Data representation and exploration with Geometric Wavelets

Abstract

Geometric Wavelets is a new multi-scale data representation technique which is useful for a variety of applications such as data compression, interpretation and anomaly detection. We have developed an interactive visualization with multiple linked views to help users quickly explore data sets and understand this novel construction. Currently the interface is being used by applied mathematicians to view results and gain new insights, speeding methods development. GEOMETRIC WAVELETSData sets such as images, documents or gene expression data may be modeled as point clouds in high-dimensional Euclidean space. In the case of images, each pixel can be thought of as one coordinate in a vector with a length equal to the number D of pixels in the image, and the intensity of each pixel corresponds to the coordinate magnitude in that pixel's direction. Real data points often have structure which has dimension d much smaller than the ambient space dimension D, for example under the well-studied case when they lie near a low-dimensional manifold M . Discovering and characterizing this lower-dimensional structure can dramatically affect the performance in tasks such as data compression, interpretation, outlier detection, classification clustering.If M is just a linear subspace, Principal Component Analysis (PCA) can discover a dictionary of d vectors which describe the data well at low computational cost. However, when M is nonlinear it is usually necessary to use random dictionaries or black box optimization, which are much more costly and in general do not yield interpretable features of the data. Geometric Wavelets [2] are multi-scale dictionary elements which are constructed directly from the data, adapt to arbitrary nonlinear manifolds, and have guarantees on the computational cost, the number of elements in the dictionary and the sparsity of the representation (as a function of an approximation error parameter). In particular they provide feature sets that may be particularly useful for data exploration, and tasks such as anomaly detection and classification.The mathematical details of the construction can be found in [1]. It proceeds in several steps: first, relationships between data points are computed with respect to a given similarity function. At the coarsest scale all data points are considered one group and global PCA is performed, yielding a d-dimensional plane fit to the data with axes in the directions of maximum variance, which we think of as a parallel of "scaling functions" in wavelet analysis. The projection of the data points onto this plane is the coarsest-scale approximation of the data. Next, the graph is split into two groups (e.g. by using METIS [3]). On each of these finer scale groups, PCA is again performed and the projection of the data points onto these two new planes will more accurately approximate M . To form a compact representation for the data at this finer scale, as in a wavelet decomposition, we only encode the differences between the original coarse projections of the data and the point...

Multi-scale geometric methods for data sets II: Geometric Multi-Resolution Analysis

“…• User interface. We are currently developing a user interface for interacting with the geometric wavelet representation of data sets [70]. • Higher order approximations.…”