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- International Journal of Computational Geometry & Applications
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Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.
We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter [Formula: see text] multiple of the [Formula: see text] norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction.
Given a closed [Formula: see text]-dimensional submanifold [Formula: see text], encapsulated in a compact domain [Formula: see text], [Formula: see text], we consider the problem of determining the intrinsic geometry of the obstacle [Formula: see text] (such as volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular [Formula: see text]-neighborhood [Formula: see text] of [Formula: see text] in [Formula: see text]. The geodesics that participate in this scattering emanate from the boundary [Formula: see text] and terminate there after a few reflections from the boundary [Formula: see text]. However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of [Formula: see text] by entering some trap there so that this ray will have infinitely many reflections from [Formula: see text]. To rule out such a possibility, we modify the geometry of a tube [Formula: see text] by building it from spherical bubbles. We need to use [Formula: see text] many bubbling tubes [Formula: see text] for detecting certain global invariants of [Formula: see text], invariants that reflect its intrinsic geometry. Thus, the words “layered scattering” are in the title.more »
Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory ManagementWe consider a stochastic inventory control problem under censored demand, lost sales, and positive lead times. This is a fundamental problem in inventory management, with significant literature establishing near optimality of a simple class of policies called “base-stock policies” as well as the convexity of long-run average cost under those policies. We consider a relatively less studied problem of designing a learning algorithm for this problem when the underlying demand distribution is unknown. The goal is to bound the regret of the algorithm when compared with the best base-stock policy. Our main contribution is a learning algorithm with a regret bound of [Formula: see text] for the inventory control problem. Here, [Formula: see text] is the fixed and known lead time, and D is an unknown parameter of the demand distribution described roughly as the expected number of time steps needed to generate enough demand to deplete one unit of inventory. Notably, our regret bounds depend linearly on L, which significantly improves the previously best-known regret bounds for this problem where the dependence on L was exponential. Our techniques utilize the convexity of the long-run average cost and a newly derived bound on the “bias” of base-stock policies to establishmore »
The goal of this paper is to study limiting behavior of a self-organized continuous flock evolving according to the 1D hydrodynamic Euler Alignment model. We provide a series of quantitative estimates that show how far the density of the limiting flock is from a uniform distribution. The key quantity that controls density distortion is the entropy [Formula: see text], and the measure of deviation from uniformity is given by a well-known conserved quantity [Formula: see text], where [Formula: see text] is velocity and [Formula: see text] is the communication operator with kernel [Formula: see text]. The cases of Lipschitz, singular geometric, and topological kernels are covered in the study.