Principal Boundary on Riemannian Manifolds

We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic metric on the manifolds. Motivated by finding an optimal boundary between the two classes, we invent a novel approach -- the principal boundary. From the perspective of classification, the principal boundary is defined as an optimal curve that moves in between the principal flows traced out from two classes of data, and at any point on the boundary, it maximizes the margin between the two classes. We estimate the boundary in quality with its direction, supervised by the two principal flows. We show that the principal boundary yields the usual decision boundary found by the support vector machine in the sense that locally, the two boundaries coincide. Some optimality and convergence properties of the random principal boundary and its population counterpart are also shown. We illustrate how to find, use and interpret the principal boundary with an application in real data.Comment: 31 pages,10 figure

Hypothesis Testing For Network Data in Functional Neuroimaging

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as "Is there a difference between the networks of these two groups of subjects?" In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry, and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fr\'echet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistical powerful, than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.Comment: 34 pages. 5 figure

An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces

Shape Dimension and Intrinsic Metric from Samples of Manifolds

We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some Euclidean space Rd. We assume that M is known to us only through a finite sample P \subset M, as is often the case in applications. The adaptive neighborhood graph is a geometric graph on P. Its complexity is at most \min{2^{O(k)n, n2}, where n = |P| and k = dim M, as opposed to the n\lceil d/2 \rceil complexity of the Delaunay triangulation, which is often used to model manifolds. We prove that we can correctly infer the connected components and the dimension of M from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is d2O(k^{7} log k) for each connected component of M. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on the intrinsic dimension k, not on the ambient dimension d. This is of particular interest if the co-dimension is high, i.e., if k is much smaller than d, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in

Barycentric Subspace Analysis on Manifolds

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).Comment: Annals of Statistics, Institute of Mathematical Statistics, A Para\^itr