[FR] low rank HMC?

[SteveBronder]

Summary:

It would be nice to have versions of HMC/NUTS that support a low-rank plus diagonal metric a la L-BFGS.

This will require the following.

1. low_rank_e_metric and low_rank_e_point like dense_e_metric.hpp and dense_e_point.hpp which would use the code from pathfinder for getting the inverse hessian from here

2. low_rank_e_nuts and adapt_low_rank_e_nuts class like diag_e_nuts.hpp and adapt_diag_e_nuts.hpp

There's more information about this in @bbbales2 thesis, and refined in a joint arXiv paper with @pourzanj and @avehtari:

• Selecting the Metric in Hamiltonian Monte Carlo.

Current Version:

v2.29.0

[betanalpha]

“Low rank” isn’t going to be precise enough for a single signature. In addition to all of the possible ranks between 1 and N there are multiple low-rank patterns that one might consider. Moreover each of these patterns would require an appropriate adaptation routine to match to the global covariance.

On Feb 17, 2022, at 12:12 PM, Steve Bronder @.***> wrote:

Summary:

Making an issue for a thing brought up in the Stan meeting. It would be nice to have a low rank method for NUTS. From the meeting we sorted out this involves adding these parts so far and will update it once we discuss it more.

low_rank_e_metric and low_rank_e_point like dense_e_metric.hpp https://github.com/stan-dev/stan/blob/develop/src/stan/mcmc/hmc/hamiltonians/dense_e_metric.hpp and dense_e_point.hpp https://github.com/stan-dev/stan/blob/develop/src/stan/mcmc/hmc/hamiltonians/dense_e_point.hpp which would use the code from pathfinder for getting the inverse hessian from here https://github.com/stan-dev/stan/blob/feature/multi-path/src/stan/services/pathfinder/single.hpp#L444 low_rank_e_nuts and adapt_low_rank_e_nuts class like diag_e_nuts.hpp https://github.com/stan-dev/stan/blob/develop/src/stan/mcmc/hmc/nuts/diag_e_nuts.hpp and adapt_diag_e_nuts.hpp https://github.com/stan-dev/stan/blob/develop/src/stan/mcmc/hmc/nuts/adapt_diag_e_nuts.hpp Description:

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Current Output:

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Current Version:

v2.29.0

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[bob-carpenter]

I believe Steve's suggesting the specific factor-based representation of a symmetric positive definite matrix as

Sigma = u' * v + diag(w)

where u and v are K x N matrices and w is an N-vector.

I agree with @betanalpha that this is a hornet's nest of design if we want to make it more general. I can think of at least two other basic structures that might be worth considering,

• block-diagonal, and
• HODLR (hier off-diag low rank).

And then we can add extra irregular sparse structure on top of this by addition. Perhaps transformed from a sparse symmetric representation using matrix exp.