# Interpreting Autoencoders
Previously we have learned about a broad scope of application of
generative models. We have seen that autoencoders can serve as powerful
generative models in the scientific context by extracting the compressed
representation of the input and using it to generate new instances of
the problem. It turns out that in the simple enough problems one can
find a meaningful interpretation of the latent representation that may
be novel enough to help us get new insight into the problem we are
analyzing.
In 2020, the group of Renato Renner considered a machine learning
perspective on one of the most historically important problems in
physics: Copernicus heliocentric system of the solar orbits. Via series
of careful and precise measurements of positions of objects in the night
sky, Copernicus conjectured that Sun is the center of the solar system
and other planets are orbiting around it. Let us now ask the following
question: is it possible to build a neural network that receives the
same observation angles Copernicus did and deduces the same conclusion
from them?
```{figure} ../../_static/lecture_specific/interpretability/copernicus.png
:name: fig:copernicus
**The Copernicus problem.** Relation between angles in heliocentric
and geocentric coordinate
system.
```
Renner group inputted into the autoencoder the angles of Mars and Sun as
observed from Earth ($\alpha_{ES}$ and $\alpha_{EM}$ in {numref}`fig:copernicus`) in certain times and asked the autoencoder
to predict the angles at other times. When analyzing the trained model
they realized that the two latent neurons included in their model are
storing information in the **heliocentric coordinates**! In particular,
one observes that the information stored in the latent space is a linear
combination of angles between Sun and Mars, $\gamma_{SM}$ and Sun and
Earth $\gamma_{SE}$. In other words, just like Copernicus, the
autoencoder has learned, that the most efficient way to store the
information given is to transform it into the heliocentric coordinate
system.
While this fascinating example is a great way to show the generative
models can be interpreted in some important cases, in general the
question of interpretability is still very much open and subject to
ongoing research. In the instances discussed earlier in this book, like
generation of molecules, where the input is compressed through several
layers of transformations requiring a complex dictionary and the
dimension of the latent space is high, interpreting latent space becomes
increasingly challenging.