RIM: Recurrent Inference Machines

This is an implementation of a Recurrent Inference Machine (see Putzky & Welling (2017)) alongside some standard neural network architectures for the type of problem RIM can solve.

Installation

To install the package, you can use pip:

pip install torch-rim

Usage

from rim import RIM, Hourglass, Unet
from torch.func import vmap

# B is the batch size
# C is the input channels
# dimensions are the spatial dimensions (e.g. [28, 28] for MNIST)

# Create a score_fn, e.g. a Gaussian likelihood score function
@vmap
def score_fn(x, y, A, Sigma): # must respect the signature (x, y, *args)
    # A is a linear forward model, Sigma is the noise covariance
    return (y - A @ x) @ Sigma.inverse() @ A

# ... or a Gaussian energy function (unnormalized log probability)
@vmap
def energy_fn(x, y, F, Sigma):
    # F is a general forward model
    return (y - F(x)) @ Sigma.inverse() @ (y - F(x))

# Create a RIM instance with the Hourglass neural network back-bone and the score function
net = Hourglass(C, dimensions=len(dimensions))
rim = RIM(dimensions, net, score_fn=score_fn)

# ... or with the energy function
rim = RIM(dimensions, net, energy_fn=energy_fn)

# Train the rim, and save its weight in checkpoints_directory
rim.fit(dataset, epochs=100, learning_rate=1e-4, checkpoints_directory=checkpoints_directory)

# Make a prediction on an observation y
x_hat = rim.predict(y, A, Sigma) # of with the signature (y, F, Sigma) with the energy_fn

Background

A RIM is a gradient-based meta-learning algorithm. It is trained not as a feed-forward neural network, but rather as an optimisation algorithm. More specifically, the RIM is given a problem instance specified by a likelihood score $\nabla_\mathbf{x} \log p(y \mid x)$, or more generally a posterior score function $\nabla_{\mathbf{x} \mid \mathbf{y}} \equiv \nabla_{\mathbf{x}} \log p(x \mid y)$, and an observation $y$ to condition said posterior.

The RIM uses this information to perform a learned gradient ascent algorithm on the posterior. This procedure will produce a MAP estimate of the parameters of interests $\mathbf{x}$ when the RIM is trained

$$\begin{align} \hat{\mathbf{x}}_{t+1} &= \hat{\mathbf{x}}_t + \mathbf{g}_\theta (\hat{\mathbf{x}}_t,\, \mathbf{y},\, \nabla_{\hat{\mathbf{x}}_t \mid \mathbf{y}},\, \mathbf{h}_t)\\\ \mathbf{h}_{t+1} &= \mathbf{g}_\theta(\hat{\mathbf{x}}_t,\, \mathbf{y},\, \nabla_{\hat{\mathbf{x}}_t \mid \mathbf{y}},\, \mathbf{h}_t) \end{align}$$

for $t \in [0, T]$. In the last equation, $\mathbf{g}_\theta$ is a neural network that act as the gradient in the gradient ascent algorithm. The second equation represent an hidden state much like modern optimisation algorithm like ADAM (Kingma & Ba (2014)) or RMSProp (Hinton (2011)) that uses hidden state to aggregate information about the trajectory of the particle during optimisation. In this case, $\mathbf{h}$ is the hidden state of a Gated Recurrent Unit (Chung et al. (2014)).

The RIM is trained using an outer loop optimisation contructed with labels $\mathbf{x}$ (the parameters of interests) and a simulator $F$:

$$\mathbf{y} = F(\mathbf{x}) + \boldsymbol{\eta}\, .$$

Equipped with a dataset of problem instances and their solutions $\mathbf{x}$

$$\mathcal{D} = \big\{\mathbf{y}^{(i)},\, \mathbf{x}^{(i)},\, \nabla_{\mathbf{x} \mid \mathbf{y}}^{(i)}\big\}_{i=1}^N\, ,$$

we can train the RIM to make it's gradient ascent trajectories as efficient as possible by minimizing a weighted mean squared loss

$$\mathcal{L}_\theta = \mathbb{E}_\mathcal{D} \left[ \sum_{t=1}^T\lVert \mathbf{w}_t(\hat{\mathbf{x}}_t - \mathbf{x})\rVert^2_2 \right]\, .$$

$\mathbf{w}_t$ weighs each MSE on the parameter trajectory.

