Doubt regarding the Hamiltonian calculations for RHVAE model.

[shikhar2333]

In the paper, Hamiltonian is defined as follows:

H(z, v) = U(z) + K(v)  = -0.5*log(det(G^-1(z)) + 0.5*v^T*v

But in the code, I see extra terms like addition of a joint probability term and a G inverse multiplied in the term for kinetic energy.
Are these 2 equations equivalent?

[clementchadebec]

Hi @shikhar2333 ,

Thank you for opening this issue. Actually, the Hamiltonian you are referring to is the one used to sample from the target distribution given by the inverse of the metric volume element post training and so it is used in the RHVAESampler instead. That is why only 0.5*log(det(G^-1(z)) appears in U(z). However, during training, the authors proposed to target another distribution which is the true posterior p(z|x) using Riemannian HMC and that is why you see those extra terms. For a derivation of the Hamiltonian used during training you can have a look to Eq. 9 page 7 in (https://arxiv.org/abs/2010.11518).

I hope this helps.

[shikhar2333]

Thanks for your response.
The link that you shared doesn't seem to work for me. Can you tell the paper title?

[clementchadebec]

Oops, sorry for that, this one should work https://arxiv.org/abs/2010.11518

[shikhar2333]

How do we ensure that \rho gets sampled from N(0, G(z))?

rho = (L @ rho.unsqueeze(-1)).squeeze(-1)

In the code I saw this particular initialization for rho. Is this done because L is the cholesky decomposition for the matrix G?

[clementchadebec]

We use the Cholesky decomposition of the metric G = LL^T allowing to sample from multivariate Gaussian distributions. Note that G(z) is a symmetric definite positive matrix

[shikhar2333]

So are M and G equivalent, since M is also defined as LL^T.

[shikhar2333]

Is E[P(x|z)] (where z~q(z|x) and E denotes the expectation value) approximated by MSE/BCE loss in the implementation?
Since maximizing log likelihood of the expectation reduces to minimizing reconstruction error L(recon_x,x)?

[clementchadebec]

Yes exactly. The MSE (resp. BCE loss) actually represents the opposite log-likelihood of a constant variance Gaussian distribution (resp. Bernoulli distribution) that are usually chosen for p(x|z) depending on the modeling of your data.

[shikhar2333]

Thanks for the confirmation.
On a side note, i was wondering whether any conditional input was required in the VAE when you trained on the MRI data, since the dimensionality of the data would be high because it's a 3D input.
Something similar to a conditional VAE.

[clementchadebec]

You're welcome :). Not really, you can down-sample a bit your input data to deal with memory issues for instance but, in theory, the VAE can handle any kind of data regardless of the dimension. If you are referring to the paper, we only down-sampled the data by a factor 2 in each dimension when dealing with 3D MRIs.

[shikhar2333]

What's the intuition for updating the metric at the end of each epoch?

[clementchadebec]

The updated metric is needed during validation.