Stanley Osher

Bio: Stanley Osher is an academic researcher from University of California, Los Angeles. The author has contributed to research in topics: Level set method & Hyperbolic partial differential equation. The author has an hindex of 114, co-authored 510 publications receiving 104028 citations. Previous affiliations of Stanley Osher include University of Minnesota & University of Innsbruck.

G-Norm Properties of Bounded Variation Regularization

Abstract: Recently Y. Meyer derived a characterization of the minimizer of the Rudin-Osher- Fatemi functional in a functional analytical framework. In statistics the discrete version of this functional is used to analyze one dimensional data and belongs to the class of nonparametric regres- sion models. In this work we generalize the functional analytical results of Meyer and apply them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multi- dimensional data. The characterization of Y. Meyer and our generalization is based on G-norm properties of the data and the minimizer. A geometric point of view of regression minimization is provided. whereDudenotes the total variation semi-norm of u and α> 0. The minimizer is called the bounded variation regularized solution. The taut-string algorithm consists in finding a string of minimal length in a tube (with radius α) around the primitive of f . The differentiated string is the taut-string reconstruction and corresponds to the minimizer of the ROF-model. Generalizing these ideas to higher dimensions is complicated by the fact that there is no obvious analog to primitives in higher space dimensions. We overcome this difficulty by solving Laplace's equation with right hand side f (i.e. integrate twice), and differentiating. The tube with radius α around the derivative of the potential specifies all functions u which satisfyu − fGs ≤ α (see also (21)). In this paper we show that the bounded variation regularized solutions (in any number of space dimensions) are contained in a tube of radius α .F or several other regression models in statistics, such as robust, quantile, and logistic regression (reformulated in a Banach space setting for analyzing multi-dimensional data) the

One-sided difference schemes and transonic flow

Abstract: Two one-sided conservation form difference approximations to a scalar one-dimensional convex conservation law are introduced. These are respectively of first- and second-order accuracy and each has the minimum possible band-width. They are nonlinearly stable, they converge only to solutions satisfying the entropy condition, and they have sharp monotone profiles. No such stable approximation of order higher than two is possible. Dimensional splitting algorithms are constructed and used to approximate the small-disturbance equation of transonic flow. These approximations are also nonlinearly stable and without nonphysical limit solutions.

ResNets Ensemble via the Feynman-Kac Formalism to Improve Natural and Robust Accuracies

Abstract: Empirical adversarial risk minimization (EARM) is a widely used mathematical framework to robustly train deep neural nets (DNNs) that are resistant to adversarial attacks. However, both natural and robust accuracies, in classifying clean and adversarial images, respectively, of the trained robust models are far from satisfactory. In this work, we unify the theory of optimal control of transport equations with the practice of training and testing of ResNets. Based on this unified viewpoint, we propose a simple yet effective ResNets ensemble algorithm to boost the accuracy of the robustly trained model on both clean and adversarial images. The proposed algorithm consists of two components: First, we modify the base ResNets by injecting a variance specified Gaussian noise to the output of each residual mapping. Second, we average over the production of multiple jointly trained modified ResNets to get the final prediction. These two steps give an approximation to the Feynman-Kac formula for representing the solution of a transport equation with viscosity, or a convection-diffusion equation. For the CIFAR10 benchmark, this simple algorithm leads to a robust model with a natural accuracy of {\bf 85.62}\% on clean images and a robust accuracy of ${\bf 57.94 \%}$ under the 20 iterations of the IFGSM attack, which outperforms the current state-of-the-art in defending against IFGSM attack on the CIFAR10. Both natural and robust accuracies of the proposed ResNets ensemble can be improved dynamically as the building block ResNet advances. The code is available at: \url{this https URL}.

A Framework for Solving Surface Partial Differential Equations for Computer Graphics Applications

Abstract: A novel framework for solving variational problems and partial dif-ferential equations for scalar and vector-valued data defined on sur-faces is introduced in this paper. The key idea is to implicitly rep-resent the surface as the level set of a higher dimensional function,and solve the surface equations in a fix ed Cartesian coordinate sys-tem using this new embedding function. This thereby eliminatesthe needforperformingcomplicatedandnot-accuratecomputationson triangulated surfaces, as it is commonly done in the graphicsand numerical analysis literature. We describe the framework andpresent examples in texture synthesis, flo w field visualization, aswell as image and vector field regularization for data defined on 3Dsurfaces.CR Categories: I.3.6 [Computer Graphics]: Methodology andTechniques—; G.1.8 [Numerical Analysis]: Partial DifferentialEquations—; I.4.10 [Image Processing and Computer Vision]: Im-age representation—;Keywords: Partial differential equations, implicit surfaces, patternformation, natural phenomena, flo w visualization, image and vectorfield regularization.

Color Texture Modeling and Color Image Decomposition in a Variational-PDE Approach

Abstract: This paper is devoted to a new variational model for color texture modeling and color image decomposition into cartoon and texture. A given image f in the RGB space is decomposed into a cartoon part and a texture part. The cartoon part is modeled by the space of vector-valued functions of bounded variation, while the texture or noise part is modeled by a space of oscillatory functions, dual in some sense of the BV space. Examples for color image decomposition, color image denoising, and color texture discrimination and segmentation will be presented.

Going deeper with convolutions

Abstract: We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Fast parallel algorithms for short-range molecular dynamics

Abstract: Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers

Abstract: Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.