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1. Regularity of the Level Set Flow

   https://doi.org/10.1002/cpa.21703  

   Colding, Tobias Holck; Minicozzi, William P (July 2017, Communications on Pure and Applied Mathematics)

   Abstract We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closedC¹manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc. 

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2. Level Set Method for Motion by Mean Curvature

   https://doi.org/10.1090/noti1439  

   Colding, Tobias Holck; Minicozzi, William P (November 2016, Notices of the American Mathematical Society)

   Abstract. Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second-order differential equations on Euclidean space. One naturally wonders, “What is the regularity of solutions?” A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties. 

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3. The Emptiness Inside: Finding Gaps, Valleys, and Lacunae with Geometric Data Analysis

   https://doi.org/10.3847/1538-3881/ac961e  

   Contardo, Gabriella; Hogg, David W.; Hunt, Jason A.; Peek, Joshua E.; Chen, Yen-Chi (October 2022, The Astronomical Journal)

   Abstract Discoveries of gaps in data have been important in astrophysics. For example, there are kinematic gaps opened by resonances in dynamical systems, or exoplanets of a certain radius that are empirically rare. A gap in a data set is a kind of anomaly, but in an unusual sense: instead of being a single outlier data point, situated far from other data points, it is a region of the space, or a set of points, that is anomalous compared to its surroundings. Gaps are both interesting and hard to find and characterize, especially when they have nontrivial shapes. We present in this paper a statistic that can be used to estimate the (local) “gappiness” of a point in the data space. It uses the gradient and Hessian of the density estimate (and thus requires a twice-differentiable density estimator). This statistic can be computed at (almost) any point in the space and does not rely on optimization; it allows us to highlight underdense regions of any dimensionality and shape in a general and efficient way. We illustrate our method on the velocity distribution of nearby stars in the Milky Way disk plane, which exhibits gaps that could originate from different processes. Identifying and characterizing those gaps could help determine their origins. We provide in an appendix implementation notes and additional considerations for finding underdensities in data, using critical points and the properties of the Hessian of the density. 7 7 A Python implementation of t methods presented here is available at https://github.com/contardog/FindTheGap . 

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4. Differentiability of Effective Fronts in the Continuous Setting in Two Dimensions

   https://doi.org/10.1093/imrn/rnad212  

   Tran, Hung V; Yu, Yifeng (September 2023, International Mathematics Research Notices)

   Abstract We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $${\mathbb{Z}}^{2}$$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics [ 4, 6]. To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the literature of partial differential equations (PDE). Combining with the sufficiency result in [ 15], our result leads to a realization type conclusion: for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior. 

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5. ADA: Adversarial Data Augmentation for Object Detection

   Behpour, Sima; Kitani, Kris; Ziebart, Brian D. (January 2019, IEEE Winter Conf. on Applications of Computer Vision)

   The use of random perturbations of ground truth data, such as random translation or scaling of bounding boxes, is a common heuristic used for data augmentation that has been shown to prevent overfitting and improve generalization. Since the design of data augmentation is largely guided by reported best practices, it is difficult to understand if those design choices are optimal. To provide a more principled perspective, we develop a game-theoretic interpretation of data augmentation in the context of object detection. We aim to find an optimal adversarial perturbations of the ground truth data (i.e., the worst case perturbations) that forces the object bounding box predictor to learn from the hardest distribution of perturbed examples for better test-time performance. We establish that the game-theoretic solution (Nash equilibrium) provides both an optimal predictor and optimal data augmentation distribution. We show that our adversarial method of training a predictor can significantly improve test-time performance for the task of object detection. On the ImageNet, Pascal VOC and MS-COCO object detection tasks, our adversarial approach improves performance by about 16%, 5%, and 2% respectively compared to the best performing data augmentation methods. 

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