Jiacheng Zhu

William Han, Jielin Qiu, Jiacheng Zhu et al. · cmu

Brain Signals, such as Electroencephalography (EEG), and human languages have been widely explored independently for many downstream tasks, however, the connection between them has not been well explored. In this study, we explore the relationship and dependency between EEG and language. To study at the representation level, we introduced \textbf{MTAM}, a \textbf{M}ultimodal \textbf{T}ransformer \textbf{A}lignment \textbf{M}odel, to observe coordinated representations between the two modalities. We used various relationship alignment-seeking techniques, such as Canonical Correlation Analysis and Wasserstein Distance, as loss functions to transfigure features. On downstream applications, sentiment analysis and relation detection, we achieved new state-of-the-art results on two datasets, ZuCo and K-EmoCon. Our method achieved an F1-score improvement of 1.7% on K-EmoCon and 9.3% on Zuco datasets for sentiment analysis, and 7.4% on ZuCo for relation detection. In addition, we provide interpretations of the performance improvement: (1) feature distribution shows the effectiveness of the alignment module for discovering and encoding the relationship between EEG and language; (2) alignment weights show the influence of different language semantics as well as EEG frequency features; (3) brain topographical maps provide an intuitive demonstration of the connectivity in the brain regions. Our code is available at \url{https://github.com/Jason-Qiu/EEG_Language_Alignment}.

Jielin Qiu, Jiacheng Zhu, William Han et al. · cmu, microsoft-research

Multimodal summarization with multimodal output (MSMO) has emerged as a promising research direction. Nonetheless, numerous limitations exist within existing public MSMO datasets, including insufficient maintenance, data inaccessibility, limited size, and the absence of proper categorization, which pose significant challenges. To address these challenges and provide a comprehensive dataset for this new direction, we have meticulously curated the \textbf{MMSum} dataset. Our new dataset features (1) Human-validated summaries for both video and textual content, providing superior human instruction and labels for multimodal learning. (2) Comprehensively and meticulously arranged categorization, spanning 17 principal categories and 170 subcategories to encapsulate a diverse array of real-world scenarios. (3) Benchmark tests performed on the proposed dataset to assess various tasks and methods, including \textit{video summarization}, \textit{text summarization}, and \textit{multimodal summarization}. To champion accessibility and collaboration, we will release the \textbf{MMSum} dataset and the data collection tool as fully open-source resources, fostering transparency and accelerating future developments. Our project website can be found at~\url{https://mmsum-dataset.github.io/}

Hanjiang Hu, Zuxin Liu, Linyi Li et al. · cmu

A vast literature shows that the learning-based visual perception model is sensitive to adversarial noises, but few works consider the robustness of robotic perception models under widely-existing camera motion perturbations. To this end, we study the robustness of the visual perception model under camera motion perturbations to investigate the influence of camera motion on robotic perception. Specifically, we propose a motion smoothing technique for arbitrary image classification models, whose robustness under camera motion perturbations could be certified. The proposed robustness certification framework based on camera motion smoothing provides tight and scalable robustness guarantees for visual perception modules so that they are applicable to wide robotic applications. As far as we are aware, this is the first work to provide robustness certification for the deep perception module against camera motions, which improves the trustworthiness of robotic perception. A realistic indoor robotic dataset with a dense point cloud map for the entire room, MetaRoom, is introduced for the challenging certifiable robust perception task. We conduct extensive experiments to validate the certification approach via motion smoothing against camera motion perturbations. Our framework guarantees the certified accuracy of 81.7% against camera translation perturbation along depth direction within -0.1m ~ 0.1m. We also validate the effectiveness of our method on the real-world robot by conducting hardware experiments on the robotic arm with an eye-in-hand camera. The code is available at https://github.com/HanjiangHu/camera-motion-smoothing.

