Metrics for matrix-valued measures via test functions

It is perhaps not widely recognized that certain common notions of distance between probability measures have an alternative dual interpretation which compares corresponding functionals against suitable families of test functions. This dual viewpoint extends in a straightforward manner to suggest metrics between matrix-valued measures. Our main interest has been in developing weakly-continuous metrics that are suitable for comparing matrix-valued power spectral density functions. To this end, and following the suggested recipe of utilizing suitable families of test functions, we develop a weakly-continuous metric that is analogous to the Wasserstein metric and applies to matrix-valued densities. We use a numerical example to compare this metric to certain standard alternatives including a different version of a matricial Wasserstein metric developed earlier

(k,q)-Compressed Sensing for dMRI with Joint Spatial-Angular Sparsity Prior

Advanced diffusion magnetic resonance imaging (dMRI) techniques, like diffusion spectrum imaging (DSI) and high angular resolution diffusion imaging (HARDI), remain underutilized compared to diffusion tensor imaging because the scan times needed to produce accurate estimations of fiber orientation are significantly longer. To accelerate DSI and HARDI, recent methods from compressed sensing (CS) exploit a sparse underlying representation of the data in the spatial and angular domains to undersample in the respective k- and q-spaces. State-of-the-art frameworks, however, impose sparsity in the spatial and angular domains separately and involve the sum of the corresponding sparse regularizers. In contrast, we propose a unified (k,q)-CS formulation which imposes sparsity jointly in the spatial-angular domain to further increase sparsity of dMRI signals and reduce the required subsampling rate. To efficiently solve this large-scale global reconstruction problem, we introduce a novel adaptation of the FISTA algorithm that exploits dictionary separability. We show on phantom and real HARDI data that our approach achieves significantly more accurate signal reconstructions than the state of the art while sampling only 2-4% of the (k,q)-space, allowing for the potential of new levels of dMRI acceleration.Comment: To be published in the 2017 Computational Diffusion MRI Workshop of MICCA

Work Distributions in 1-D Fermions and Bosons with Dual Contact Interactions

We extend the well-known static duality \cite{girardeau1960relationship, cheon1999fermion} between 1-D Bosons and 1-D Fermions to the dynamical version. By utilizing this dynamical duality we find the duality of non-equilibrium work distributions between interacting 1-D bosonic (Lieb-Liniger model) and 1-D fermionic (Cheon-Shigehara model) systems with dual contact interactions. As a special case, the work distribution of the Tonks-Girardeau (TG) gas is identical to that of 1-D free fermionic system even though their momentum distributions are significantly different. In the classical limit, the work distributions of Lieb-Liniger models (Cheon-Shigehara models) with arbitrary coupling strength converge to that of the 1-D noninteracting distinguishable particles, although their elemetary excitations (quasi-particles) obey different statistics, e.g. the Bose-Einstein, the Fermi-Dirac and the fractional statistics. We also present numerical results of the work distributions of Lieb-Liniger model with various coupling strengths, which demonstrate the convergence of work distributions in the classical limit.Comment: 8 pages, 2 figure, 2 table

Distances and Riemannian metrics for multivariate spectral densities

We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close" power spectra are quadratic, and hence, they induce a differential-geometric structure. We study the corresponding Riemannian metrics and, for a particular case, provide explicit formulae for the corresponding geodesics and geodesic distances. The close connection between the geometry of power spectra and the geometry of the Fisher-Rao metric is noted.Comment: 21 pages, 8 figure