We are in the throes of a data deluge. The vast amount of information generated by data sources positioned across the globe is poised to overwhelm current state-of-the-art data processing algorithms. Processing this information in any meaningful fashion is as difficult as searching for a tiny needle in a haystack.
Fortunately, some exciting recent developments in signal processing/machine learning can help counter this bleak possibility. The key insight is that despite its apparent high dimensionality, the aggregate data can instead be described using simple, low-dimensional geometric models. This geometric structure of the data not only enables efficient algorithms for extracting useful information from the data, but also lends itself to elegant analysis that characterizes the fundamental limits of these algorithms.
union-of-subspace models for signals/images
sub-Nyquist sampling of bilinear models
model-based compressive sensing
Proposes a general framework for sub-Nyquist sampling and recovery
sparse signals modeled as arising from a union of
subspaces. Develops theory and algorithms that have been successfully used in a wide range of applications.
manifold models for signal/image ensembles
image synthesis and modeling
Explores the use of optical flow in learning certain classes of image
manifolds. Applications include
image synthesis, parameter estimation, and charting.
multi-manifold signal recovery
Introduces and analyzes an algorithm for sampling and recovery of signals
that are the linear sum of manifold-modeled components.
Develops a convex method for designing sub-Nyquist measurement kernels adapted to
data from low-dimensional manifolds, and/or adapted to a specified
signal processing task (note the distinction from adaptive sampling,
also referred to as active sensing).
compressive classification and parameter estimation
Enables methods for efficient multi-manifold data fusion for
classification/parameter estimation. Relevant for networks of
high-resolution static cameras observing a common scene.
nonparametric models for images
Develops a highly efficient algorithm for the upsampling of
edges and textures in natural photos. Our algorithm can crisply upscale a
200x200 image by a factor of 4x in less than a minute.