This paper describes a novel approach to change-point detection when the observed high-dimensional data may have missing elements. The performance of classical methods for change-point detection typically scales poorly with the dimensionality of the data, so that a large number of observations are collected after the true change-point before it can be reliably detected. Furthermore, missing components in the observed data handicap conventional approaches. The proposed method addresses these challenges by modeling the dynamic distribution underlying the data as lying close to a time-varying low-dimensional submanifold embedded within the ambient observation space. Specifically, streaming data is used to track a submanifold approximation, measure deviations from this approximation, and calculate a series of statistics of the deviations for detecting when the underlying manifold has changed in a sharp or unexpected manner. The approach described in this paper leverages several recent results in the field of high-dimensional data analysis, including subspace tracking with missing data, multiscale analysis techniques for point clouds, online optimization, and change-point detection performance analysis. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.
Our approach is also closely related to Geometric Multi-Resolution Analysis (GMRA) @cite_19 , which was developed for analyzing intrinsically low-dimensional point clouds in high-dimensional spaces. The basic idea of GMRA is to first iteratively partition a dataset to form a multiscale collection of subsets of the data, then find a low-rank approximation for the data in each subset, and finally efficiently encode the difference between the low-rank approximations at different scales. This approach is a batch method without a straightforward extension to online settings.
@cite_19
2001613026 2094248329 2949671016 2229640315
This paper reports a novel mutual information (MI) based algorithm for automatic registration of unstructured 3D point clouds comprised of co-registered 3D lidar and camera imagery. The proposed method provides a robust and principled framework for fusing the complementary information obtained from these two different sensing modalities. High-dimensional features are extracted from a training set of textured point clouds (scans) and hierarchical k-means clustering is used to quantize these features into a set of codewords. Using this codebook, any new scan can be represented as a collection of codewords. Under the correct rigid-body transformation aligning two overlapping scans, the MI between the codewords present in the scans is maximized. We apply a James-Stein-type shrinkage estimator to estimate the true MI from the marginal and joint histograms of the codewords extracted from the scans. Experimental results using scans obtained by a vehicle equipped with a 3D laser scanner and an omnidirectional camera are used to validate the robustness of the proposed algorithm over a wide range of initial conditions. We also show that the proposed method works well with 3D data alone. A hierarchical Bayes Gaussian Mixture Model (HBGMM) with adaptive sparsity is proposed.We impose our hierarchical Bayes prior on the entries of precision matrices in GMM to encourage sparsity.Our adaptive sparsity can reduce the dimensionality of the parameters in GMM to be estimated.Our HBGMM method does not involve any hyperparameters to be estimated, i.e., it is hyperparameter-free. Gaussian Mixture Model (GMM) has been widely used in statistics for its great flexibility. However, parameter estimation for GMM with high dimensionality is a challenge because of the large number of parameters and the lack of observation data. In this paper, we propose an effective method named hierarchical Bayes based Adaptive Sparsity in Gaussian Mixture Model (ASGMM) to estimate the parameters in a GMM by incorporating a two-layer hierarchical Bayes based adaptive sparsity prior. The prior we impose on the precision matrices can encourage sparsity and hence reduce the dimensionality of the parameters to be estimated. In contrast to the l 1 -norm penalty or Laplace prior, our approach does not involve any hyperparameters that must be tuned, and the sparsity adapts to the observation data. The proposed method is achieved by three steps: first, we formulate an adaptive hierarchical Bayes model of the precision matrices in the GMM with a Jeffrey's noninformative hyperprior, which expresses scale-invariance and, more importantly, is hyperparameter-free and unbiased. Second, we perform a Cholesky decomposition on the precision matrices to impose the positive definite property. Finally, we exploit the expectation maximization (EM) algorithm to obtain the final estimated parameters in the GMM. Experimental results on synthetic and real-world datasets demonstrate that ASGMM cannot only adapt the sparsity of high-dimensional data with small estimated error, but also achieve better clustering performance comparing with several classical methods. Three-dimensional geometric data offer an excellent domain for studying representation learning and generative modeling. In this paper, we look at geometric data represented as point clouds. We introduce a deep AutoEncoder (AE) network with state-of-the-art reconstruction quality and generalization ability. The learned representations outperform existing methods on 3D recognition tasks and enable shape editing via simple algebraic manipulations, such as semantic part editing, shape analogies and shape interpolation, as well as shape completion. We perform a thorough study of different generative models including GANs operating on the raw point clouds, significantly improved GANs trained in the fixed latent space of our AEs, and Gaussian Mixture Models (GMMs). To quantitatively evaluate generative models we introduce measures of sample fidelity and diversity based on matchings between sets of point clouds. Interestingly, our evaluation of generalization, fidelity and diversity reveals that GMMs trained in the latent space of our AEs yield the best results overall. Large data sets arise in a wide variety of applications and are often modeled as samples from a probability distribution in high-dimensional space. It is sometimes assumed that the support of such probability distribution is well approximated by a set of low intrinsic dimension, perhaps even a low-dimensional smooth manifold. Samples are often corrupted by high-dimensional noise. We are interested in developing tools for studying the geometry of such high-dimensional data sets. In particular, we present here a multiscale transform that maps high-dimensional data as above to a set of multiscale coefficients that are compressible sparse under suitable assumptions on the data. We think of this as a geometric counterpart to multi-resolution analysis in wavelet theory: whereas wavelets map a signal (typically low dimensional, such as a one-dimensional time series or a two-dimensional image) to a set of multiscale coefficients, the geometric wavelets discussed here map points in a high-dimensional point cloud to a multiscale set of coefficients. The geometric multi-resolution analysis (GMRA) we construct depends on the support of the probability distribution, and in this sense it fits with the paradigm of dictionary learning or data-adaptive representations, albeit the type of representation we construct is in fact mildly nonlinear, as opposed to standard linear representations. Finally, we apply the transform to a set of synthetic and real-world data sets.