Carol Cuesta-Lazaro (IAIFI Fellow)

in collaboration with Siddarth Mishra-Sharma

Generative models for the Large Scale Structure

ML for Large Scale Structure:

Carol's wish list

Generative models

Learn p(x)

Evaluate the likelihood of a 3D map, as a function of the parameters of interest

1

Combine different galaxy properties (such as velocities and positions)

2

Sample 3D maps from the posterior distribution

3

p(
)
|
\mathrm{Cosmology}

Explicit Density

Implicit Density

Tractable Density

Approximate Density

Normalising flows

Variational Autoencoders

Diffusion models

The zoo of generative models

z_T
z_{0}
z_{1}
z_{2}
p(z_{t-1}|z_t)
p(z_t|z_{t-1})

Reverse diffusion: Denoise previous step

Forward diffusion: Add Gaussian noise (fixed)

Diffusion models

A person half Yoda half Gandalf

z_T
z_{0}
z_{1}

Diffusion on point clouds

z_{2}
q_\theta(z_{t-1}|z_t)
p(z_t|z_{t-1})

Reverse diffusion: Denoise previous step

Forward diffusion: Add Gaussian noise (fixed)

Cosmology

p(x,y,z, v_x, v_y, v_z, M_h|\Omega_m, \sigma_8)

Halo Mass Function

Velocity

PDF

Mean pairwise velocity

\log p(\boldsymbol{x}) =
\geq \mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{z}\mid\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{z})}{q_{\boldsymbol{\phi}}(\boldsymbol{z}\mid\boldsymbol{x})}\right]
\mathbb{E}_{q_{\boldsymbol{\phi}}(\boldsymbol{z}\mid\boldsymbol{x})}\left[\log\frac{p(\boldsymbol{x}, \boldsymbol{z})}{q_{\boldsymbol{\phi}}(\boldsymbol{z}\mid\boldsymbol{x})}\right] + \mathcal{D}_{\text{KL}}(q_{\boldsymbol{\phi}}(\boldsymbol{z}\mid\boldsymbol{x}) \mid\mid p(\boldsymbol{z}\mid\boldsymbol{x}))

Evidence Lower Bound

Distance to true posterior

q_\theta(z|x) \approx p(z|x)

Find

1. ELBO is a lower bound of the evidence

2. Maximising ELBO = Minimising KL

Maximise ELBO to maximise ev/likelihood

Maximise ELBO to approximate true posterior

+ Galaxy formation

+ Observational systematics (Cut-sky, Fiber collisions)

+ Lightcone, Redshift Space Distortions....

Forward Model

N-body simulations

Observations

By carol cuesta

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