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The miniaturization of inertial measurement units (IMUs) facilitates their widespread use in a growing number of application domains. Orientation estimation is a prerequisite for most further data processing steps in inertial motion tracking, such as position/velocity estimation, joint angle estimation, and 3D visualization. Errors in the estimated orientations severely affect all further processing steps. Recent systematic comparisons of existing algorithms show that out-of-the-box accuracy is often low and that application-specific tuning is required to obtain high accuracy. In the present work, we propose and extensively evaluate a quaternion-based orientation estimation algorithm that is based on a novel approach of filtering the acceleration measurements in an almost-inertial frame and that includes extensions for gyroscope bias estimation and magnetic disturbance rejection, as well as a variant for offline data processing. In contrast to all existing work, we perform an extensive evaluation, using a large collection of publicly available datasets and eight literature methods for comparison. The proposed method consistently outperforms all literature methods and achieves an average RMSE of 2.9{\deg}, while the errors obtained with literature methods range from 5.3{\deg} to 16.7{\deg}. Since the evaluation was performed with one single fixed parametrization across a very diverse dataset collection, we conclude that the proposed method provides unprecedented out-of-the-box performance for a broad range of motions, sensor hardware, and environmental conditions. This gain in orientation estimation accuracy is expected to advance the field of IMU-based motion analysis and provide performance benefits in numerous applications. The provided open-source implementation makes it easy to employ the proposed method.

The stochastic mirror descent (SMD) algorithm is a general class of training algorithms, which includes the celebrated stochastic gradient descent (SGD), as a special case. It utilizes a mirror potential to influence the implicit bias of the training algorithm. In this paper we explore the performance of the SMD iterates on mean-field ensemble models. Our results generalize earlier ones obtained for SGD on such models. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial differential equation to which the continuous time process converges in the asymptotic regime of large networks. The impact of the mirror potential appears through a multiplicative term that is equal to the inverse of its Hessian and which can be interpreted as defining a gradient flow over an appropriately defined Riemannian manifold. We provide numerical simulations which allow us to study and characterize the effect of the mirror potential on the performance of networks trained with SMD for some binary classification problems.

Learning-based methods in robotics hold the promise of generalization, but what can be done if a learned policy does not generalize to a new situation? In principle, if an agent can at least evaluate its own success (i.e., with a reward classifier that generalizes well even when the policy does not), it could actively practice the task and finetune the policy in this situation. We study this problem in the setting of industrial insertion tasks, such as inserting connectors in sockets and setting screws. Existing algorithms rely on precise localization of the connector or socket and carefully managed physical setups, such as assembly lines, to succeed at the task. But in unstructured environments such as homes or even some industrial settings, robots cannot rely on precise localization and may be tasked with previously unseen connectors. Offline reinforcement learning on a variety of connector insertion tasks is a potential solution, but what if the robot is tasked with inserting previously unseen connector? In such a scenario, we will still need methods that can robustly solve such tasks with online practice. One of the main observations we make in this work is that, with a suitable representation learning and domain generalization approach, it can be significantly easier for the reward function to generalize to a new but structurally similar task (e.g., inserting a new type of connector) than for the policy. This means that a learned reward function can be used to facilitate the finetuning of the robot's policy in situations where the policy fails to generalize in zero shot, but the reward function generalizes successfully. We show that such an approach can be instantiated in the real world, pretrained on 50 different connectors, and successfully finetuned to new connectors via the learned reward function. Videos can be viewed at //sites.google.com/view/learningonthejob

The area under the ROC curve (AUC) is one of the most widely used performance measures for classification models in machine learning. However, it summarizes the true positive rates (TPRs) over all false positive rates (FPRs) in the ROC space, which may include the FPRs with no practical relevance in some applications. The partial AUC, as a generalization of the AUC, summarizes only the TPRs over a specific range of the FPRs and is thus a more suitable performance measure in many real-world situations. Although partial AUC optimization in a range of FPRs had been studied, existing algorithms are not scalable to big data and not applicable to deep learning. To address this challenge, we cast the problem into a non-smooth difference-of-convex (DC) program for any smooth predictive functions (e.g., deep neural networks), which allowed us to develop an efficient approximated gradient descent method based on the Moreau envelope smoothing technique, inspired by recent advances in non-smooth DC optimization. To increase the efficiency of large data processing, we used an efficient stochastic block coordinate update in our algorithm. Our proposed algorithm can also be used to minimize the sum of ranked range loss, which also lacks efficient solvers. We established a complexity of $\tilde O(1/\epsilon^6)$ for finding a nearly $\epsilon$-critical solution. Finally, we numerically demonstrated the effectiveness of our proposed algorithms for both partial AUC maximization and sum of ranked range loss minimization.

Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of certain partial differential equations (PDEs) using conventional techniques where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the "plug-and-play" integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multi-scale coupling of finite element FOMs in solid mechanics, we present a new extension of the Schwarz formulation that enables ROM-FOM and ROM-ROM coupling in nonlinear solid mechanics. In order to maintain efficiency, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting approach. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.

An important issue in medical image processing is to be able to estimate not only the performances of algorithms but also the precision of the estimation of these performances. Reporting precision typically amounts to reporting standard-error of the mean (SEM) or equivalently confidence intervals. However, this is rarely done in medical image segmentation studies. In this paper, we aim to estimate what is the typical confidence that can be expected in such studies. To that end, we first perform experiments for Dice metric estimation using a standard deep learning model (U-net) and a classical task from the Medical Segmentation Decathlon. We extensively study precision estimation using both Gaussian assumption and bootstrapping (which does not require any assumption on the distribution). We then perform simulations for other test set sizes and performance spreads. Overall, our work shows that small test sets lead to wide confidence intervals ($\sim$6 points of Dice for 20 samples) and that, in order to obtain a confidence interval narrower than 2, it is necessary to have at least 200 test samples.

Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.

We present an alternating least squares type numerical optimization scheme to estimate conditionally-independent mixture models in $\mathbb{R}^n$, with minimal additional distributional assumptions. Following the method of moments, we tackle a coupled system of low-rank tensor decomposition problems. The steep costs associated with high-dimensional tensors are avoided, through the development of specialized tensor-free operations. Numerical experiments illustrate the performance of the algorithm and its applicability to various models and applications. In many cases the results exhibit improved reliability over the expectation-maximization algorithm, with similar time and storage costs. We also provide some supporting theory, establishing identifiability and local linear convergence.

We consider optimization problems in which the goal is find a $k$-dimensional subspace of $\mathbb{R}^n$, $k<<n$, which minimizes a convex and smooth loss. Such problems generalize the fundamental task of principal component analysis (PCA) to include robust and sparse counterparts, and logistic PCA for binary data, among others. This problem could be approached either via nonconvex gradient methods with highly-efficient iterations, but for which arguing about fast convergence to a global minimizer is difficult or, via a convex relaxation for which arguing about convergence to a global minimizer is straightforward, but the corresponding methods are often inefficient in high dimensions. In this work we bridge these two approaches under a strict complementarity assumption, which in particular implies that the optimal solution to the convex relaxation is unique and is also the optimal solution to the original nonconvex problem. Our main result is a proof that a natural nonconvex gradient method which is \textit{SVD-free} and requires only a single QR-factorization of an $n\times k$ matrix per iteration, converges locally with a linear rate. We also establish linear convergence results for the nonconvex projected gradient method, and the Frank-Wolfe method when applied to the convex relaxation.

The training of high-dimensional regression models on comparably sparse data is an important yet complicated topic, especially when there are many more model parameters than observations in the data. From a Bayesian perspective, inference in such cases can be achieved with the help of shrinkage prior distributions, at least for generalized linear models. However, real-world data usually possess multilevel structures, such as repeated measurements or natural groupings of individuals, which existing shrinkage priors are not built to deal with. We generalize and extend one of these priors, the R2D2 prior by Zhang et al. (2020), to linear multilevel models leading to what we call the R2D2M2 prior. The proposed prior enables both local and global shrinkage of the model parameters. It comes with interpretable hyperparameters, which we show to be intrinsically related to vital properties of the prior, such as rates of concentration around the origin, tail behavior, and amount of shrinkage the prior exerts. We offer guidelines on how to select the prior's hyperparameters by deriving shrinkage factors and measuring the effective number of non-zero model coefficients. Hence, the user can readily evaluate and interpret the amount of shrinkage implied by a specific choice of hyperparameters. Finally, we perform extensive experiments on simulated and real data, showing that our inference procedure for the prior is well calibrated, has desirable global and local regularization properties and enables the reliable and interpretable estimation of much more complex Bayesian multilevel models than was previously possible.