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large scale variational Gaussian state-space models

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# eXponential FAmily Dynamical Systems (XFADS): Large-scale nonlinear Gaussian state-space modeling
Approximate inference targeted at variational Gaussian state-space models with dense covariance matrix approximations.  For more details, see our paper: [Dowling, Zhao, Park. 2024](https://arxiv.org/abs/2403.01371) \[7\]

## Introduction
A LowRankNonlinearStateSpaceModel object is used to perform inference in a state-space graphical model specified by,

``` math
\begin{matrix}p(y_{1:T}, z_{1:T}) = p_{\theta}(z_1) p(y_1 | z_1) \prod p_{\psi}(y_t | z_t) p_{\theta}(z_t | z_{t-1}) \\ where \\ p_{\theta}(z_1) = N(m_0, Q_0) \\ p_{\theta}(z_t | z_{t-1}) = N(m_{\theta}(z_{t-1}), Q)\end{matrix}
```
<br>

Specification of a LowRankNonlinearStateSpaceModel requires:

1. dynamics_mod: a nn.Module with a `mean_fn` attribute that specifies $m_{\theta}(\cdot)$ (e.g. some neural network function) 
2. initial_c_pdf: a nn.Module that specifies the initial mean, $m_0$, and initial state-noise $Q_0$
3. likelihood_pdf: a nn.Module that implements a `get_ell(y, z)` method that returns $\log p_{\psi}(y|z)$
4. local_encoder: a nn.Module whose `forward` function maps time-instantaneous observations, $y_t$, to a natural parameter update, $\alpha_t$
5. backward_encoder: a nn.Module whose `forward` function maps sequences of local natural parameter updates, $\alpha_{1:T}$, to backward natural parameter updates, $\beta_{1:T}$
6. nl_filter: a nn.Module whose forward method takes natural parameter representation of observations, $\tilde{\lambda}_{1:T}$, and performs approximate posterior inference for the specified nonlinear dynamical system

## Parameter descriptions
There are several parameters that can be configured to trade off expressivity/speed/generalization properties of the model and inference algorithm that we detail below.  Some are specific to the case of neural data modeled using a Poisson observation model with log-linear readout of the latent state.

1. n_latents: (int) total number of latent variables
2. n_latents_read: (int) total number of latent variables accessed by the observation model
3. rank_local: (int) the rank of the local precision update -- $A(y_t) A(y_t)^{\top}$
4. rank_backward: (int) the rank of the backward precision update -- $B(\alpha_{t+1:T}) B(\alpha_{t+1:T})^{\top}$
5. p_mask_a: (float) the probability of setting $\alpha_t$ to $0$
6. p_mask_b: (float) the probability of setting $\beta_t$ to $0$
7. p_mask_apb: (float) the probability of setting $\tilde{\lambda}_t = \alpha_t + \beta_t$ to $0$
8. p_mask_y_in: (float) the probability of masking data fed as input to the inference networks
9. use_cd: (bool) whether to use the coordinated dropout technique of [Keshtkaran and Pandarinath, 2019](https://arxiv.org/abs/1908.07896) \[4\] -- in which case `p_mask_y_in` specifies the coordinated dropout rate

Setting `p_mask_a` is equivalent to masking *actual* observations, $y_t$; this strategy was used in the context of structured VAE's for linear dynamical systems in [Zhao, and Linderman. 2023](https://arxiv.org/abs/2305.16543) \[5\] to promote learning dynamics more adept at prediction (and thus generating more realistic data).  Setting `p_mask_b` is equivalent to masking `pseudo` observations, $\tilde{y}_t$ -- this helps to regularize both the local/backward encoders required.

## Installation
1. Install miniconda or anaconda
This is just to leverage `conda` for managing the Python environment. You can still use the IDE or code editor of your choice.
2. Clone the repo
   ```
   git clone https://github.com/catniplab/xfads
   ```
3. Create a Conda environment, and install the dependencies
   Make sure you are in the project directory i.e., the same directory as `pyproject.toml`, and run:
   ```
   conda create -n xfads python=3.11
   conda activate xfads
   pip install -e .
   ```

## Getting started with the examples:
The set of examples in this codebase covers the primary functioning of the graphical state-space model of XFADS. The code is structured in a modular way that allows the users to change and plug in their own definitions of the classes that structure the elements of the model, i.e. the dynamics function, the likelihood density, the amortization network, etc.

