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Mathematics Colloquium: Data and Succinctness
3:30 pm; Neill 5W
Kevin Vixie, Rick Chartrand & Tom Asaki
Abstract: In this talk we will give brief snapshots from three pieces of our research program: geometric analysis, nonconvex compressive sensing, and mixed-variable optimization. A unifying theme is that the ability to recover meaningful information from very few data, or to process very large data sets, can depend on how the information is represented. The pixel or voxel representation of objects in images is inefficient; sometimes, a succinct representation can be what allows a challenging scientific problem to be solved. Geometric measure theory gives us deep and precise insights into the nature of sets, functions, and measures that are the focus of image and shape analysis methods. Compressive sensing lets us recover images and signals perfectly from very few data, and nonvonvex methods reduce requirements even further. Optimization search and surrogate methods that are robust to mixtures of categorical, discrete, and continuous variables are necessary for tractable solutions of many real inverse problems. The applications of these ideas are many. Medical diagnoses can be possible with a few X-rays instead of a high-dose CT scan. Hyperspectral imagery can eliminate the data bottleneck by simply measuring less. Video streams can be reduced from terapixels to their shape content. Objects can be found in noisy, cluttered scenes by looking for natural, low dimensional signatures that are invariant or largely immune to noise and clutter. In dealing with the challenges of data at the extremes of large and small, we find that parsimony, and not industry, is the immediate cause of the increase of knowledge.