Documentation improvements (inc. lengthscale explanation) and Matern1…

…2Kernel alias (#213)

* various edits for clarity and typos
* remove reference to not-yet-implemented feature (#38)
* adds Matern12Kernel as alias for ExponentialKernel (in line with the explicitly defined Matern32Kernel and Matern52Kernel) and gives all aliases docstrings
* incorporates the lengthscales explanation from #212.

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

docs/make.jl

@@ -16,7 +16,7 @@ makedocs(
              "User Guide" => "userguide.md",
              "Examples"=>"example.md",
              "Kernel Functions"=>"kernels.md",
-             "Transform"=>"transform.md",
+             "Input Transforms"=>"transform.md",
              "Metrics"=>"metrics.md",
              "Theory"=>"theory.md",
              "Custom Kernels"=>"create_kernel.md",

docs/src/create_kernel.md

@@ -2,9 +2,9 @@
 
 KernelFunctions.jl contains the most popular kernels already but you might want to make your own!
 
-Here are a few ways depending on how complicated your kernel is :
+Here are a few ways depending on how complicated your kernel is:
 
-### SimpleKernel for kernels function depending on a metric
+### SimpleKernel for kernel functions depending on a metric
 
 If your kernel function is of the form `k(x, y) = f(d(x, y))` where `d(x, y)` is a `PreMetric`,
 you can construct your custom kernel by defining `kappa` and `metric` for your kernel.
@@ -20,15 +20,15 @@ KernelFunctions.metric(::MyKernel) = SqEuclidean()
 ### Kernel for more complex kernels
 
 If your kernel does not satisfy such a representation, all you need to do is define `(k::MyKernel)(x, y)` and inherit from `Kernel`.
-For example we recreate here the `NeuralNetworkKernel`
+For example, we recreate here the `NeuralNetworkKernel`:
 
 ```julia
 struct MyKernel <: KernelFunctions.Kernel end
 
 (::MyKernel)(x, y) = asin(dot(x, y) / sqrt((1 + sum(abs2, x)) * (1 + sum(abs2, y))))
 ```
 
-Note that `BaseKernel` do not use `Distances.jl` and can therefore be a bit slower.
+Note that the fallback implementation of the base `Kernel` evaluation does not use `Distances.jl` and can therefore be a bit slower.
 
 ### Additional Options
 
@@ -37,7 +37,7 @@ Finally there are additional functions you can define to bring in more features:
  - `KernelFunctions.dim(x::MyDataType)`: by default the dimension of the inputs will only be checked for vectors of type `AbstractVector{<:Real}`. If you want to check the dimensionality of your inputs, dispatch the `dim` function on your datatype. Note that `0` is the default.
  - `dim` is called within `KernelFunctions.validate_inputs(x::MyDataType, y::MyDataType)`, which can instead be directly overloaded if you want to run special checks for your input types.
  - `kernelmatrix(k::MyKernel, ...)`: you can redefine the diverse `kernelmatrix` functions to eventually optimize the computations.
- - `Base.print(io::IO, k::MyKernel)`: if you want to specialize the printing of your kernel
+ - `Base.print(io::IO, k::MyKernel)`: if you want to specialize the printing of your kernel.
 
 KernelFunctions uses [Functors.jl](https://github.com/FluxML/Functors.jl) for specifying trainable kernel parameters
 in a way that is compatible with the [Flux ML framework](https://github.com/FluxML/Flux.jl).

docs/src/index.md

@@ -1,6 +1,6 @@
 # KernelFunctions.jl
 
-Model agnostic kernel functions compatible with automatic differentiation
+Model-agnostic kernel functions compatible with automatic differentiation
 
 **KernelFunctions.jl** is a general purpose kernel package.
 It aims at providing a flexible framework for creating kernels and manipulating them.

docs/src/kernels.md

@@ -4,7 +4,7 @@
 
 # Base Kernels
 
-These are the basic kernels without any transformation of the data. They are the building blocks of KernelFunctions
+These are the basic kernels without any transformation of the data. They are the building blocks of KernelFunctions.
 
