An illustration of kernel trick in SVM. Here the kernel is given by:
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ((x_{1},x_{2}))=(x_{1},x_{2},x_{1}^{2}+x_{2}^{2}).}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>ϕ<!-- ϕ --></mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>,</mo><msubsup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \phi ((x_{1},x_{2}))=(x_{1},x_{2},x_{1}^{2}+x_{2}^{2}).}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43e2cce0b0c530c2174ba47988b2d2a77d0b5d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:30.804ex; height:3.176ex;" alt="{\displaystyle \phi ((x_{1},x_{2}))=(x_{1},x_{2},x_{1}^{2}+x_{2}^{2}).}"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\phi (\mathbf {x} )\cdot \phi (\mathbf {y} )=\mathbf {x} \cdot \mathbf {y} +\parallel \mathbf {x} \parallel ^{2}\parallel \mathbf {y} \parallel ^{2}.}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>K</mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">y</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ<!-- ϕ --></mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mo stretchy="false">)</mo><mo>⋅<!-- ⋅ --></mo><mi>ϕ<!-- ϕ --></mi><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">y</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><mo>⋅<!-- ⋅ --></mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">y</mi></mrow><mo>+</mo><mo>∥<!-- ∥ --></mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">x</mi></mrow><msup><mo>∥<!-- ∥ --></mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>∥<!-- ∥ --></mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">y</mi></mrow><msup><mo>∥<!-- ∥ --></mo><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\phi (\mathbf {x} )\cdot \phi (\mathbf {y} )=\mathbf {x} \cdot \mathbf {y} +\parallel \mathbf {x} \parallel ^{2}\parallel \mathbf {y} \parallel ^{2}.}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9533a723f3efa5705dbff59f5234edb1a570aa6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:44.581ex; height:3.176ex;" alt="{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\phi (\mathbf {x} )\cdot \phi (\mathbf {y} )=\mathbf {x} \cdot \mathbf {y} +\parallel \mathbf {x} \parallel ^{2}\parallel \mathbf {y} \parallel ^{2}.}"></span></dd></dl>
<a href="//commons.wikimedia.org/wiki/User:Shiyu_Ji" title="User:Shiyu Ji">Shiyu Ji</a>
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