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Machine Learning: Difference between revisions
→Kernel Trick
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Since we only use <math>\langle x, z\rangle</math> then we only need <math>\phi(x)^T\phi(z)</math> to simulate a non-linear processing of the data. | Since we only use <math>\langle x, z\rangle</math> then we only need <math>\phi(x)^T\phi(z)</math> to simulate a non-linear processing of the data. | ||
A kernel <math>K(x,z)</math> is a function that can be expressed as <math>K(x,z)=\phi(x)^T\phi(z)</math> for some function <math>\phi</math | A kernel <math>K(x,z)</math> is a function that can be expressed as <math>K(x,z)=\phi(x)^T\phi(z)</math> for some function <math>\phi</math>. | ||
Ideally, our kernel function should be able to be computed without having to compute the actual <math>\phi</math> and the full dot product. | |||
====Identifying if a function is a kernel==== | ====Identifying if a function is a kernel==== | ||
Basic check: | Basic check: | ||
Since the kernel is an inner-product between <math>\phi(x), \phi(z)</math>, it should satisfy the axioms of inner products, namely <math>K(x,z)=K(z,x)</math>, otherwise it is not a kernel. | Since the kernel is an inner-product between <math>\phi(x), \phi(z)</math>, it should satisfy the axioms of inner products, namely <math>K(x,z)=K(z,x)</math>, otherwise it is not a kernel. | ||
====Mercer's Theorem==== | ====Mercer's Theorem==== | ||
Let our kernel function be <math>K(z,x)</math>. | Let our kernel function be <math>K(z,x)</math>. | ||