Deprecate use of Mahalanobis distance

.github/workflows/ci.yml

@@ -62,7 +62,7 @@ jobs:
       - name: Format check
         run: |
           CHANGED="$(git diff --name-only)"
-          if [ ! -z $CHANGED ]; then
+          if [ ! -z "$CHANGED" ]; then
             >&2 echo "Some files have not been formatted !!!"
             echo "$CHANGED"
             exit 1

docs/src/api.md

@@ -35,7 +35,6 @@ NeuralNetworkKernel
 LinearKernel
 PolynomialKernel
 PiecewisePolynomialKernel
-MahalanobisKernel
 RationalQuadraticKernel
 GammaRationalQuadraticKernel
 spectral_mixture_kernel

docs/src/kernels.md

@@ -153,19 +153,22 @@ where $r$ has the same dimension as $x$ and $r_i > 0$.
 
 ## Piecewise Polynomial Kernel
 
-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
+The [`PiecewisePolynomialKernel`](@ref) of degree $v \in \{0,1,2,3\}$ is defined for
+inputs $x, x' \in \mathbb{R}^d$ of dimension $d$ as
 ```math
-  k(x,x'; P, V) = \max(1 - \sqrt{x^\top P x'}, 0)^{j + V} f_V(\sqrt{x^\top P x'}, j),
+k(x, x'; v) = \max(1 - \|x - x'\|, 0)^{\alpha} f_{v,d}(\|x - x'\|),
 ```
-where $j = \lfloor \frac{D}{2}\rfloor + V + 1$, and $f_V$ are polynomials defined as follows:
+where $\alpha = \lfloor \frac{d}{2}\rfloor + 2v + 1$, and $f_{v,d}$ are polynomials of
+degree $v$ given by
 ```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.
+f_{0,d}(r) &= 1, \\
+f_{1,d}(r) &= 1 + (j + 1) r, \\
+f_{2,d}(r) &= 1 + (j + 2) r + ((j^2 + 4j + 3) / 3) r^2, \\
+f_{3,d}(r) &= 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 $j = \lfloor \frac{d}{2}\rfloor + v + 1$.
 
 ## Polynomial Kernels
 

docs/src/userguide.md

@@ -23,6 +23,22 @@ For example, a squared exponential kernel is created by
       k = 3.0 * SqExponentialKernel()
     ```
 
+!!! tip "How do I use a Mahalanobis kernel?"
+    The `MahalanobisKernel(; P=P)`, defined by
+    ```math
+    k(x, x'; P) = \exp{\big(- (x - x')^\top P (x - x')\big)}
+    ```
+    for a positive definite matrix $P = Q^\top Q$, is deprecated. Instead you can
+    use a squared exponential kernel together with a [`LinearTransform`](@ref) of
+    the inputs:
+    ```julia
+    k = transform(SqExponentialKernel(), LinearTransform(sqrt(2) .* Q))
+    ```
+    Analogously, you can combine other kernels such as the
+    [`PiecewisePolynomialKernel`](@ref) with a [`LinearTransform`](@ref) of the
+    inputs to obtain a kernel that is a function of the Mahalanobis distance
+    between inputs.
+
 ## Using a kernel function
 
 To evaluate the kernel function on two vectors you simply call the kernel object:

src/KernelFunctions.jl

@@ -31,7 +31,7 @@ export FBMKernel
 export MaternKernel, Matern12Kernel, Matern32Kernel, Matern52Kernel
 export LinearKernel, PolynomialKernel
 export RationalQuadraticKernel, GammaRationalQuadraticKernel
-export MahalanobisKernel, GaborKernel, PiecewisePolynomialKernel
+export GaborKernel, PiecewisePolynomialKernel
 export PeriodicKernel, NeuralNetworkKernel
 export KernelSum, KernelProduct
 export TransformedKernel, ScaledKernel
@@ -90,7 +90,6 @@ include(joinpath("basekernels", "exponential.jl"))
 include(joinpath("basekernels", "exponentiated.jl"))
 include(joinpath("basekernels", "fbm.jl"))
 include(joinpath("basekernels", "gabor.jl"))
-include(joinpath("basekernels", "maha.jl"))
 include(joinpath("basekernels", "matern.jl"))
 include(joinpath("basekernels", "nn.jl"))
 include(joinpath("basekernels", "periodic.jl"))
@@ -118,6 +117,8 @@ include("zygote_adjoints.jl")
 
 include("test_utils.jl")
 
+include("deprecated.jl")
+
 function __init__()
     @require Kronecker = "2c470bb0-bcc8-11e8-3dad-c9649493f05e" begin
         include(joinpath("matrix", "kernelkroneckermat.jl"))

src/basekernels/piecewisepolynomial.jl

@@ -1,56 +1,101 @@
 """
-    PiecewisePolynomialKernel{V}(maha::AbstractMatrix)
+    PiecewisePolynomialKernel(; degree::Int=0, dim::Int)
+    PiecewisePolynomialKernel{degree}(dim::Int)
 
-Piecewise Polynomial covariance function with compact support, V = 0,1,2,3.
-The kernel functions are 2V times continuously differentiable and the corresponding
-processes are hence V times mean-square differentiable. The kernel function is:
+Piecewise polynomial kernel of degree `degree` for inputs of dimension `dim` with support in
+the unit ball.
+
+# Definition
+
+For inputs ``x, x' \\in \\mathbb{R}^d`` of dimension ``d``, the piecewise polynomial kernel
+of degree ``v \\in \\{0,1,2,3\\}`` is defined as
+```math
+k(x, x'; v) = \\max(1 - \\|x - x'\\|, 0)^{\\alpha(v,d)} f_{v,d}(\\|x - x'\\|),
+```
+where ``\\alpha(v, d) = \\lfloor \\frac{d}{2}\\rfloor + 2v + 1`` and ``f_{v,d}`` are
+polynomials of degree ``v`` given by
 ```math
-    κ(x, y) = max(1 - r, 0)^(j + V) * f(r, j) with j = floor(D / 2) + V + 1
+\\begin{aligned}
+f_{0,d}(r) &= 1, \\\\
+f_{1,d}(r) &= 1 + (j + 1) r, \\\\
+f_{2,d}(r) &= 1 + (j + 2) r + ((j^2 + 4j + 3) / 3) r^2, \\\\
+f_{3,d}(r) &= 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 `r` is the Mahalanobis distance mahalanobis(x,y) with `maha` as the metric.
+where ``j = \\lfloor \\frac{d}{2}\\rfloor + v + 1``.
+
+The kernel is ``2v`` times continuously differentiable and the corresponding Gaussian
+process is hence ``v`` times mean-square differentiable.
 """
-struct PiecewisePolynomialKernel{V,A<:AbstractMatrix{<:Real}} <: SimpleKernel
-    maha::A
-    j::Int
-    function PiecewisePolynomialKernel{V}(maha::AbstractMatrix{<:Real}) where {V}
-        V in (0, 1, 2, 3) || error("Invalid parameter V=$(V). Should be 0, 1, 2 or 3.")
-        LinearAlgebra.checksquare(maha)
-        j = div(size(maha, 1), 2) + V + 1
-        return new{V,typeof(maha)}(maha, j)
+struct PiecewisePolynomialKernel{D,C<:Tuple} <: SimpleKernel
+    alpha::Int
+    coeffs::C
+
+    function PiecewisePolynomialKernel{D}(dim::Int) where {D}
+        dim > 0 || error("number of dimensions has to be positive")
+        j = div(dim, 2) + D + 1
+        alpha = j + D
+        coeffs = piecewise_polynomial_coefficients(Val(D), j)
+        return new{D,typeof(coeffs)}(alpha, coeffs)
     end
 end
 
-function PiecewisePolynomialKernel(; v::Integer=0, maha::AbstractMatrix{<:Real})
-    return PiecewisePolynomialKernel{v}(maha)
-end
+# TODO: remove `maha` keyword argument in next breaking release
+function PiecewisePolynomialKernel(; v::Int=-1, degree::Int=v, maha=nothing, dim::Int=-1)
+    if v != -1
+        Base.depwarn(
+            "keyword argument `v` is deprecated, use `degree` instead",
+            :PiecewisePolynomialKernel,
+        )
+    end
 
-# Have to reconstruct the type parameter
-# See also https://github.com/FluxML/Functors.jl/issues/3#issuecomment-626747663
-function Functors.functor(::Type{<:PiecewisePolynomialKernel{V}}, x) where {V}
-    function reconstruct_kernel(xs)
-        return PiecewisePolynomialKernel{V}(xs.maha)
+    if maha !== nothing
+        Base.depwarn(
+            "keyword argument `maha` is deprecated, use a `LinearTransform` instead",
+            :PiecewisePolynomialKernel,
+        )
+        dim = size(maha, 1)
+        return transform(
+            PiecewisePolynomialKernel{degree}(dim), LinearTransform(cholesky(maha).U)
+        )
+    else
+        return PiecewisePolynomialKernel{degree}(dim)
     end
-    return (maha=x.maha,), reconstruct_kernel
 end
 
-_f(κ::PiecewisePolynomialKernel{0}, r, j) = 1
-_f(κ::PiecewisePolynomialKernel{1}, r, j) = 1 + (j + 1) * r
-_f(κ::PiecewisePolynomialKernel{2}, r, j) = 1 + (j + 2) * r + (j^2 + 4 * j + 3) / 3 * r .^ 2
-function _f(κ::PiecewisePolynomialKernel{3}, r, j)
-    return 1 +
-           (j + 3) * r +
-           (6 * j^2 + 36j + 45) / 15 * r .^ 2 +
-           (j^3 + 9 * j^2 + 23j + 15) / 15 * r .^ 3
+piecewise_polynomial_coefficients(::Val{0}, ::Int) = (1,)
+piecewise_polynomial_coefficients(::Val{1}, j::Int) = (1, j + 1)
+piecewise_polynomial_coefficients(::Val{2}, j::Int) = (1, j + 2, (j^2 + 4 * j)//3 + 1)
+function piecewise_polynomial_coefficients(::Val{3}, j::Int)
+    return (1, j + 3, (2 * j^2 + 12 * j)//5 + 3, (j^3 + 9 * j^2 + 23 * j)//15 + 1)
+end
+function piecewise_polynomial_coefficients(::Val{D}, ::Int) where {D}
+    return error("invalid degree $D, only 0, 1, 2, or 3 are supported")
 end
 
-function kappa(κ::PiecewisePolynomialKernel{V}, r) where {V}
-    return max(1 - r, 0)^(κ.j + V) * _f(κ, r, κ.j)
+# `evalpoly` is not available on Julia < 1.4
+@static if VERSION < v"1.4"
+    @generated function _evalpoly(r, coeffs)
+        N = length(coeffs.parameters)
+        return quote
+            return @evalpoly(r, $((:(coeffs[$i]) for i in 1:N)...))
+        end
+    end
+else
+    _evalpoly(r, coeffs) = evalpoly(r, coeffs)
 end
 
-metric(κ::PiecewisePolynomialKernel) = Mahalanobis(κ.maha)
+kappa(κ::PiecewisePolynomialKernel, r) = max(1 - r, 0)^κ.alpha * _evalpoly(r, κ.coeffs)
+
+metric(::PiecewisePolynomialKernel) = Euclidean()
 
-function Base.show(io::IO, κ::PiecewisePolynomialKernel{V}) where {V}
+function Base.show(io::IO, κ::PiecewisePolynomialKernel{D}) where {D}
     return print(
-        io, "Piecewise Polynomial Kernel (v = ", V, ", size(maha) = ", size(κ.maha), ")"
+        io,
+        "Piecewise Polynomial Kernel (degree = ",
+        D,
+        ", ⌊dim/2⌋ = ",
+        κ.alpha - 1 - 2 * D,
+        ")",
     )
 end

src/deprecated.jl

@@ -0,0 +1,9 @@
+# TODO: remove tests when removed
+@deprecate MahalanobisKernel(; P::AbstractMatrix{<:Real}) transform(
+    SqExponentialKernel(), LinearTransform(sqrt(2) .* cholesky(P).U)
+)
+
+# TODO: remove keyword argument `maha` when removed
+@deprecate PiecewisePolynomialKernel{V}(A::AbstractMatrix{<:Real}) where {V} transform(
+    PiecewisePolynomialKernel{V}(size(A, 1)), LinearTransform(cholesky(A).U)
+)

test/basekernels/maha.jl

@@ -1,6 +1,5 @@
 @testset "maha" begin
     rng = MersenneTwister(123456)
-    x = 2 * rand(rng)
     D_in = 3
     v1 = rand(rng, D_in)
     v2 = rand(rng, D_in)
@@ -9,40 +8,10 @@
     P = Matrix(Cholesky(U, 'U', 0))
     @assert isposdef(P)
 
