Human mobility analysis at urban-scale requires models to represent the complex nature of human movements, which in turn are affected by accessibility to nearby points of interest, underlying socioeconomic factors of a place, and local transport choices for people living in a geographic region. In this work, we represent human mobility and the associated flow of movements as a grapyh. Graph-based approaches for mobility analysis are still in their early stages of adoption and are actively being researched. The challenges of graph-based mobility analysis are multifaceted - the lack of sufficiently high-quality data to represent flows at high spatial and teporal resolution whereas, limited computational resources to translate large voluments of mobility data into a network structure, and scaling issues inherent in graph models etc. The current study develops a methodology by embedding graphs into a continuous space, which alleviates issues related to fast graph matching, graph time-series modeling, and visualization of mobility dynamics. Through experiments, we demonstrate how mobility data collected from taxicab trajectories could be transformed into network structures and patterns of mobility flow changes, and can be used for downstream tasks reporting approx 40% decrease in error on average in matched graphs vs unmatched ones.

Clinical assessments for neuromuscular disorders, such as Spinal Muscular Atrophy (SMA) and Duchenne Muscular Dystrophy (DMD), continue to rely on subjective measures to monitor treatment response and disease progression. We introduce a novel method using wearable sensors to objectively assess motor function during daily activities in 19 patients with DMD, 9 with SMA, and 13 age-matched controls. Pediatric movement data is complex due to confounding factors such as limb length variations in growing children and variability in movement speed. Our approach uses Shape-based Principal Component Analysis to align movement trajectories and identify distinct kinematic patterns, including variations in motion speed and asymmetry. Both DMD and SMA cohorts have individuals with motor function on par with healthy controls. Notably, patients with SMA showed greater activation of the motion asymmetry pattern. We further combined projections on these principal components with partial least squares (PLS) to identify a covariation mode with a canonical correlation of r = 0.78 (95% CI: [0.34, 0.94]) with muscle fat infiltration, the Brooke score (a motor function score), and age-related degenerative changes, proposing a novel motor function index. This data-driven method can be deployed in home settings, enabling better longitudinal tracking of treatment efficacy for children with neuromuscular disorders.

Wasserstein distances form a family of metrics on spaces of probability measures that have recently seen many applications. However, statistical analysis in these spaces is complex due to the nonlinearity of Wasserstein spaces. One potential solution to this problem is Linear Optimal Transport (LOT). This method allows one to find a Euclidean embedding, called LOT embedding, of measures in some Wasserstein spaces, but some information is lost in this embedding. So, to understand whether statistical analysis relying on LOT embeddings can make valid inferences about original data, it is helpful to quantify how well these embeddings describe that data. To answer this question, we present a decomposition of the Fr\'echet variance of a set of measures in the 2-Wasserstein space, which allows one to compute the percentage of variance explained by LOT embeddings of those measures. We then extend this decomposition to the Fused Gromov-Wasserstein setting. We also present several experiments that explore the relationship between the dimension of the LOT embedding, the percentage of variance explained by the embedding, and the classification accuracy of machine learning classifiers built on the embedded data. We use the MNIST handwritten digits dataset, IMDB-50000 dataset, and Diffusion Tensor MRI images for these experiments. Our results illustrate the effectiveness of low dimensional LOT embeddings in terms of the percentage of variance explained and the classification accuracy of models built on the embedded data.

The success of deep learning models deployed in the real world depends critically on their ability to generalize well across diverse data domains. Here, we address a fundamental challenge with selective classification during automated diagnosis with domain-shifted medical images. In this scenario, models must learn to avoid making predictions when label confidence is low, especially when tested with samples far removed from the training set (covariate shift). Such uncertain cases are typically referred to the clinician for further analysis and evaluation. Yet, we show that even state-of-the-art domain generalization approaches fail severely during referral when tested on medical images acquired from a different demographic or using a different technology. We examine two benchmark diagnostic medical imaging datasets exhibiting strong covariate shifts: i) diabetic retinopathy prediction with retinal fundus images and ii) multilabel disease prediction with chest X-ray images. We show that predictive uncertainty estimates do not generalize well under covariate shifts leading to non-monotonic referral curves, and severe drops in performance (up to 50%) at high referral rates (>70%). We evaluate novel combinations of robust generalization and post hoc referral approaches, that rescue these failures and achieve significant performance improvements, typically >10%, over baseline methods. Our study identifies a critical challenge with referral in domain-shifted medical images and finds key applications in reliable, automated disease diagnosis.

