Baba C. Vemuri

Xiran Fan, Chun-Hao Yang, Baba C. Vemuri

Hyperbolic spaces have been quite popular in the recent past for representing hierarchically organized data. Further, several classification algorithms for data in these spaces have been proposed in the literature. These algorithms mainly use either hyperplanes or geodesics for decision boundaries in a large margin classifiers setting leading to a non-convex optimization problem. In this paper, we propose a novel large margin classifier based on horospherical decision boundaries that leads to a geodesically convex optimization problem that can be optimized using any Riemannian gradient descent technique guaranteeing a globally optimal solution. We present several experiments depicting the competitive performance of our classifier in comparison to SOTA.

Sofia Ivolgina, P. Thomas Fletcher, Baba C. Vemuri

Batch normalization (BN) is a ubiquitous operation in deep neural networks used primarily to achieve stability and regularization during network training. BN involves feature map centering and scaling using sample means and variances, respectively. Since these statistics are being estimated across the feature maps within a batch, this problem is ideally suited for the application of Stein's shrinkage estimation, which leads to a better, in the mean-squared-error sense, estimate of the mean and variance of the batch. In this paper, we prove that the Stein shrinkage estimator for the mean and variance dominates over the sample mean and variance estimators in the presence of adversarial attacks when modeling these attacks using sub-Gaussian distributions. This facilitates and justifies the application of Stein shrinkage to estimate the mean and variance parameters in BN and use it in image classification (segmentation) tasks with and without adversarial attacks. We present SOTA performance results using this Stein corrected batch norm in a standard ResNet architecture applied to the task of image classification using CIFAR-10 data, 3D CNN on PPMI (neuroimaging) data and image segmentation using HRNet on Cityscape data with and without adversarial attacks.

Gianfranco Cortes, Xiaoda Qu, Baba C. Vemuri

Nonrigid registration is vital to medical image analysis but remains challenging for diffusion MRI (dMRI) due to its high-dimensional, orientation-dependent nature. While classical methods are accurate, they are computationally demanding, and deep neural networks, though efficient, have been underexplored for nonrigid dMRI registration compared to structural imaging. We present a novel, deep learning framework for model-free, nonrigid registration of raw diffusion MRI data that does not require explicit reorientation. Unlike previous methods relying on derived representations such as diffusion tensors or fiber orientation distribution functions, in our approach, we formulate the registration as an equivariant diffeomorphism of position-and-orientation space. Central to our method is an $\mathsf{SE}(3)$-equivariant UNet that generates velocity fields while preserving the geometric properties of a raw dMRI's domain. We introduce a new loss function based on the maximum mean discrepancy in Fourier space, implicitly matching ensemble average propagators across images. Experimental results on Human Connectome Project dMRI data demonstrate competitive performance compared to state-of-the-art approaches, with the added advantage of bypassing the overhead for estimating derived representations. This work establishes a foundation for data-driven, geometry-aware dMRI registration directly in the acquisition space.

Gianfranco Cortes, Yue Yu, Robin Chen et al.

With the advent of group equivariant convolutions in deep networks literature, spherical CNNs with $\mathsf{SO}(3)$-equivariant layers have been developed to cope with data that are samples of signals on the sphere $S^2$. One can implicitly obtain $\mathsf{SO}(3)$-equivariant convolutions on $S^2$ with significant efficiency gains by explicitly requiring gauge equivariance w.r.t. $\mathsf{SO}(2)$. In this paper, we build on this fact by introducing a higher order generalization of the gauge equivariant convolution, whose implementation is dubbed a gauge equivariant Volterra network (GEVNet). This allows us to model spatially extended nonlinear interactions within a given receptive field while still maintaining equivariance to global isometries. We prove theoretical results regarding the equivariance and construction of higher order gauge equivariant convolutions. Then, we empirically demonstrate the parameter efficiency of our model, first on computer vision benchmark data (e.g. spherical MNIST), and then in combination with a convolutional kernel network (CKN) on neuroimaging data. In the neuroimaging data experiments, the resulting two-part architecture (CKN + GEVNet) is used to automatically discriminate between patients with Lewy Body Disease (DLB), Alzheimer's Disease (AD) and Parkinson's Disease (PD) from diffusion magnetic resonance images (dMRI). The GEVNet extracts micro-architectural features within each voxel, while the CKN extracts macro-architectural features across voxels. This compound architecture is uniquely poised to exploit the intra- and inter-voxel information contained in the dMRI data, leading to improved performance over the classification results obtained from either of the individual components.

