A Paired Autoencoder Framework for Inverse Problems via Bayes Risk Minimization

TL;DR

A paired autoencoder framework is considered, where two autoencoders are used to efficiently represent the input and target spaces separately and optimal mappings are learned between latent spaces, thus enabling forward and inverse surrogate mappings.

Abstract

In this work, we describe a new data-driven approach for inverse problems that exploits technologies from machine learning, in particular autoencoder network structures. We consider a paired autoencoder framework, where two autoencoders are used to efficiently represent the input and target spaces separately and optimal mappings are learned between latent spaces, thus enabling forward and inverse surrogate mappings. We focus on interpretations using Bayes risk and empirical Bayes risk minimization, and we provide various theoretical results and connections to existing works on low-rank matrix approximations. Similar to end-to-end approaches, our paired approach creates a surrogate model for forward propagation and regularized inversion. However, our approach outperforms existing approaches in scenarios where training data for unsupervised learning are readily available but training pairs for supervised learning are scarce. Furthermore, we show that cheaply computable evaluation metrics are available through this framework and can be used to predict whether the solution for a new sample should be predicted well.

Figures (9)

• Figure 1: End-to-end encoder-decoder network for inversion. The network is mapping input vector ${\bf b}$ to output vector ${\bf x}$. The encoder maps the input vector ${\bf b}$ to the latent variable ${\bf z}$ and the decoder maps this latent variable ${\bf z}$ to the output ${\bf x}$.
• Figure 1: PAIR network mapping. Two autoencoders are used to compress vectors ${\bf x}$ and ${\bf b}$ (on the top and bottom respectively), where the corresponding latent spaces are represented using variables ${\bf z}_{\bf x}$ and ${\bf z}_{\bf b}$. Mappings between latent spaces are denoted as $m$ and $m^\dagger$. The PAIR network provides both a data-driven inverse mapping $d_{\bf x} \circ m^\dagger \circ e_{\bf b}$ and a data-driven forward surrogate approximation $d_{\bf b} \circ m \circ e_{\bf x}$.
• Figure 1: Linear PAIR results for CT. Relative error is averaged over the 2,000 testing images according to the following: X autoencoder, $||\widehat{{\bf D}}_{\bf x}\widehat{{\bf E}}_{\bf x} {\bf x} - {\bf x}||_2/||{\bf x}||_2$; B autoencoder, $||\widehat{{\bf D}}_{\bf b} \widehat{{\bf E}}_{\bf b} {\bf b} - {\bf b}||_2/||{\bf b}||_2$; PAIR Forward, $||\widehat{{\bf P}} {\bf x} - {\bf b}||_2/||{\bf b}||_2$; PAIR Inverse, $||\widehat{{\bf P}}^\dagger {\bf b} - {\bf x} ||_2/||{\bf x}||_2$; TSVD Forward, $||{\bf U}_{{\bf A},r} {\boldsymbol{\Sigma}}_{{\bf A},r} {\bf V}^\top_{{\bf A},r}{\bf x} - {\bf b}||_2/||{\bf b}||_2$; and TSVD Inverse, $||{\bf V}_{{\bf A},r} {\boldsymbol{\Sigma}}_{{\bf A},r}^{-\top} {\bf U}_{{\bf A},r}^\top {\bf b}- {\bf x}||_2/||{\bf x}||_2$.
• Figure 2: An illustration for one example image pair from the testing set for the CT example (true image shown in the top right corner). The top two rows contain reconstructed sinograms and phantoms from the autoencoders for ${\bf b}$ and ${\bf x}$ respectively, for different latent dimensions. The bottom two rows contain reconstructions for the PAIR forward surrogate and the PAIR inverse surrogate respectively, for different latent dimensions.
• Figure 3: Example images inverted through the PAIR network. For each sample from the test set, the top row shows a blurred input digit, the second row shows the predicted reconstruction, the third row shows the true original target image and the fourth row shows the absolute pixel-wise error between the true and predicted.

Theorems & Definitions (7)

• Theorem 3.1
• Theorem 3.2
• Proof 1
• Theorem 3.3
• Proof 2: Proof of \ref{['thm:pair']}
• Proposition 3.4
• Proof 3