Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_\theta$ (with $\theta$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincar\'e type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_\theta)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Br\'ezis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space $B^\alpha_{\infty,\infty}(\mathbb{T}^d_\theta)$ for $\alpha>0$ is the quantum analogue of the usual Zygmund class of order $\alpha$. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple $(L_p(\mathbb{T}^d_\theta), \, W^k_p(\mathbb{T}^d_\theta))$, which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix $\theta$, so coincide with those on the corresponding spaces on the usual $d$-torus

Radar-on-Lidar: metric radar localization on prior lidar maps

Radar and lidar, provided by two different range sensors, each has pros and cons of various perception tasks on mobile robots or autonomous driving. In this paper, a Monte Carlo system is used to localize the robot with a rotating radar sensor on 2D lidar maps. We first train a conditional generative adversarial network to transfer raw radar data to lidar data, and achieve reliable radar points from generator. Then an efficient radar odometry is included in the Monte Carlo system. Combining the initial guess from odometry, a measurement model is proposed to match the radar data and prior lidar maps for final 2D positioning. We demonstrate the effectiveness of the proposed localization framework on the public multi-session dataset. The experimental results show that our system can achieve high accuracy for long-term localization in outdoor scenes

Sparse Recovery via Differential Inclusions

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, \emph{i.e.} finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics \emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to \emph{Linearized Bregman iteration}, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal $l_2$-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.Comment: In Applied and Computational Harmonic Analysis, 201

LocNet: Global localization in 3D point clouds for mobile vehicles

Global localization in 3D point clouds is a challenging problem of estimating the pose of vehicles without any prior knowledge. In this paper, a solution to this problem is presented by achieving place recognition and metric pose estimation in the global prior map. Specifically, we present a semi-handcrafted representation learning method for LiDAR point clouds using siamese LocNets, which states the place recognition problem to a similarity modeling problem. With the final learned representations by LocNet, a global localization framework with range-only observations is proposed. To demonstrate the performance and effectiveness of our global localization system, KITTI dataset is employed for comparison with other algorithms, and also on our long-time multi-session datasets for evaluation. The result shows that our system can achieve high accuracy.Comment: 6 pages, IV 2018 accepte