Xiran Fan, Chun-Hao Yang, Baba C. Vemuri

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this ensuing nested hyperbolic space representation with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Monami Banerjee, Rudrasis Chakraborty, Jose Bouza et al.

Convolutional neural networks have been highly successful in image-based learning tasks due to their translation equivariance property. Recent work has generalized the traditional convolutional layer of a convolutional neural network to non-Euclidean spaces and shown group equivariance of the generalized convolution operation. In this paper, we present a novel higher order Volterra convolutional neural network (VolterraNet) for data defined as samples of functions on Riemannian homogeneous spaces. Analagous to the result for traditional convolutions, we prove that the Volterra functional convolutions are equivariant to the action of the isometry group admitted by the Riemannian homogeneous spaces, and under some restrictions, any non-linear equivariant function can be expressed as our homogeneous space Volterra convolution, generalizing the non-linear shift equivariant characterization of Volterra expansions in Euclidean space. We also prove that second order functional convolution operations can be represented as cascaded convolutions which leads to an efficient implementation. Beyond this, we also propose a dilated VolterraNet model. These advances lead to large parameter reductions relative to baseline non-Euclidean CNNs. To demonstrate the efficacy of the VolterraNet performance, we present several real data experiments involving classification tasks on spherical-MNIST, atomic energy, Shrec17 data sets, and group testing on diffusion MRI data. Performance comparisons to the state-of-the-art are also presented.

Chun-Hao Yang, Baba C. Vemuri

In the recent past, nested structures in Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested spheres. In this paper, we propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifolds. Common examples of homogeneous Riemannian manifolds include the $n$-sphere, the Stiefel manifold, the Grassmann manifold and many others. In particular, we focus on applying the proposed framework to the Grassmann manifold, giving rise to the nested Grassmannians (NG). An important application in which Grassmann manifolds are encountered is planar shape analysis. Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grass-mann manifold. Some salient features of our framework are: (i) it explicitly exploits the geometry of the homogeneous Riemannian manifolds and (ii) the nested lower-dimensional submanifolds need not be geodesic. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively. The proposed algorithms are compared with PGA via simulation studies and real data experiments and are shown to achieve a higher ratio of expressed variance compared to PGA.

Jose J. Bouza, Chun-Hao Yang, David Vaillancourt et al.

Geometric deep learning has attracted significant attention in recent years, in part due to the availability of exotic data types for which traditional neural network architectures are not well suited. Our goal in this paper is to generalize convolutional neural networks (CNN) to the manifold-valued image case which arises commonly in medical imaging and computer vision applications. Explicitly, the input data to the network is an image where each pixel value is a sample from a Riemannian manifold. To achieve this goal, we must generalize the basic building block of traditional CNN architectures, namely, the weighted combinations operation. To this end, we develop a tangent space combination operation which is used to define a convolution operation on manifold-valued images that we call, the Manifold-Valued Convolution (MVC). We prove theoretical properties of the MVC operation, including equivariance to the action of the isometry group admitted by the manifold and characterizing when compositions of MVC layers collapse to a single layer. We present a detailed description of how to use MVC layers to build full, multi-layer neural networks that operate on manifold-valued images, which we call the MVC-net. Further, we empirically demonstrate superior performance of the MVC-nets in medical imaging and computer vision tasks.

Rudrasis Chakraborty, Jose Bouza, Jonathan Manton et al.

Deep neural networks have become the main work horse for many tasks involving learning from data in a variety of applications in Science and Engineering. Traditionally, the input to these networks lie in a vector space and the operations employed within the network are well defined on vector-spaces. In the recent past, due to technological advances in sensing, it has become possible to acquire manifold-valued data sets either directly or indirectly. Examples include but are not limited to data from omnidirectional cameras on automobiles, drones etc., synthetic aperture radar imaging, diffusion magnetic resonance imaging, elastography and conductance imaging in the Medical Imaging domain and others. Thus, there is need to generalize the deep neural networks to cope with input data that reside on curved manifolds where vector space operations are not naturally admissible. In this paper, we present a novel theoretical framework to generalize the widely popular convolutional neural networks (CNNs) to high dimensional manifold-valued data inputs. We call these networks, ManifoldNets. In ManifoldNets, convolution operation on data residing on Riemannian manifolds is achieved via a provably convergent recursive computation of the weighted Fréchet Mean (wFM) of the given data, where the weights makeup the convolution mask, to be learned. Further, we prove that the proposed wFM layer achieves a contraction mapping and hence ManifoldNet does not need the non-linear ReLU unit used in standard CNNs. We present experiments, using the ManifoldNet framework, to achieve dimensionality reduction by computing the principal linear subspaces that naturally reside on a Grassmannian. The experimental results demonstrate the efficacy of ManifoldNets in the context of classification and reconstruction accuracy.

