Feasibility-based Fixed Point Networks

Heaton, Howard; Fung, Samy Wu; Gibali, Aviv; Yin, Wotao

doi:10.48550/arxiv.2104.14090

Cited by 4 publications

(5 citation statements)

References 98 publications

(116 reference statements)

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“…To understand the dynamic systems interpretation of ResNet [14], the prediction models like time-dependent PDEs [39,40] and ODEs [10,17,26,51] were also categorized in unfolding networks. For instance, in [39], the forward Euler scheme was considered as the temporal discretization of the evolution PDE and resulted in iterative algorithms.…”

Section: Related Workmentioning

confidence: 99%

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IDmUNet: A new image decomposition induced network for sparse feature segmentation

Ren¹,

Gao²,

Wu³

et al. 2022

Preprint

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UNet and its variants are among the most popular methods for medical image segmentation. Despite their successes in task generality, most of them consider little mathematical modeling behind specific applications. In this paper, we focus on the sparse feature segmentation task and make a taskoriented network design, in which the target objects are sparsely distributed and the background is hard to be mathematically modeled. We start from an image decomposition model with sparsity regularization, and propose a deep unfolding network, namely IDNet, based on an iterative solver, scaled alternating direction method of multipliers (scaled-ADMM). The IDNet splits raw inputs into double feature layers. Then a new task-oriented segmentation network is constructed, dubbed as IDmUNet, based on the proposed IDNets and a mini-UNet. Because of the sparsity prior and deep unfolding method in the structure design, this IDmUNet combines the advantages of mathematical modeling and data-driven approaches. Firstly, our approach has mathematical interpretability and can achieve favorable performance with far fewer learnable parameters. Secondly, our IDmUNet is robust in a simple end-to-end training with explainable behaviors. In the experiments of retinal vessel segmentation (RVS), IDmUNet produces the state-of-the-art results with only 0.07m parameters, whereas SA-UNet, one of the latest variants of UNet, contains 0.54m and the original UNet 31.04m. Moreover, the training procedure of our network converges faster without overfitting phenomenon. This decomposition-based network construction strategy can be generalized to other problems with mathematically clear targets and complicated unclear backgrounds.

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Section: Related Workmentioning

confidence: 99%

“…For instance, in [39], the forward Euler scheme was considered as the temporal discretization of the evolution PDE and resulted in iterative algorithms. Moreover, there were also networks with explicit depth [17,26], for which alternative back-propagation procedures based on ODE-theory were proposed, instead of the chain rule.…”

Section: Related Workmentioning

confidence: 99%

IDmUNet: A new image decomposition induced network for sparse feature segmentation

Ren¹,

Gao²,

Wu³

et al. 2022

Preprint

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show abstract

“…Discussion Comparisons of our method (Implicit L2O) with U-Net, 38 F-FPNs, 32 and total variation (TV) Minimization are given in Figure 8 and Table 3. Table 3 shows the average PSNR and SSIM reconstructions.…”

Section: Model Designmentioning

confidence: 99%

“…visible staircasing effects 17;59 ), which shows novelty of certificates as these features are intuitive, yet prior methods to quantify this were, to our knowledge, unknown. 32 U-Net was trained with filtered backprojection as in prior work. 38 Three properties are used to check trustworthiness: box constraints, compliance with measurements (i.e.…”

Section: Model Designmentioning

confidence: 99%

Explainable AI via Learning to Optimize

Heaton¹,

Fung²

2022

Preprint

Self Cite

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Indecipherable black boxes are common in machine learning (ML), but applications increasingly require explainable artificial intelligence (XAI). The core of XAI is to establish transparent and interpretable data-driven algorithms. This work provides concrete tools for XAI in situations where prior knowledge must be encoded and untrustworthy inferences flagged. We use the "learn to optimize" (L2O) methodology wherein each inference solves a data-driven optimization problem. Our L2O models are straightforward to implement, directly encode prior knowledge, and yield theoretical guarantees (e.g. satisfaction of constraints). We also propose use of interpretable certificates to verify whether model inferences are trustworthy. Numerical examples are provided in the applications of dictionary-based signal recovery, CT imaging, and arbitrage trading of cryptoassets. 1

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“…Instead, we employ the Jacobian-free backpropagation (JFB) of [21], which consists of replacing J −1 Θ in ( 14) with the identity matrix. This substitution yields a preconditioned gradient and is effective for training in image classification [21] and data-driven CT reconstructions [26]. Importantly, computing this preconditioned gradient only requires backpropagating through a single application of T Θ (i.e.…”

Section: Constraint Decouplingmentioning

confidence: 99%

Learn to Predict Equilibria via Fixed Point Networks

Heaton¹,

McKenzie²,

Li³

et al. 2021

Preprint

Self Cite

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Systems of interacting agents can often be modeled as contextual games, where the context encodes additional information, beyond the control of any agent (e.g. weather for traffic and fiscal policy for market economies). In such systems, the most likely outcome is given by a Nash equilibrium. In many practical settings, only game equilibria are observed, while the optimal parameters for a game model are unknown. This work introduces Nash Fixed Point Networks (N-FPNs), a class of implicit-depth neural networks that output Nash equilibria of contextual games. The N-FPN architecture fuses data-driven modeling with provided constraints. Given equilibrium observations of a contextual game, N-FPN parameters are learnt to predict equilibria outcomes given only the context. We present an end-to-end training scheme for N-FPNs that is simple and memory efficient to implement with existing autodifferentiation tools. N-FPNs also exploit a novel constraint decoupling scheme to avoid costly projections. Provided numerical examples show the efficacy of N-FPNs on atomic and non-atomic games (e.g. traffic routing). 2 * Equal contribution 2 Codes are available on Github: github.com/howardheaton/nash_fixed_point_networks Preprint. Under review.

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Feasibility-based Fixed Point Networks

Cited by 4 publications

References 98 publications

IDmUNet: A new image decomposition induced network for sparse feature segmentation

IDmUNet: A new image decomposition induced network for sparse feature segmentation

Explainable AI via Learning to Optimize

Learn to Predict Equilibria via Fixed Point Networks

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