Statistics on the (compact) Stiefel manifold: Theory and Applications

A Stiefel manifold of the compact type is often encountered in many fields of Engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Fr\'echet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based non-recursive counter part as well as the stochastic gradient descent based method prevalent in literature

ManifoldNorm: Extending normalizations on Riemannian Manifolds

Many measurements in computer vision and machine learning manifest as non-Euclidean data samples. Several researchers recently extended a number of deep neural network architectures for manifold valued data samples. Researchers have proposed models for manifold valued spatial data which are common in medical image processing including processing of diffusion tensor imaging (DTI) where images are fields of $3\times 3$ symmetric positive definite matrices or representation in terms of orientation distribution field (ODF) where the identification is in terms of field on hypersphere. There are other sequential models for manifold valued data that recently researchers have shown to be effective for group difference analysis in study for neuro-degenerative diseases. Although, several of these methods are effective to deal with manifold valued data, the bottleneck includes the instability in optimization for deeper networks. In order to deal with these instabilities, researchers have proposed residual connections for manifold valued data. One of the other remedies to deal with the instabilities including gradient explosion is to use normalization techniques including {\it batch norm} and {\it group norm} etc.. But, so far there is no normalization techniques applicable for manifold valued data. In this work, we propose a general normalization techniques for manifold valued data. We show that our proposed manifold normalization technique have special cases including popular batch norm and group norm techniques. On the experimental side, we focus on two types of manifold valued data including manifold of symmetric positive definite matrices and hypersphere. We show the performance gain in one synthetic experiment for moving MNIST dataset and one real brain image dataset where the representation is in terms of orientation distribution field (ODF)

Dictionary Learning and Sparse Coding on Statistical Manifolds

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.Comment: arXiv admin note: substantial text overlap with arXiv:1604.0693

Generative Adversarial Network based Autoencoder: Application to fault detection problem for closed loop dynamical systems

Fault detection problem for closed loop uncertain dynamical systems, is investigated in this paper, using different deep learning based methods. Traditional classifier based method does not perform well, because of the inherent difficulty of detecting system level faults for closed loop dynamical system. Specifically, acting controller in any closed loop dynamical system, works to reduce the effect of system level faults. A novel Generative Adversarial based deep Autoencoder is designed to classify datasets under normal and faulty operating conditions. This proposed network performs significantly well when compared to any available classifier based methods, and moreover, does not require labeled fault incorporated datasets for training purpose. Finally, this aforementioned network's performance is tested on a high complexity building energy system dataset.Comment: 9 pages, 2 figure

A GMM based algorithm to generate point-cloud and its application to neuroimaging

Recent years have witnessed the emergence of 3D medical imaging techniques with the development of 3D sensors and technology. Due to the presence of noise in image acquisition, registration researchers focused on an alternative way to represent medical images. An alternative way to analyze medical imaging is by understanding the 3D shapes represented in terms of point-cloud. Though in the medical imaging community, 3D point-cloud processing is not a ``go-to'' choice, it is a ``natural'' way to capture 3D shapes. However, as the number of samples for medical images are small, researchers have used pre-trained models to fine-tune on medical images. Furthermore, due to different modality in medical images, standard generative models can not be used to generate new samples of medical images. In this work, we use the advantage of point-cloud representation of 3D structures of medical images and propose a Gaussian mixture model-based generation scheme. Our proposed method is robust to outliers. Experimental validation has been performed to show that the proposed scheme can generate new 3D structures using interpolation techniques, i.e., given two 3D structures represented as point-clouds, we can generate point-clouds in between. We have also generated new point-clouds for subjects with and without dementia and show that the generated samples are indeed closely matched to the respective training samples from the same class

An "augmentation-free" rotation invariant classification scheme on point-cloud and its application to neuroimaging

Recent years have witnessed the emergence and increasing popularity of 3D medical imaging techniques with the development of 3D sensors and technology. However, achieving geometric invariance in the processing of 3D medical images is computationally expensive but nonetheless essential due to the presence of possible errors caused by rigid registration techniques. An alternative way to analyze medical imaging is by understanding the 3D shapes represented in terms of point-cloud. Though in the medical imaging community, 3D point-cloud processing is not a "go-to" choice, it is a canonical way to preserve rotation invariance. Unfortunately, due to the presence of discrete topology, one can not use the standard convolution operator on point-cloud. To the best of our knowledge, the existing ways to do "convolution" can not preserve the rotation invariance without explicit data augmentation. Therefore, we propose a rotation invariant convolution operator by inducing topology from hypersphere. Experimental validation has been performed on publicly available OASIS dataset in terms of classification accuracy between subjects with (without) dementia, demonstrating the usefulness of our proposed method in terms of model complexity, classification accuracy, and last but most important invariance to rotations.Comment: arXiv admin note: text overlap with arXiv:1910.13050 and arXiv:1911.0170

ManifoldNet: A Deep Network Framework for Manifold-valued Data

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 Fr\'{e}chet 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

SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold

Complex-valued data is ubiquitous in signal and image processing applications, and complex-valued representations in deep learning have appealing theoretical properties. While these aspects have long been recognized, complex-valued deep learning continues to lag far behind its real-valued counterpart. We propose a principled geometric approach to complex-valued deep learning. Complex-valued data could often be subject to arbitrary complex-valued scaling; as a result, real and imaginary components could co-vary. Instead of treating complex values as two independent channels of real values, we recognize their underlying geometry: We model the space of complex numbers as a product manifold of non-zero scaling and planar rotations. Arbitrary complex-valued scaling naturally becomes a group of transitive actions on this manifold. We propose to extend the property instead of the form of real-valued functions to the complex domain. We define convolution as weighted Fr\'echet mean on the manifold that is equivariant to the group of scaling/rotation actions, and define distance transform on the manifold that is invariant to the action group. The manifold perspective also allows us to define nonlinear activation functions such as tangent ReLU and G-transport, as well as residual connections on the manifold-valued data. We dub our model SurReal, as our experiments on MSTAR and RadioML deliver high performance with only a fractional size of real-valued and complex-valued baseline models.Comment: 12 pages, accepted to TNNLS journa

An Online Riemannian PCA for Stochastic Canonical Correlation Analysis

We present an efficient stochastic algorithm (RSG+) for canonical correlation analysis (CCA) using a reparametrization of the projection matrices. We show how this reparametrization (into structured matrices), simple in hindsight, directly presents an opportunity to repurpose/adjust mature techniques for numerical optimization on Riemannian manifolds. Our developments nicely complement existing methods for this problem which either require $O(d^3)$ time complexity per iteration with $O(\frac{1}{\sqrt{t}})$ convergence rate (where $d$ is the dimensionality) or only extract the top $1$ component with $O(\frac{1}{t})$ convergence rate. In contrast, our algorithm offers a strict improvement for this classical problem: it achieves $O(d^2k)$ runtime complexity per iteration for extracting the top $k$ canonical components with $O(\frac{1}{t})$ convergence rate. While the paper primarily focuses on the formulation and technical analysis of its properties, our experiments show that the empirical behavior on common datasets is quite promising. We also explore a potential application in training fair models where the label of protected attribute is missing or otherwise unavailable

A CNN for homogneous Riemannian manifolds with applications to Neuroimaging

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