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The distributed filtering problem sequentially estimates a global state variable using observations from a network of local sensors with different measurement models. In this work, we introduce a novel methodology for distributed nonlinear filtering by combining techniques from transportation of measures, dimensionality reduction, and consensus algorithms. We illustrate our methodology on a satellite pose estimation problem from a network of direct and indirect observers. The numerical results serve as a proof of concept, offering new venues for theoretical and applied research in the domain of distributed filtering., Comment: 7 pages, 2 figures

The Wasserstein distance from optimal mass transport (OMT) is a powerful mathematical tool with numerous applications that provides a natural measure of the distance between two probability distributions. Several methods to incorporate OMT into widely used probabilistic models, such as Gaussian or Gaussian mixture, have been developed to enhance the capability of modeling complex multimodal densities of real datasets. However, very few studies have explored the OMT problems in a reproducing kernel Hilbert space (RKHS), wherein the kernel trick is utilized to avoid the need to explicitly map input data into a high-dimensional feature space. In the current study, we propose a Wasserstein-type metric to compute the distance between two Gaussian mixtures in a RKHS via the kernel trick, i.e., kernel Gaussian mixture models., Comment: 17 pages, 5 figures, 2 tables

In this paper we extend recent developments in computational optimal transport to the setting of Riemannian manifolds. In particular, we show how to learn optimal transport maps from samples that relate probability distributions defined on manifolds. Specializing these maps for sampling conditional probability distributions provides an ensemble approach for solving nonlinear filtering problems defined on such geometries. The proposed computational methodology is illustrated with examples of transport and nonlinear filtering on Lie groups, including the circle $S^1$, the special Euclidean group $SE(2)$, and the special orthogonal group $SO(3)$., Comment: 6 pages, 7 figures

The purpose of this work is to develop a framework to calibrate signed datasets so as to be consistent with specified marginals by suitably extending the Schr\"odinger-Fortet-Sinkhorn paradigm. Specifically, we seek to revise sign-indefinite multi-dimensional arrays in a way that the updated values agree with specified marginals. Our approach follows the rationale in Schr\"odinger's problem, aimed at updating a "prior" probability measure to agree with marginal distributions. The celebrated Sinkhorn's algorithm (established earlier by R.\ Fortet) that solves Schr\"odinger's problem found early applications in calibrating contingency tables in statistics and, more recently, multi-marginal problems in machine learning and optimal transport. Herein, we postulate a sign-indefinite prior in the form of a multi-dimensional array, and propose an optimization problem to suitably update this prior to ensure consistency with given marginals. The resulting algorithm generalizes the Sinkhorn algorithm in that it amounts to iterative scaling of the entries of the array along different coordinate directions. The scaling is multiplicative but also, in contrast to Sinkhorn, inverse-multiplicative depending on the sign of the entries. Our algorithm reduces to the classical Sinkhorn algorithm when the entries of the prior are positive., Comment: 11 pages, 4 figures

Biological networks often encapsulate promotion/inhibition as signed edge-weights of a graph. Nodes may correspond to genes assigned expression levels (mass) of respective proteins. The promotion/inhibition nature of co-expression between nodes is encoded in the sign of the corresponding entry of a sign-indefinite adjacency matrix, though the strength of such co-expression (i.e., the precise value of edge weights) cannot typically be directly measured. Herein we address the inverse problem to determine network edge-weights based on a sign-indefinite adjacency and expression levels at the nodes. While our motivation originates in gene networks, the framework applies to networks where promotion/inhibition dictates a stationary mass distribution at the nodes. In order to identify suitable edge-weights we adopt a framework of ``negative probabilities,'' advocated by P.\ Dirac and R.\ Feynman, and we set up a likelihood formalism to obtain values for the sought edge-weights. The proposed optimization problem can be solved via a generalization of the well-known Sinkhorn algorithm; in our setting the Sinkhorn-type ``diagonal scalings'' are multiplicative or inverse-multiplicative, depending on the sign of the respective entries in the adjacency matrix, with value computed as the positive root of a quadratic polynomial., Comment: 6 pages

As a generalization of the optimal mass transport (OMT) approach of Benamou and Brenier's, the regularized optimal mass transport (rOMT) formulates a transport problem from an initial mass configuration to another with the optimality defined by the total kinetic energy, but subject to an advection-diffusion constraint equation. Both rOMT and the Benamou and Brenier's formulation require the total initial and final masses to be equal; mass is preserved during the entire transport process. However, for many applications, e.g., in dynamic image tracking, this constraint is rarely if ever satisfied. Therefore, we propose to employ an unbalanced version of rOMT to remove this constraint together with a detailed numerical solution procedure and applications to analyzing fluid flows in the brain., Comment: 16 pages, 5 figures

In this paper, we combine nonlinear diffusion with the regularized optimal mass transport (rOMT) model. As we will demonstrate, this new approach provides further insights into certain applications of fluid flow analysis in the brain. From the point of view of image processing, the anisotropic diffusion method, based on Perona-Malik, explicitly considers edge information. Applied to rOMT analysis of glymphatic transport based on DCE-MRI data, this new framework appears to capture a larger advection-dominant volume.