 
 ## Constant Kernels
@@ -86,21 +86,20 @@ The [`FBMKernel`](@ref) is defined as
   k(x,x';h) =  \frac{|x|^{2h} + |x'|^{2h} - |x-x'|^{2h}}{2},
 ```
 
-where $h$ is the [Hurst index](https://en.wikipedia.org/wiki/Hurst_exponent#Generalized_exponent) and $0<h<1$.
+where $h$ is the [Hurst index](https://en.wikipedia.org/wiki/Hurst_exponent#Generalized_exponent) and $0 < h < 1$.
 
 ## Gabor Kernel
 
 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.
+where $l_i > 0$ is the lengthscale and $p_i > 0$ is the period.
 
-## Matern Kernels
+## Matérn Kernels
 
-### Matern Kernel
+### General Matérn Kernel
 
 The [`MaternKernel`](@ref) is defined as
 
@@ -110,15 +109,23 @@ The [`MaternKernel`](@ref) is defined as
 
 where $\nu > 0$.
 
-### Matern 3/2 Kernel
+### Matérn 1/2 Kernel
+
+The Matérn 1/2 kernel is defined as
+```math
+  k(x,x') = \exp\left(-|x-x'|\right),
+```
+equivalent to the Exponential kernel. `Matern12Kernel` is an alias for [`ExponentialKernel`](@ref).
+
+### Matérn 3/2 Kernel
 
 The [`Matern32Kernel`](@ref) is defined as
 
 ```math
   k(x,x') = \left(1+\sqrt{3}|x-x'|\right)\exp\left(\sqrt{3}|x-x'|\right).
 ```
 
-### Matern 5/2 Kernel
+### Matérn 5/2 Kernel
 
 The [`Matern52Kernel`](@ref) is defined as
 
@@ -128,7 +135,7 @@ The [`Matern52Kernel`](@ref) is defined as
 
 ## Neural Network Kernel
 
-The [`NeuralNetworkKernel`](@ref) (as in the kernel for an infinitely wide neural network interpretated as a Gaussian process) is defined as
+The [`NeuralNetworkKernel`](@ref) (as in the kernel for an infinitely wide neural network interpreted as a Gaussian process) is defined as
 
 ```math
   k(x, x') = \arcsin\left(\frac{\langle x, x'\rangle}{\sqrt{(1+\langle x, x\rangle)(1+\langle x',x'\rangle)}}\right).
@@ -142,19 +149,23 @@ The [`PeriodicKernel`](@ref) is defined as
   k(x,x';r) = \exp\left(-0.5 \sum_i (sin (π(x_i - x'_i))/r_i)^2\right),
 ```
 
-where $r$ has the same dimension as $x$ and $r_i >0$.
+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 $x, x'\in \mathbb{R}^D$, a positive-definite matrix $P \in \mathbb{R}^{D \times D}$, 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 - \sqrt{x^\top P x'}, 0)^{j + V} f_V(\sqrt{x^\top P x'}, j),
+```
+where $j = \lfloor \frac{D}{2}\rfloor + V + 1$, and $f_V$ are polynomials defined as follows:
+```math
+\begin{aligned}
+    f_0(r, j) &= 1, \\
+    f_1(r, j) &= 1 + (j + 1) r, \\
+    f_2(r, j) &= 1 + (j + 2) r + ((j^2 + 4j + 3) / 3) r^2, \\
+    f_3(r, j) &= 1 + (j + 3) r + ((6 j^2 + 36j + 45) / 15) r^2 + ((j^3 + 9 j^2 + 23j + 15) / 15) r^3.
+\end{aligned}
 ```
-where $x\in \mathbb{R}^D$, $V \in \{0,1,2,3\} and $P$ is a positive definite matrix.
-$f$ is a piecewise polynomial (see source code).
 
 ## Polynomial Kernels
 
@@ -166,7 +177,7 @@ The [`LinearKernel`](@ref) is defined as
   k(x,x';c) = \langle x,x'\rangle + c,
 ```
 
-where $c \in \mathbb{R}$
+where $c \in \mathbb{R}$.
 
 ### Polynomial Kernel
 
@@ -176,7 +187,7 @@ The [`PolynomialKernel`](@ref) is defined as
   k(x,x';c,d) = \left(\langle x,x'\rangle + c\right)^d,
 ```
 
-where $c \in \mathbb{R}$ and $d>0$
+where $c \in \mathbb{R}$ and $d>0$.
 
