The Stackelberg prediction game (SPG) is a popular model for characterizing strategic interactions between a learner and an adversarial data provider. Although optimization problems in SPGs are often NP-hard, a notable special case involving the least squares loss (SPG-LS) has gained significant research attention recently, (Bishop et al. 2020; Wang et al. 2021; Wang et al. 2022). The latest state-of-the-art method for solving the SPG-LS problem is the spherically constrained least squares reformulation (SCLS) method proposed in the work of Wang et al. (2022). However, the lack of theoretical analysis on the error of the SCLS method limits its large-scale applications. In this paper, we investigate the estimation error between the learner obtained by the SCLS method and the actual learner. Specifically, we reframe the estimation error of the SCLS method as a Primary Optimization ($\textbf{PO}$) problem and utilize the Convex Gaussian min-max theorem (CGMT) to transform the $\textbf{PO}$ problem into an Auxiliary Optimization ($\textbf{AO}$) problem. Subsequently, we provide a theoretical error analysis for the SCLS method based on this simplified $\textbf{AO}$ problem. This analysis not only strengthens the theoretical framework of the SCLS method but also confirms the reliability of the learner produced by it. We further conduct experiments to validate our theorems, and the results are in excellent agreement with our theoretical predictions.

Sparse ridge regression problems play a significant role across various domains. To solve sparse ridge regression, Liu et al. (2023) recently propose an advanced algorithm, Scalable Optimal $K$-Sparse Ridge Regression (OKRidge), which is both faster and more accurate than existing approaches. However, the absence of theoretical analysis on the error of OKRidge impedes its large-scale applications. In this paper, we reframe the estimation error of OKRidge as a Primary Optimization ($\textbf{PO}$) problem and employ the Convex Gaussian min-max theorem (CGMT) to simplify the $\textbf{PO}$ problem into an Auxiliary Optimization ($\textbf{AO}$) problem. Subsequently, we provide a theoretical error analysis for OKRidge based on the $\textbf{AO}$ problem. This error analysis improves the theoretical reliability of OKRidge. We also conduct experiments to verify our theorems and the results are in excellent agreement with our theoretical findings.

Although deep neural networks (DNN) have achieved great success, their applications in safety-critical areas are hindered due to their vulnerability to adversarial attacks. Some recent works have accordingly proposed to enhance the robustness of DNN from a dynamic system perspective. Following this line of inquiry, and inspired by the asymptotic stability of the general nonautonomous dynamical system, we propose to make each clean instance be the asymptotically stable equilibrium points of a slowly time-varying system in order to defend against adversarial attacks. We present a theoretical guarantee that if a clean instance is an asymptotically stable equilibrium point and the adversarial instance is in the neighborhood of this point, the asymptotic stability will reduce the adversarial noise to bring the adversarial instance close to the clean instance. Motivated by our theoretical results, we go on to propose a nonautonomous neural ordinary differential equation (ASODE) and place constraints on its corresponding linear time-variant system to make all clean instances act as its asymptotically stable equilibrium points. Our analysis suggests that the constraints can be converted to regularizers in implementation. The experimental results show that ASODE improves robustness against adversarial attacks and outperforms state-of-the-art methods.

In the automobile industry, recent years have witnessed a growing interest in developing self-parking systems. For such systems, how to accurately and efficiently detect and localize the parking-slots defined by regular line segments near the vehicle is a key and still unresolved issue. In fact, kinds of unfavorable factors, such as the diversity of ground materials, changes in illumination conditions, and unpredictable shadows caused by nearby trees, make the vision-based parking-slot detection much harder than it looks. In this paper, we attempt to solve this issue to some extent and our contributions are twofold. First, we propose a novel DCNN (Deep Convolutional Neural Networks) based parking-slot detection approach, namely DeepPS, which takes the surround-view image as the input. There are two key steps in DeepPS, identifying all the marking-points on the input image and classifying local image patterns formed by pairs of markingpoints. We formulate both of them as learning problems, which can be solved naturally by modern DCNN models. Second, to facilitate the study of vision-based parking-slot detection, a largescale labeled dataset is established. This dataset is the largest in this field, comprising 12,165 surround-view images collected from typical indoor and outdoor parking sites. For each image, the marking-points and parking-slots are carefully labeled. The efficacy and efficiency of DeepPS have been corroborated on our collected dataset. To make our results fully reproducible, all the relevant source codes and the dataset have been made publicly available at https://cslinzhang.github.io/deepps/.

Recent years have witnessed a growing interest in developing automatic parking systems in the field of intelligent vehicle. However, how to effectively and efficiently locating parking-slots using a vision-based system is still an unresolved issue. In this paper, we attempt to fill this research gap to some extent and our contributions are twofold. Firstly, to facilitate the study of vision-based parking-slot detection, a large-scale parking-slot image database is established. For each image in this database, the marking-points and parking-slots are carefully labelled. Such a database can serve as a benchmark to design and validate parking-slot detection algorithms. Secondly, a learning based parking-slot detection approach is proposed. With this approach, given a test image, the marking-points will be detected at first and then the valid parking-slots can be inferred. Its efficacy and efficiency have been corroborated on our database. The labeled database and the source codes are publicly available at http://sse.tongji.edu.cn/linzhang/ps/index.htm.