Block-Simultaneous Direction Method of Multipliers: A proximal primal-dual splitting algorithm for nonconvex problems with multiple constraints

We introduce a generalization of the linearized Alternating Direction Method of Multipliers to optimize a real-valued function $f$ of multiple arguments with potentially multiple constraints $g_\circ$ on each of them. The function $f$ may be nonconvex as long as it is convex in every argument, while the constraints $g_\circ$ need to be convex but not smooth. If $f$ is smooth, the proposed Block-Simultaneous Direction Method of Multipliers (bSDMM) can be interpreted as a proximal analog to inexact coordinate descent methods under constraints. Unlike alternative approaches for joint solvers of multiple-constraint problems, we do not require linear operators $L$ of a constraint function $g(L\ \cdot)$ to be invertible or linked between each other. bSDMM is well-suited for a range of optimization problems, in particular for data analysis, where $f$ is the likelihood function of a model and $L$ could be a transformation matrix describing e.g. finite differences or basis transforms. We apply bSDMM to the Non-negative Matrix Factorization task of a hyperspectral unmixing problem and demonstrate convergence and effectiveness of multiple constraints on both matrix factors. The algorithms are implemented in python and released as an open-source package.Comment: 13 pages, 4 figure

SCARLET: Source separation in multi-band images by Constrained Matrix Factorization

We present the source separation framework SCARLET for multi-band images, which is based on a generalization of the Non-negative Matrix Factorization to alternative and several simultaneous constraints. Our approach describes the observed scene as a mixture of components with compact spatial support and uniform spectra over their support. We present the algorithm to perform the matrix factorization and introduce constraints that are useful for optical images of stars and distinct stellar populations in galaxies, in particular symmetry and monotonicity with respect to the source peak position. We also derive the treatment of correlated noise and convolutions with band-dependent point spread functions, rendering our approach applicable to coadded images observed under variable seeing conditions. SCARLET thus yields a PSF-matched photometry measurement with an optimally chosen weight function given by the mean morphology in all available bands. We demonstrate the performance of SCARLET for deblending crowded extragalactic scenes and on an AGN jet -- host galaxy separation problem in deep 5-band imaging from the Hyper Suprime-Cam Stategic Survey Program. Using simulations with prominent crowding we show that SCARLET yields superior results to the HSC-SDSS deblender for the recovery of total fluxes, colors, and morphologies. Due to its non-parametric nature, a conceptual limitation of SCARLET is its sensitivity to undetected sources or multiple stellar population within detected sources, but an iterative strategy that adds components at the location of significant residuals appears promising. The code is implemented in Python with C++ extensions and is available at https://github.com/fred3m/scarletComment: accepted by Astronomy & Computin

Planetary Construction Zones in Occultation: Discovery of an Extrasolar Ring System Transiting a Young Sun-like Star and Future Prospects for Detecting Eclipses by Circumsecondary and Circumplanetary Disks

The large relative sizes of circumstellar and circumplanetary disks imply that they might be seen in eclipse in stellar light curves. We estimate that a survey of ~10^4 young (~10 Myr old) post-accretion pre-MS stars monitored for ~10 years should yield at least a few deep eclipses from circumplanetary disks and disks surrounding low mass companion stars. We present photometric and spectroscopic data for a pre-MS K5 star (1SWASP J140747.93-394542.6), a newly discovered ~0.9 Msun member of the ~16 Myr-old Upper Cen-Lup subgroup of Sco-Cen at a kinematic distance of 128 pc. SuperWASP and ASAS light curves for this star show a remarkably long, deep, and complex eclipse event centered on 29 April 2007. At least 5 multi-day dimming events of >0.5 mag are identified, with a >3.3 mag deep eclipse bracketed by two pairs of ~1 mag eclipses symmetrically occurring +-12 days and +-26 days before and after. Hence, significant dimming of the star was taking place on and off over at least a ~54 day period in 2007, and a strong >1 mag dimming event occurred over a ~12 day span. We place a firm lower limit on the period of 850 days (i.e. the orbital radius of the eclipser must be >1.7 AU and orbital velocity must be <22 km/s). The shape of the light curve is similar to the lop-sided eclipses of the Be star EE Cep. We suspect that this new star is being eclipsed by a low-mass object orbited by a dense inner disk, girded by at least 3 dusty rings of lower optical depth. Between these rings are at least two annuli of near-zero optical depth (i.e. gaps), possibly cleared out by planets or moons, depending on the nature of the secondary. For possible periods in the range 2.33-200 yr, the estimated total ring mass is ~8-0.4 Mmoon (if the rings have optical opacity similar to Saturn's rings), and the edge of the outermost detected ring has orbital radius ~0.4-0.09 AU.Comment: Astronomical Journal, in press, 13 figure