RESTORING LATENTLY-LOST MEANING IN POPULATION-DYNAMICAL GALAXY - PowerPoint PPT Presentation

) ∈ ℝ $'$% * ! ∈ ℝ $% & '$% ( RESTORING LATENTLY-LOST MEANING IN POPULATION-DYNAMICAL GALAXY DECOMPOSITIONS Prashin Jethwa – University of Vienna Ling Zhu (SHAO), Adriano Poci (ESO/Macquarie), Jesus Falcon-Barroso (IAC), Glenn Van deVen (Uni. Vienna)

RECENT ACRETION

Belokurov et al 2017 ANCIENT MERGER OF THE MILKY WAY • Gaia decomposes inner stellar halo of the MW, revealing a major merger: • ancient (7-10 Gyr) • massive (M * ~ 10 9 ) • radial orbit • Detection required combination of kinematics and stellar populations • Can we extend this beyond the MW…? Gaia collaboration 2018

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION Spectral stellar population recovery in EAGLE simulation True Recovered Boeker et al 2019

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION observed kinematics library of orbits recovered orbit distribution Zhu et al 2018

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION data noise covariance IFU data + ,, . /0 ~2 3, Σ model spectrum of population 8 model velocities of orbits 9 3 = 6 7 8, 9 : ,; 8 ∗ =(. 2D , , ; 9) d8d9 distribution function A 8, 9 encodes all quantities of interest inverse problem: B 7 + )

OUR CURRENT APPROACH TO POPULATION-DYNAMICAL DECOMPOSITION data + ,, . /0 (3) extract observed All steps formulated as linear problems • (1) extract population maps D = EF + G velocity maps D = ∑ I J(,, 8 K )! I e.g. step 3: • 7 8 (8; . /0 ) = [J 1 , … , O p ] F 7 = (C; . /0 ) = E F Constraints F ≽ 0 • (2) find orbits (4) find mapping from Dimensions: dim (E) = S, T • orbits to populations S ~10 V'W T ~10 X'Y 7 9 (9) & 8 ( 9 ) 7 9 (9)

maps extracted from IFU data-cube APPLICATION TO NGC 3115 Poci et al 19 surface density data + ,, . /0 (3) extract observed (1) extract mean velocity population maps velocity maps 7 8 (8; . /0 ) 7 = (C; . /0 ) velocity dispersion (2) find orbits (4) find mapping from orbits to populations h 3 7 9 (9) & 8 ( 9 ) 7 9 (9) ~ skewness

observed maps modelled maps APPLICATION TO NGC 3115 Poci et al 19 data + ,, . /0 (3) extract observed (1) extract population maps velocity maps 7 8 (8; . /0 ) 7 = (C; . /0 ) (2) find orbits (4) find mapping from recovered orbital orbits to populations distribution 7 9 (9) & 8 ( 9 ) 7 9 (9)

observed maps age metallicity APPLICATION TO NGC 3115 Poci et al 19 data + ,, . /0 (3) extract observed (1) extract population maps velocity maps 7 8 (8; . /0 ) 7 = (C; . /0 ) (2) find orbits (4) find mapping from orbits to populations 7 9 (9) & 8 ( 9 ) 7 9 (9)

observed maps age metallicity APPLICATION TO NGC 3115 Poci et al 19 data + ,, . /0 (3) extract observed (1) extract population maps velocity maps 7 8 (8; . /0 ) 7 = (C; . /0 ) (2) find orbits (4) find mapping from orbits to populations 7 9 (9) & 8 ( 9 ) 7 9 (9) modelled maps

THE BUILD-UP OF NGC 3115’S STELLAR DISK vertical velocity dispersion in disk Poci et al 19

MERGED SATELLITES? • Clumpy orbit distribution à merged satellites? • Unclear… depends on regularization recovered orbital distribution of NGC 3115

THE NEED FOR REGULARISATION unregularized recovery

THE NEED FOR REGULARISATION better fit to data better smoothness

POSTERIOR ON DECOMPOSITION- WEIGHTS truth best MAP recovery median of posterior distribution TIME: seconds ~ 30 minutes

THE NEED FOR DIMENSIONALITY REDUCTION 1. T o detect accreted satellites, we need to understand uncertainties in decomposition • à we want the posterior on the decomposition weights à for speed, we must reduce dimension of parameter space F ∈ ℝ $% & '$% ( • 2. May help us tackle the full problem • i.e. invert the full generative model rather than current step-by-step approach • Strategy: • go to low dimensional latent space à sample posterior à de-project to original space

LINEAR DIMENSIONALITY REDUCTION (A.K.A. matrix factorization) TRANSFORMED PROBLEM ORIGINAL PROBLEM • Factorise E = `a b • D ~ 2 EF, Z • Dims: S, T → S, d d, T where d ≪ T • P F ∝ 2 ], ^ 2 9 • Underlying probabilistic model: . f ~2 a g f , h g~2 ], 9 h = di diag (l 1 , … , l S ) • Regression becomes: D ~ 2 EF, Z → D ~ 2 `m , Z + h m = a b F • Sample posterior P(m|D ) then invert: F = pm p = ps pseudo-in inverse of of a b

