Estimating Nuisance Parameters in Inverse Problems

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.Comment: 16 pages, 5 figure

Visualization of Tonal Harmony for Jazz Lead Sheets

Jazz improvisation is the extemporaneous expression of melody, and musicians commonly base their performances on chord progressions given by lead sheets. It is standard practice to commit a progression to memory by analyzing it for common patterns. This paper presents a visualization design intended to help reduce the amount of cognitive work needed to assimilate a song’s chords and harmonic patterns. It does this using color, shapes, and glyphs as visual variables to convey meaning about tonal centers, chord functions, and harmonic structure