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Prove the following functions are valid kernels, or disprove it, z. y R¹. 1. k(x, y) = (x + 2)² 2. k(z,y)=(z-y-1)³ Hint: A kernel is considered valid when it satisfies Mercer's theorem, i.e., Vr,... € R, KER is a positive semi-definite matriz, where its element at the i-th row and j-th column is equal to k(,,,), namely Kisk(r.), i, j€ {1,...,n). This property can also be shown by finding a function (), such that k(x,y)= o(r)o(v).

Prove the following functions are valid kernels, or disprove it, z. y R¹. 1. k(x, y) = (x + 2)² 2. k(z,y)=(z-y-1)³ Hint: A kernel is considered valid when it satisfies Mercer's theorem, i.e., Vr,... € R, KER is a positive semi-definite matriz, where its element at the i-th row and j-th column is equal to k(,,,), namely Kisk(r.), i, j€ {1,...,n). This property can also be shown by finding a function (), such that k(x,y)= o(r)o(v).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the following functions are valid kernels, or disprove it, x, y € R¹. 1. k(x, y) = (x y + 2)² 2. k(x,y)=(x-y-1)³ Hint: A kernel is considered valid when it satisfies Mercer's theorem, i.e., V₁, 2,..., n Rd, KER is a positive semi-definite matria, where its element at the i-th row and j-th column is equal to k(r.a), namely Kjk(,), i, je {1,...,n). This property can also be shown by finding a function (), such that k(x, y) = o(x)o(y).
Transcribed Image Text:Prove the following functions are valid kernels, or disprove it, x, y € R¹. 1. k(x, y) = (x y + 2)² 2. k(x,y)=(x-y-1)³ Hint: A kernel is considered valid when it satisfies Mercer's theorem, i.e., V₁, 2,..., n Rd, KER is a positive semi-definite matria, where its element at the i-th row and j-th column is equal to k(r.a), namely Kjk(,), i, je {1,...,n). This property can also be shown by finding a function (), such that k(x, y) = o(x)o(y).
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