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

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)

! ∈ ℝ$%&'$%( ) ∈ ℝ$'$%*

RECENT ACRETION

ANCIENT MERGER OF THE MILKY WAY

Gaia collaboration 2018 Belokurov et al 2017

• Gaia decomposes inner stellar halo of the

MW, revealing a major merger:

• ancient (7-10 Gyr)
• massive (M* ~ 109)
• radial orbit
• Detection required combination of

kinematics and stellar populations

• Can we extend this beyond the MW…?

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION

Boeker et al 2019 Spectral stellar population recovery in EAGLE simulation True Recovered

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION

library of orbits

• bserved kinematics

recovered orbit distribution Zhu et al 2018

EXTERNAL GALAXIES: POPULATION-DYNAMICAL DECOMPOSITION

+ ,, ./0 ~2 3, Σ 3 = 6 7 8, 9 : ,; 8 ∗ =(.2D , , ; 9) d8d9 distribution function A 8, 9 encodes all quantities of interest inverse problem: B 7 + )

data noise covariance IFU data model spectrum of population 8 model velocities of orbits 9

OUR CURRENT APPROACH TO POPULATION-DYNAMICAL DECOMPOSITION

79(9) & 8(9) (1) extract velocity maps data + ,, ./0 78(8; ./0) 7=(C; ./0) 79(9) (4) find mapping from

• rbits to populations

(3) extract observed population maps (2) find orbits

• All steps formulated as linear problems

D = EF + G

• e.g. step 3:

D = ∑I J(,, 8K )!I = [J1, … , Op] F = E F

• Constraints

F ≽ 0

• Dimensions:

dim(E) = S, T S ~10V'W T ~10X'Y

APPLICATION TO NGC 3115

(2) find orbits 79(9) & 8(9) (1) extract velocity maps data + ,, ./0 78(8; ./0) 7=(C; ./0) 79(9) (4) find mapping from

• rbits to populations

(3) extract observed population maps

Poci et al 19 surface density maps extracted from IFU data-cube mean velocity velocity dispersion h3 ~ skewness

APPLICATION TO NGC 3115

(2) find orbits 79(9) & 8(9) (1) extract velocity maps data + ,, ./0 78(8; ./0) 7=(C; ./0) 79(9) (4) find mapping from

• rbits to populations

(3) extract observed population maps

Poci et al 19

• bserved maps

modelled maps recovered orbital distribution

APPLICATION TO NGC 3115

(2) find orbits 79(9) & 8(9) (1) extract velocity maps data + ,, ./0 78(8; ./0) 7=(C; ./0) 79(9) (4) find mapping from

• rbits to populations

(3) extract observed population maps

Poci et al 19

• bserved maps

age metallicity

APPLICATION TO NGC 3115

(2) find orbits 79(9) & 8(9) (1) extract velocity maps data + ,, ./0 78(8; ./0) 7=(C; ./0) 79(9) (4) find mapping from

• rbits to populations

(3) extract observed population maps

Poci et al 19

• bserved maps

age metallicity 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

better fit to data

THE NEED FOR REGULARISATION

better smoothness

POSTERIOR ON DECOMPOSITION- WEIGHTS

best MAP recovery truth TIME: seconds ~ 30 minutes median of posterior distribution

THE NEED FOR DIMENSIONALITY REDUCTION

1. T

• 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

ORIGINAL PROBLEM

• D ~ 2 EF, Z
• P F ∝ 2 ], ^29
• Factorise E = `ab
• Dims:

S, T → S, d d, T where d ≪ T

• Underlying probabilistic model:

.f ~2 a gf , h g~2 ], 9 h = di diag(l1, … , lS)

• Regression becomes:

D ~ 2 EF, Z → D ~ 2 `m , Z + h m = abF

• Sample posterior P(m|D) then invert:

F = pm p = ps pseudo-in inverse of

• f ab

TRANSFORMED PROBLEM

LINEAR DIMENSIONALITY REDUCTION

(A.K.A. matrix factorization)

ORIGINAL PROBLEM

• D ~ 2 EF, Z
• P F ∝ 2 ], ^29
• Factorise E = `ab
• Dims:

S, T → S, d d, T where d ≪ T

• Underlying probabilistic model:

