Equivariant normalizing flows and their application to cosmology
Carolina Cuesta-Lazaro
April 2022 - IAIFI JC
Simulated Data
Data
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500655/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500656/pasted-from-clipboard.png)
Prior
Posterior
Forwards
Inverse
Cosmological parameters
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500687/density.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500694/pasted-from-clipboard.png)
EARLY UNIVERSE
LATE UNIVERSE
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500706/ps.png)
Normalizing flows: Generative models and density estimators
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500831/gaussian.png)
VAE,GAN ...
Gaussianization
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500823/gaussian.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500823/gaussian.png)
Data space
Latent space
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9500776/pasted-from-clipboard.png)
Maximize the data likelihood
NeuralNet
f must be invertible
J efficient to compute
1-D
n-D
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9462724/5a660530eace967f8e026bb3.png)
Equivariance
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501218/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501218/pasted-from-clipboard.png)
Invariance
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501218/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501218/pasted-from-clipboard.png)
Equivariant
Invariant
Equivariant
Invariant
Challenge: Expressive + Invertible + Equivariant
1. Continuous time Normalizing flows
ODE solutions are invertible!
z = odeint(self.phi, x, [0, 1])
torchdiffeq
solving the ODE might introduce error in estimating p(x)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501307/pasted-from-clipboard.png)
Image Credit: https://arxiv.org/abs/1810.01367
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501351/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501358/pasted-from-clipboard.png)
Equivariant? GNNs
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501388/pasted-from-clipboard.png)
1. Invertible but expressive
2. Equivariant to E(n)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501388/pasted-from-clipboard.png)
E(n) equivariant normalizing flows
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9462724/5a660530eace967f8e026bb3.png)
Cosmological simulations -> Millions of particles!
Solution: Density on mesh + Convolutions in Fourier space
1-D functions learned from data
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9503312/splines.png)
Cubic splines (8 spline points)
Monotonic rational quadratic splines
(8 spline points)
Loss Function
Generative: Maximize likelihood
Discriminative: target the posterior
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501540/discriminative_generative.png)
Gaussian Random Field:
The Power spectrum is an optimal summary statistic
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501527/grf.png)
Analytical likelihood
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501527/grf.png)
Flow likelihood
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501531/recover_pk.png)
Non-Gaussian N-body simulations
1. Inference
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501532/posteriors.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501538/summary_statistics.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501565/samples.png)
Non-Gaussian N-body simulations
2. Sampling
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/9501526/tests.png)
- Can we quantify the full information content? Can normalizing flows extract all the information there is about cosmology?
- Can the latent space be the initial conditions for the N-body sim?
- Are current models to embed symmetries too constraining?
- Model misspecification?
- Does dimensionality reduction help with interpretability?
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By carol cuesta
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