Hanjiang Hu, Zuxin Liu, Linyi Li et al. · cmu

Deep learning-based visual perception models lack robustness when faced with camera motion perturbations in practice. The current certification process for assessing robustness is costly and time-consuming due to the extensive number of image projections required for Monte Carlo sampling in the 3D camera motion space. To address these challenges, we present a novel, efficient, and practical framework for certifying the robustness of 3D-2D projective transformations against camera motion perturbations. Our approach leverages a smoothing distribution over the 2D pixel space instead of in the 3D physical space, eliminating the need for costly camera motion sampling and significantly enhancing the efficiency of robustness certifications. With the pixel-wise smoothed classifier, we are able to fully upper bound the projection errors using a technique of uniform partitioning in camera motion space. Additionally, we extend our certification framework to a more general scenario where only a single-frame point cloud is required in the projection oracle. Through extensive experimentation, we validate the trade-off between effectiveness and efficiency enabled by our proposed method. Remarkably, our approach achieves approximately 80% certified accuracy while utilizing only 30% of the projected image frames. The code is available at https://github.com/HanjiangHu/pixel-wise-smoothing.

Zuxin Liu, Zijian Guo, Haohong Lin et al. · cmu

This paper presents a comprehensive benchmarking suite tailored to offline safe reinforcement learning (RL) challenges, aiming to foster progress in the development and evaluation of safe learning algorithms in both the training and deployment phases. Our benchmark suite contains three packages: 1) expertly crafted safe policies, 2) D4RL-styled datasets along with environment wrappers, and 3) high-quality offline safe RL baseline implementations. We feature a methodical data collection pipeline powered by advanced safe RL algorithms, which facilitates the generation of diverse datasets across 38 popular safe RL tasks, from robot control to autonomous driving. We further introduce an array of data post-processing filters, capable of modifying each dataset's diversity, thereby simulating various data collection conditions. Additionally, we provide elegant and extensible implementations of prevalent offline safe RL algorithms to accelerate research in this area. Through extensive experiments with over 50000 CPU and 800 GPU hours of computations, we evaluate and compare the performance of these baseline algorithms on the collected datasets, offering insights into their strengths, limitations, and potential areas of improvement. Our benchmarking framework serves as a valuable resource for researchers and practitioners, facilitating the development of more robust and reliable offline safe RL solutions in safety-critical applications. The benchmark website is available at \url{www.offline-saferl.org}.

Jielin Qiu, William Han, Jiacheng Zhu et al. · cmu

Recent advancements in Large Language Models (LLMs) have drawn increasing attention since the learned embeddings pretrained on large-scale datasets have shown powerful ability in various downstream applications. However, whether the learned knowledge by LLMs can be transferred to clinical cardiology remains unknown. In this work, we aim to bridge this gap by transferring the knowledge of LLMs to clinical Electrocardiography (ECG). We propose an approach for cardiovascular disease diagnosis and automatic ECG diagnosis report generation. We also introduce an additional loss function by Optimal Transport (OT) to align the distribution between ECG and language embedding. The learned embeddings are evaluated on two downstream tasks: (1) automatic ECG diagnosis report generation, and (2) zero-shot cardiovascular disease detection. Our approach is able to generate high-quality cardiac diagnosis reports and also achieves competitive zero-shot classification performance even compared with supervised baselines, which proves the feasibility of transferring knowledge from LLMs to the cardiac domain.

Yihang Yao, Zuxin Liu, Zhepeng Cen et al. · cmu

Safe reinforcement learning (RL) focuses on training reward-maximizing agents subject to pre-defined safety constraints. Yet, learning versatile safe policies that can adapt to varying safety constraint requirements during deployment without retraining remains a largely unexplored and challenging area. In this work, we formulate the versatile safe RL problem and consider two primary requirements: training efficiency and zero-shot adaptation capability. To address them, we introduce the Conditioned Constrained Policy Optimization (CCPO) framework, consisting of two key modules: (1) Versatile Value Estimation (VVE) for approximating value functions under unseen threshold conditions, and (2) Conditioned Variational Inference (CVI) for encoding arbitrary constraint thresholds during policy optimization. Our extensive experiments demonstrate that CCPO outperforms the baselines in terms of safety and task performance while preserving zero-shot adaptation capabilities to different constraint thresholds data-efficiently. This makes our approach suitable for real-world dynamic applications.