.f ~2 a gf , h g~2 ], 9 h = di diag(l1, … , lS)

• Regression becomes:

D ~ 2 EF, Z → D ~ 2 `m , Z + h m = abF

• Sample posterior P(m|D) then invert:

F = pm p = ps pseudo-in inverse of

• f ab

TRANSFORMED PROBLEM

LINEAR DIMENSIONALITY REDUCTION

(A.K.A. matrix factorization) Bayesian Latent Factor Regression - West 2003

ORIGINAL PROBLEM

• D ~ 2 EF, Z
• P F ∝ 2 ], ^29
• Factorise E = `ab
• Dims:

S, T → S, d d, T where d ≪ T

• Underlying probabilistic model:

.f ~2 a gf , h g~2 ], 9 h = di diag(l1, … , lS)

• Regression becomes:

D ~ 2 EF, Z → D ~ 2 `m , Z + h m = abF

• Sample posterior P(m|D) then invert:

F = pm p = ps pseudo-in inverse of

• f ab

TRANSFORMED PROBLEM

LINEAR DIMENSIONALITY REDUCTION

(A.K.A. matrix factorization) everything normal à covariances sum strategy for picking q: high enough that h ≪ Z Bayesian Latent Factor Regression - West 2003

ORIGINAL PROBLEM

• D ~ 2 EF, Z
• P F ∝ 2 ], ^29
• Factorise E = `ab
• Dims:

S, T → S, d d, T where d ≪ T

• Underlying probabilistic model:

.f ~2 a gf , h g~2 ], 9 h = di diag(l1, … , lS)

• Regression becomes:

D ~ 2 EF, Z → D ~ 2 `m , Z + h m = abF

• Sample posterior P(m|D) then invert:

F = pm p = pseudo-inverse of ab

TRANSFORMED PROBLEM

LINEAR DIMENSIONALITY REDUCTION

(A.K.A. matrix factorization) Bayesian Latent Factor Regression - West 2003

ORIGINAL PROBLEM

• D ~ 2 EF, Z
• P F ∝ 2 EF, Z
• Factorise E = `ab
• Dims:

S, T → S, d d, T where d ≪ T

• Underlying probabilistic model:

.f ~2 a gf , h g~2 ], 9 h = di diag(l1, … , lS)

• Regression becomes:

D ~ 2 EF, Z → D ~ 2 `m , Z + h m = abF

• Sample posterior P(m|D) then invert:

F = pm p = pseudo-inverse of ab

TRANSFORMED PROBLEM

LINEAR DIMENSIONALITY REDUCTION

(A.K.A. matrix factorization) P F ∝ z2 ], ^29 if F ≽ 0 0 otherwise but our problem has positivity constraints no guarantee that least-square inverse satisfies constraints Bayesian Latent Factor Regression - West 2003

DEPROJECTING WITH POSITIVITY CONSTRAINTS

• The least-squares inverse is incomplete i.e.

F = pm where p = pseudo−inverse of ab

• We can add any vector  from null-space of ab, i.e.

F = pm + Ä where Å = orthogonal complement of p in ℝÖ

• T
• 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:

argmin

F FFb subject to

F ≽ 0 and à ab F = 1 m

D ~ 2 EF, Z à D ~ 2 `m , Z + h , , m = abF

DEPROJECTING WITH POSITIVITY CONSTRAINTS

• The least-squares inverse is incomplete i.e.

F = pm where p = pseudo−inverse of ab

• We can add any vector  from null-space of ab, i.e.

F = pm + Ä where Å = orthogonal complement of p in ℝÖ

• T
• 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:

argmin

F FFb subject to

F ≽ 0 and à ab F = 1 m

D ~ 2 EF, Z à D ~ 2 `m , Z + h , , m = abF infinitely many solutions – so pick the one which minimizes L2 norm find ä rather than sample 

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

best MAP recovery truth median of full posterior median of latent posterior TIME: seconds ~ 30 minutes 1 minute

SUMMARY

• Population–dynamical galaxy decompositions
• A detailed look into the distant pasts of nearby galaxies
• Application to NGC 3115

30 minutes 1 minute

• To progress, dimensionality reduction:
• adapted linear decomposition for posterior deprojection for

positivity constraints

• proof-of-concept of efficacy/efficiency: