arXiv subjects

Allen Tannenbaum

Publications and source records attributed to Allen Tannenbaum.

At least 19 recordsLinked to original sources

arXiv2023

Distributed Nonlinear Filtering using Triangular Transport Maps

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.

cs.MA↗

arXiv2023

Optimal Transport for Kernel Gaussian Mixture Models

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.

cs.LG↗

arXiv2023

Computational Optimal Transport and Filtering on Riemannian manifolds

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 $SO(3)$.

math.OC↗

arXiv2023

Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm

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.

stat.ML↗

arXiv2023

Promotion/Inhibition Effects in Networks: A Model with Negative Probabilities

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.

q-bio.MN↗

arXiv2022

Regularized Optimal Mass Transport with Nonlinear Diffusion

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.

physics.flu-dyn↗

arXiv2022

Optimal transport for vector Gaussian mixture models

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.

stat.ML↗

arXiv2022

Wasserstein Image Local Analysis: Histogram of Orientations, Smoothing and Edge Detection

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.

eess.IV↗

arXiv2020

Reproducible and Interpretable Spiculation Quantification for Lung Cancer Screening

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 $ρ= 0.44$) 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.

cs.CV↗

arXiv2020

A Vectorial Approach to Unbalanced Optimal Mass Transport

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.

math.OC↗

arXiv2020

An Analytical Approach for Insulin-like Growth Factor Receptor 1 and Mammalian Target of Rapamycin Blockades in Ewing Sarcoma

We present preliminary results that quantify network robustness and fragility of Ewing sarcoma (ES), a rare pediatric bone cancer that often exhibits de novo or acquired drug resistance. By identifying novel proteins or pathways susceptible to drug targeting, this formalized approach promises to improve preclinical drug development and may lead to better treatment outcomes. Toward that end, our network modeling focused upon the IGF-1R-PI3K-Akt-mTOR pathway, which is of proven importance in ES. The clinical response and proteomic networks of drug-sensitive parental cell lines and their drug-resistant counterparts were assessed using two small molecule inhibitors for IGF-1R (OSI-906 and NVP-ADW-742) and an mTOR inhibitor (mTORi), MK8669, such that protein-to-protein expression networks could be generated for each group. For the first time, mathematical modeling proves that drug resistant ES samples exhibit higher degrees of overall network robustness (e.g., the ability of a system to withstand random perturbations to its network configuration) to that of their untreated or short-term (72-hour) treated samples. This was done by leveraging previous work, which suggests that Ricci curvature, a key geometric feature of a given network, is positively correlated to increased network robustness. More importantly, given that Ricci curvature is a local property of the system, it is capable of resolving pathway fragility. In this note, we offer some encouraging yet limited insights in terms of system-level robustness of ES and lay the foundation for scope of future work in which a complete study will be conducted.

q-bio.MN↗

arXiv2020

Multimarginal Wasserstein Barycenter for Stain Normalization and Augmentation

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.

eess.IV↗

arXiv2020

Fisher-Rao Regularized Transport Analysis of the Glymphatic System and Waste Drainage

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 Schrödinger 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.

physics.flu-dyn↗

arXiv2019

Kernel Wasserstein Distance

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.

cs.LG↗

arXiv2018

GlymphVIS: Visualizing Glymphatic Transport Pathways Using Regularized Optimal Transport

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.

cs.CV↗

arXiv2018

Affine Differential Invariants for Invariant Feature Point Detection

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.

cs.CV↗

arXiv2018

Stochastic control and non-equilibrium thermodynamics: fundamental limits

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.

cond-mat.stat-mech↗

arXiv2018

Relaxed Schroedinger bridges and robust network routing

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.

math.OC↗