Gustavo K. Rohde

Abu Hasnat Mohammad Rubaiyat, Shiying Li, Xuwang Yin et al.

This paper presents a new end-to-end signal classification method using the signed cumulative distribution transform (SCDT). We adopt a transport-based generative model to define the classification problem. We then make use of mathematical properties of the SCDT to render the problem easier in transform domain, and solve for the class of an unknown sample using a nearest local subspace (NLS) search algorithm in SCDT domain. Experiments show that the proposed method provides high accuracy classification results while being data efficient, robust to out-of-distribution samples, and competitive in terms of computational complexity with respect to the deep learning end-to-end classification methods. The implementation of the proposed method in Python language is integrated as a part of the software package PyTransKit (https://github.com/rohdelab/PyTransKit).

Sumati Thareja, Gustavo Rohde, Rocio Diaz Martin et al.

We describe a method for signal parameter estimation using the signed cumulative distribution transform (SCDT), a recently introduced signal representation tool based on optimal transport theory. The method builds upon signal estimation using the cumulative distribution transform (CDT) originally introduced for positive distributions. Specifically, we show that Wasserstein-type distance minimization can be performed simply using linear least squares techniques in SCDT space for arbitrary signal classes, thus providing a global minimizer for the estimation problem even when the underlying signal is a nonlinear function of the unknown parameters. Comparisons to current signal estimation methods using $L_p$ minimization shows the advantage of the method.

Le Gong, Shiying Li, Naqib Sad Pathan et al.

Here we describe a new image representation technique based on the mathematics of transport and optimal transport. The method relies on the combination of the well-known Radon transform for images and a recent signal representation method called the Signed Cumulative Distribution Transform. The newly proposed method generalizes previous transport-related image representation methods to arbitrary functions (images), and thus can be used in more applications. We describe the new transform, and some of its mathematical properties and demonstrate its ability to partition image classes with real and simulated data. In comparison to existing transport transform methods, as well as deep learning-based classification methods, the new transform more accurately represents the information content of signed images, and thus can be used to obtain higher classification accuracies. The implementation of the proposed method in Python language is integrated as a part of the software package PyTransKit, available on Github.

Rocio Diaz Martin, Ivan Medri, Yikun Bai et al.

The optimal transport problem for measures supported on non-Euclidean spaces has recently gained ample interest in diverse applications involving representation learning. In this paper, we focus on circular probability measures, i.e., probability measures supported on the unit circle, and introduce a new computationally efficient metric for these measures, denoted as Linear Circular Optimal Transport (LCOT). The proposed metric comes with an explicit linear embedding that allows one to apply Machine Learning (ML) algorithms to the embedded measures and seamlessly modify the underlying metric for the ML algorithm to LCOT. We show that the proposed metric is rooted in the Circular Optimal Transport (COT) and can be considered the linearization of the COT metric with respect to a fixed reference measure. We provide a theoretical analysis of the proposed metric and derive the computational complexities for pairwise comparison of circular probability measures. Lastly, through a set of numerical experiments, we demonstrate the benefits of LCOT in learning representations of circular measures.

Shiying Li, Abu Hasnat Mohammad Rubaiyat, Gustavo K. Rohde

Transport-based metrics and related embeddings (transforms) have recently been used to model signal classes where nonlinear structures or variations are present. In this paper, we study the geodesic properties of time series data with a generalized Wasserstein metric and the geometry related to their signed cumulative distribution transforms in the embedding space. Moreover, we show how understanding such geometric characteristics can provide added interpretability to certain time series classifiers, and be an inspiration for more robust classifiers.

Mohammad Shifat E Rabbi, Abu Hasnat Mohammad Rubaiyat, Yan Zhuang et al.