The Histogram of Oriented Gradient is a widely used image feature, which describes local image directionality based on numerical differentiation. Due to its ill-posed nature, small noise may lead to large errors. Conventional HOG may fail to produce meaningful directionality results in the presence of noise, which is common in medical radiographic imaging. We approach the directionality problem from a novel perspective by the use of the optimal transport map of a local image patch to a uni-color patch of its mean. We decompose the transport map into sub-work costs in different directions. We evaluated the ability of the optimal transport to quantify tumor heterogeneity from brain MRI images of patients with glioblastoma multiforme from the TCIA. By considering the entropy difference of the extracted local directionality within tumor regions, we found that patients with higher entropy in their images, had statistically significant worse overall survival (p $=0.008$), which indicates that tumors exhibiting flows in many directions may be more malignant, perhaps reflecting high tumor histologic grade, a reflection of histologic disorganization. We also explored the possibility of solving classical image processing problems such as smoothing and edge detection via optimal transport. By looking for a 2-color patch with minimum transport distance to a local patch, we derive a nonlinear shock filter, which preserves edges. Moreover, we found that the color difference of the computed 2-color patch indicates whether there is a large change in color, i.e., an edge in the given patch. In summary, we expand the usefulness of optimal transport as an image local analysis tool, to extract robust measures of imaging tumor heterogeneity for outcomes prediction as well as image pre-processing. Because of its robust nature, we find it offers several advantages over the classical approaches.

Regularized optimal mass transport (rOMT) problem adds a diffusion term to the continuity equation in the original dynamic formulation of the optimal mass transport (OMT) problem proposed by Benamou and Brenier. We show that the rOMT model serves as a powerful tool in computational fluid dynamics (CFD) for visualizing fluid flows in the glymphatic system. In the present work, we describe how to modify the previous numerical method for efficient implementation, resulting in a significant reduction in computational runtime. Numerical results applied to synthetic and real-data are provided.

Vector-valued Gaussian mixtures form an important special subset of vector-valued distributions. In general, vector-valued distributions constitute natural representations for physical entities, which can mutate or transit among alternative manifestations distributed in a given space. A key example is color imagery. In this note, we vectorize the Gaussian mixture model and study several different optimal mass transport related problems associated to such models. The benefits of using vector Gaussian mixture for optimal mass transport include computational efficiency and the ability to preserve structure.

Unbalanced optimal mass transport (OMT) seeks to remove the conservation of mass constraint by adding a source term to the standard continuity equation in the Benamou-Brenier formulation of OMT. In this note, we show how the addition of the source fits into the vector-valued OMT framework.

Variations in hematoxylin and eosin (H&E) stained images (due to clinical lab protocols, scanners, etc) directly impact the quality and accuracy of clinical diagnosis, and hence it is important to control for these variations for a reliable diagnosis. In this work, we present a new approach based on the multimarginal Wasserstein barycenter to normalize and augment H&E stained images given one or more references. Specifically, we provide a mathematically robust way of naturally incorporating additional images as intermediate references to drive stain normalization and augmentation simultaneously. The presented approach showed superior results quantitatively and qualitatively as compared to state-of-the-art methods for stain normalization. We further validated our stain normalization and augmentations in the nuclei segmentation task on a publicly available dataset, achieving state-of-the-art results against competing approaches., Comment: To appear in MICCAI 2020

The Wasserstein distance is a powerful metric based on the theory of optimal transport. It gives a natural measure of the distance between two distributions with a wide range of applications. In contrast to a number of the common divergences on distributions such as Kullback-Leibler or Jensen-Shannon, it is (weakly) continuous, and thus ideal for analyzing corrupted data. To date, however, no kernel methods for dealing with nonlinear data have been proposed via the Wasserstein distance. In this work, we develop a novel method to compute the L2-Wasserstein distance in a kernel space implemented using the kernel trick. The latter is a general method in machine learning employed to handle data in a nonlinear manner. We evaluate the proposed approach in identifying computerized tomography (CT) slices with dental artifacts in head and neck cancer, performing unsupervised hierarchical clustering on the resulting Wasserstein distance matrix that is computed on imaging texture features extracted from each CT slice. Our experiments show that the kernel approach outperforms classical non-kernel approaches in identifying CT slices with artifacts., Comment: 10 pages, 4 figures

