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 KernelMatrixAlignment

Source Notebook

Compute the alignment similarity between two matrices

Contributed by: Felipe Amorim

ResourceFunction["KernelMatrixAlignment"][k1,k2]

computes the alignment similarity between matrices k1 and k2.

Details and Options

Kernel alignment was first introduced by Cristianini et al. as a measure of similarity between two (kernel) matrices and is defined as , where 〈k1,k2〉F is the Frobenius product between k1 and k2.
KernelAlignment retuns normilized values between -1 and 1.
In practice, centered alignment is often preferred (see Cortes et al.), where the kernel matrices are centered before computing the alignment score. The centered kernel matrix associated with K, denoted Kc, is defined as Kc⩵H.K.H, where . The centered alignment can be returned using the "Centered" option:
"Centered"Falsewhether the kernel matrices are centered before computing the alignment

Examples

Basic Examples (2) 

Calculate the alignment between two matrices:

In[1]:=
ResourceFunction["KernelMatrixAlignment"][\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", RowBox[{"-", "2"}]}, {"13", "4"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"4", "2"}, {"3", "10"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)] // N
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The alignment between identical matrices is 1:

In[2]:=
k = {{1, 2}, {3, 4}}; ResourceFunction["KernelMatrixAlignment"][k, k]
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Scope (2) 

Create two kernel matrices:

In[4]:=
n = 4; k1 = RandomReal[1, {n, n}]; k2 = RandomReal[1, {n, n}];

Show scaling invariance:

In[5]:=
ResourceFunction["KernelMatrixAlignment"][k1, k2] == ResourceFunction["KernelMatrixAlignment"][10 k1, k2]
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Options (2) 

Create two kernel matrices:

In[6]:=
n = 4; base = ConstantArray[10, {n, n}]; k1 = base + RandomReal[0.1, {n, n}]; k2 = base + RandomReal[0.1, {n, n}];

Use the "Centered" option to return the centered alignment value:

In[7]:=
ResourceFunction["KernelMatrixAlignment"][k1, k2, "Centered" -> True]
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Applications (5) 

Start with some 2-dimensional data:

In[8]:=
cluster1 = RandomVariate[MultinormalDistribution[{0, 0}, IdentityMatrix[2]], 200]; cluster2 = MapThread[{#1 Cos[#2], #1 Sin[#2]} &, {RandomReal[{4, 5}, 200], RandomReal[{0, 2 Pi}, 200]}]; Short[data = <|"center" -> cluster1, "circle" -> cluster2|>]
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Define a radial basis function (RBF) kernel:

In[11]:=
rbf[data_, \[Gamma]_] := Outer[Exp[-\[Gamma]*Norm[#1 - #2]^2] &, data, data, 1]

Evaluate the kernel on the data for three different γ values:

In[12]:=
With[{points = Flatten[Values@data, 1]}, kernel1 = rbf[points, 1]; kernel2 = rbf[points, 0.05]; kernel3 = rbf[points, 0.0001];]

Compare the alignment values between kernel 1 and the other two:

In[13]:=
ResourceFunction["KernelMatrixAlignment"][kernel1, #, "Centered" -> True] & /@ {kernel2, kernel3}
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Larger alignment values correspond to more similar decision boundaries induced by the kernels:

In[14]:=
With[{gammaValues = {1, 0.05, 0.0001}}, GraphicsGrid[{Table[ResourceFunction[ ResourceObject[<|"Name" -> "DecisionBoundaryPlot", "ShortName" -> "DecisionBoundaryPlot", "UUID" -> "17642d9f-3153-49b8-91d4-c9b9057f07c1", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Plot the decision boundaries of a classifier", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$83347f65c9ab4655a5c610ad84d80aab`DecisionBoundaryPlot", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/451ca251-7093-43ff-8331-ec5af61752e4"]|>, ResourceSystemBase -> Automatic]][data, Method -> {"SupportVectorMachine", "KernelType" -> "RadialBasisFunction", "GammaScalingParameter" -> gamma}, PlotLabel -> \[Gamma] == gamma], {gamma, gammaValues}]}]]
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Properties and Relations (2) 

Create two kernel matrices with different amounts of noise:

In[15]:=
n = 4; base = ConstantArray[10, {n, n}]; k1 = base + RandomReal[0.1, {n, n}]; k2 = base + RandomReal[0.1, {n, n}];

Use the "Centered" option to return the centered alignment value:

In[16]:=
ResourceFunction["KernelMatrixAlignment"][k1, k2, "Centered" -> True]
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Compare this to the default uncentered alignment:

In[17]:=
ResourceFunction["KernelMatrixAlignment"][k1, k2, "Centered" -> False]
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The uncentered alignment is close to 1 because both matrices share a large constant component, even though their noise structures are unrelated.

Define two kernel matrices:

In[18]:=
n = 3; k1 = RandomReal[1, {n, n}]; k2 = RandomReal[1, {n, n}];

Use the "Centered" option to return the centered alignment:

In[19]:=
ResourceFunction["KernelMatrixAlignment"][k1, k2, "Centered" -> True]
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Setting "Centered"->True is equivalent to centering the two kernel matrices before calculating the alignment. The matrices can be centered with the resource function CenteringMatrix as follows:

In[20]:=
mCentered[k_] := ResourceFunction[ ResourceObject[<|"Name" -> "CenteringMatrix", "ShortName" -> "CenteringMatrix", "UUID" -> "6fdd6ee1-1cc0-4402-aaf2-327452d9ea54", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Generate an n-sized centering matrix", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$80cab4aa62e44f20939542fca9bdab83`CenteringMatrix", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/168c1480-9031-4800-97a6-2e4e40a3c2b7"]|>, ResourceSystemBase -> Automatic]][Length[k]] . k . ResourceFunction[ ResourceObject[<|"Name" -> "CenteringMatrix", "ShortName" -> "CenteringMatrix", "UUID" -> "6fdd6ee1-1cc0-4402-aaf2-327452d9ea54", "ResourceType" -> "Function", "Version" -> "1.0.0", "Description" -> "Generate an n-sized centering matrix", "RepositoryLocation" -> URL[ "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"], "SymbolName" -> "FunctionRepository`$80cab4aa62e44f20939542fca9bdab83`CenteringMatrix", "FunctionLocation" -> CloudObject[ "https://www.wolframcloud.com/obj/168c1480-9031-4800-97a6-2e4e40a3c2b7"]|>, ResourceSystemBase -> Automatic]][Length[k]]

Confirm that the centered kernel has zero row and column means (double centered):

In[21]:=
Chop@{Mean[mCentered[k1]], Mean[Transpose@mCentered[k1]]}
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After centering the matrices explicitly, the alignment value is the same as above:

In[22]:=
ResourceFunction["KernelMatrixAlignment"][mCentered[k1], mCentered[k2]]
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Possible Issues (1) 

Kernel matrices must be of equal size:

In[23]:=
ResourceFunction["KernelMatrixAlignment"][\!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "1", "1"}, {"2", "2", "2"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\), \!\(\* TagBox[ RowBox[{"(", "", GridBox[{ {"1", "2"}, {"2", "3"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]\)] // N
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Publisher

Felipe Amorim

Version History

  • 1.0.0 – 27 February 2026

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