-    k = MahalanobisKernel(; P=P)
-
-    @test kappa(k, x) == exp(-x)
+    k = @test_deprecated MahalanobisKernel(; P=P)
+    @test k isa TransformedKernel{SqExponentialKernel,<:LinearTransform}
+    @test k.transform.A ≈ sqrt(2) .* U
     @test k(v1, v2) ≈ exp(-sqmahalanobis(v1, v2, P))
-    @test kappa(ExponentialKernel(), x) == kappa(k, x)
-    @test repr(k) == "Mahalanobis Kernel (size(P) = $(size(P)))"
-
-    M1, M2 = rand(rng, 3, 2), rand(rng, 3, 2)
-
-    function FiniteDifferences.to_vec(dist::SqMahalanobis)
-        return vec(dist.qmat), x -> SqMahalanobis(reshape(x, size(dist.qmat)...))
-    end
-    a = rand()
-
-    function test_mahakernel(U::UpperTriangular, v1::AbstractVector, v2::AbstractVector)
-        return MahalanobisKernel(; P=Array(U' * U))(v1, v2)
-    end
-
-    @test all(
-        FiniteDifferences.j′vp(FDM, test_mahakernel, a, U, v1, v2)[1] .≈
-        UpperTriangular(Zygote.pullback(test_mahakernel, U, v1, v2)[2](a)[1]),
-    )
-
-    function test_sqmaha(U::UpperTriangular, v1::AbstractVector, v2::AbstractVector)
-        return SqMahalanobis(Array(U' * U))(v1, v2)
-    end
-
-    @test all(
-        FiniteDifferences.j′vp(FDM, test_sqmaha, a, U, v1, v2)[1] .≈
-        UpperTriangular(Zygote.pullback(test_sqmaha, U, v1, v2)[2](a)[1]),
-    )
-
-    # test_ADs(U -> MahalanobisKernel(P=Array(U' * U)), U, ADs=[:Zygote])
-    @test_broken "Nothing passes (problem with Mahalanobis distance in Distances)"
 
     # Standardised tests.
     @testset "ColVecs" begin
@@ -57,5 +26,4 @@
         x2 = RowVecs(randn(2, D_in))
         TestUtils.test_interface(k, x0, x1, x2)
     end
-    test_params(k, (P,))
 end

test/basekernels/piecewisepolynomial.jl

@@ -3,22 +3,32 @@
     v1 = rand(D)
     v2 = rand(D)
     maha = Matrix{Float64}(I, D, D)
-    v = 3
-    k = PiecewisePolynomialKernel{v}(maha)
+    degree = 3
 
-    k2 = PiecewisePolynomialKernel(; v=v, maha=maha)
+    k = PiecewisePolynomialKernel(; degree=degree, dim=D)
+    k2 = PiecewisePolynomialKernel{degree}(D)
+    k3 = @test_deprecated PiecewisePolynomialKernel{degree}(maha)
+    k4 = @test_deprecated PiecewisePolynomialKernel(; degree=degree, maha=maha)
+    k5 = @test_deprecated PiecewisePolynomialKernel(; v=degree, dim=D)
+    k6 = @test_deprecated PiecewisePolynomialKernel(; v=degree, maha=maha)
 
-    @test k2(v1, v2) ≈ k(v1, v2) atol = 1e-5
+    @test k2(v1, v2) == k(v1, v2)
+    @test k3(v1, v2) ≈ k(v1, v2)
+    @test k4(v1, v2) ≈ k(v1, v2)
+    @test k5(v1, v2) ≈ k(v1, v2)
+    @test k6(v1, v2) ≈ k(v1, v2)
 
     @test_throws ErrorException PiecewisePolynomialKernel{4}(maha)
+    @test_throws ErrorException PiecewisePolynomialKernel{4}(D)
+    @test_throws ErrorException PiecewisePolynomialKernel{degree}(-1)
 
-    @test repr(k) == "Piecewise Polynomial Kernel (v = $(v), size(maha) = $(size(maha)))"
+    @test repr(k) ==
+          "Piecewise Polynomial Kernel (degree = $(degree), ⌊dim/2⌋ = $(div(D, 2)))"
 
     # Standardised tests.
     TestUtils.test_interface(k, ColVecs{Float64}; dim_in=2)
     TestUtils.test_interface(k, RowVecs{Float64}; dim_in=2)
-    # test_ADs(maha-> PiecewisePolynomialKernel(v=2, maha = maha), maha)
-    @test_broken "Nothing passes (problem with Mahalanobis distance in Distances)"
+    test_ADs(() -> PiecewisePolynomialKernel{degree}(D))
 
-    test_params(k, (maha,))
+    test_params(k, ())
 end