There exist growing interests in intelligent systems for numerous medical imaging, image processing, and computer vision applications, such as face recognition, medical diagnosis, character recognition, and self-driving cars, among others. These applications usually require solving complex classification problems involving complex images with unknown data generative processes. In addition to recent successes of the current classification approaches relying on feature engineering and deep learning, several shortcomings of them, such as the lack of robustness, generalizability, and interpretability, have also been observed. These methods often require extensive training data, are computationally expensive, and are vulnerable to out-of-distribution samples, e.g., adversarial attacks. Recently, an accurate, data-efficient, computationally efficient, and robust transport-based classification approach has been proposed, which describes a generative model-based problem formulation and closed-form solution for a specific category of classification problems. However, all these approaches lack mechanisms to detect test samples outside the class distributions used during training. In real-world settings, where the collected training samples are unable to exhaust or cover all classes, the traditional classification schemes are unable to handle the unseen classes effectively, which is especially an important issue for safety-critical systems, such as self-driving and medical imaging diagnosis. In this work, we propose a method for detecting out-of-class distributions based on the distribution of sliced-Wasserstein distance from the Radon Cumulative Distribution Transform (R-CDT) subspace. We tested our method on the MNIST and two medical image datasets and reported better accuracy than the state-of-the-art methods without an out-of-class distribution detection procedure.

Hongyu Kan, Kristofor Pas, Ivan Medri et al.

Transport-Based Morphometry (TBM) has emerged as a new framework for 3D medical image analysis. By embedding images into a transport domain via invertible transformations, TBM facilitates effective classification, regression, and other tasks using transport-domain features. Crucially, the inverse mapping enables the projection of analytic results back into the original image space, allowing researchers to directly interpret clinical features associated with model outputs in a spatially meaningful way. To facilitate broader adoption of TBM in clinical imaging research, we present 3D-TBM, a tool designed for morphological analysis of 3D medical images. The framework includes data preprocessing, computation of optimal transport embeddings, and analytical methods such as visualization of main transport directions, together with techniques for discerning discriminating directions and related analysis methods. We also provide comprehensive documentation and practical tutorials to support researchers interested in applying 3D-TBM in their own medical imaging studies. The source code is publicly available through PyTransKit.

Mohammad Shifat E Rabbi, Yan Zhuang, Shiying Li et al.

Deep convolutional neural networks (CNNs) are broadly considered to be state-of-the-art generic end-to-end image classification systems. However, they are known to underperform when training data are limited and thus require data augmentation strategies that render the method computationally expensive and not always effective. Rather than using a data augmentation strategy to encode invariances as typically done in machine learning, here we propose to mathematically augment a nearest subspace classification model in sliced-Wasserstein space by exploiting certain mathematical properties of the Radon Cumulative Distribution Transform (R-CDT), a recently introduced image transform. We demonstrate that for a particular type of learning problem, our mathematical solution has advantages over data augmentation with deep CNNs in terms of classification accuracy and computational complexity, and is particularly effective under a limited training data setting. The method is simple, effective, computationally efficient, non-iterative, and requires no parameters to be tuned. Python code implementing our method is available at https://github.com/rohdelab/mathematical_augmentation. Our method is integrated as a part of the software package PyTransKit, which is available at https://github.com/rohdelab/PyTransKit.

Soheil Kolouri, Navid Naderializadeh, Gustavo K. Rohde et al.

We present Wasserstein Embedding for Graph Learning (WEGL), a novel and fast framework for embedding entire graphs in a vector space, in which various machine learning models are applicable for graph-level prediction tasks. We leverage new insights on defining similarity between graphs as a function of the similarity between their node embedding distributions. Specifically, we use the Wasserstein distance to measure the dissimilarity between node embeddings of different graphs. Unlike prior work, we avoid pairwise calculation of distances between graphs and reduce the computational complexity from quadratic to linear in the number of graphs. WEGL calculates Monge maps from a reference distribution to each node embedding and, based on these maps, creates a fixed-sized vector representation of the graph. We evaluate our new graph embedding approach on various benchmark graph-property prediction tasks, showing state-of-the-art classification performance while having superior computational efficiency. The code is available at https://github.com/navid-naderi/WEGL.

Mohammad Shifat-E-Rabbi, Xuwang Yin, Abu Hasnat Mohammad Rubaiyat et al.

