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.

/pdf/going-deeper-with-convolutions-1yobw2o2ds.pdf

Going deeper with convolutions

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 …

I and i

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.

Fast parallel algorithms for short-range molecular dynamics

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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.

/pdf/distributed-optimization-and-statistical-learning-via-the-8s9gtddw0d.pdf

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

Inspired by the nonlocal methods in image processing and the point integral method, we introduce a novel weighted nonlocal Laplacian method to compute a continuous interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted nonlocal Laplacian is a reliable and efficient interpolation method. In addition, it is fast and easy to implement.

Weighted Nonlocal Laplacian on Interpolation from Sparse Data

0. Introduction. In this paper, we obtain some new inequalities for integrals of convex functions of the curvature (resp. Wulff curvature) of convex plane curves. We also show that the difference between the two sides of our inequalities are monotone decreasing as the region flows under the unit-speed outward normal (resp. Wulff) flow. Given a bounded plane region K, the unit-speed outward normal flow has been highly studied, and is of interest in many applied problems, e.g. combustion. If instead K grows by varying the outward normal speed to be a function 7(0) of the direction of the unit normal, one has the WulfF flow, which is also of considerable interest, e.g. in studying the growth of crystals [O-M]. When the region K is convex, there is a simple closed-form expression which describes these flows, and the region converges to a disk in the first case and a Wulff shape in the second. The area of the region when the initial region K is convex is a polynomial in t, known respectively as the Steiner polynomial or WulfF-Steiner polynomial. A novel feature of our approach is that we study the roots of the Steiner and WulffSteiner polynomials, which occur at negative values of t. The classical isoperimetric inequality in both cases states that these polynomials have (negative) real roots ti >t2, and that they are distinct if and only if K is not a disc (respectively not a Wulff shape). Bonnesen's inequality states that the inradius and outradius r^ and re lie in the interval [—£1, —£2], and in the open interval if K is not a disk (respectively not a Wulff shape). Our inequalities are most naturally stated and proved in terms of the roots ti and £2We feel that this is a potentially quite fruitful approach to studying convex bodies in higher dimensions. In the context of this new approach, a very natural link between the outward normal and Wulff flows and the curvature integrals of the region appears. In important cases, the quantities that our inequalities state are positive are shown to be monotone decreasing as the region evolves under the flow. Particularly suggestive is the fact that the entropy of the curvature (respectively Wulff curvature) is bounded above in terms of the area and is monotone decreasing with time. The inequalities themselves are quite fascinating. It came as a surprise to us that there are interesting new things to be said about convex plane curves. We state our inequalities here for arbitrary smooth bounded convex plane regions K in the curvature case, and leave the Wulff case to the body of the paper. One of them is due to Gage [G], whose result was a source of inspiration to us. Gage's result is

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0003/AJM-1999-0003-0003-a005.pdf

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

: This paper studies the total variation regularization model with an L1 fidelity term (TV-L1) for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry sub-problems. Using this result we show that the TV-L1 model is able to separate image features according to their scales, where the scale is analytically defined by the G-value. A number of other properties including the geometric and morphological invariance of the TV-L1 model are also proved and their applications discussed.

/pdf/the-total-variation-regularized-l1-model-for-multiscale-1jyka1jo7q.pdf

The Total Variation Regularized L1 Model for Multiscale Decomposition

A class of conservative difference approximations for the steady full potential equation was presented. They are, in general, easier to program than the usual density biasing algorithms, and in fact, differ only slightly from them. Rigorous proof indicated that these new schemes satisfied a new discrete entropy inequality, which ruled out expansion shocks, and that they have sharp, steady, discrete shocks. A key tool in the analysis is the construction of a new entropy inequality for the full potential equation itself. Results of some numerical experiments using the new schemes are presented.

/pdf/entropy-condition-satisfying-approximations-for-the-full-12jsvh0fsq.pdf

Entropy Condition Satisfying Approximations for the Full Potential Equation of Transonic Flow

We present a proof of a conjecture made in the fleld of crystal growth. Namely, for an initial state consisting of any number of growing crystals moving outwards with normal velocity given to be ∞(~), for ~ the unit outwards normal, then the asymptotic growth shape is a Wulfi crystal, appropriately scaled in time. This shape minimizes the surface energy, which is the surface integral of ∞(~), for a given volume. The proof works in any number of dimensions. Additionally, we develop a new approach for obtaining the Wulfi shape by minimizing the surface energy divided by the enclosed volume to the 1 power in R d . We show that if we evolve a convex surface (not enclosing a Wulfi shape) under the motion described above, that the quantity to be minimized strictly decreases to its minimum as time increases. We have thus discovered a link between this surface evolution and this (generally nonconvex) energy minimization. A generalized Huyghen's principle is obtained. Finally, given the asymptotic shape we also obtain the associated (unique) convex ∞(~). The key technical tool is the level set method and the theory and characterization of viscosity solutions to Hamilton-Jacobi equations. 1. Introduction. The study of an anisotropic crystal growing in a melt gives rise to an equation relating the normal velocity of the motion to both the orientation of the crystal and to its curvature. In this paper we present a very simple and straightforward proof of a statement frequently made in the crystal literature { see e.g. Chernov (4), p. 215. Namely, in the case when the outwards normal velocity is equal to ∞(~), for ∞ the surface tension and ~ the unit outwards normal, then the asymptotic growth shape is precisely the celebrated Wulfi crystal, appropriately scaled in time. This shape minimizes the surface energy for a given volume. The proof, which works in any number of space dimensions, is constructive, giving asymptotic in time estimates. Moreover, the initial state can consist of any number of growing crystals, some of which may even contain holes. As time increases the individuals will merge into a growing crystal whose asymptotic limit is a single Wulfi shape. The associated energy function need not be convex. Thus singularities in the shape, i.e. jumps in the normal direction, may develop not only in time but also in the asymptotic limit. Facets and other jumps in normal direction can be characterized precisely with the help of the theory of viscosity solutions (5,6). Numerical results using the localized level set method (21) and the high order essentially nonoscillatory approximations to Hamilton-Jacobi equations developed in (14,15) validate our theoretical results. We shall discuss this in future work with D. Peng. In a parallel work, Osher, Merriman, Zhao and Peng, (13), have developed a connection between the static shape of crystalline materials in the plane, and the propagation of shock waves. They show that there is a precise sense in which any two dimensional crystalline form can be described in terms of rarefaction waves and contact discontinuities.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1997/0001/0003/AJM-1997-0001-0003-a006.pdf

Stanley Osher

Papers

Weighted Nonlocal Laplacian on Interpolation from Sparse Data

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

The Total Variation Regularized L1 Model for Multiscale Decomposition

Entropy Condition Satisfying Approximations for the Full Potential Equation of Transonic Flow

The wulff shape as the asymptotic limit of a growing crystalline interface