Existing Optimal Transport (OT) methods mainly derive the optimal transport plan/matching under the criterion of transport cost/distance minimization, which may cause incorrect matching in some cases. In many applications, annotating a few matched keypoints across domains is reasonable or even effortless in annotation burden. It is valuable to investigate how to leverage the annotated keypoints to guide the correct matching in OT. In this paper, we propose a novel KeyPoint-Guided model by ReLation preservation (KPG-RL) that searches for the optimal matching (i.e., transport plan) guided by the keypoints in OT. To impose the keypoints in OT, first, we propose a mask-based constraint of the transport plan that preserves the matching of keypoint pairs. Second, we propose to preserve the relation of each data point to the keypoints to guide the matching. The proposed KPG-RL model can be solved by Sinkhorn's algorithm and is applicable even when distributions are supported in different spaces. We further utilize the relation preservation constraint in the Kantorovich Problem and Gromov-Wasserstein model to impose the guidance of keypoints in them. Meanwhile, the proposed KPG-RL model is extended to the partial OT setting. Moreover, we deduce the dual formulation of the KPG-RL model, which is solved using deep learning techniques. Based on the learned transport plan from dual KPG-RL, we propose a novel manifold barycentric projection to transport source data to the target domain. As applications, we apply the proposed KPG-RL model to the heterogeneous domain adaptation and image-to-image translation. Experiments verified the effectiveness of the proposed approach.
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Autonomous ultrasound (US) scanning has attracted increased attention, and it has been seen as a potential solution to overcome the limitations of conventional US examinations, such as inter-operator variations. However, it is still challenging to autonomously and accurately transfer a planned scan trajectory on a generic atlas to the current setup for different patients, particularly for thorax applications with limited acoustic windows. To address this challenge, we proposed a skeleton graph-based non-rigid registration to adapt patient-specific properties using subcutaneous bone surface features rather than the skin surface. To this end, the self-organization mapping is successively used twice to unify the input point cloud and extract the key points, respectively. Afterward, the minimal spanning tree is employed to generate a tree graph to connect all extracted key points. To appropriately characterize the rib cartilage outline to match the source and target point cloud, the path extracted from the tree graph is optimized by maximally maintaining continuity throughout each rib. To validate the proposed approach, we manually extract the US cartilage point cloud from one volunteer and seven CT cartilage point clouds from different patients. The results demonstrate that the proposed graph-based registration is more effective and robust in adapting to the inter-patient variations than the ICP (distance error mean/SD: 5.0/1.9 mm vs 8.6/6.7 mm on seven CTs).
We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morris. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives.
Noisy marginals are a common form of confidentiality-protecting data release and are useful for many downstream tasks such as contingency table analysis, construction of Bayesian networks, and even synthetic data generation. Privacy mechanisms that provide unbiased noisy answers to linear queries (such as marginals) are known as matrix mechanisms. We propose ResidualPlanner, a matrix mechanism for marginals with Gaussian noise that is both optimal and scalable. ResidualPlanner can optimize for many loss functions that can be written as a convex function of marginal variances (prior work was restricted to just one predefined objective function). ResidualPlanner can optimize the accuracy of marginals in large scale settings in seconds, even when the previous state of the art (HDMM) runs out of memory. It even runs on datasets with 100 attributes in a couple of minutes. Furthermore ResidualPlanner can efficiently compute variance/covariance values for each marginal (prior methods quickly run out of memory, even for relatively small datasets).
We derive optimal statistical decision rules for discrete choice problems when payoffs depend on a partially-identified parameter $\theta$ and the decision maker can use a point-identified parameter $P$ to deduce restrictions on $\theta$. Leading examples include optimal treatment choice under partial identification and optimal pricing with rich unobserved heterogeneity. Our optimal decision rules minimize the maximum risk or regret over the identified set of payoffs conditional on $P$ and use the data efficiently to learn about $P$. We discuss implementation of optimal decision rules via the bootstrap and Bayesian methods, in both parametric and semiparametric models. We provide detailed applications to treatment choice and optimal pricing. Using a limits of experiments framework, we show that our optimal decision rules can dominate seemingly natural alternatives. Our asymptotic approach is well suited for realistic empirical settings in which the derivation of finite-sample optimal rules is intractable.
Nonlinear model predictive control (NMPC) solves a multivariate optimization problem to estimate the system's optimal control inputs in each control cycle. Such optimization is made more difficult by several factors, such as nonlinearities inherited in the system, highly coupled inputs, and various constraints related to the system's physical limitations. These factors make the optimization to be non-convex and hard to solve traditionally. Genetic algorithm (GA) is typically used extensively to tackle such optimization in several application domains because it does not involve differential calculation or gradient evaluation in its solution estimation. However, the size of the search space in which the GA searches for the optimal control inputs is crucial for the applicability of the GA with systems that require fast response. This paper proposes an approach to accelerate the genetic optimization of NMPC by learning optimal search space size. The proposed approach trains a multivariate regression model to adaptively predict the best smallest search space in every control cycle. The estimated best smallest size of search space is fed to the GA to allow for searching the optimal control inputs within this search space. The proposed approach not only reduces the GA's computational time but also improves the chance of obtaining the optimal control inputs in each cycle. The proposed approach was evaluated on two nonlinear systems and compared with two other genetic-based NMPC approaches implemented on the GPU of a Nvidia Jetson TX2 embedded platform in a processor-in-the-loop (PIL) fashion. The results show that the proposed approach provides a 39-53\% reduction in computational time. Additionally, it increases the convergence percentage to the optimal control inputs within the cycle's time by 48-56\%, resulting in a significant performance enhancement. The source code is available on GitHub.
