Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Learning Non-Linear Combinations of Kernels

Abstract: This paper studies the general problem of learning kernels based on a polynomial combination of base kernels. We analyze this problem in the case of regression and the kernel ridge regression algorithm. We examine the corresponding learning kernel optimization problem, show how that minimax problem can be reduced to a simpler minimization problem, and prove that the global solution of this problem always lies on the boundary. We give a projection-based gradient descent algorithm for solving the optimization problem, shown empirically to converge in few iterations. Finally, we report the results of extensive experiments with this algorithm using several publicly available datasets demonstrating the effectiveness of our technique.

Object recognition by affine invariant matching

Abstract: Novel techniques are described for model-based recognition of 3-D objects from unknown viewpoints using single-gray-scale images. The objects in the scene may be overlapping and partially occluded. Efficient matching algorithms, which assume affine approximation to the perspective viewing transformation, are proposed. The study is currently restricted to flat rigid 3-D objects. Point, line and curve matching algorithms are presented. The study especially emphasizes the curve matching problem. Experimental results are included. >

The mechanics of slithering locomotion

Abstract: In this experimental and theoretical study, we investigate the slithering of snakes on flat surfaces. Previous studies of slithering have rested on the assumption that snakes slither by pushing laterally against rocks and branches. In this study, we develop a theoretical model for slithering locomotion by observing snake motion kinematics and experimentally measuring the friction coefficients of snakeskin. Our predictions of body speed show good agreement with observations, demonstrating that snake propulsion on flat ground, and possibly in general, relies critically on the frictional anisotropy of their scales. We have also highlighted the importance of weight distribution in lateral undulation, previously difficult to visualize and hence assumed uniform. The ability to redistribute weight, clearly of importance when appendages are airborne in limbed locomotion, has a much broader generality, as shown by its role in improving limbless locomotion.

Statistical mechanics of classical particles with logarithmic interactions

Abstract: The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second-order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger-Moser inequality known from differential geometry – in its sharp form.

Deep Neural Networks Improve Radiologists’ Performance in Breast Cancer Screening

Abstract: We present a deep convolutional neural network for breast cancer screening exam classification, trained, and evaluated on over 200000 exams (over 1000000 images). Our network achieves an AUC of 0.895 in predicting the presence of cancer in the breast, when tested on the screening population. We attribute the high accuracy to a few technical advances. 1) Our network’s novel two-stage architecture and training procedure, which allows us to use a high-capacity patch-level network to learn from pixel-level labels alongside a network learning from macroscopic breast-level labels. 2) A custom ResNet-based network used as a building block of our model, whose balance of depth and width is optimized for high-resolution medical images. 3) Pretraining the network on screening BI-RADS classification, a related task with more noisy labels. 4) Combining multiple input views in an optimal way among a number of possible choices. To validate our model, we conducted a reader study with 14 readers, each reading 720 screening mammogram exams, and show that our model is as accurate as experienced radiologists when presented with the same data. We also show that a hybrid model, averaging the probability of malignancy predicted by a radiologist with a prediction of our neural network, is more accurate than either of the two separately. To further understand our results, we conduct a thorough analysis of our network’s performance on different subpopulations of the screening population, the model’s design, training procedure, errors, and properties of its internal representations. Our best models are publicly available at https://github.com/nyukat/breast_cancer_classifier .