 
 ## Rational Quadratic
@@ -223,43 +234,41 @@ where $i\in\{-1,0,1,2,3\}$ and coefficients $a_i$, $b_i$ are fixed and residuals
 
 ### Transformed Kernel
 
-The [`TransformedKernel`](@ref) is a kernel where input are transformed via a function `f`
+The [`TransformedKernel`](@ref) is a kernel where inputs are transformed via a function `f`:
 
 ```math
-  k(x,x';f,\widetile{k}) = \widetilde{k}(f(x),f(x')),
+  k(x,x';f,\widetilde{k}) = \widetilde{k}(f(x),f(x')),
 ```
-
-Where $\widetilde{k}$ is another kernel and $f$ is an arbitrary mapping.
+where $\widetilde{k}$ is another kernel and $f$ is an arbitrary mapping.
 
 ### Scaled Kernel
 
 The [`ScaledKernel`](@ref) is defined as
 
 ```math
-  k(x,x';\sigma^2,\widetilde{k}) = \sigma^2\widetilde{k}(x,x')
+  k(x,x';\sigma^2,\widetilde{k}) = \sigma^2\widetilde{k}(x,x') ,
 ```
-
-Where $\widetilde{k}$ is another kernel and $\sigma^2 > 0$.
+where $\widetilde{k}$ is another kernel and $\sigma^2 > 0$.
 
 ### Kernel Sum
 
-The [`KernelSum`](@ref) is defined as a sum of kernels
+The [`KernelSum`](@ref) is defined as a sum of kernels:
 
 ```math
   k(x, x'; \{k_i\}) = \sum_i k_i(x, x').
 ```
 
-### KernelProduct
+### Kernel Product
 
-The [`KernelProduct`](@ref) is defined as a product of kernels
+The [`KernelProduct`](@ref) is defined as a product of kernels:
 
 ```math
   k(x,x';\{k_i\}) = \prod_i k_i(x,x').
 ```
 
 ### Tensor Product
 
-The [`TensorProduct`](@ref) is defined as :
+The [`TensorProduct`](@ref) is defined as:
 
 ```math
   k(x,x';\{k_i\}) = \prod_i k_i(x_i,x'_i)

docs/src/metrics.md

@@ -1,16 +1,19 @@
 # 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 all kernels do not rely on distances metrics respecting all the definitions. That's why  additional metrics come with the package such as `DotProduct` (`<x,y>`) and `Delta` (`δ(x,y)`).
-Note that every `SimpleKernel` must have a defined metric defined as :
+`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`.
+
+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 distance just implement the following :
+If you want to create a new "metric" just implement the following:
 
 ```julia
 struct Delta <: Distances.PreMetric

docs/src/transform.md

@@ -1,9 +1,9 @@
-# Transform
+# Input Transforms
 
 `Transform` is the object that takes care of transforming the input data before distances are being computed. It can be as standard as `IdentityTransform` returning the same input, or multiplying the data by a scalar with `ScaleTransform` or by a vector with `ARDTransform`.
-There is a more general `Transform`: `FunctionTransform` that uses a function and apply it on each vector via `mapslices`.
-You can also create a pipeline of `Transform` via `TransformChain`. For example `LowRankTransform(rand(10,5))∘ScaleTransform(2.0)`.
+There is a more general `Transform`: `FunctionTransform` that uses a function and applies it on each vector via `mapslices`.
+You can also create a pipeline of `Transform` via `TransformChain`. For example, `LowRankTransform(rand(10,5))∘ScaleTransform(2.0)`.
 
-One apply a transformation on a matrix or a vector via `KernelFunctions.apply(t::Transform,v::AbstractVecOrMat)`
+A transformation `t` can be applied to a matrix or a vector `v` via `KernelFunctions.apply(t, v)`.
 
-Check the list on the [API page](@ref Transforms)
+Check the full list of provided transforms on the [API page](@ref Transforms).