Monocular 3D human pose estimation from RGB images has attracted significant attention in recent years. However, recent models depend on supervised training with 3D pose ground truth data or known pose priors for their target domains. 3D pose data is typically collected with motion capture devices, severely limiting their applicability. In this paper, we present a heuristic weakly supervised 3D human pose (HW-HuP) solution to estimate 3D poses in when no ground truth 3D pose data is available. HW-HuP learns partial pose priors from 3D human pose datasets and uses easy-to-access observations from the target domain to estimate 3D human pose and shape in an optimization and regression cycle. We employ depth data for weak supervision during training, but not inference. We show that HW-HuP meaningfully improves upon state-of-the-art models in two practical settings where 3D pose data can hardly be obtained: human poses in bed, and infant poses in the wild. Furthermore, we show that HW-HuP retains comparable performance to cutting-edge models on public benchmarks, even when such models train on 3D pose data.
Transformer-based methods have demonstrated superior performance for monocular 3D object detection recently, which predicts 3D attributes from a single 2D image. Most existing transformer-based methods leverage visual and depth representations to explore valuable query points on objects, and the quality of the learned queries has a great impact on detection accuracy. Unfortunately, existing unsupervised attention mechanisms in transformer are prone to generate low-quality query features due to inaccurate receptive fields, especially on hard objects. To tackle this problem, this paper proposes a novel ``Supervised Scale-constrained Deformable Attention'' (SSDA) for monocular 3D object detection. Specifically, SSDA presets several masks with different scales and utilizes depth and visual features to predict the local feature for each query. Imposing the scale constraint, SSDA could well predict the accurate receptive field of a query to support robust query feature generation. What is more, SSDA is assigned with a Weighted Scale Matching (WSM) loss to supervise scale prediction, which presents more confident results as compared to the unsupervised attention mechanisms. Extensive experiments on ``KITTI'' demonstrate that SSDA significantly improves the detection accuracy especially on moderate and hard objects, yielding SOTA performance as compared to the existing approaches. Code will be publicly available at //github.com/mikasa3lili/SSD-MonoDETR.
We demonstrate the relevance of an algorithm called generalized iterative scaling (GIS) or simultaneous multiplicative algebraic reconstruction technique (SMART) and its rescaled block-iterative version (RBI-SMART) in the field of optimal transport (OT). Many OT problems can be tackled through the use of entropic regularization by solving the Schr\"odinger problem, which is an information projection problem, that is, with respect to the Kullback--Leibler divergence. Here we consider problems that have several affine constraints. It is well-known that cyclic information projections onto the individual affine sets converge to the solution. In practice, however, even these individual projections are not explicitly available in general. In this paper, we exchange them for one GIS iteration. If this is done for every affine set, we obtain RBI-SMART. We provide a convergence proof using an interpretation of these iterations as two-step affine projections in an equivalent problem. This is done in a slightly more general setting than RBI-SMART, since we use a mix of explicitly known information projections and GIS iterations. We proceed to specialize this algorithm to several OT applications. First, we find the measure that minimizes the regularized OT divergence to a given measure under moment constraints. Second and third, the proposed framework yields an algorithm for solving a regularized martingale OT problem, as well as a relaxed version of the barycentric weak OT problem. Finally, we show an approach from the literature for unbalanced OT problems.
Direct policy optimization in reinforcement learning is usually solved with policy-gradient algorithms, which optimize policy parameters via stochastic gradient ascent. This paper provides a new theoretical interpretation and justification of these algorithms. First, we formulate direct policy optimization in the optimization by continuation framework. The latter is a framework for optimizing nonconvex functions where a sequence of surrogate objective functions, called continuations, are locally optimized. Second, we show that optimizing affine Gaussian policies and performing entropy regularization can be interpreted as implicitly optimizing deterministic policies by continuation. Based on these theoretical results, we argue that exploration in policy-gradient algorithms consists in computing a continuation of the return of the policy at hand, and that the variance of policies should be history-dependent functions adapted to avoid local extrema rather than to maximize the return of the policy.
The Dirichlet process has been pivotal to the development of Bayesian nonparametrics, allowing one to learn the law of the observations through closed-form expressions. Still, its learning mechanism is often too simplistic and many generalizations have been proposed to increase its flexibility, a popular one being the class of normalized completely random measures. Here we investigate a simple yet fundamental matter: will a different prior actually guarantee a different learning outcome? To this end, we develop a new framework for assessing the merging rate of opinions based on three leading pillars: i) the investigation of identifiability of completely random measures; ii) the measurement of their discrepancy through a novel optimal transport distance; iii) the establishment of general techniques to conduct posterior analyses, unravelling both finite-sample and asymptotic behaviour of the distance as the number of observations grows. Our findings provide neat and interpretable insights on the impact of popular Bayesian nonparametric priors, avoiding the usual restrictive assumptions on the data-generating process.
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