Sparse Modeling Heuristics for Parameter Estimation - Applications in Statistical Signal Processing

This thesis examines sparse statistical modeling on a range of applications in audio modeling, audio localizations, DNA sequencing, and spectroscopy. In the examined cases, the resulting estimation problems are computationally cumbersome, both as one often suffers from a lack of model order knowledge for this form of problems, but also due to the high dimensionality of the parameter spaces, which typically also yield optimization problems with numerous local minima.
In this thesis, these problems are treated using sparse modeling heuristics, with the resulting criteria being solved using convex relaxations, inspired from disciplined convex programming ideas, to maintain tractability. The contributions to audio modeling and estimation focus on the estimation of the fundamental frequency of harmonically related sinusoidal signals, which is commonly used model for, e.g., voiced speech or tonal audio. We examine both the problems of estimating multiple audio sources assuming the expected harmonic structure, as well as the problem of robustness to the often occurring inharmonic structure, such that the higher order sinusoidal components deviate in an unknown way from the expected multiples of the fundamental frequency. This is a problem commonly occurring for, for instance, string instruments, which, if not properly accounted for, will degrade the performance of most pitch estimators noticeably. We also consider the problem of localizing audio sources in an unknown and possibly reverberant acoustic environment, allowing for simultaneous localization of far-field and near-field signals. The DNA sequencing contribution, presented in the more general setting of arbitrary categorical sequences, is inspired by the problem of identifying segments in the genome, which are characterized by the highly periodic behavior of the sequence. In each of the contributions, an appropriate computationally efficient algorithm is proposed. Specifically for the sparse models, alternating directions method of multipliers and cyclic coordinate descent implementations are suggested, since the proposed convex criteria are in practice easier to solve than the standard interior point solvers would suggest. The suggested methods are in all cases compared with previously proposed algorithms and/or measured data, as appropriate.