We propose a probabilistic enhancement of standard kernel Support Vector Machines for binary classification, in order to address the case when, along with given data sets, a description of uncertainty (e.g., error bounds) may be available on each datum. In the present paper, we specifically consider Gaussian distributions to model uncertainty. Thereby, our data consist of pairs $(x_i,\Sigma_i)$, $i\in\{1,\ldots,N\}$, along with an indicator $y_i\in\{-1,1\}$ to declare membership in one of two categories for each pair. These pairs may be viewed to represent the mean and covariance, respectively, of random vectors $\xi_i$ taking values in a suitable linear space (typically $\mathbb R^n$). Thus, our setting may also be viewed as a modification of Support Vector Machines to classify distributions, albeit, at present, only Gaussian ones. We outline the formalism that allows computing suitable classifiers via a natural modification of the standard "kernel trick." The main contribution of this work is to point out a suitable kernel function for applying Support Vector techniques to the setting of uncertain data for which a detailed uncertainty description is also available (herein, "Gaussian points")., Comment: 6 pages, 6 figures

Tumorigenesis is a complex process that is heterogeneous and affected by numerous sources of variability. This study presents a stochastic extension of a biologically grounded tumor growth model, referred to as the Norton-Simon-Massagu\'e (NSM) tumor growth model. We first study the uncontrolled version of the model where the effect of chemotherapeutic drug agent is absent. Conditions on the model's parameters are derived to guarantee the positivity of the tumor volume and hence the validity of the proposed stochastic NSM model. To calibrate the proposed model we utilize a maximum likelihood-based estimation algorithm and population mixed-effect modeling formulation. The algorithm is tested by fitting previously published tumor volume mice data. Then, we study the controlled version of the model which includes the effect of chemotherapy treatment. Analysis of the influence of adding the control drug agent into the model and how sensitive it is to the stochastic parameters is performed both in open-loop and closed-loop viewpoints through different numerical simulations.

In this work, a unified representation of all the time-varying dynamics is accomplished with a Lagrangian framework for analyzing Fisher-Rao regularized dynamical optimal mass transport (OMT) derived flows. While formally equivalent to the Eulerian based Schr\"odinger bridge OMT regularization scheme, the Fisher-Rao approach allows a simple and interpretable methodology for studying the flows of interest in the present work. The advantage of the proposed Lagrangian technique is that the time-varying particle trajectories and attributes are displayed in a single visualization. This provides a natural capability to identify and distinguish flows under different conditions. The Lagrangian analysis applied to the glymphatic system (brain waste removal pathway associated with Alzheimer's Disease) successfully captures known flows and distinguishes between flow patterns under two different anesthetics, providing deeper insights into altered states of waste drainage., Comment: To appear in MICCAI 2020

Spiculations are important predictors of lung cancer malignancy, which are spikes on the surface of the pulmonary nodules. In this study, we proposed an interpretable and parameter-free technique to quantify the spiculation using area distortion metric obtained by the conformal (angle-preserving) spherical parameterization. We exploit the insight that for an angle-preserved spherical mapping of a given nodule, the corresponding negative area distortion precisely characterizes the spiculations on that nodule. We introduced novel spiculation scores based on the area distortion metric and spiculation measures. We also semi-automatically segment lung nodule (for reproducibility) as well as vessel and wall attachment to differentiate the real spiculations from lobulation and attachment. A simple pathological malignancy prediction model is also introduced. We used the publicly-available LIDC-IDRI dataset pathologists (strong-label) and radiologists (weak-label) ratings to train and test radiomics models containing this feature, and then externally validate the models. We achieved AUC$=$0.80 and 0.76, respectively, with the models trained on the 811 weakly-labeled LIDC datasets and tested on the 72 strongly-labeled LIDC and 73 LUNGx datasets; the previous best model for LUNGx had AUC$=$0.68. The number-of-spiculations feature was found to be highly correlated (Spearman's rank correlation coefficient $\rho = 0.44$) with the radiologists' spiculation score. We developed a reproducible and interpretable, parameter-free technique for quantifying spiculations on nodules. The spiculation quantification measures was then applied to the radiomics framework for pathological malignancy prediction with reproducible semi-automatic segmentation of nodule. Using our interpretable features (size, attachment, spiculation, lobulation), we were able to achieve higher performance than previous models., Comment: Computer Methods and Programs in Biomedicine (Journal Extension of MICCAI ShapeMI 2018 Workshop paper). **First two authors contributed equally to this work

The glymphatic system (GS) is a transit passage that facilitates brain metabolic waste removal and its dysfunction has been associated with neurodegenerative diseases such as Alzheimer's disease. The GS has been studied by acquiring temporal contrast enhanced magnetic resonance imaging (MRI) sequences of a rodent brain, and tracking the cerebrospinal fluid injected contrast agent as it flows through the GS. We present here a novel visualization framework, GlymphVIS, which uses regularized optimal transport (OT) to study the flow behavior between time points at which the images are taken. Using this regularized OT approach, we can incorporate diffusion, handle noise, and accurately capture and visualize the time varying dynamics in GS transport. Moreover, we are able to reduce the registration mean-squared and infinity-norm error across time points by up to a factor of 5 as compared to the current state-of-the-art method. Our visualization pipeline yields flow patterns that align well with experts' current findings of the glymphatic system., Comment: MICCAI 2018, 8 pages

Image feature points are detected as pixels which locally maximize a detector function, two commonly used examples of which are the (Euclidean) image gradient and the Harris-Stephens corner detector. A major limitation of these feature detectors are that they are only Euclidean-invariant. In this work we demonstrate the application of a 2D affine-invariant image feature point detector based on differential invariants as derived through the equivariant method of moving frames. The fundamental equi-affine differential invariants for 3D image volumes are also computed.

We consider damped stochastic systems in a controlled (time-varying) quadratic potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermodynamics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via and the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting., Comment: 20 pages, 5 figures

We consider network routing under random link failures with a desired final distribution. We provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is a maximum entropy problem for path distributions with an extra terminal cost. We show that the unique solution may be obtained solving a generalized Schroedinger system. An iterative algorithm to compute the solution is provided. It contracts the Hilbert metric with contraction ratio less than 1/2 leading to extremely fast convergence., Comment: 16 pages, 1 figure

In the present work, we study the properties of biological networks by applying analogous notions of fundamental concepts in Riemannian geometry and optimal mass transport to discrete networks described by weighted graphs. Specifically, we employ possible generalizations of the notion of Ricci curvature on Riemannian manifold to discrete spaces in order to infer certain robustness properties of the networks of interest. We compare three possible discrete notions of Ricci curvature (Olivier Ricci curvature, Bakry-\'Emery Ricci curvature, and Forman Ricci curvature) on some model and biological networks. While the exact relationship of each of the three definitions of curvature to one another is still not known, they do yield similar results on our biological networks of interest. These notions are initially defined on positively weighted graphs; however, Forman-Ricci curvature can also be defined on directed positively weighted networks. We will generalize this notion of directed Forman Ricci curvature on the network to a form that also considers the signs of the directions (e. g., activator and repressor in transcription networks). We call this notion the \emph{signed-control Ricci curvature}. Given real biological networks are almost always directed and possess positive and negative controls, the notion of signed-control curvature can elucidate the network-based study of these complex networks. Finally, we compare the results of these notions of Ricci curvature on networks to some known network measures, namely, betweenness centrality and clustering coefficients on graphs., Comment: 23 pages

We present an optimal mass transport framework on the space of Gaussian mixture models, which are widely used in statistical inference. Our method leads to a natural way to compare, interpolate and average Gaussian mixture models. Basically, we study such models on a certain submanifold of probability densities with certain structure. Different aspects of this framework are discussed and several examples are presented to illustrate the results. This method represents our first attempt to study optimal transport problems for more general probability densities with structures., Comment: 14 pages, 10 figures

We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color images processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming (SQP). By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergent rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases., Comment: 18 pages, 4 figures

In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular, a "dual of the dual" formulation of $W_1$. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation., Comment: 8 pages

We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. The present paper builds on [13] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient-robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge) - an iterated bridge invariance property and the invariance of the most probable paths. These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a "temperature" parameter, tends to the solution of the "most economical" but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example., Comment: 18 pages

The classical Monge-Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the "Wasserstein distance." In this note, we formulate a natural matrix counterpart of the MK problem for positive definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimization problem, strong duality, an analogue of the Poincare-Wirtinger inequality, and a Lax-Hopf-Oleinik type result., Comment: 23 pages

In this note, we propose an unbalanced version of the quantum mechanical version of optimal mass transport that was based on the Lindblad equation. We formulate a natural interpolation framework between density matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1610.03041