我们引入了一种新的神经信号模型，设计用于有效的大型信号的高分辨率表示。我们的多尺度隐式神经表示（矿工）中的关键创新是通过拉普拉斯金字塔的内部表示，它提供了信号的稀疏多尺度分解，可捕获跨尺度的信号的正交部分。我们通过用小型MLP在每个尺度上代表金字塔的小差异斑块来利用拉普拉斯金字塔的优势。这使网络能够适应从粗尺度到细尺度的能力增加，仅代表具有强信号能量的信号的一部分。每个MLP的参数是从粗到细节优化的，从而在更粗糙的尺度下更快地近似，从而最终是一个非常快速的训练过程。我们将矿工应用于一系列大规模信号表示任务，包括吉吉像素图像和非常大的点云，并证明它需要少于参数的25％，33％的内存足迹和10％的计算时间和10％竞争技术（例如橡子）达到相同的表示准确性。

We present VoxelMorph, a fast learning-based framework for deformable, pairwise medical image registration. Traditional registration methods optimize an objective function for each pair of images, which can be time-consuming for large datasets or rich deformation models. In contrast to this approach, and building on recent learning-based methods, we formulate registration as a function that maps an input image pair to a deformation field that aligns these images. We parameterize the function via a convolutional neural network (CNN), and optimize the parameters of the neural network on a set of images. Given a new pair of scans, VoxelMorph rapidly computes a deformation field by directly evaluating the function. In this work, we explore two different training strategies. In the first (unsupervised) setting, we train the model to maximize standard image matching objective functions that are based on the image intensities. In the second setting, we leverage auxiliary segmentations available in the training data. We demonstrate that the unsupervised model's accuracy is comparable to state-of-the-art methods, while operating orders of magnitude faster. We also show that VoxelMorph trained with auxiliary data improves registration accuracy at test time, and evaluate the effect of training set size on registration. Our method promises to speed up medical image analysis and processing pipelines, while facilitating novel directions in learning-based registration and its applications. Our code is freely available at http://voxelmorph.csail.mit.edu.

This work builds on the models and concepts presented in part 1 to learn approximate dictionary representations of Koopman operators from data. Part I of this paper presented a methodology for arguing the subspace invariance of a Koopman dictionary. This methodology was demonstrated on the state-inclusive logistic lifting (SILL) basis. This is an affine basis augmented with conjunctive logistic functions. The SILL dictionary's nonlinear functions are homogeneous, a norm in data-driven dictionary learning of Koopman operators. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary. In a widely used algorithm, Extended Dynamic Mode Decomposition, the dictionary functions are drawn from a fixed class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators. Part 2 of this paper will extend this explanation by demonstrating the subspace invariant of heterogeneous dictionaries and presenting a head-to-head numerical comparison of deepDMD and low-parameter heterogeneous dictionary learning.

Cloud computing holds the promise of reduced costs through economies of scale. To realize this promise, cloud computing vendors typically solve sequential resource allocation problems, where customer workloads are packed on shared hardware. Virtual machines (VM) form the foundation of modern cloud computing as they help logically abstract user compute from shared physical infrastructure. Traditionally, VM packing problems are solved by predicting demand, followed by a Model Predictive Control (MPC) optimization over a future horizon. We introduce an approximate formulation of an industrial VM packing problem as an MILP with soft-constraints parameterized by the predictions. Recently, predict-and-optimize (PnO) was proposed for end-to-end training of prediction models by back-propagating the cost of decisions through the optimization problem. But, PnO is unable to scale to the large prediction horizons prevalent in cloud computing. To tackle this issue, we propose the Predict-and-Critic (PnC) framework that outperforms PnO with just a two-step horizon by leveraging reinforcement learning. PnC jointly trains a prediction model and a terminal Q function that approximates cost-to-go over a long horizon, by back-propagating the cost of decisions through the optimization problem \emph{and from the future}. The terminal Q function allows us to solve a much smaller two-step horizon optimization problem than the multi-step horizon necessary in PnO. We evaluate PnO and the PnC framework on two datasets, three workloads, and with disturbances not modeled in the optimization problem. We find that PnC significantly improves decision quality over PnO, even when the optimization problem is not a perfect representation of reality. We also find that hardening the soft constraints of the MILP and back-propagating through the constraints improves decision quality for both PnO and PnC.