Rudrasis Chakraborty, Chun-Hao Yang, Baba C. Vemuri

Deep networks have gained immense popularity in Computer Vision and other fields in the past few years due to their remarkable performance on recognition/classification tasks surpassing the state-of-the art. One of the keys to their success lies in the richness of the automatically learned features. In order to get very good accuracy, one popular option is to increase the depth of the network. Training such a deep network is however infeasible or impractical with moderate computational resources and budget. The other alternative to increase the performance is to learn multiple weak classifiers and boost their performance using a boosting algorithm or a variant thereof. But, one of the problems with boosting algorithms is that they require a re-training of the networks based on the misclassified samples. Motivated by these problems, in this work we propose an aggregation technique which combines the output of multiple weak classifiers. We formulate the aggregation problem using a mixture model fitted to the trained classifier outputs. Our model does not require any re-training of the `weak' networks and is computationally very fast (takes $<30$ seconds to run in our experiments). Thus, using a less expensive training stage and without doing any re-training of networks, we experimentally demonstrate that it is possible to boost the performance by $12\%$. Furthermore, we present experiments using hand-crafted features and improved the classification performance using the proposed aggregation technique. One of the major advantages of our framework is that our framework allows one to combine features that are very likely to be of distinct dimensions since they are extracted using different networks/algorithms. Our experimental results demonstrate a significant performance gain from the use of our aggregation technique at a very small computational cost.

Rudrasis Chakraborty, Chun-Hao Yang, Xingjian Zhen et al.

In a number of disciplines, the data (e.g., graphs, manifolds) to be analyzed are non-Euclidean in nature. Geometric deep learning corresponds to techniques that generalize deep neural network models to such non-Euclidean spaces. Several recent papers have shown how convolutional neural networks (CNNs) can be extended to learn with graph-based data. In this work, we study the setting where the data (or measurements) are ordered, longitudinal or temporal in nature and live on a Riemannian manifold -- this setting is common in a variety of problems in statistical machine learning, vision and medical imaging. We show how recurrent statistical recurrent network models can be defined in such spaces. We give an efficient algorithm and conduct a rigorous analysis of its statistical properties. We perform extensive numerical experiments demonstrating competitive performance with state of the art methods but with significantly less number of parameters. We also show applications to a statistical analysis task in brain imaging, a regime where deep neural network models have only been utilized in limited ways.

Rudrasis Chakraborty, Monami Banerjee, Baba C. Vemuri

Convolutional neural networks are ubiquitous in Machine Learning applications for solving a variety of problems. They however can not be used in their native form when the domain of the data is commonly encountered manifolds such as the sphere, the special orthogonal group, the Grassmanian, the manifold of symmetric positive definite matrices and others. Most recently, generalization of CNNs to data domains such as the 2-sphere has been reported by some research groups, which is referred to as the spherical CNNs (SCNNs). The key property of SCNNs distinct from CNNs is that they exhibit the rotational equivariance property that allows for sharing learned weights within a layer. In this paper, we theoretically generalize the CNNs to Riemannian homogeneous manifolds, that include but are not limited to the aforementioned example manifolds. Our key contributions in this work are: (i) A theorem stating that linear group equivariance systems are fully characterized by correlation of functions on the domain manifold and vice-versa. This is fundamental to the characterization of all linear group equivariant systems and parallels the widely used result in linear system theory for vector spaces. (ii) As a corrolary, we prove the equivariance of the correlation operation to group actions admitted by the input domains which are Riemannian homogeneous manifolds. (iii) We present the first end-to-end deep network architecture for classification of diffusion magnetic resonance image (dMRI) scans acquired from a cohort of 44 Parkinson Disease patients and 50 control/normal subjects. (iv) A proof of concept experiment involving synthetic data generated on the manifold of symmetric positive definite matrices is presented to demonstrate the applicability of our network to other types of domains.

