Transcription of 1 Reproducing Kernel Hilbert Spaces - People
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1 Reproducing Kernel Hilbert Spaces - People
CS281B/Stat241B (Spring 2008) Statistical Learning TheoryLecture: 7 Reproducing Kernel Hilbert SpacesLecturer: Peter BartlettScribe: Chunhui Gu1 Reproducing Kernel Hilbert Hilbert Space and KernelAn inner product u,v can be1. a usual dot product: u,v =v w= iviwi2. a Kernel product: u,v =k(v,w) = (v) (w) (where (u) may have infinite dimensions)However, an inner product , must satisfy the following conditions:1. Symmetry u,v = v,u u,v X2. Bilinearity u+ v,w = u,w + v,w u,v,w X, , R3. Positive definiteness u,u 0, u X u,u = 0 u= 0 Now we can define the notion of a Hilbert Spaceis an inner product space that is complete and separable with respect to thenorm defined by the inner of Hilbert Spaces include:1. The vector spaceRnwith a,b =a b, the vector dot product The spacel2of square summable sequences, with inner product x,y = i=1xiyi3. The spaceL2of square integrable functions ( , sf(x)2dx < ), with inner product f,g = sf(x)g(x) ( , ) is areproducing kernelof a Hilbert spaceHif f H,f(x) = k(x, ),f( ).
1. khas the reproducing property, i.e., f(x) = hf(·),k(·,x)i 2. kspans H = span{k(·,x) : x∈ X} 1.3 Mercer’s Theorem Another way to characterize a symmetric positive semi-definite kernel kis via the Mercer’s Theorem. Theorem 1.1 (Mercer’s). Suppose kis a continuous positive semi-definite kernel on a compact set X, and the integral ...
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