This paper focuses on the statistical analysis of shapes of data objects called shape graphs, a set of nodes connected by articulated curves with arbitrary shapes. A critical need here is a constrained registration of points (nodes to nodes, edges to edges) across objects. This, in turn, requires optimization over the permutation group, made challenging by differences in nodes (in terms of numbers, locations) and edges (in terms of shapes, placements, and sizes) across objects. This paper tackles this registration problem using a novel neural-network architecture and involves an unsupervised loss function developed using the elastic shape metric for curves. This architecture results in (1) state-of-the-art matching performance and (2) an order of magnitude reduction in the computational cost relative to baseline approaches. We demonstrate the effectiveness of the proposed approach using both simulated data and real-world 2D and 3D shape graphs. Code and data will be made publicly available after review to foster research.

This paper investigates the challenge of learning image manifolds, specifically pose manifolds, of 3D objects using limited training data. It proposes a DNN approach to manifold learning and for predicting images of objects for novel, continuous 3D rotations. The approach uses two distinct concepts: (1) Geometric Style-GAN (Geom-SGAN), which maps images to low-dimensional latent representations and maintains the (first-order) manifold geometry. That is, it seeks to preserve the pairwise distances between base points and their tangent spaces, and (2) uses Euler's elastica to smoothly interpolate between directed points (points + tangent directions) in the low-dimensional latent space. When mapped back to the larger image space, the resulting interpolations resemble videos of rotating objects. Extensive experiments establish the superiority of this framework in learning paths on rotation manifolds, both visually and quantitatively, relative to state-of-the-art GANs and VAEs.

Empirically multidimensional discriminator (critic) output can be advantageous, while a solid explanation for it has not been discussed. In this paper, (i) we rigorously prove that high-dimensional critic output has advantage on distinguishing real and fake distributions; (ii) we also introduce an square-root velocity transformation (SRVT) block which further magnifies this advantage. The proof is based on our proposed maximal p-centrality discrepancy which is bounded above by p-Wasserstein distance and perfectly fits the Wasserstein GAN framework with high-dimensional critic output n. We have also showed when n = 1, the proposed discrepancy is equivalent to 1-Wasserstein distance. The SRVT block is applied to break the symmetric structure of high-dimensional critic output and improve the generalization capability of the discriminator network. In terms of implementation, the proposed framework does not require additional hyper-parameter tuning, which largely facilitates its usage. Experiments on image generation tasks show performance improvement on benchmark datasets.

Unsupervised clustering of curves according to their shapes is an important problem with broad scientific applications. The existing model-based clustering techniques either rely on simple probability models (e.g., Gaussian) that are not generally valid for shape analysis or assume the number of clusters. We develop an efficient Bayesian method to cluster curve data using an elastic shape metric that is based on joint registration and comparison of shapes of curves. The elastic-inner product matrix obtained from the data is modeled using a Wishart distribution whose parameters are assigned carefully chosen prior distributions to allow for automatic inference on the number of clusters. Posterior is sampled through an efficient Markov chain Monte Carlo procedure based on the Chinese restaurant process to infer (1) the posterior distribution on the number of clusters, and (2) clustering configuration of shapes. This method is demonstrated on a variety of synthetic data and real data examples on protein structure analysis, cell shape analysis in microscopy images, and clustering of shaped from MPEG7 database.

While signal estimation under random amplitudes, phase shifts, and additive noise is studied frequently, the problem of estimating a deterministic signal under random time-warpings has been relatively unexplored. We present a novel framework for estimating the unknown signal that utilizes the action of the warping group to form an equivalence relation between signals. First, we derive an estimator for the equivalence class of the unknown signal using the notion of Karcher mean on the quotient space of equivalence classes. This step requires the use of Fisher-Rao Riemannian metric and a square-root representation of signals to enable computations of distances and means under this metric. Then, we define a notion of the center of a class and show that the center of the estimated class is a consistent estimator of the underlying unknown signal. This estimation algorithm has many applications: (1)registration/alignment of functional data, (2) separation of phase/amplitude components of functional data, (3) joint demodulation and carrier estimation, and (4) sparse modeling of functional data. Here we demonstrate only (1) and (2): Given signals are temporally aligned using nonlinear warpings and, thus, separated into their phase and amplitude components. The proposed method for signal alignment is shown to have state of the art performance using Berkeley growth, handwritten signatures, and neuroscience spike train data.

We present a geometric approach to statistical shape analysis of closed curves in images. The basic idea is to specify a space of closed curves satisfying given constraints, and exploit the differential geometry of this space to solve optimization and inference problems. We demonstrate this approach by: (i) defining and computing statistics of observed shapes, (ii) defining and learning a parametric probability model on shape space, and (iii) designing a binary hypothesis test on this space.