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.
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.
With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively.
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.
To this end, we present a provably convergent recursive computation of the wFM of the given data, where the weights makeup the convolution mask, to be learned.
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.
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.
We show how recurrent statistical recurrent network models can be defined in such 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.
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).
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.
Finally, by using existing algorithms for recursive Frechet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few PGs.
We have demonstrated competitive performance of our proposed online subspace algorithm method on one synthetic and two real data sets.
Statistical machine learning models that operate on manifold-valued data are being extensively studied in vision, motivated by applications in activity recognition, feature tracking and medical imaging.
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.
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 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).
Probability density functions (PDFs) are fundamental "objects" in mathematics with numerous applications in computer vision, machine learning and medical imaging.
In the limit as the number of samples tends to infinity, we prove that GiFME converges to the FM (this is called the weak consistency result on the Grassmann manifold).
In this paper, we propose a new intrinsic recursive filter on the product manifold of shape and orientation.
In this paper, we use the well known Riemannian framework never before used for point cloud matching, and present a novel matching algorithm.
Then, the problem of point set registration is reformulated as the problem of aligning two Gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized.