Suppose we have n kernel functions Kj(,), j = 1, ..., n such that there are...
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Suppose we have n kernel functions Kj(·,·), j = 1, ..., n such that there are n implicit high-dimensional feature maps ; : Rd → RD RD j = 1, ... , n that satisfies Vx, z E Rd, Kj(x, z) = (Þ; (x), Þj(z)), where (0;(x), Þj(z)) = Σº, Þ(x); Þ(z); is the dot product (a.k.a. inner product) in the D-dimensional space. (.) i=1 denotes the i-th dimension of the j-th mapped feature. j Is the summation of those kernel functions, that is, K(x, z) = Σ; K;(x, z), still a valid kernel function? If yes, prove that. If no, please explain why (You can attach images for your derivation) Suppose we have n kernel functions Kj(·,·), j = 1, ..., n such that there are n implicit high-dimensional feature maps ; : Rd → RD RD j = 1, ... , n that satisfies Vx, z E Rd, Kj(x, z) = (Þ; (x), Þj(z)), where (0;(x), Þj(z)) = Σº, Þ(x); Þ(z); is the dot product (a.k.a. inner product) in the D-dimensional space. (.) i=1 denotes the i-th dimension of the j-th mapped feature. j Is the summation of those kernel functions, that is, K(x, z) = Σ; K;(x, z), still a valid kernel function? If yes, prove that. If no, please explain why (You can attach images for your derivation)
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Related Book For
Introduction to Algorithms
ISBN: 978-0262033848
3rd edition
Authors: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest
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