We present a new supervised image classification method applicable to a broad class of image deformation models. The method makes use of the previously described Radon Cumulative Distribution Transform (R-CDT) for image data, whose mathematical properties are exploited to express the image data in a form that is more suitable for machine learning. While certain operations such as translation, scaling, and higher-order transformations are challenging to model in native image space, we show the R-CDT can capture some of these variations and thus render the associated image classification problems easier to solve. The method -- utilizing a nearest-subspace algorithm in R-CDT space -- is simple to implement, non-iterative, has no hyper-parameters to tune, is computationally efficient, label efficient, and provides competitive accuracies to state-of-the-art neural networks for many types of classification problems. In addition to the test accuracy performances, we show improvements (with respect to neural network-based methods) in terms of computational efficiency (it can be implemented without the use of GPUs), number of training samples needed for training, as well as out-of-distribution generalization. The Python code for reproducing our results is available at https://github.com/rohdelab/rcdt_ns_classifier.

Mohammad Shifat E Rabbi, Naqib Sad Pathan, Shiying Li et al.

Learning from point sets is an essential component in many computer vision and machine learning applications. Native, unordered, and permutation invariant set structure space is challenging to model, particularly for point set classification under spatial deformations. Here we propose a framework for classifying point sets experiencing certain types of spatial deformations, with a particular emphasis on datasets featuring affine deformations. Our approach employs the Linear Optimal Transport (LOT) transform to obtain a linear embedding of set-structured data. Utilizing the mathematical properties of the LOT transform, we demonstrate its capacity to accommodate variations in point sets by constructing a convex data space, effectively simplifying point set classification problems. Our method, which employs a nearest-subspace algorithm in the LOT space, demonstrates label efficiency, non-iterative behavior, and requires no hyper-parameter tuning. It achieves competitive accuracies compared to state-of-the-art methods across various point set classification tasks. Furthermore, our approach exhibits robustness in out-of-distribution scenarios where training and test distributions vary in terms of deformation magnitudes.

Yan Zhuang, Shiying Li, Mohammad Shifat-E-Rabbi et al.

We present a new method for face recognition from digital images acquired under varying illumination conditions. The method is based on mathematical modeling of local gradient distributions using the Radon Cumulative Distribution Transform (R-CDT). We demonstrate that lighting variations cause certain types of deformations of local image gradient distributions which, when expressed in R-CDT domain, can be modeled as a subspace. Face recognition is then performed using a nearest subspace in R-CDT domain of local gradient distributions. Experiment results demonstrate the proposed method outperforms other alternatives in several face recognition tasks with challenging illumination conditions. Python code implementing the proposed method is available, which is integrated as a part of the software package PyTransKit.

Akram Aldroubi, Rocio Diaz Martin, Ivan Medri et al.

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $\overline{\mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.

Xuwang Yin, Shiying Li, Gustavo K. Rohde

We study a new approach to learning energy-based models (EBMs) based on adversarial training (AT). We show that (binary) AT learns a special kind of energy function that models the support of the data distribution, and the learning process is closely related to MCMC-based maximum likelihood learning of EBMs. We further propose improved techniques for generative modeling with AT, and demonstrate that this new approach is capable of generating diverse and realistic images. Aside from having competitive image generation performance to explicit EBMs, the studied approach is stable to train, is well-suited for image translation tasks, and exhibits strong out-of-distribution adversarial robustness. Our results demonstrate the viability of the AT approach to generative modeling, suggesting that AT is a competitive alternative approach to learning EBMs.

Akram Aldroubi, Shiying Li, Gustavo K. Rohde

A relatively new set of transport-based transforms (CDT, R-CDT, LOT) have shown their strength and great potential in various image and data processing tasks such as parametric signal estimation, classification, cancer detection among many others. It is hence worthwhile to elucidate some of the mathematical properties that explain the successes of these transforms when they are used as tools in data analysis, signal processing or data classification. In particular, we give conditions under which classes of signals that are created by algebraic generative models are transformed into convex sets by the transport transforms. Such convexification of the classes simplify the classification and other data analysis and processing problems when viewed in the transform domain. More specifically, we study the extent and limitation of the convexification ability of these transforms under an algebraic generative modeling framework. We hope that this paper will serve as an introduction to these transforms and will encourage mathematicians and other researchers to further explore the theoretical underpinnings and algorithmic tools that will help understand the successes of these transforms and lay the groundwork for further successful applications.