Citations

If you find this work useful, please consider citing

Putzky & Welling (2017)

@article{Putzky2017,
    author = {Putzky, Patrick and Welling, Max},
    journal = {arXiv e-prints},
    archivePrefix = {arXiv},
    eprint = {0902.0885},
    primaryClass = {cs.NE},
    title = {{Recurrent Inference Machines for Solving Inverse Problems}},
    url = {http://arxiv.org/abs/1706.04008},
    year = {2017}
}

You might also find some of the following applications of the RIM useful

Adam et al. (2023) (Gravitational lensing)

@article{Adam2023,
       author = {{Adam}, Alexandre and {Perreault-Levasseur}, Laurence and {Hezaveh}, Yashar and {Welling}, Max},
        title = "{Pixelated Reconstruction of Foreground Density and Background Surface Brightness in Gravitational Lensing Systems Using Recurrent Inference Machines}",
     keywords = {Convolutional neural networks, Astronomical simulations, Nonparametric inference},
      journal = {\apj},
         year = 2023,
        month = jul,
       volume = {951},
       number = {1},
          eid = {6},
        pages = {6},
          doi = {10.3847/1538-4357/accf84},
archivePrefix = {arXiv},
       eprint = {2301.04168},
 primaryClass = {astro-ph.IM},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2023ApJ...951....6A},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

Morningstar et al. (2019) (Gravitational lensing)

@article{Morningstar2019,
       author = {{Morningstar}, Warren R. and {Perreault Levasseur}, Laurence and {Hezaveh}, Yashar D. and {Blandford}, Roger and {Marshall}, Phil and {Putzky}, Patrick and {Rueter}, Thomas D. and {Wechsler}, Risa and {Welling}, Max},
        title = "{Data-driven Reconstruction of Gravitationally Lensed Galaxies Using Recurrent Inference Machines}",
      journal = {\apj},
     keywords = {Convolutional neural networks, Strong gravitational lensing, Neural networks, Astrophysics - Instrumentation and Methods for Astrophysics, Astrophysics - Cosmology and Nongalactic Astrophysics, Astrophysics - Astrophysics of Galaxies},
         year = 2019,
        month = sep,
       volume = {883},
       number = {1},
          eid = {14},
        pages = {14},
          doi = {10.3847/1538-4357/ab35d7},
archivePrefix = {arXiv},
       eprint = {1901.01359},
 primaryClass = {astro-ph.IM},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2019ApJ...883...14M},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

Morningstar et al. (2018) (Interferometric imaging)

@ARTICLE{Morningstar2018,
       author = {{Morningstar}, Warren R. and {Hezaveh}, Yashar D. and {Perreault Levasseur}, Laurence and {Blandford}, Roger D. and {Marshall}, Philip J. and {Putzky}, Patrick and {Wechsler}, Risa H.},
        title = "{Analyzing interferometric observations of strong gravitational lenses with recurrent and convolutional neural networks}",
      journal = {arXiv e-prints},
     keywords = {Astrophysics - Instrumentation and Methods for Astrophysics, Astrophysics - Cosmology and Nongalactic Astrophysics},
         year = 2018,
        month = jul,
          eid = {arXiv:1808.00011},
        pages = {arXiv:1808.00011},
          doi = {10.48550/arXiv.1808.00011},
archivePrefix = {arXiv},
       eprint = {1808.00011},
 primaryClass = {astro-ph.IM},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv180800011M},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

Lønning et al. (2019) (MRI imaging)

@article{Lonning2019,
    author = {L{\o}nning, Kai and Putzky, Patrick and Sonke, Jan Jakob and Reneman, Liesbeth and Caan, Matthan W.A. and Welling, Max},
    doi = {10.1016/j.media.2019.01.005},
    issn = {13618423},
    journal = {Medical Image Analysis},
    keywords = {Deep learning,Inverse problems,MRI,Reconstruction},
    month = {apr},
    pages = {64--78},
    pmid = {30703579},
    publisher = {Elsevier},
    title = {{Recurrent inference machines for reconstructing heterogeneous MRI data}},
    volume = {53},
    year = {2019}
}

Modi et al. (2021) (Initial conditions of the Universe)

@ARTICLE{Modi2021,
       author = {{Modi}, Chirag and {Lanusse}, Fran{\c{c}}ois and {Seljak}, Uro{\v{s}} and {Spergel}, David N. and {Perreault-Levasseur}, Laurence},
        title = "{CosmicRIM : Reconstructing Early Universe by Combining Differentiable Simulations with Recurrent Inference Machines}",
      journal = {arXiv e-prints},
archivePrefix = {arXiv},
       eprint = {2104.12864},
 primaryClass = {astro-ph.CO},
     keywords = {Astrophysics - Cosmology and Nongalactic Astrophysics},
         year = 2021,
        month = apr,
          eid = {arXiv:2104.12864},
          doi = {10.48550/arXiv.2104.12864},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2021arXiv210412864M},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

License

This package is licensed under the MIT License.