We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure shape of the minimizers.
In the anisotropic setting, optimal transport techniques have been used with great success (see, e.g. @cite_41 ). However, usually convexity of the extremizers is required (e.g. to guarantee the necessary regularity on the transport map). Other techniques, such as the selection principle, often require understanding the spectrum of the relevant energy linearized around the extremizers (to obtain estimates like Fuglede's, @cite_4 ). In the generality we consider here, there is very little one can say about the structure of the extremizers (i.e. isoperimetric regions) or symmetry of the underlying space. This lack of knowledge is our primary technical obstacle.
@cite_41 @cite_4
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In this paper we address the problem of dense depth map estimation from sparse noisy range data to reconstruct large heterogeneous outdoor scenes. We propose a surface inpainting solution through energy minimisation with an adaptive selection of surface regularisers among a set of well known convex and non-convex regularisers. In fact, the selection of norm is pivotal with respect to the intrinsic surface characteristics. Our goal is to show how dense interpolation of sparse range data can be leveraged of more exotic and non-convex regularisers such as the log and logTGV [1] which can better capture the scene geometry. In contrast to state of the art solutions, we do not restrict ourselves to this set of norms, instead we search for the most apt norm for each semantically segmented part of the scene. Our energy model selection use Bayesian optimisation to learn the best choice of free parameters. This results in an adaptive model selection and the generalisation of well studied regularisation norms. We conclude with a detailed experimental analysis of our approach using a basis of four norms over a set of challenging outdoor scenes. Entropic regularization is quickly emerging as a new standard in optimal transport (OT). It enables to cast the OT computation as a differentiable and unconstrained convex optimization problem, which can be efficiently solved using the Sinkhorn algorithm. However, entropy keeps the transportation plan strictly positive and therefore completely dense, unlike unregularized OT. This lack of sparsity can be problematic in applications where the transportation plan itself is of interest. In this paper, we explore regularizing the primal and dual OT formulations with a strongly convex term, which corresponds to relaxing the dual and primal constraints with smooth approximations. We show how to incorporate squared @math -norm and group lasso regularizations within that framework, leading to sparse and group-sparse transportation plans. On the theoretical side, we bound the approximation error introduced by regularizing the primal and dual formulations. Our results suggest that, for the regularized primal, the approximation error can often be smaller with squared @math -norm than with entropic regularization. We showcase our proposed framework on the task of color transfer. The formation of secure transportation corridors, where cargoes and shipments from points of entry can be dispatched safely to highly sensitive and secure locations, is a high national priority. One of the key tasks of the program is the detection of anomalous cargo based on sensor readings in truck weigh stations. Due to the high variability, dimensionality, and or noise content of sensor data in transportation corridors, appropriate feature representation is crucial to the success of anomaly detection methods in this domain. In this paper, we empirically investigate the usefulness of manifold embedding methods for feature representation in anomaly detection problems in the domain of transportation corridors. We focus on both linear methods, such as multi-dimensional scaling (MDS), as well as nonlinear methods, such as locally linear embedding (LLE) and isometric feature mapping (ISOMAP). Our study indicates that such embedding methods provide a natural mechanism for keeping anomalous points away from the dense normal regions in the embedding of the data. We illustrate the efficacy of manifold embedding methods for anomaly detection through experiments on simulated data as well as real truck data from weigh stations. This paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains. Our main contribution is to show that optimal transportation can be made tractable over large domains used in graphics, such as images and triangle meshes, improving performance by orders of magnitude compared to previous work. To this end, we approximate optimal transportation distances using entropic regularization. The resulting objective contains a geodesic distance-based kernel that can be approximated with the heat kernel. This approach leads to simple iterative numerical schemes with linear convergence, in which each iteration only requires Gaussian convolution or the solution of a sparse, pre-factored linear system. We demonstrate the versatility and efficiency of our method on tasks including reflectance interpolation, color transfer, and geometry processing.