Soheil Kolouri, Xuwang Yin, Gustavo K. Rohde

Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as operators on probability distributions for observed data, we show that the distribution of outputs for any node in a neural network can be interpreted as a nonlinear projection along hypersurfaces defined by level surfaces over the input data space. We utilize these descriptions to provide new interpretation for phenomena such as nonlinearity, pooling, activation functions, and adversarial examples in neural network-based learning problems.

Xuwang Yin, Soheil Kolouri, Gustavo K. Rohde

The vulnerabilities of deep neural networks against adversarial examples have become a significant concern for deploying these models in sensitive domains. Devising a definitive defense against such attacks is proven to be challenging, and the methods relying on detecting adversarial samples are only valid when the attacker is oblivious to the detection mechanism. In this paper we propose a principled adversarial example detection method that can withstand norm-constrained white-box attacks. Inspired by one-versus-the-rest classification, in a K class classification problem, we train K binary classifiers where the i-th binary classifier is used to distinguish between clean data of class i and adversarially perturbed samples of other classes. At test time, we first use a trained classifier to get the predicted label (say k) of the input, and then use the k-th binary classifier to determine whether the input is a clean sample (of class k) or an adversarially perturbed example (of other classes). We further devise a generative approach to detecting/classifying adversarial examples by interpreting each binary classifier as an unnormalized density model of the class-conditional data. We provide comprehensive evaluation of the above adversarial example detection/classification methods, and demonstrate their competitive performances and compelling properties.

Soheil Kolouri, Kimia Nadjahi, Umut Simsekli et al.

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We also show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed distances. Finally, we compare the numerical performance of the proposed distances on several generative modeling tasks, including SW flows and SW auto-encoders.

Soheil Kolouri, Phillip E. Pope, Charles E. Martin et al.

In this paper we study generative modeling via autoencoders while using the elegant geometric properties of the optimal transport (OT) problem and the Wasserstein distances. We introduce Sliced-Wasserstein Autoencoders (SWAE), which are generative models that enable one to shape the distribution of the latent space into any samplable probability distribution without the need for training an adversarial network or defining a closed-form for the distribution. In short, we regularize the autoencoder loss with the sliced-Wasserstein distance between the distribution of the encoded training samples and a predefined samplable distribution. We show that the proposed formulation has an efficient numerical solution that provides similar capabilities to Wasserstein Autoencoders (WAE) and Variational Autoencoders (VAE), while benefiting from an embarrassingly simple implementation.

Liam Cattell, Gustavo K. Rohde

In many scientific fields imaging is used to relate a certain physical quantity to other dependent variables. Therefore, images can be considered as a map from a real-world coordinate system to the non-negative measurements being acquired. In this work we describe an approach for simultaneous modeling and inference of such data, using the mathematics of optimal transport. To achieve this, we describe a numerical implementation of the linear optimal transport transform, based on the solution of the Monge-Ampere equation, which uses Brenier's theorem to characterize the solution of the Monge functional as the derivative of a convex potential function. We use our implementation of the transform to compute a curl-free mapping between two images, and show that it is able to match images with lower error that existing methods. Moreover, we provide theoretical justification for properties of the linear optimal transport framework observed in the literature, including a theorem for the linear separation of data classes. Finally, we use our optimal transport method to empirically demonstrate that the linear separability theorem holds, by rendering non-linearly separable data as linearly separable following transform to transport space.

