paper added

paper/paper.md

@@ -68,6 +68,8 @@ aas-journal: Astrophysical Journal <- The name of the AAS journal.
     - Memory efficiency compared to normalising flows for the same tasks (one network conditioned on 't' compared to many sub-flows + faster than CNFs)
 
     - implemented in JAX, equinox and diffrax
+
+    - likelihood weighting (maximum likelihood training of SBGMs)
 -->
 
 Diffusion-based generative models are a method for density estimation and sampling from high-dimensional distributions. A sub-class of these models, score-based diffusion generatives models (SBGMs), permit exact-likelihood estimation via a change-of-variables associated with the forward diffusion process. Diffusion models allow fitting generative models to high-dimensional data in a more efficient way than normalising flows since only one neural network model parameterises the diffusion process as opposed to a stack of networks in typical normalising flow architectures.
@@ -103,7 +105,7 @@ scientific explorations of forthcoming data releases from the *Gaia* mission
 # Mathematics
 
 <!-- What is diffusion -->
-Diffusion models model the reverse of a forward diffusion process on samples of data $\boldsymbol{x}$ by adding a sequence of noisy perturbations. In \autoref{fig:sde_ode}
+Diffusion models model the reverse of a forward diffusion process on samples of data $\boldsymbol{x}$ by adding a sequence of noisy perturbations. 
 
 <!-- What is a diffusion model -->
 Score-based diffusion models model the forward diffusion process with Stochastic Differential Equations (SDEs) of the form
@@ -138,6 +140,8 @@ $$
 
 where $\lambda(t)$ is an arbitrary scalar weighting function, chosen to weight certain times - usually near $t=0$ where the data has only a small amount of noise added.
 
+In Figure \autoref{fig:sde_ode} the forward and reverse diffusion processes are shown for a toy problem with their corresponding SDE and ODE paths.
+
 The reverse SDE may be solved with Euler-Murayama sampling (or other annealed Langevin sampling methods) which is featured in the code. 
 
 However, many of the applications of generative models depend on being able to calculate the likelihood of data. In [1] it is shown that any SDE may be converted into an ordinary differential equation (ODE) without changing the distributions, defined by the SDE, from which the noise is sampled from in the diffusion process. This ODE is known as the probability flow ODE and is written