近几十年来，科学和工程的可用数据数量的重大增长彻底改变了。然而，尽管现在收集和存储数据的空前很容易，但通过补充每个功能的标签来标记数据仍然是具有挑战性的。标签过程需要专家知识或乏味且耗时的说明任务包括用诊断X射线标记X射线，具有蛋白质类型的蛋白质序列，其主题的文本，通过其情感推文或视频通过其类型的视频。在这些和许多其他示例中，由于成本和时间限制，只能手动标记一些功能。我们如何才能最好地将标签信息从少数昂贵的标签功能到大量未标记的标签信息传播？这是半监督学习（SSL）提出的问题。本文概述了基于图的贝叶斯SSL的最新基础发展，这是一种使用功能之间的相似性的标签传播概率框架。 SSL是一个活跃的研究领域，对现有文献的彻底回顾超出了本文的范围。我们的重点将放在我们自己的研究中得出的主题，这些主题说明了对基于图的贝叶斯SSL的统计准确性和计算效率进行严格研究的广泛数学工具和思想。

We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint. We perform the Burer-Monteiro factorization and consider a particular Riemannian quotient geometry in a search space that has a total space equipped with the Euclidean metric. When the original objective f satisfies standard restricted strong convexity and smoothness properties, we characterize the global landscape of the factorized objective under the Riemannian quotient geometry. We show the entire search space can be divided into three regions: (R1) the region near the target parameter of interest, where the factorized objective is geodesically strongly convex and smooth; (R2) the region containing neighborhoods of all strict saddle points; (R3) the remaining regions, where the factorized objective has a large gradient. To our best knowledge, this is the first global landscape analysis of the Burer-Monteiro factorized objective under the Riemannian quotient geometry. Our results provide a fully geometric explanation for the superior performance of vanilla gradient descent under the Burer-Monteiro factorization. When f satisfies a weaker restricted strict convexity property, we show there exists a neighborhood near local minimizers such that the factorized objective is geodesically convex. To prove our results we provide a comprehensive landscape analysis of a matrix factorization problem with a least squares objective, which serves as a critical bridge. Our conclusions are also based on a result of independent interest stating that the geodesic ball centered at Y with a radius 1/3 of the least singular value of Y is a geodesically convex set under the Riemannian quotient geometry, which as a corollary, also implies a quantitative bound of the convexity radius in the Bures-Wasserstein space. The convexity radius obtained is sharp up to constants.

在（特殊的）平滑样条问题中，一个人考虑了二次数据保真惩罚和拉普拉斯正则化的变异问题。可以通过用聚拉普拉斯的正规机构代替拉普拉斯的常规机构来获得较高的规律性。该方法很容易适应图，在这里，我们考虑在完全监督的，非参数，噪声损坏的回归问题中图形多拉普拉斯正则化。特别是，给定一个数据集$ \ {x_i \} _ {i = 1}^n $和一组嘈杂的标签$ \ {y_i \} _ {i = 1}^n \ subset \ subset \ mathbb {r}令$ u_n：\ {x_i \} _ {i = 1}^n \ to \ mathbb {r} $是由数据保真项组成的能量的最小化器，由数据保真术语和适当缩放的图形poly-laplacian项组成。当$ y_i = g（x_i）+\ xi_i $，对于IID噪声$ \ xi_i $，并使用几何随机图，我们在大型中识别（高概率）$ u_n $ to $ g $的收敛速率数据限制$ n \ to \ infty $。此外，我们的速率（到对数）与通常的平滑样条模型中已知的收敛速率相吻合。

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

我们在非参数二进制分类的一个对抗性训练问题之间建立了等价性，以及规范器是非识别范围功能的正则化风险最小化问题。由此产生的正常风险最小化问题允许在图像分析和基于图形学习中常常研究的$ L ^ 1 + $（非本地）$ \ Operatorvers {TV} $的精确凸松弛。这种重构揭示了丰富的几何结构，这反过来允许我们建立原始问题的最佳解决方案的一系列性能，包括存在最小和最大解决方案（以合适的意义解释），以及常规解决方案的存在（也以合适的意义解释）。此外，我们突出了对抗性训练和周长最小化问题的联系如何为涉及周边/总变化的正规风险最小化问题提供一种新颖的直接可解释的统计动机。我们的大部分理论结果与用于定义对抗性攻击的距离无关。

