Reproducing kernel Hilbert space RKHS is composed of functions, which use the "kernel trick" in Hilbert space to map a set of data into a high-dimensional space, which is the reproducible kernel Hilbert space.

Reproducing kernel Hilbert space concept

Under certain conditions, we can find the unique reproducing kernel function K corresponding to this Hilbert space, which satisfies the following points:

• For any fixed x0 belongs to X, there is K(x, x0) as a function of X that belongs to H;
• For any x belongs to X, f (y) belongs to H, f ( x ) ≤ f ( y ) , K ( y , x ) > H, then K ( x , y ) is called the reproducing kernel of H, and H is the Hilbert space with K ( x , y ) as the reproducing kernel, abbreviated as the reproducing kernel Hilbert space.

Hilbert space definition process

Vector space → Inner product space → Normed vector space → Metric space → Banach space → Hilbert space

• Vector Space: A collection of vectors that satisfy addition and scalar multiplication operations
• Normed Vector Space: A vector space that defines the length of a vector
• Metric Space: A set that defines the distance between two points
• Banach space: a complete normed vector space
• Inner Product Space: refers to the vector space on which the inner product operation can be performed on the domain.
• Hilbert Space: When an inner product space satisfies that the norm space can be derived through the inner product space and is complete, then this inner product space is a Hilbert space.

Two Theorems of RKHS

• A Hilbert space H is a reproducing kernel Hilbert space if and only if it has a reproducing kernel;
• For a given reproducing kernel Hilbert space, its reproducing kernel is unique.

Related terms: Hilbert space, reproducing kernel