Haohong Lin, Wenhao Ding, Jian Chen et al. · cmu

Offline model-based reinforcement learning (MBRL) enhances data efficiency by utilizing pre-collected datasets to learn models and policies, especially in scenarios where exploration is costly or infeasible. Nevertheless, its performance often suffers from the objective mismatch between model and policy learning, resulting in inferior performance despite accurate model predictions. This paper first identifies the primary source of this mismatch comes from the underlying confounders present in offline data for MBRL. Subsequently, we introduce \textbf{B}ilin\textbf{E}ar \textbf{CAUS}al r\textbf{E}presentation~(BECAUSE), an algorithm to capture causal representation for both states and actions to reduce the influence of the distribution shift, thus mitigating the objective mismatch problem. Comprehensive evaluations on 18 tasks that vary in data quality and environment context demonstrate the superior performance of BECAUSE over existing offline RL algorithms. We show the generalizability and robustness of BECAUSE under fewer samples or larger numbers of confounders. Additionally, we offer theoretical analysis of BECAUSE to prove its error bound and sample efficiency when integrating causal representation into offline MBRL.

Jielin Qiu, Jiacheng Zhu, Mengdi Xu et al.

Multimedia summarization with multimodal output can play an essential role in real-world applications, i.e., automatically generating cover images and titles for news articles or providing introductions to online videos. In this work, we propose a multimodal hierarchical multimedia summarization (MHMS) framework by interacting visual and language domains to generate both video and textual summaries. Our MHMS method contains video and textual segmentation and summarization module, respectively. It formulates a cross-domain alignment objective with optimal transport distance which leverages cross-domain interaction to generate the representative keyframe and textual summary. We evaluated MHMS on three recent multimodal datasets and demonstrated the effectiveness of our method in producing high-quality multimodal summaries.

Jiacheng Zhu, Jielin Qiu, Zhuolin Yang et al.

There has been an increased interest in applying deep neural networks to automatically interpret and analyze the 12-lead electrocardiogram (ECG). The current paradigms with machine learning methods are often limited by the amount of labeled data. This phenomenon is particularly problematic for clinically-relevant data, where labeling at scale can be time-consuming and costly in terms of the specialized expertise and human effort required. Moreover, deep learning classifiers may be vulnerable to adversarial examples and perturbations, which could have catastrophic consequences, for example, when applied in the context of medical treatment, clinical trials, or insurance claims. In this paper, we propose a physiologically-inspired data augmentation method to improve performance and increase the robustness of heart disease detection based on ECG signals. We obtain augmented samples by perturbing the data distribution towards other classes along the geodesic in Wasserstein space. To better utilize domain-specific knowledge, we design a ground metric that recognizes the difference between ECG signals based on physiologically determined features. Learning from 12-lead ECG signals, our model is able to distinguish five categories of cardiac conditions. Our results demonstrate improvements in accuracy and robustness, reflecting the effectiveness of our data augmentation method.

Jielin Qiu, Jiacheng Zhu, Mengdi Xu et al.

Multimedia summarization with multimodal output (MSMO) is a recently explored application in language grounding. It plays an essential role in real-world applications, i.e., automatically generating cover images and titles for news articles or providing introductions to online videos. However, existing methods extract features from the whole video and article and use fusion methods to select the representative one, thus usually ignoring the critical structure and varying semantics. In this work, we propose a Semantics-Consistent Cross-domain Summarization (SCCS) model based on optimal transport alignment with visual and textual segmentation. In specific, our method first decomposes both video and article into segments in order to capture the structural semantics, respectively. Then SCCS follows a cross-domain alignment objective with optimal transport distance, which leverages multimodal interaction to match and select the visual and textual summary. We evaluated our method on three recent multimodal datasets and demonstrated the effectiveness of our method in producing high-quality multimodal summaries.

Jiacheng Zhu, Gregory Darnell, Agni Kumar et al.

Heart rate variability (HRV) is a practical and noninvasive measure of autonomic nervous system activity, which plays an essential role in cardiovascular health. However, using HRV to assess physiology status is challenging. Even in clinical settings, HRV is sensitive to acute stressors such as physical activity, mental stress, hydration, alcohol, and sleep. Wearable devices provide convenient HRV measurements, but the irregularity of measurements and uncaptured stressors can bias conventional analytical methods. To better interpret HRV measurements for downstream healthcare applications, we learn a personalized diurnal rhythm as an accurate physiological indicator for each individual. We develop Physiological Multitask-Learning (PhysioMTL) by harnessing Optimal Transport theory within a Multitask-learning (MTL) framework. The proposed method learns an individual-specific predictive model from heterogeneous observations, and enables estimation of an optimal transport map that yields a push forward operation onto the demographic features for each task. Our model outperforms competing MTL methodologies on unobserved predictive tasks for synthetic and two real-world datasets. Specifically, our method provides remarkable prediction results on unseen held-out subjects given only $20\%$ of the subjects in real-world observational studies. Furthermore, our model enables a counterfactual engine that generates the effect of acute stressors and chronic conditions on HRV rhythms.

Jiacheng Zhu, Jielin Qiu, Aritra Guha et al.

We propose to study and promote the robustness of a model as per its performance through the interpolation of training data distributions. Specifically, (1) we augment the data by finding the worst-case Wasserstein barycenter on the geodesic connecting subpopulation distributions of different categories. (2) We regularize the model for smoother performance on the continuous geodesic path connecting subpopulation distributions. (3) Additionally, we provide a theoretical guarantee of robustness improvement and investigate how the geodesic location and the sample size contribute, respectively. Experimental validations of the proposed strategy on \textit{four} datasets, including CIFAR-100 and ImageNet, establish the efficacy of our method, e.g., our method improves the baselines' certifiable robustness on CIFAR10 up to $7.7\%$, with $16.8\%$ on empirical robustness on CIFAR-100. Our work provides a new perspective of model robustness through the lens of Wasserstein geodesic-based interpolation with a practical off-the-shelf strategy that can be combined with existing robust training methods.

Peide Huang, Mengdi Xu, Jiacheng Zhu et al.

Curriculum Reinforcement Learning (CRL) aims to create a sequence of tasks, starting from easy ones and gradually learning towards difficult tasks. In this work, we focus on the idea of framing CRL as interpolations between a source (auxiliary) and a target task distribution. Although existing studies have shown the great potential of this idea, it remains unclear how to formally quantify and generate the movement between task distributions. Inspired by the insights from gradual domain adaptation in semi-supervised learning, we create a natural curriculum by breaking down the potentially large task distributional shift in CRL into smaller shifts. We propose GRADIENT, which formulates CRL as an optimal transport problem with a tailored distance metric between tasks. Specifically, we generate a sequence of task distributions as a geodesic interpolation (i.e., Wasserstein barycenter) between the source and target distributions. Different from many existing methods, our algorithm considers a task-dependent contextual distance metric and is capable of handling nonparametric distributions in both continuous and discrete context settings. In addition, we theoretically show that GRADIENT enables smooth transfer between subsequent stages in the curriculum under certain conditions. We conduct extensive experiments in locomotion and manipulation tasks and show that our proposed GRADIENT achieves higher performance than baselines in terms of learning efficiency and asymptotic performance.

Yaru Niu, Shiyu Jin, Zeqing Zhang et al.

In this work, we first formulate the problem of robotic water scooping using goal-conditioned reinforcement learning. This task is particularly challenging due to the complex dynamics of fluids and the need to achieve multi-modal goals. The policy is required to successfully reach both position goals and water amount goals, which leads to a large convoluted goal state space. To overcome these challenges, we introduce Goal Sampling Adaptation for Scooping (GOATS), a curriculum reinforcement learning method that can learn an effective and generalizable policy for robot scooping tasks. Specifically, we use a goal-factorized reward formulation and interpolate position goal distributions and amount goal distributions to create curriculum throughout the learning process. As a result, our proposed method can outperform the baselines in simulation and achieves 5.46% and 8.71% amount errors on bowl scooping and bucket scooping tasks, respectively, under 1000 variations of initial water states in the tank and a large goal state space. Besides being effective in simulation environments, our method can efficiently adapt to noisy real-robot water-scooping scenarios with diverse physical configurations and unseen settings, demonstrating superior efficacy and generalizability. The videos of this work are available on our project page: https://sites.google.com/view/goatscooping.

