/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Medical image analysis

PURPOSE: In diffusion MRI, a technique known as diffusion spectrum imaging reconstructs the propagator with a discrete Fourier transform, from a Cartesian sampling of the diffusion signal. Alternatively, it is possible to directly reconstruct the orientation distribution function in q-ball imaging, providing so-called high angular resolution diffusion imaging. In between these two techniques, acquisitions on several spheres in q-space offer an interesting trade-off between the angular resolution and the radial information gathered in diffusion MRI. A careful design is central in the success of multishell acquisition and reconstruction techniques. METHODS: The design of acquisition in multishell is still an open and active field of research, however. In this work, we provide a general method to design multishell acquisition with uniform angular coverage. This method is based on a generalization of electrostatic repulsion to multishell. RESULTS: We evaluate the impact of our method using simulations, on the angular resolution in one and two bundles of fiber configurations. Compared to more commonly used radial sampling, we show that our method improves the angular resolution, as well as fiber crossing discrimination. DISCUSSION: We propose a novel method to design sampling schemes with optimal angular coverage and show the positive impact on angular resolution in diffusion MRI.

https://hal.inria.fr/hal-00821688/document

Design of multishell sampling schemes with uniform coverage in diffusion MRI

We derive generalization bounds for learning algorithms based on their robustness: the property that if a testing sample is "similar" to a training sample, then the testing error is close to the training error. This provides a novel approach, different from complexity or stability arguments, to study generalization of learning algorithms. One advantage of the robustness approach, compared to previous methods, is the geometric intuition it conveys. Consequently, robustness-based analysis is easy to extend to learning in non-standard setups such as Markovian samples or quantile loss. We further show that a weak notion of robustness is both sufficient and necessary for generalizability, which implies that robustness is a fundamental property that is required for learning algorithms to work.

Robustness and Generalization.

Optimal real-time Q-ball imaging using regularized Kalman filtering with incremental orientation sets

In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networks. The adversarial robustness regularization combines adversarial training, which we show to be equivalent to Total Variation regularization, with Lipschitz regularization. We demonstrate empirically that the regularized models are more robust, and that gradient norms of images can be used for attack detection.

Lipschitz regularized Deep Neural Networks generalize and are adversarially robust

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Holder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.

https://iopscience.iop.org/article/10.1088/1361-6544/aae949/pdf

The game theoretic p-Laplacian and semi-supervised learning with few labels

Generalization of deep neural networks (DNNs) is an open problem which, if solved, could impact the reliability and verification of deep neural network architectures. In this paper, we show that if the usual fidelity term used in training DNNs is augmented by a Lipschitz regularization term, then the networks converge and generalize. The convergence is in the limit as the number of data points, n → ∞, while also allowing the network to grow as needed to fit the data. Two regimes are identified: in the case of clean labels, we prove convergence to the label function which corresponds to zero loss, in the case of corrupted labels which we prove convergence to a regularized label function which is the solution of a limiting variational problem. In both cases, a convergence rate is also provided.

Lipschitz regularized Deep Neural Networks converge and generalize

We study the consistency of Lipschitz learning on graphs in the limit of infinite unlabeled data and finite labeled data. Previous work has conjectured that Lipschitz learning is well-posed in this...

https://epubs.siam.org/doi/pdf/10.1137/18M1199241

Jeff Calder

Papers

Optimal real-time Q-ball imaging using regularized Kalman filtering with incremental orientation sets

Lipschitz regularized Deep Neural Networks generalize and are adversarially robust

The game theoretic p-Laplacian and semi-supervised learning with few labels

Lipschitz regularized Deep Neural Networks converge and generalize

Consistency of Lipschitz Learning with Infinite Unlabeled Data and Finite Labeled Data