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.

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.

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.

Assessing the critical view of safety in laparoscopic cholecystectomy requires accurate identification and localization of key anatomical structures, reasoning about their geometric relationships to one another, and determining the quality of their exposure. In this work, we propose to capture each of these aspects by modeling the surgical scene with a disentangled latent scene graph representation, which we can then process using a graph neural network. Unlike previous approaches using graph representations, we explicitly encode in our graphs semantic information such as object locations and shapes, class probabilities and visual features. We also incorporate an auxiliary image reconstruction objective to help train the latent graph representations. We demonstrate the value of these components through comprehensive ablation studies and achieve state-of-the-art results for critical view of safety prediction across multiple experimental settings.

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。

为有效语义分割和特别是视频语义分割构建模型的主要障碍是缺乏大型和良好的注释数据集。这种瓶颈在高度专业化的和监管领域特别禁止，例如医学和手术，视频语义细分可能具有重要应用，但数据和专家注释是稀缺的。在这些设置中，可以在培训期间利用时间线索和解剖结构来提高性能。在这里，我们呈现时间限制的神经网络（TCNN），是用于外科视频的视频语义分割的半监督框架。在这项工作中，我们表明AutoEncoder网络可用于有效地提供空间和时间监控信号来培训深度学习模型。我们在新推出的腹腔镜胆囊切除术文程序，内测序和对CADIS，CADIS的公共数据集的适应时测试我们的方法。我们证明，可以利用预测面罩的较低尺寸表示，以在稀疏标记的数据集中提供一致的改进，这些数据集在推理时间不具有额外的计算成本。此外，TCNN框架是模型无关的，可以与其他模型设计选择结合使用，具有最小的额外复杂性。

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.