This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

Graph convolution operator of the GCN model is originally motivated from a localized first-order approximation of spectral graph convolutions.This work stands on a different view; establishing a connection between graph convolution and graph-regularized PCA. Based on this connection, GCN architecture, shaped by stacking graph convolution layers, shares a close relationship with stacking graph-regularized PCA (GPCA). We empirically demonstrate that the unsupervised embeddings by GPCA paired with a logistic regression classifier achieves similar performance to GCN on semi-supervised node classification tasks. Further, we capitalize on the discovered relationship to design an effective initialization strategy for GCN based on stacking GPCA.

Matrix factorization methods are extensively employed to understand complex data. In this paper, we introduce the cross-product penalized component analysis (XCAN), a sparse matrix factorization based on the optimization of a loss function that allows a trade-off between variance maximization and structural preservation. The approach is based on previous developments, notably (i) the Sparse Principal Component Analysis (SPCA) framework based on the LASSO, (ii) extensions of SPCA to constrain both modes of the factorization, like co-clustering or the Penalized Matrix Decomposition (PMD), and (iii) the Group-wise Principal Component Analysis (GPCA) method. The result is a flexible modeling approach that can be used for data exploration in a large variety of problems. We demonstrate its use with applications from different disciplines.

Variables in many massive high-dimensional data sets are structured, arising for example from measurements on a regular grid as in imaging and time series or from spatial-temporal measurements as in climate studies. Classical multivariate techniques ignore these structural relationships often resulting in poor performance. We propose a generalization of the singular value decomposition (SVD) and principal components analysis (PCA) that is appropriate for massive data sets with structured variables or known two-way dependencies. By finding the best low rank approximation of the data with respect to a transposable quadratic norm, our decomposition, entitled the Generalized least squares Matrix Decomposition (GMD), directly accounts for structural relationships. As many variables in high-dimensional settings are often irrelevant or noisy, we also regularize our matrix decomposition by adding two-way penalties to encourage sparsity or smoothness. We develop fast computational algorithms using our methods to perform generalized PCA (GPCA), sparse GPCA, and functional GPCA on massive data sets. Through simulations and a whole brain functional MRI example we demonstrate the utility of our methodology for dimension reduction, signal recovery, and feature selection with high-dimensional structured data.

We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes. Starting from a given or random initial condition, we use normalized gradient descent to update the coefficients of a time varying polynomial whose degree is the number of hyperplanes and whose derivatives at a trajectory give an estimate of the vector normal to the hyperplane containing that trajectory. As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. The segmentation of the trajectories is then obtained by clustering their associated normal vectors. The final result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. We test our algorithm on the segmentation of dynamic scenes containing rigid motions and dynamic textures, e.g., a bird floating on water. Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e.g., periodic motion) directly from the temporal evolution of the estimated polynomial coefficients. Our experiments also show that our method can deal with appearing and disappearing motions in the scene.

We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes. Starting from a given or random initial condition, we use normalized gradient descent to update the coefficients of a time varying polynomial whose degree is the number of hyperplanes and whose derivatives at a trajectory give an estimate of the vector normal to the hyperplane containing that trajectory. As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. The segmentation of the trajectories is then obtained by clustering their associated normal vectors. The final result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. We test our algorithm on the segmentation of dynamic scenes containing rigid motions and dynamic textures, e.g., a bird floating on water. Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e.g., periodic motion) directly from the temporal evolution of the estimated polynomial coefficients. Our experiments also show that our method can deal with appearing and disappearing motions in the scene.

We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes.Starting from a given or random initial condition, we use normalized gradientdescent to update the coefficients of a time varying polynomial whose degree is the number of hyperplanes and whose derivatives at a trajectory give an estimate of the vector normal to the hyperplane containing that trajectory. As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. The segmentation of the trajectories is then obtained by clustering their associated normal vectors. The final result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. We test our algorithm on the segmentation of dynamic scenes containing rigid motions anddynamic textures, e.g., a bird floating on water. Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e.g., periodic motion) directly from the temporal evolution of the estimated polynomial coefficients. Our experiments also show that our method can deal with appearing and disappearing motions in the scene.