Rudrasis Chakraborty, Monami Banerjee, Baba C. Vemuri

In this paper, we propose a novel information theoretic framework for dictionary learning (DL) and sparse coding (SC) on a statistical manifold (the manifold of probability distributions). Unlike the traditional DL and SC framework, our new formulation does not explicitly incorporate any sparsity inducing norm in the cost function being optimized but yet yields sparse codes. Our algorithm approximates the data points on the statistical manifold (which are probability distributions) by the weighted Kullback-Leibeler center/mean (KL-center) of the dictionary atoms. The KL-center is defined as the minimizer of the maximum KL-divergence between itself and members of the set whose center is being sought. Further, we prove that the weighted KL-center is a sparse combination of the dictionary atoms. This result also holds for the case when the KL-divergence is replaced by the well known Hellinger distance. From an applications perspective, we present an extension of the aforementioned framework to the manifold of symmetric positive definite matrices (which can be identified with the manifold of zero mean gaussian distributions), $\mathcal{P}_n$. We present experiments involving a variety of dictionary-based reconstruction and classification problems in Computer Vision. Performance of the proposed algorithm is demonstrated by comparing it to several state-of-the-art methods in terms of reconstruction and classification accuracy as well as sparsity of the chosen representation.

Rudrasis Chakraborty, Søren Hauberg, Baba C. Vemuri

Principal Component Analysis (PCA) and Kernel Principal Component Analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional Reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by $K$-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.

Rudrasis Chakraborty, Monami Banerjee, Victoria Crawford et al.

In this work, we propose a novel information theoretic framework for dictionary learning (DL) and sparse coding (SC) on a statistical manifold (the manifold of probability distributions). Unlike the traditional DL and SC framework, our new formulation {\it does not explicitly incorporate any sparsity inducing norm in the cost function but yet yields SCs}. Moreover, we extend this framework to the manifold of symmetric positive definite matrices, $\mathcal{P}_n$. Our algorithm approximates the data points, which are probability distributions, by the weighted Kullback-Leibeler center (KL-center) of the dictionary atoms. The KL-center is the minimizer of the maximum KL-divergence between the unknown center and members of the set whose center is being sought. Further, {\it we proved that this KL-center is a sparse combination of the dictionary atoms}. Since, the data reside on a statistical manifold, the data fidelity term can not be as simple as in the case of the vector-space data. We therefore employ the geodesic distance between the data and a sparse approximation of the data element. This cost function is minimized using an acceleterated gradient descent algorithm. An extensive set of experimental results show the effectiveness of our proposed framework. We present several experiments involving a variety of classification problems in Computer Vision applications. Further, we demonstrate the performance of our algorithm by comparing it to several state-of-the-art methods both in terms of classification accuracy and sparsity.

Rudrasis Chakraborty, Dohyung Seo, Baba C. Vemuri

Manifold-valued datasets are widely encountered in many computer vision tasks. A non-linear analog of the PCA, called the Principal Geodesic Analysis (PGA) suited for data lying on Riemannian manifolds was reported in literature a decade ago. Since the objective function in PGA is highly non-linear and hard to solve efficiently in general, researchers have proposed a linear approximation. Though this linear approximation is easy to compute, it lacks accuracy especially when the data exhibits a large variance. Recently, an alternative called exact PGA was proposed which tries to solve the optimization without any linearization. For general Riemannian manifolds, though it gives better accuracy than the original (linearized) PGA, for data that exhibit large variance, the optimization is not computationally efficient. In this paper, we propose an efficient exact PGA for constant curvature Riemannian manifolds (CCM-EPGA). CCM-EPGA differs significantly from existing PGA algorithms in two aspects, (i) the distance between a given manifold-valued data point and the principal submanifold is computed analytically and thus no optimization is required as in existing methods. (ii) Unlike the existing PGA algorithms, the descent into codimension-1 submanifolds does not require any optimization but is accomplished through the use of the Rimeannian inverse Exponential map and the parallel transport operations. We present theoretical and experimental results for constant curvature Riemannian manifolds depicting favorable performance of CCM-EPGA compared to existing PGA algorithms. We also present data reconstruction from principal components and directions which has not been presented in literature in this setting.