address review comments

Author: st--

docs/src/kernels.md

@@ -93,8 +93,7 @@ where $h$ is the [Hurst index](https://en.wikipedia.org/wiki/Hurst_exponent#Gene
 The [`GaborKernel`](@ref) is defined as
 
 ```math
-  k(x,x'; l,p) =& h(x-x';l,p)\\
-  h(u;l,p) =& \exp\left(-\cos\left(\pi \sum_i \frac{u_i}{p_i}\right)\sum_i \frac{u_i^2}{l_i^2}\right),
+  k(x,x'; l,p) = \exp\left(-\cos\left(\pi \sum_i \frac{x_i - x'_i}{p_i}\right)\sum_i \frac{(x_i - x'_i)^2}{l_i^2}\right),
 ```
 where $l_i > 0$ is the lengthscale and $p_i > 0$ is the period.
 
@@ -154,14 +153,12 @@ where $r$ has the same dimension as $x$ and $r_i > 0$.
 
 ## Piecewise Polynomial Kernel
 
-The [`PiecewisePolynomialKernel`](@ref) is defined as
-
+The [`PiecewisePolynomialKernel`](@ref) is defined for $x\in \mathbb{R}^D$ and $V \in \{0,1,2,3\}$ as
 ```math
-  k(x,x'; P, V) =& \max(1 - r, 0)^{j + V} f(r, j),\\
-  r =& x^\top P x',\\
-  j =& \lfloor \frac{D}{2}\rfloor + V + 1,
+  k(x,x'; P, V) &= \max(1 - r, 0)^{j + V} f(r, j),
 ```
-where $x\in \mathbb{R}^D$, $V \in \{0,1,2,3\} and $P$ is a positive-definite matrix.
+where $r = x^\top P x'$ (with $P$ a positive-definite matrix),
+$j = \lfloor \frac{D}{2}\rfloor + V + 1$, and
 $f$ is a piecewise polynomial (see source code).
 
 ## Polynomial Kernels

docs/src/metrics.md

@@ -1,14 +1,16 @@
 # Metrics
 
-KernelFunctions.jl relies on [Distances.jl](https://github.com/JuliaStats/Distances.jl) for computing the pairwise matrix.
-To do so a distance measure is needed for each kernel. Two very common ones can already be used: `SqEuclidean` and `Euclidean`.
-However, not all kernels rely on distance metrics respecting all the definitions as in Distances.jl. For this reason, KernelFunctions.jl provides additional "metrics" such as `DotProduct` ($\langle x, y \rangle$) and `Delta` ($\delta(x,y)$).
+`SimpleKernel` implementations rely on [Distances.jl](https://github.com/JuliaStats/Distances.jl) for efficiently computing the pairwise matrix.
+This requires a distance measure or metric, such as the commonly used `SqEuclidean` and `Euclidean`.
 
-Note that every `SimpleKernel` must have a metric, specified as
+The metric used by a given kernel type is specified as
 ```julia
-    KernelFunctions.metric(::CustomKernel) = SqEuclidean()
+KernelFunctions.metric(::CustomKernel) = SqEuclidean()
 ```
 
+However, there are kernels that can be implemented efficiently using "metrics" that do not respect all the definitions expected by Distances.jl. For this reason, KernelFunctions.jl provides additional "metrics" such as `DotProduct` ($\langle x, y \rangle$) and `Delta` ($\delta(x,y)$).
+
+
 ## Adding a new metric
 
 If you want to create a new "metric" just implement the following: