consists of individual particles).
The true pattern (i.e. the many-particle wavefunction) is smooth. The issue is that the pattern depends on the positions of every electron in the atom. The variational principle gives us a measure of the goodness of the wavefunction, but it doesn't give us a way to find consistent sets of positions. We have to rely on numerical methods to find self-consistent solutions for the set of differential equations, but it's ludicrously expensive to try to sample the solution space given the dimensionality of that space.
It's really difficult to solve large systems ...
This thread is intended to provide a space for 'crazy' ideas. Ideas that spontaneously come to mind (and feel great), ideas you long wanted to tell but never found the place and time for and also for ideas you think should be obvious and simple - but nobody ever mentions them.
This thread itself is such an idea. Or rather the tangent of such an idea which I post below as a seed for this thread.
Rules for this thread:
If this should become a regular thread I suggest the following :