Zhenghan Fang*, Sam Buchanan*, Jeremias Sulam

IMSI Computational Imaging Workshop

Wed, 7 Aug, 2024

*Equal contribution.

What's in a Prior?

Learned Proximal Networks for Inverse Problems

Inverse Problems

x
y
  • Computational imaging

Measure

Inversion

y=Ax+v

Inverse Problems

MAP estimate

\min_x \tfrac{1}{2}\|y - Ax \|_2^2 + {\color{darkorange} R}(x)
{\color{darkorange} R} = -\log p_x

Prox. Grad. Desc.

x_{k+1} = {\color{darkorange}\mathrm{prox}_{\eta R}}(x_k - \eta A^H(A x_k - y))

Deep neural networks

{\color{darkorange}\mathrm{prox}_{R}} (z) = \argmin_u \tfrac{1}{2} \|u-z\|_2^2 + R(u)

Proximal Operator

x_{k+1} = \quad {\color{red}f_{\theta}} \quad \ (x_k - \eta A^H(A x_k - y))

Plug-n-Play

Questions and Contributions

1. When is a neural network \(f_\theta\) a proximal operator?

2. What's the prior \(R\) learned by the neural network \(f_\theta\)?

3. How can we learn the prox of the true prior?

4. Convergence guarantees for PnP

Prior Landscape

Proximal Matching Loss

Learned proximal networks (LPN)

neural networks for prox. operators

Questions and Contributions

1. When is a neural network \(f_\theta\) a proximal operator?

2. What's the prior \(R\) learned by the neural network \(f_\theta\)?

3. How can we learn the prox of the true prior?

4. Convergence guarantees for PnP

Prior Landscape

Proximal Matching Loss

Learned proximal networks (LPN)

neural networks for prox. operators

Compressed Sensing

Acknowledgements and References

Fang, Buchanan, Sulam. What's in a Prior? Learned Proximal Networks for Inverse Problems. ICLR 2024.

Sam Buchanan

TTIC

Jeremias Sulam

JHU

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