In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be obtained by sampling a distribution on a union of manifolds $\mathcal{M} = \mathcal{M}_1 \cup\dots \cup \mathcal{M}_N$ that may intersect with each other and that may have different dimensions. We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their corresponding graph Laplacians to capture the right geometric information to solve the MMC problem. Precisely, we provide high probability error bounds for the spectral approximation of a tensorized Laplacian on $\mathcal{M}$ with a suitable graph Laplacian built from the observations; the recovered tensorized Laplacian contains all geometric information of all the individual underlying manifolds. We provide an example of a family of similarity graphs, which we call annular proximity graphs with angle constraints, satisfying these sufficient conditions. We contrast our family of graphs with other constructions in the literature based on the alignment of tangent planes. Extensive numerical experiments expand the insights that our theory provides on the MMC problem.

This article formulates a generic representation of a path-following controller operating under contained motion, which was developed in the context of surgical robotics. It reports two types of constrained motion: i) Bilateral Constrained Motion, also called Remote Center Motion (RCM), and ii) Unilaterally Constrained Motion (UCM). In the first case, the incision hole has almost the same diameter as the robotic tool. In contrast, in the second state, the diameter of the incision orifice is larger than the tool diameter. The second case offers more space where the surgical instrument moves freely without constraints before touching the incision wall. The proposed method combines two tasks that must operate hierarchically: i) respect the RCM or UCM constraints formulated by equality or inequality, respectively, and ii) perform a surgical assignment, e.g., scanning or ablation expressed as a 3D path-following task. The proposed methods and materials were tested first on our simulator that mimics realistic conditions of middle ear surgery, and then on an experimental platform. Different validation scenarios were carried out experimentally to assess quantitatively and qualitatively each developed approach. Although ultimate precision was not the goal of this work, our concept is validated with enough accuracy (inferior to 100 micrometres) for ear surgery.

Several self-supervised representation learning methods have been proposed for reinforcement learning (RL) with rich observations. For real-world applications of RL, recovering underlying latent states is crucial, particularly when sensory inputs contain irrelevant and exogenous information. In this work, we study how information bottlenecks can be used to construct latent states efficiently in the presence of task-irrelevant information. We propose architectures that utilize variational and discrete information bottlenecks, coined as RepDIB, to learn structured factorized representations. Exploiting the expressiveness bought by factorized representations, we introduce a simple, yet effective, bottleneck that can be integrated with any existing self-supervised objective for RL. We demonstrate this across several online and offline RL benchmarks, along with a real robot arm task, where we find that compressed representations with RepDIB can lead to strong performance improvements, as the learned bottlenecks help predict only the relevant state while ignoring irrelevant information.

To face the dependency on fossil fuels and limit carbon emissions, fuel cells are a very promising technology and appear to be a key candidate to tackle the increase of the energy demand and promote the energy transition. To meet future needs for both transport and stationary applications, the time to market of fuel cell stacks must be drastically reduced. Here, a new concept to shorten their development time by introducing a disruptive and highefficiency data augmentation approach based on artificial intelligence is presented. Our results allow reducing the testing time before introducing a product on the market from a thousand to a few hours. The innovative concept proposed here can support engineering and research tasks during the fuel cell development process to achieve decreased development costs alongside a reduced time to market.

A learned system uses machine learning (ML) internally to improve performance. We can expect such systems to be vulnerable to some adversarial-ML attacks. Often, the learned component is shared between mutually-distrusting users or processes, much like microarchitectural resources such as caches, potentially giving rise to highly-realistic attacker models. However, compared to attacks on other ML-based systems, attackers face a level of indirection as they cannot interact directly with the learned model. Additionally, the difference between the attack surface of learned and non-learned versions of the same system is often subtle. These factors obfuscate the de-facto risks that the incorporation of ML carries. We analyze the root causes of potentially-increased attack surface in learned systems and develop a framework for identifying vulnerabilities that stem from the use of ML. We apply our framework to a broad set of learned systems under active development. To empirically validate the many vulnerabilities surfaced by our framework, we choose 3 of them and implement and evaluate exploits against prominent learned-system instances. We show that the use of ML caused leakage of past queries in a database, enabled a poisoning attack that causes exponential memory blowup in an index structure and crashes it in seconds, and enabled index users to snoop on each others' key distributions by timing queries over their own keys. We find that adversarial ML is a universal threat against learned systems, point to open research gaps in our understanding of learned-systems security, and conclude by discussing mitigations, while noting that data leakage is inherent in systems whose learned component is shared between multiple parties.