LINEAR DIMENSIONALITY REDUCTION (A.K.A. matrix factorization) TRANSFORMED PROBLEM ORIGINAL PROBLEM • Factorise E = `a b • D ~ 2 EF, Z • Dims: S, T → S, d d, T where d ≪ T • P F ∝ 2 ], ^ 2 9 • Underlying probabilistic model: . f ~2 a g f , h g~2 ], 9 h = di diag (l 1 , … , l S ) • Regression becomes: D ~ 2 EF, Z → D ~ 2 `m , Z + h m = a b F • Sample posterior P(m|D ) then invert: F = pm p = ps pseudo-in inverse of of a b Bayesian Latent Factor Regression - West 2003

LINEAR DIMENSIONALITY REDUCTION (A.K.A. matrix factorization) TRANSFORMED PROBLEM ORIGINAL PROBLEM • Factorise E = `a b • D ~ 2 EF, Z • Dims: S, T → S, d d, T where d ≪ T • P F ∝ 2 ], ^ 2 9 • Underlying probabilistic model: . f ~2 a g f , h everything normal g~2 ], 9 à covariances sum h = di diag (l 1 , … , l S ) • Regression becomes: D ~ 2 EF, Z → D ~ 2 `m , Z + h m = a b F strategy for picking q: • Sample posterior P(m|D ) then invert: high enough that F = pm h ≪ Z p = ps pseudo-in inverse of of a b Bayesian Latent Factor Regression - West 2003

LINEAR DIMENSIONALITY REDUCTION (A.K.A. matrix factorization) TRANSFORMED PROBLEM ORIGINAL PROBLEM • Factorise E = `a b • D ~ 2 EF, Z • Dims: S, T → S, d d, T where d ≪ T • P F ∝ 2 ], ^ 2 9 • Underlying probabilistic model: . f ~2 a g f , h g~2 ], 9 h = di diag (l 1 , … , l S ) • Regression becomes: D ~ 2 EF, Z → D ~ 2 `m , Z + h m = a b F • Sample posterior P(m|D ) then invert: F = pm p = pseudo-inverse of a b Bayesian Latent Factor Regression - West 2003

LINEAR DIMENSIONALITY REDUCTION (A.K.A. matrix factorization) TRANSFORMED PROBLEM ORIGINAL PROBLEM • Factorise E = `a b • D ~ 2 EF, Z • Dims: S, T → S, d d, T where d ≪ T • P F ∝ 2 EF, Z but our • Underlying probabilistic model: . f ~2 a g f , h problem has P F ∝ z2 ], ^ 2 9 i f F ≽ 0 g~2 ], 9 positivity 0 otherwise h = di diag (l 1 , … , l S ) constraints • Regression becomes: D ~ 2 EF, Z → D ~ 2 `m , Z + h m = a b F no guarantee that • Sample posterior P(m|D ) then invert: least-square F = pm inverse satisfies p = pseudo-inverse of a b constraints Bayesian Latent Factor Regression - West 2003

, m = a b F D ~ 2 EF, Z D ~ 2 `m , Z + h , à DEPROJECTING WITH POSITIVITY CONSTRAINTS • The least-squares inverse is incomplete i.e. F = pm where p = pseudo−inverse of a b • We can add any vector  from null-space of a b , i.e. F = pm + Ä where Å = orthogonal complement of p in ℝ Ö • T o invert, for each m ~ P(m|D ) we need to find a  such that F satisfies positivity constraints • Finding  à solving a quadratic programming problem: F FF b subject to argmin 1 à F ≽ 0 and a b F = m

, m = a b F D ~ 2 EF, Z D ~ 2 `m , Z + h , à DEPROJECTING WITH POSITIVITY CONSTRAINTS • The least-squares inverse is incomplete i.e. F = pm where p = pseudo−inverse of a b • We can add any vector  from null-space of a b , i.e. F = pm + Ä find ä where Å = orthogonal complement of p in ℝ Ö rather than sample  • T o invert, for each m ~ P(m|D ) we need to find a  such that F satisfies positivity constraints • Finding  à solving a quadratic programming problem: infinitely many F FF b subject to argmin solutions – so pick 1 à the one which F ≽ 0 and a b F = m minimizes L2 norm

EXAMPLE • In practice: • use SVD to factorise E (i.e. PCA) • pick latent dimension d so that variance PCA << noise variance i.e. h ≪ Z • sample m ~ P(m|D ) • for each sampled m , solve quadratic programming problem to get F

POSTERIOR ON DECOMPOSITION- WEIGHTS truth best MAP recovery median of median of full posterior latent posterior 1 minute TIME: seconds ~ 30 minutes

• Population–dynamical galaxy decompositions • A detailed look into the distant pasts of nearby galaxies • Application to NGC 3115 SUMMARY • To progress, dimensionality reduction: adapted linear decomposition for posterior deprojection for • positivity constraints proof-of-concept of efficacy/efficiency: • 30 minutes 1 minute