Jielin Qiu, Jiacheng Zhu, Shiqi Liu et al. · cmu

Automated interpretation of electrocardiograms (ECG) has garnered significant attention with the advancements in machine learning methodologies. Despite the growing interest, most current studies focus solely on classification or regression tasks, which overlook a crucial aspect of clinical cardio-disease diagnosis: the diagnostic report generated by experienced human clinicians. In this paper, we introduce a novel approach to ECG interpretation, leveraging recent breakthroughs in Large Language Models (LLMs) and Vision-Transformer (ViT) models. Rather than treating ECG diagnosis as a classification or regression task, we propose an alternative method of automatically identifying the most similar clinical cases based on the input ECG data. Also, since interpreting ECG as images is more affordable and accessible, we process ECG as encoded images and adopt a vision-language learning paradigm to jointly learn vision-language alignment between encoded ECG images and ECG diagnosis reports. Encoding ECG into images can result in an efficient ECG retrieval system, which will be highly practical and useful in clinical applications. More importantly, our findings could serve as a crucial resource for providing diagnostic services in underdeveloped regions.

Jielin Qiu, Peide Huang, Makiya Nakashima et al.

Self-supervised learning is crucial for clinical imaging applications, given the lack of explicit labels in healthcare. However, conventional approaches that rely on precise vision-language alignment are not always feasible in complex clinical imaging modalities, such as cardiac magnetic resonance (CMR). CMR provides a comprehensive visualization of cardiac anatomy, physiology, and microstructure, making it challenging to interpret. Additionally, CMR reports require synthesizing information from sequences of images and different views, resulting in potentially weak alignment between the study and diagnosis report pair. To overcome these challenges, we propose \textbf{CMRformer}, a multimodal learning framework to jointly learn sequences of CMR images and associated cardiologist's reports. Moreover, one of the major obstacles to improving CMR study is the lack of large, publicly available datasets. To bridge this gap, we collected a large \textbf{CMR dataset}, which consists of 13,787 studies from clinical cases. By utilizing our proposed CMRformer and our collected dataset, we achieved remarkable performance in real-world clinical tasks, such as CMR image retrieval and diagnosis report retrieval. Furthermore, the learned representations are evaluated to be practically helpful for downstream applications, such as disease classification. Our work could potentially expedite progress in the CMR study and lead to more accurate and effective diagnosis and treatment.

Haoran Zhang, Aparna Balagopalan, Nassim Oufattole et al.

Large repositories of image-caption pairs are essential for the development of vision-language models. However, these datasets are often extracted from noisy data scraped from the web, and contain many mislabeled instances. In order to improve the reliability of downstream models, it is important to identify and filter images with incorrect captions. However, beyond filtering based on image-caption embedding similarity, no prior works have proposed other methods to filter noisy multimodal data, or concretely assessed the impact of noisy captioning data on downstream training. In this work, we propose, theoretically justify, and empirically validate LEMoN, a method to identify label errors in image-caption datasets. Our method leverages the multimodal neighborhood of image-caption pairs in the latent space of contrastively pretrained multimodal models to automatically identify label errors. Through empirical evaluations across eight datasets and twelve baselines, we find that LEMoN outperforms the baselines by over 3% in label error detection, and that training on datasets filtered using our method improves downstream captioning performance by more than 2 BLEU points over noisy training.

Zilan Wang, Junfeng Guo, Jiacheng Zhu et al.

Recent advances in large-scale text-to-image (T2I) diffusion models have enabled a variety of downstream applications, including style customization, subject-driven personalization, and conditional generation. As T2I models require extensive data and computational resources for training, they constitute highly valued intellectual property (IP) for their legitimate owners, yet making them incentive targets for unauthorized fine-tuning by adversaries seeking to leverage these models for customized, usually profitable applications. Existing IP protection methods for diffusion models generally involve embedding watermark patterns and then verifying ownership through generated outputs examination, or inspecting the model's feature space. However, these techniques are inherently ineffective in practical scenarios when the watermarked model undergoes fine-tuning, and the feature space is inaccessible during verification ((i.e., black-box setting). The model is prone to forgetting the previously learned watermark knowledge when it adapts to a new task. To address this challenge, we propose SleeperMark, a novel framework designed to embed resilient watermarks into T2I diffusion models. SleeperMark explicitly guides the model to disentangle the watermark information from the semantic concepts it learns, allowing the model to retain the embedded watermark while continuing to be adapted to new downstream tasks. Our extensive experiments demonstrate the effectiveness of SleeperMark across various types of diffusion models, including latent diffusion models (e.g., Stable Diffusion) and pixel diffusion models (e.g., DeepFloyd-IF), showing robustness against downstream fine-tuning and various attacks at both the image and model levels, with minimal impact on the model's generative capability. The code is available at https://github.com/taco-group/SleeperMark.

