Exploring Photometric Redshifts as an Optimization Problem: An Ensemble MCMC and Simulated Annealing-Driven Template-Fitting Approach

Using a grid of $\sim 2$ million elements ($\Delta z = 0.005$) adapted from COSMOS photometric redshift (photo-z) searches, we investigate the general properties of template-based photo-z likelihood surfaces. We find these surfaces are filled with numerous local minima and large degeneracies that generally confound rapid but "greedy" optimization schemes, even with additional stochastic sampling methods. In order to robustly and efficiently explore these surfaces, we develop BAD-Z [Brisk Annealing-Driven Redshifts (Z)], which combines ensemble Markov Chain Monte Carlo (MCMC) sampling with simulated annealing to sample arbitrarily large, pre-generated grids in approximately constant time. Using a mock catalog of 384,662 objects, we show BAD-Z samples $\sim 40$ times more efficiently compared to a brute-force counterpart while maintaining similar levels of accuracy. Our results represent first steps toward designing template-fitting photo-z approaches limited mainly by memory constraints rather than computation time.Comment: 14 pages, 8 figures; submitted to MNRAS; comments welcom

Parallel Deterministic and Stochastic Global Minimization of Functions with Very Many Minima

The optimization of three problems with high dimensionality and many local minima are investigated under five different optimization algorithms: DIRECT, simulated annealing, Spall’s SPSA algorithm, the KNITRO package, and QNSTOP, a new algorithm developed at Indiana University

Distribution on Warp Maps for Alignment of Open and Closed Curves

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using model- or optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in $\mathbb{R}^d, d=1,2,3$, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of $[0,1]$ and the unit circle $\mathbb{S}^1$ that is (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on $[0,1]$, the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided