Eunseong Bae, Wolfgang Polonik
Under the assumption that data lie on a compact (unknown) manifold without boundary, we derive finite sample bounds for kernel smoothing and its (first and second) derivatives, and we establish asymptotic normality through Berry-Esseen type bounds. Special cases include kernel density estimation, kernel regression and the heat kernel signature. Connections to the graph Laplacian are also discussed.
Olympio Hacquard, Krishnakumar Balasubramanian, Gilles Blanchard et al.
We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.
Wanli Qiao, Wolfgang Polonik
The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences, by which we mean that the algorithms can asymptotically recover the full ridge set. We consider the new algorithms as alternatives to the Subspace Constrained Mean Shift (SCMS) algorithm for which no such theoretical guarantees are known.
Rushil Anirudh, Jayaraman J. Thiagarajan, Irene Kim et al.
We present an approach to model time series data from resting state fMRI for autism spectrum disorder (ASD) severity classification. We propose to adopt kernel machines and employ graph kernels that define a kernel dot product between two graphs. This enables us to take advantage of spatio-temporal information to capture the dynamics of the brain network, as opposed to aggregating them in the spatial or temporal dimension. In addition to the conventional similarity graphs, we explore the use of L1 graph using sparse coding, and the persistent homology of time delay embeddings, in the proposed pipeline for ASD classification. In our experiments on two datasets from the ABIDE collection, we demonstrate a consistent and significant advantage in using graph kernels over traditional linear or non linear kernels for a variety of time series features.