Tensor methods for high-dimensional analysis and visualization

Abstract

Most visual computing domains are witnessing a steady growth in sheer data set size, complexity, and dimensionality. Flexible and scalable mathematical models that can efficiently compress, process, store, manipulate, retrieve and visualize such data sets are therefore of paramount importance, especially for higher dimensions. In this context, tensor decompositions constitute a powerful mathematical framework for compactly representing and operating on both dense and sparse data. Initially proposed as an extension of the concept of matrix decomposition for three and more dimensions, they have found various applications in data-intensive machine learning and high-dimensional signal processing. This thesis aims to help bridge these aspects and tackle modern visual computing challenges under the paradigm of a common representation format, namely tensors. Many kinds of data admit a natural representation as higher-order tensors and/or can be parametrized, learned, or interpolated in the form of compact tensor models. Numerous tools that are native and unique to said decompositions exist for analysis and visualization, and such tools can be exploited as soon as the known ground-truth is abstracted into this kind of reduced representation. To this end we develop a volume compression algorithm tailored to high reduction rates in visualization applications; we explore compressed-domain processing possibilities including multiresolution convolution, derivation, integration and summed area tables; we produce visualization diagrams directly from compressed tensors via interactive reconstruction; and we propose sensitivity analysis algorithms for model interpretation and knowledge discovery. Emphasis is placed on compactness and interactivity and is addressed via careful tensor format selection and model building, as well as a range of auxiliary technical tools including out-of-core memory management, adaptive quantization, parallelized multilinear algebra operations, and others. We conclude that the models chosen result in a viable and fruitful toolbox for data of diverse origin, size, dimensionality, resolution, and sparsity.