Topic

This project aims to generate galaxy images based on data provided by the Hubble Space Telescope (HST). To do so, we are implementing an unsupervised machine learning technique called a Variational Autoencoder (Vae) based on statistical Bayesian inference. The trained Vae model allows us to decode a random latent variable $z$ sampled from a normal distribution $\mathcal{N}(0;1)$ into a realistic galaxy image. For further information on the general Vae model, please read : Auto-encoding Variational Bayes, Kingma and Welling, 2014.

We used two datasets :

• 47 955 galaxies from Hubble's famous Deep Field image (the images have
  128 $\times$ 128 pixels)
• 81 499 galaxies and their associated redshifts from the Cosmic Survey (the images have 158 $\times$ 158 pixels)

For each dataset, we developped a $\beta$-Vae architecture. For galaxy image generation, we found that the reconstruction performance of our model is better with low value for $\beta$. In our case, we fixed this hyperparameter to $\beta = 0.1$ during the whole training process.

Based on Kihyuk Sohn's paper, we even implemented a Conditional Vae on the second dataset with the redshifts of each galaxy. In the end, our conditional vae is able to generate galaxy structures for a specific redshift. We can even do an interpolation of the same galaxy structure for different redshifts.

Models architecture

In the folder Models architecture, you will find the details of the different models used.

First, two disentangled Vae models (one per dataset) with almost the same architecture (just a few changes made to the convolutional layers arguments due to the image size difference for each dataset). The models can take as input either a value or an array for the hyperparameter $\beta$. It can be interesting if you want to make $\beta$ change during the training process (i.e. $\beta$ increasing over each epoch). After experimentation of different behaviour for $\beta$, we concluded that we obtain the best model by fixing this hyperparameter to $\beta = 0.1$. We observe the same thibg for the conditional Vaes.

Then, three Conditional Vae models (for the second dataset):

• cvae: new input created by concatenation of the redshifts to the galaxy images into a second channel which is fed to the CNN. Then, we concatenate the redshifts to the latent variable $z$ into a second channel. The final output is the reconstructed galaxy image.

• cvae2: concatenation of the redshifts to the output of the encoder's CNN and to the latent variable $z$ before decoding.

• fancy_cvae: similar to cvae but the final output is a prediction of the galaxy images and the redshifts.

The performance of these architectures is really similar. The unique noteworthy difference is the training time which is shorter by 20 sec/epoch for cvae2 compared to the others, so I would recommend using the cvae2 architecture for conditioned galaxy image generation.

Notebooks

In the folder notebooks, you will find all the code related to each model's training and the evaluation of its performance:

• Loss
• Image reconstruction
• Image generation
• Latent space visualization

How to improve the current model?

Normalizing flows

Currently, we are generating images by giving some random samples of a Gaussian $\mathcal{N}(0,I)$ to the decoder. Since we are working with a disentangled conditional vae, we don't really know if these samples give us a good representation of our latent space. In fact, they are very unlikely to give us a good representation when $\beta$ is close to 0. We can visualize the latent space with 2d histograms to see if it's the case but it would be great to have a method to be sure we sample directly from the latent space. That's why the next step would be to implement a Normalizing flow to learn an invertible transformation from the rather "complex" probability distribution learnt by the vae to a simple Normal distribution. This would help the decoder during the generative process.

Bigger dataset

Training our model on a bigger dataset (314 000 galaxy images instead of 81 500).

Fine-tuning of the model

We could maybe improve the results with some fine-tuning of the hyperparameters of the model or with different machine learning approaches that are not implemented yet (e.g. learning rate decay).