Soheil Kolouri, Gustavo K. Rohde, Heiko Hoffmann

Gaussian mixture models (GMM) are powerful parametric tools with many applications in machine learning and computer vision. Expectation maximization (EM) is the most popular algorithm for estimating the GMM parameters. However, EM guarantees only convergence to a stationary point of the log-likelihood function, which could be arbitrarily worse than the optimal solution. Inspired by the relationship between the negative log-likelihood function and the Kullback-Leibler (KL) divergence, we propose an alternative formulation for estimating the GMM parameters using the sliced Wasserstein distance, which gives rise to a new algorithm. Specifically, we propose minimizing the sliced-Wasserstein distance between the mixture model and the data distribution with respect to the GMM parameters. In contrast to the KL-divergence, the energy landscape for the sliced-Wasserstein distance is more well-behaved and therefore more suitable for a stochastic gradient descent scheme to obtain the optimal GMM parameters. We show that our formulation results in parameter estimates that are more robust to random initializations and demonstrate that it can estimate high-dimensional data distributions more faithfully than the EM algorithm.

Shinjini Kundu, Soheil Kolouri, Kirk I Erickson et al.

Disease in the brain is often associated with subtle, spatially diffuse, or complex tissue changes that may lie beneath the level of gross visual inspection, even on magnetic resonance imaging (MRI). Unfortunately, current computer-assisted approaches that examine pre-specified features, whether anatomically-defined (i.e. thalamic volume, cortical thickness) or based on pixelwise comparison (i.e. deformation-based methods), are prone to missing a vast array of physical changes that are not well-encapsulated by these metrics. In this paper, we have developed a technique for automated pattern analysis that can fully determine the relationship between brain structure and observable phenotype without requiring any a priori features. Our technique, called transport-based morphometry (TBM), is an image transformation that maps brain images losslessly to a domain where they become much more separable. The new approach is validated on structural brain images of healthy older adult subjects where even linear models for discrimination, regression, and blind source separation enable TBM to independently discover the characteristic changes of aging and highlight potential mechanisms by which aerobic fitness may mediate brain health later in life. TBM is a generative approach that can provide visualization of physically meaningful shifts in tissue distribution through inverse transformation. The proposed framework is a powerful technique that can potentially elucidate genotype-structural-behavioral associations in myriad diseases.

Matthew Thorpe, Serim Park, Soheil Kolouri et al.

Transport based distances, such as the Wasserstein distance and earth mover's distance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the $TL^p$ distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The framework allows the application of existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled, non-positive one and two-dimensional signals, and color transfer.

Soheil Kolouri, Serim Park, Matthew Thorpe et al.

Transport-based techniques for signal and data analysis have received increased attention recently. Given their abilities to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications including content-based retrieval, cancer detection, image super-resolution, and statistical machine learning, to name a few, and shown to produce state of the art in several applications. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here we provide an overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications.

Soheil Kolouri, Se Rim Park, Gustavo K. Rohde

Invertible image representation methods (transforms) are routinely employed as low-level image processing operations based on which feature extraction and recognition algorithms are developed. Most transforms in current use (e.g. Fourier, Wavelet, etc.) are linear transforms, and, by themselves, are unable to substantially simplify the representation of image classes for classification. Here we describe a nonlinear, invertible, low-level image processing transform based on combining the well known Radon transform for image data, and the 1D Cumulative Distribution Transform proposed earlier. We describe a few of the properties of this new transform, and with both theoretical and experimental results show that it can often render certain problems linearly separable in transform space.

Soheil Kolouri, Yang Zou, Gustavo K. Rohde

Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on alternative formulations of the optimal transport have allowed for faster solutions to the problem and has revamped its practical applications in machine learning. In this paper, we exploit the widely used kernel methods and provide a family of provably positive definite kernels based on the Sliced Wasserstein distance and demonstrate the benefits of these kernels in a variety of learning tasks. Our work provides a new perspective on the application of optimal transport flavored distances through kernel methods in machine learning tasks.

Se Rim Park, Soheil Kolouri, Shinjini Kundu et al.

Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern identification that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT) is invertible, with well defined forward and inverse operations. We show that it can be useful in `parsing out' variations (confounds) that are `Lagrangian' (displacement and intensity variations) by converting these to `Eulerian' (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.