Diffusion maps for changing data

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion map. As the data changes over the parameter space, the low dimensional embedding changes as well. We give a way to go between these embeddings, and furthermore, map them all into a common space, allowing one to track the evolution of X in its intrinsic geometry. A global diffusion distance is also defined, which gives a measure of the global behavior of the data over the parameter space. Approximation theorems in terms of randomly sampled data are presented, as are potential applications.Comment: 38 pages. 9 figures. To appear in Applied and Computational Harmonic Analysis. v2: Several minor changes beyond just typos. v3: Minor typo corrected, added DO

Construction of embedded fMRI resting state functional connectivity networks using manifold learning

We construct embedded functional connectivity networks (FCN) from benchmark resting-state functional magnetic resonance imaging (rsfMRI) data acquired from patients with schizophrenia and healthy controls based on linear and nonlinear manifold learning algorithms, namely, Multidimensional Scaling (MDS), Isometric Feature Mapping (ISOMAP) and Diffusion Maps. Furthermore, based on key global graph-theoretical properties of the embedded FCN, we compare their classification potential using machine learning techniques. We also assess the performance of two metrics that are widely used for the construction of FCN from fMRI, namely the Euclidean distance and the lagged cross-correlation metric. We show that the FCN constructed with Diffusion Maps and the lagged cross-correlation metric outperform the other combinations

Functional diffusion maps

Nowadays many real-world datasets can be considered as functional, in the sense that the processes which generate them are continuous. A fundamental property of this type of data is that in theory they belong to an infinite-dimensional space. Although in practice we usually receive finite observations, they are still high-dimensional and hence dimensionality reduction methods are crucial. In this vein, the main state-of-the-art method for functional data analysis is Functional PCA. Nevertheless, this classic technique assumes that the data lie in a linear manifold, and hence it could have problems when this hypothesis is not fulfilled. In this research, attention has been placed on a non-linear manifold learning method: Diffusion Maps. The article explains how to extend this multivariate method to functional data and compares its behavior against Functional PCA over different simulated and real example

Kernelized Diffusion maps

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a graph kernel approach, however this local average construction is known to be cursed by the high-dimension d. In this article, we build a different estimator of the Laplacian, via a reproducing kernel Hilbert space method, which adapts naturally to the regularity of the problem. We provide non-asymptotic statistical rates proving that the kernel estimator we build can circumvent the curse of dimensionality. Finally we discuss techniques (Nystr\"om subsampling, Fourier features) that enable to reduce the computational cost of the estimator while not degrading its overall performance.Comment: 19 pages, 1 Figur