cosmetic fixed -- resolve linter complaints

Author: sunxd3

example/targets/banana.jl

@@ -15,20 +15,19 @@ $(FIELDS)
 The banana distribution is obtained by applying a transformation ϕ to a multivariate normal 
 distribution ``\\mathcal{N}(0, \\text{diag}(var, 1, 1, …, 1))``. The transformation ϕ is defined as
 ```math
-\phi(x_1, … , x_p) = (x_1, x_2 - B x_1^² + \text{var}*B, x_3, … , x_p)
-````
+\\phi(x_1, … , x_p) = (x_1, x_2 - B x_1^² + \\text{var}*B, x_3, … , x_p)
+```
 which has a unit Jacobian determinant.
 
 Hence the density "fb" of a p-dimensional banana distribution is given by
 ```math
-fb(x_1, \dots, x_p) = \exp\left[ -\frac{1}{2}\frac{x_1^2}{\text{var}} -
-\frac{1}{2}(x_2 + B x_1^2 - \text{var}*B)^2 - \frac{1}{2}(x_3^2 + x_4^2 + \dots
-+ x_p^2) \right] / Z,
+fb(x_1, \\dots, x_p) = \\exp\\left[ -\\frac{1}{2}\\frac{x_1^2}{\\text{var}} -
+\\frac{1}{2}(x_2 + B x_1^2 - \\text{var}*B)^2 - \\frac{1}{2}(x_3^2 + x_4^2 + \\dots
++ x_p^2) \\right] / Z,
 ```
 where "B" is the "banananicity" constant, determining the curvature of a banana, and   
 ``Z = \\sqrt{\\text{var} * (2\\pi)^p)}`` is the normalization constant.
 
-
 # Reference
 
 Gareth O. Roberts and Jeffrey S. Rosenthal

example/targets/cross.jl

@@ -11,13 +11,13 @@ The Cross distribution is a 2-dimension 4-component Gaussian distribution with a
 shape that is symmetric about the y- and x-axises. The mixture is defined as
 
 ```math
-\begin{aligned}
+\\begin{aligned}
 p(x) =
-& 0.25 \mathcal{N}(x | (0, \mu), (\sigma, 1)) + \\
-& 0.25 \mathcal{N}(x | (\mu, 0), (1, \sigma)) + \\
-& 0.25 \mathcal{N}(x | (0, -\mu), (\sigma, 1)) + \\
-& 0.25 \mathcal{N}(x | (-\mu, 0), (1, \sigma)))
-\end{aligned}
+& 0.25 \\mathcal{N}(x | (0, \\mu), (\\sigma, 1)) + \\\\
+& 0.25 \\mathcal{N}(x | (\\mu, 0), (1, \\sigma)) + \\\\
+& 0.25 \\mathcal{N}(x | (0, -\\mu), (\\sigma, 1)) + \\\\
+& 0.25 \\mathcal{N}(x | (-\\mu, 0), (1, \\sigma))
+\\end{aligned}
 ```
 
 where ``μ`` and ``σ`` are the mean and standard deviation of the Gaussian components, 

example/targets/neal_funnel.jl

@@ -13,14 +13,13 @@ $(FIELDS)
 The Neal's Funnel distribution is a p-dimensional distribution with a funnel shape, 
 originally proposed by Radford Neal in [2]. 
 The marginal distribution of ``x_1`` is Gaussian with mean "μ" and standard
-deviation "σ". The conditional distribution of ``x_2, \dots, x_p | x_1`` are independent 
+deviation "σ". The conditional distribution of ``x_2, \\dots, x_p | x_1`` are independent 
 Gaussian distributions with mean 0 and standard deviation ``\\exp(x_1/2)``. 
 The generative process is given by
 ```math
-x_1 \sim \mathcal{N}(\mu, \sigma^2), \quad x_2, \ldots, x_p \sim \mathcal{N}(0, \exp(x_1))
+x_1 \\sim \\mathcal{N}(\\mu, \\sigma^2), \\quad x_2, \\ldots, x_p \\sim \\mathcal{N}(0, \\exp(x_1))
 ```
 
-
 # Reference
 [1] Stan User’s Guide: 
 https://mc-stan.org/docs/2_18/stan-users-guide/reparameterization-section.html#ref-Neal:2003

example/targets/warped_gaussian.jl

@@ -13,13 +13,12 @@ $(FIELDS)
 The banana distribution is obtained by applying a transformation ϕ to a 2-dimensional normal 
 distribution ``\\mathcal{N}(0, diag(\\sigma_1, \\sigma_2))``. The transformation ϕ(x) is defined as
 ```math
-ϕ(x_1, x_2) = (r*\cos(\theta + r/2), r*\sin(\theta + r/2)), 
+\\phi(x_1, x_2) = (r*\\cos(\\theta + r/2), r*\\sin(\\theta + r/2)), 
 ```
-where ``r = \\sqrt{x\_1^2 + x_2^2}``, ``\\theta = \\atan(x₂, x₁)``, 
-and "atan(y, x) ∈ [-π, π]" is the angle, in radians, between the positive x axis and the 
+where ``r = \\sqrt{x_1^2 + x_2^2}``, ``\\theta = \\atan(x_2, x_1)``, 
+and ``\\atan(y, x) \\in [-\\pi, \\pi]`` is the angle, in radians, between the positive x axis and the 
 ray to the point "(x, y)". See page 18. of [1] for reference.
 
-
 # Reference
 [1] Zuheng Xu, Naitong Chen, Trevor Campbell
 "MixFlows: principled variational inference via mixed flows."

src/NormalizingFlows.jl

@@ -24,7 +24,6 @@ Train the given normalizing flow `flow` by calling `optimize`.
 - `flow`: normalizing flow to be trained, we recommend to define flow as `<:Bijectors.TransformedDistribution` 
 - `args...`: additional arguments for `vo`
 
-
 # Keyword Arguments
 - `max_iters::Int=1000`: maximum number of iterations
 - `optimiser::Optimisers.AbstractRule=Optimisers.ADAM()`: optimiser to compute the steps