Jiacheng Zhu, Kristjan Greenewald, Kimia Nadjahi et al.

Parameter-efficient fine-tuning optimizes large, pre-trained foundation models by updating a subset of parameters; in this class, Low-Rank Adaptation (LoRA) is particularly effective. Inspired by an effort to investigate the different roles of LoRA matrices during fine-tuning, this paper characterizes and leverages unexpected asymmetry in the importance of low-rank adapter matrices. Specifically, when updating the parameter matrices of a neural network by adding a product $BA$, we observe that the $B$ and $A$ matrices have distinct functions: $A$ extracts features from the input, while $B$ uses these features to create the desired output. Based on this observation, we demonstrate that fine-tuning $B$ is inherently more effective than fine-tuning $A$, and that a random untrained $A$ should perform nearly as well as a fine-tuned one. Using an information-theoretic lens, we also bound the generalization of low-rank adapters, showing that the parameter savings of exclusively training $B$ improves the bound. We support our conclusions with experiments on RoBERTa, BART-Large, LLaMA-2, and ViTs.

Yiqing Liang, Jielin Qiu, Wenhao Ding et al. · cmu

Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a powerful paradigm for post-training large language models (LLMs), achieving state-of-the-art performance on tasks with structured, verifiable answers. Applying RLVR to Multimodal LLMs (MLLMs) presents significant opportunities but is complicated by the broader, heterogeneous nature of vision-language tasks that demand nuanced visual, logical, and spatial capabilities. As such, training MLLMs using RLVR on multiple datasets could be beneficial but creates challenges with conflicting objectives from interaction among diverse datasets, highlighting the need for optimal dataset mixture strategies to improve generalization and reasoning. We introduce a systematic post-training framework for Multimodal LLM RLVR, featuring a rigorous data mixture problem formulation and benchmark implementation. Specifically, (1) We developed a multimodal RLVR framework for multi-dataset post-training by curating a dataset that contains different verifiable vision-language problems and enabling multi-domain online RL learning with different verifiable rewards; (2) We proposed a data mixture strategy that learns to predict the RL fine-tuning outcome from the data mixture distribution, and consequently optimizes the best mixture. Comprehensive experiments showcase that multi-domain RLVR training, when combined with mixture prediction strategies, can significantly boost MLLM general reasoning capacities. Our best mixture improves the post-trained model's accuracy on out-of-distribution benchmarks by an average of 5.24% compared to the same model post-trained with uniform data mixture, and by a total of 20.74% compared to the pre-finetuning baseline.

Jielin Qiu, Jiacheng Zhu, Mengdi Xu et al.

In this paper, we focus on a new method of data augmentation to solve the data imbalance problem within imbalanced ECG datasets to improve the robustness and accuracy of heart disease detection. By using Optimal Transport, we augment the ECG disease data from normal ECG beats to balance the data among different categories. We build a Multi-Feature Transformer (MF-Transformer) as our classification model, where different features are extracted from both time and frequency domains to diagnose various heart conditions. Learning from 12-lead ECG signals, our model is able to distinguish five categories of cardiac conditions. Our results demonstrate 1) the classification models' ability to make competitive predictions on five ECG categories; 2) improvements in accuracy and robustness reflecting the effectiveness of our data augmentation method.

Jiacheng Zhu, Aritra Guha, Dat Do et al.

We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. { Since probability measures in infinite dimensional spaces generally lack absolute continuity (that is, with respect to non-degenerate Gaussian measures), the Monge map in the standard optimal transport theory for finite dimensional spaces may not exist. Our approach to the optimal transport problem in infinite dimensions is by a suitable regularization technique -- we restrict the class of transport maps to be a Hilbert-Schmidt space of operators.} To this end, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and examine the functional properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.