New technologies and the availability of geospatial data have drawn attention to spatio-temporal biases present in society. For example: the COVID-19 pandemic highlighted disparities in the availability of broadband service and its role in the digital divide; the environmental justice movement in the United States has raised awareness to health implications for minority populations stemming from historical redlining practices; and studies have found varying quality and coverage in the collection and sharing of open-source geospatial data. Despite the extensive literature on machine learning (ML) fairness, few algorithmic strategies have been proposed to mitigate such biases. In this paper we highlight the unique challenges for quantifying and addressing spatio-temporal biases, through the lens of use cases presented in the scientific literature and media. We envision a roadmap of ML strategies that need to be developed or adapted to quantify and overcome these challenges -- including transfer learning, active learning, and reinforcement learning techniques. Further, we discuss the potential role of ML in providing guidance to policy makers on issues related to spatial fairness.

We consider the problem of multi-agent navigation and collision avoidance when observations are limited to the local neighborhood of each agent. We propose InforMARL, a novel architecture for multi-agent reinforcement learning (MARL) which uses local information intelligently to compute paths for all the agents in a decentralized manner. Specifically, InforMARL aggregates information about the local neighborhood of agents for both the actor and the critic using a graph neural network and can be used in conjunction with any standard MARL algorithm. We show that (1) in training, InforMARL has better sample efficiency and performance than baseline approaches, despite using less information, and (2) in testing, it scales well to environments with arbitrary numbers of agents and obstacles.

Rates of missing data often depend on record-keeping policies and thus may change across times and locations, even when the underlying features are comparatively stable. In this paper, we introduce the problem of Domain Adaptation under Missingness Shift (DAMS). Here, (labeled) source data and (unlabeled) target data would be exchangeable but for different missing data mechanisms. We show that when missing data indicators are available, DAMS can reduce to covariate shift. Focusing on the setting where missing data indicators are absent, we establish the following theoretical results for underreporting completely at random: (i) covariate shift is violated (adaptation is required); (ii) the optimal source predictor can perform worse on the target domain than a constant one; (iii) the optimal target predictor can be identified, even when the missingness rates themselves are not; and (iv) for linear models, a simple analytic adjustment yields consistent estimates of the optimal target parameters. In experiments on synthetic and semi-synthetic data, we demonstrate the promise of our methods when assumptions hold. Finally, we discuss a rich family of future extensions.

数据驱动的湍流建模正在经历数据科学算法和硬件开发后的兴趣激增。我们讨论了一种使用可区分物理范式的方法，该方法将已知的物理学与机器学习结合起来，以开发汉堡湍流的闭合模型。我们将1D汉堡系统视为一种原型测试问题，用于建模以对流为主的湍流问题中未解决的术语。我们训练一系列模型，这些模型在后验损失函数上结合了不同程度的物理假设，以测试模型在一系列系统参数（包括粘度，时间和网格分辨率）上的疗效。我们发现，以部分微分方程形式的归纳偏差的约束模型包含已知物理或现有闭合方法会产生高度数据效率，准确和可推广的模型，并且表现优于最先进的基准。以物理信息形式添加结构还为模型带来了一定程度的解释性，可能为封闭建模的未来提供了垫脚石。

机器学习（ML）是指根据大量数据预测有意义的输出或对复杂系统进行分类的计算机算法。 ML应用于各个领域，包括自然科学，工程，太空探索甚至游戏开发。本文的重点是在化学和生物海洋学领域使用机器学习。在预测全球固定氮水平，部分二氧化碳压力和其他化学特性时，ML的应用是一种有前途的工具。机器学习还用于生物海洋学领域，可从各种图像（即显微镜，流车和视频记录器），光谱仪和其他信号处理技术中检测浮游形式。此外，ML使用其声学成功地对哺乳动物进行了分类，在特定的环境中检测到濒临灭绝的哺乳动物和鱼类。最重要的是，使用环境数据，ML被证明是预测缺氧条件和有害藻华事件的有效方法，这是对环境监测的重要测量。此外，机器学习被用来为各种物种构建许多对其他研究人员有用的数据库，而创建新算法将帮助海洋研究界更好地理解海洋的化学和生物学。