We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial SLE2 path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be Holder continuous in the capacity parameterization, assuming its driving term is Holder continuous. We also briefly discuss the case when the curves are a priori known to be Holder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.
To implement these ideas in a particular setting we need to show that the assumptions we used are satisfied uniformly in @math , with high probability in terms of @math . If a convergence rate for the driving terms (or martingale observable in rough domains) is known, then we believe it is possible to derive the remaining required information from existing results in the literature on discrete models without too much effort, and we derive the needed SLE derivative estimates, from estimates in @cite_12 , in this paper. Indeed, as already mentioned, the event that the geometric condition fails implies annuli crossing events that are reasonably well-understood for the models known to converge to SLE.
@cite_12
2118154107 2026158992 2035707149 2903103701
We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a 0 on the complementary arc, then the convergence is to SLE(4; a λ - 1, b λ - 1), a variant of SLE(4). In this paper, we formally present a novel estimation method, referred to as the Stochastic Learning Weak Estimator (SLWE), which yields the estimate of the parameters of a binomial distribution, where the convergence of the estimate is weak, i.e. with regard to the first and second moments. The estimation is based on the principles of stochastic learning. The mean of the final estimate is independent of the scheme's learning coefficient, @l, and both the variance of the final distribution and the speed decrease with @l. Similar results are true for the multinomial case, except that the equations transform from being of a scalar type to be of a vector type. Amazingly enough, the speed of the latter only depends on the same parameter, @l, which turns out to be the only non-unity eigenvalue of the underlying stochastic matrix that determines the time-dependence of the estimates. An empirical analysis on synthetic data shows the advantages of the scheme for non-stationary distributions. The paper also briefly reports (without detailed explanation) conclusive results that demonstrate the superiority of SLWE in pattern-recognition-based data compression, where the underlying data distribution is non-stationary. Finally, and more importantly, the paper includes the results of two pattern recognition exercises, the first of which involves artificial data, and the second which involves the recognition of the types of data that are present in news reports of the Canadian Broadcasting Corporation (CBC). The superiority of the SLWE in both these cases is demonstrated. We consider the optimal mass transportation problem in @math with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures. In this paper, we make the first attempt to build a framework to simultaneously estimate semantic parts, shape, translation, and orientation of cars from single street view. Our framework contains three major contributions. Firstly, a novel domain adaptation approach based on the class consistency loss is developed to transfer our part segmentation model from the synthesized images to the real images. Secondly, we propose a novel network structure that leverages part-level features from street views and 3D losses for pose and shape estimation. Thirdly, we construct a high quality dataset that contains more than 300 different car models with physical dimensions and part-level annotations based on global and local deformations. We have conducted experiments on both synthesized data and real images. Our results show that the domain adaptation approach can bring 35.5 percentage point performance improvement in terms of mean intersection-over-union score (mIoU) comparing with the baseline network using domain randomization only. Our network for translation and orientation estimation achieves competitive performance on highly complex street views (e.g., 11 cars per image on average). Moreover, our network is able to reconstruct a list of 3D car models with part-level details from street views, which could benefit various applications such as fine-grained car recognition, vehicle re-identification, and traffic simulation.