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.

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.

Principal Component Analysis (PCA) is a widely popular dimensionality reduction technique for vector-valued inputs. In the past decade, a nonlinear generalization of PCA, called the Principal Geodesic Analysis (PGA) was developed to tackle data that lie on a smooth manifold. PGA suffers from the same problem as PCA in that, in both the methods, each Principal Component (PC) is a linear combination of the original variables. This makes it very difficult to interpret the PCs especially in high dimensions. This lead to the introduction of sparse PCA (SPCA) in the vector-space input case. In this paper, we present a novel generalization of SPCA, called sparse exact PGA (SEPGA) that can cope with manifold-valued input data and respect the intrinsic geometry of the underlying manifold. Sparsity has the advantage of not only easy interpretability but also computational efficiency. We achieve this by formulating the PGA problem as a minimization of the projection error in conjunction with sparsity constraints enforced on the principal vectors post isomorphic mapping to Rm, where m is the dimension of the manifold on which the data reside. Further, for constant curvature smooth manifolds, we use analytic formulae for the projection error leading to an efficient solution to the SEPGA problem. We present extensive experimental results demonstrating the performance of SEPGA in achieving very good sparse principal components without sacrificing the accuracy of reconstruction. This makes SEPGA accurate and efficient in representing manifold-valued data.

Regression is an essential tool in Statistical analysis of data with many applications in Computer Vision, Machine Learning, Medical Imaging and various disciplines of Science and Engineering. Linear and nonlinear regression in a vector space setting has been well studied in literature. However, generalizations to manifold-valued data are only recently gaining popularity. With the exception of a few, most existing methods of regression for manifold valued data are limited to geodesic regression which is a generalization of the linear regression in vector-spaces. In this paper, we present a novel nonlinear kernel-based regression method that is applicable to manifold valued data. Our method is applicable to cases when the independent and dependent variables in the regression model are both manifold-valued or one is manifold-valued and the other is vector or scalar valued. Further, unlike most methods, our method does not require any imposed ordering on the manifold-valued data. The performance of our model is tested on a large number of real data sets acquired from Alzhiemers and movement disorder (Parkinsons and Essential Tremor) patients. We present an extensive set of results along with statistical validation and comparisons.

Probability density functions (PDFs) are fundamental objects in mathematics with numerous applications in computer vision, machine learning and medical imaging. The feasibility of basic operations such as computing the distance between two PDFs and estimating a mean of a set of PDFs is a direct function of the representation we choose to work with. In this paper, we study the Gaussian mixture model (GMM) representation of the PDFs motivated by its numerous attractive features. (1) GMMs are arguably more interpretable than, say, square root parameterizations (2) the model complexity can be explicitly controlled by the number of components and (3) they are already widely used in many applications. The main contributions of this paper are numerical algorithms to enable basic operations on such objects that strictly respect their underlying geometry. For instance, when operating with a set of K component GMMs, a first order expectation is that the result of simple operations like interpolation and averaging should provide an object that is also a K component GMM. The literature provides very little guidance on enforcing such requirements systematically. It turns out that these tasks are important internal modules for analysis and processing of a field of ensemble average propagators (EAPs), common in diffusion weighted magnetic resonance imaging. We provide proof of principle experiments showing how the proposed algorithms for interpolation can facilitate statistical analysis of such data, essential to many neuroimaging studies. Separately, we also derive interesting connections of our algorithm with functional spaces of Gaussians, that may be of independent interest.

Features selected by a supervised/ unsupervised technique often include redundant or correlated features. While use of correlated features may result in an increase in the design and decision making cost, removing redundancy completely can make the system vulnerable to measurement errors. Most feature selection schemes do not account for redundancy at all, while a few supervised methods try to discard correlated features. We propose a novel unsupervised feature selection scheme (UFeSCoR), which not only discards irrelevant features, but also selects features with controlled redundancy. Here, the number of selected features can also be directed. Our algorithm optimizes an objective function, which tries to select a specified number of features, with a controlled level of redundancy, such that the topology of the original data set can be maintained in the reduced dimension. Here, we have used Sammon's error as a measure of preservation of topology. We demonstrate the effectiveness of the algorithm in terms of choosing relevant features, controlling redundancy, and selecting a given number of features using several data sets. We make a comparative study with five unsupervised feature selection methods. Our results reveal that the proposed method can select useful features with controlled redundancy.