Implicit neural networks are a general class of learning models that replace the layers in traditional feedforward models with implicit algebraic equations. Compared to traditional learning models, implicit networks offer competitive performance and reduced memory consumption. However, they can remain brittle with respect to input adversarial perturbations. This paper proposes a theoretical and computational framework for robustness verification of implicit neural networks; our framework blends together mixed monotone systems theory and contraction theory. First, given an implicit neural network, we introduce a related embedded network and show that, given an infinity-norm box constraint on the input, the embedded network provides an infinity-norm box overapproximation for the output of the original network. Second, using infinity-matrix measures, we propose sufficient conditions for well-posedness of both the original and embedded system and design an iterative algorithm to compute the infinity-norm box robustness margins for reachability and classification problems. Third, of independent value, we show that employing a suitable relative classifier variable in our analysis will lead to tighter bounds on the certified adversarial robustness in classification problems. Finally, we perform numerical simulations on a Non-Euclidean Monotone Operator Network (NEMON) trained on the MNIST dataset. In these simulations, we compare the accuracy and run time of our mixed monotone contractive approach with the existing robustness verification approaches in the literature for estimating the certified adversarial robustness.
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Neural radiosity
We introduce Neural Radiosity, an algorithm to solve the rendering equation by minimizing the norm of its residual, similar as in classical radiosity techniques. Traditional basis functions used in radiosity, such as piecewise polynomials or meshless basis functions are typically limited to representing isotropic scattering from diffuse surfaces. Instead, we propose to leverage neural networks to represent the full four-dimensional radiance distribution, directly optimizing network parameters to minimize the norm of the residual. Our approach decouples solving the rendering equation from rendering (perspective) images similar as in traditional radiosity techniques, and allows us to efficiently synthesize arbitrary views of a scene. In addition, we propose a network architecture using geometric learnable features that improves convergence of our solver compared to previous techniques. Our approach leads to an algorithm that is simple to implement, and we demonstrate its effectiveness on a variety of scenes with diffuse and non-diffuse surfaces.
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- Award ID(s):
- 2126407
- PAR ID:
- 10385733
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 40
- Issue:
- 6
- ISSN:
- 0730-0301
- Page Range / eLocation ID:
- 1 to 11
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3478513.3480569