## Walk-through
Now, for simplicity, and to get a grasp of the wheel, it's recommended to go through the examples of applying XFADS to some of the benchmarking datasets, before reconfiguring for yours.
- `lda_example` A simple linear dynamical system.
- `vdp_example` Adding a bit of non-linearity; training XFADS on data synthesized from the Vanderpool dynamical oscillator.
Then, some real experimental data:
- `monkey_reaching` Approximating the posterior of the latents and inferring the underlying dynamics of the `mc_maze` dataset.
- `monkey_timing` similarly, for the `dmfc_rsg` dataset.\

If you feel comfortable dabbling with XFADS by directly applying it to neural data, you can jump directly to the `monkey_reaching` example

## Example configuration
LSVS was designed with custom configurations in mind so that depending on the problem, `dynamics_mod`, `initial_c_pdf`, `likelihood_pdf`, `local_encoder`, and `backward_encoder` can be configured as desired.  We include some general classes in `ssm_modules/encoders`, `ssm_modules/likelihoods`, and `ssm_modules/dynamics` that should be sufficient for a wide range of problems.  Below is an example configuration.
```
    """likelihood pdf"""
    C = torch.nn.Linear(cfg.n_latents, n_neurons_obs, device=cfg.device)
    likelihood_pdf = PoissonLikelihood(C, n_neurons_obs, cfg.bin_sz, device=cfg.device)

"""dynamics module"""
    Q = torch.ones(cfg.n_latents, device=cfg.device)
    dynamics_fn = utils.build_gru_dynamics_function(cfg.n_latents, cfg.n_hidden_dynamics, device=cfg.device)
    dynamics_mod = DenseGaussianNonlinearDynamics(dynamics_fn, cfg.n_latents, Q, device=cfg.device)

"""initial condition"""
    Q0 = torch.ones(cfg.n_latents, device=cfg.device)
    m0 = torch.zeros(cfg.n_latents, device=cfg.device)
    initial_condition_pdf = DenseGaussianInitialCondition(cfg.n_latents, m0, Q0, device=cfg.device)

"""local/backward encoder"""
    backward_encoder = BackwardEncoderLRMvn(cfg.n_latents_read, cfg.n_hidden_backward, cfg.n_latents, cfg.rank_local,
                                            cfg.rank_backward, device=cfg.device)
    local_encoder = LocalEncoderLRMvn(cfg.n_latents, n_neurons_enc, cfg.n_hidden_local, cfg.n_latents,
                                      cfg.rank_local, device=cfg.device, dropout=cfg.p_local_dropout)
    nl_filter = NonlinearFilter(dynamics_mod, initial_condition_pdf, device=cfg.device)

"""sequence vae"""
    ssm = LowRankNonlinearStateSpaceModel(dynamics_mod, likelihood_pdf, initial_condition_pdf, backward_encoder, local_encoder, nl_filter, device=cfg.device)

"""lightning"""
    seq_vae = LightningNlbNonlinearSSM(ssm, cfg)

```

## Acknowledgements and references
The structure of the code and configuration management was heavily inspired by the excellently written `lfads-torch` package at [https://github.com/arsedler9/lfads-torch](https://github.com/arsedler9/lfads-torch) as described in [Sedler and Pandarinath, 2023](https://arxiv.org/abs/2309.01230) \[6\].

For neural latents benchmark experiments, we use reformatted versions of the mc_maze_small \[1\], mc_maze_medium \[2\], and mc_maze large \[3\] datasets.

\[1\] Churchland, Mark; Kaufman, Matthew (2022) MC_Maze_Small: macaque primary motor and dorsal premotor cortex spiking activity during delayed reaching (Version 0.220113.0408) [Data set]. DANDI archive. https://doi.org/10.48324/dandi.000140/0.220113.0408

\[2\] Churchland, Mark; Kaufman, Matthew (2022) MC_Maze_Medium: macaque primary motor and dorsal premotor cortex spiking activity during delayed reaching (Version 0.220113.0408) [Data set]. DANDI archive. https://doi.org/10.48324/dandi.000139/0.220113.0408

\[3\] Churchland, Mark; Kaufman, Matthew (2022) MC_Maze_Large: macaque primary motor and dorsal premotor cortex spiking activity during delayed reaching (Version 0.220113.0407) [Data set]. DANDI archive. https://doi.org/10.48324/dandi.000138/0.220113.0407

\[4\] Keshtkaran, Mohammad Reza, and Chethan Pandarinath. "Enabling hyperparameter optimization in sequential autoencoders for spiking neural data." Advances in neural information processing systems 32 (2019).

\[5\] Zhao, Yixiu, and Scott Linderman. "Revisiting structured variational autoencoders." International Conference on Machine Learning. PMLR, 2023.

\[6\] Sedler, Andrew R., and Chethan Pandarinath. "lfads-torch: A modular and extensible implementation of latent factor analysis via dynamical systems." arXiv preprint arXiv:2309.01230 (2023).

\[7\] Dowling, Matthew, Yuan Zhao, and Il Memming Park. "Large-scale variational Gaussian state-space models." arXiv preprint arXiv:2403.01371 (2024).

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