Reproducing kernel Hilbert space


In functional analysis, a reproducing kernel Hilbert space is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions and in the RKHS are close in norm, i.e., is small, then and are also pointwise close, i.e., is small for all. The reverse does not need to be true.
It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Note that L2 spaces are not Hilbert spaces of functions, but rather Hilbert spaces of equivalence classes of functions. However, there are RKHSs in which the norm is an L2-norm, such as the space of band-limited functions.
An RKHS is associated with a kernel that reproduces every function in the space in the sense that for any in the set on which the functions are defined, "evaluation at " can be performed by taking an inner product with a function determined by the kernel. Such a reproducing kernel exists if and only if every evaluation functional is continuous.
The reproducing kernel was first introduced in the 1907 work of Stanisław Zaremba concerning boundary value problems for harmonic and biharmonic functions. James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of Gábor Szegő, Stefan Bergman, and Salomon Bochner. The subject was eventually systematically developed in the early 1950s by Nachman Aronszajn and Stefan Bergman.
These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics. Reproducing kernel Hilbert spaces are particularly important in the field of statistical learning theory because of the celebrated representer theorem which states that every function in an RKHS that minimises an empirical risk functional can be written as a linear combination of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the empirical risk minimization problem from an infinite dimensional to a finite dimensional optimization problem.
For ease of understanding, we provide the framework for real-valued Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important examples of reproducing kernel Hilbert spaces that are spaces of analytic functions.

Definition

Let be an arbitrary set and a Hilbert space of real-valued functions on. The evaluation functional over the Hilbert space of functions is a linear functional that evaluates each function at a point,
We say that H is a reproducing kernel Hilbert space if, for all in, is continuous at any in or, equivalently, if is a bounded operator on, i.e. there exists some M > 0 such that
While property is the weakest condition that ensures both the existence of an inner product and the evaluation of every function in at every point in the domain, it does not lend itself to easy application in practice. A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional can be represented by taking the inner product of with a function in . This function is the so-called reproducing kernel for the Hilbert space from which the RKHS takes its name. More formally, the Riesz representation theorem implies that for all in there exists a unique element of with the reproducing property,
Since is itself a function defined on with values in the field and as
is in we have that
where is the element in associated to.
This allows us to define the reproducing kernel of as a function by
From this definition it is easy to see that is both symmetric and positive definite, i.e.
for any The Moore–Aronszajn theorem is a sort of converse to this: if a function satisfies these conditions then there is a Hilbert space of functions on for which it is a reproducing kernel.

Example

The space of bandlimited continuous functions is a RKHS, as we now show. Formally, fix some cutoff frequency and define the Hilbert space
where is the set of continuous functions, and is the Fourier transform of.
From the Fourier inversion theorem, we have
It then follows by the Cauchy–Schwarz inequality and Parseval's theorem that, for all,
This inequality shows that the evaluation functional is bounded, proving that is indeed a RKHS.
The kernel function in this case is given by
To see this, we first note that the Fourier transform of defined above is given by
which is a consequence of the time-shifting property of the Fourier transform. Consequently, using Plancherel's theorem, we have
Thus we obtain the reproducing property of the kernel.
Note that in this case is the "bandlimited version" of the Dirac delta function, and that converges to in the weak sense as the cutoff frequency tends to infinity.

Moore–Aronszajn theorem

We have seen how a reproducing kernel Hilbert space defines a reproducing kernel function that is both symmetric and positive definite. The Moore–Aronszajn theorem goes in the other direction; it states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. The theorem first appeared in Aronszajn's Theory of Reproducing Kernels, although he attributes it to E. H. Moore.
Proof. For all x in X, define Kx = K. Let H0 be the linear span of. Define an inner product on H0 by
which implies.
The symmetry of this inner product follows from the symmetry of K and the non-degeneracy follows from the fact that K is positive definite.
Let H be the completion of H0 with respect to this inner product. Then H consists of functions of the form
Now we can check the reproducing property :
To prove uniqueness, let G be another Hilbert space of functions for which K is a reproducing kernel. For any x and y in X, implies that
By linearity, on the span of. Then because G is complete and contains H0 and hence contains its completion.
Now we need to prove that every element of G is in H. Let be an element of G. Since H is a closed subspace of G, we can write where and. Now if then, since K is a reproducing kernel of G and H:
where we have used the fact that belongs to H so that its inner product with in G is zero.
This shows that in G and concludes the proof.

Integral operators and Mercer's theorem

We may characterize a symmetric positive definite kernel via the integral operator using Mercer's theorem and obtain an additional view of the RKHS. Let be a compact space equipped with a strictly positive finite Borel measure and a continuous, symmetric, and positive definite function. Define the integral operator as
where is the space of square integrable functions with respect to.
Mercer's theorem states that the spectral decomposition of the integral operator of yields a series representation of in terms of the eigenvalues and eigenfunctions of. This then implies that is a reproducing kernel so that the corresponding RKHS can be defined in terms of these eigenvalues and eigenfunctions. We provide the details below.
Under these assumptions is a compact, continuous, self-adjoint, and positive operator. The spectral theorem for self-adjoint operators implies that there is an at most countable decreasing sequence such that and
, where the form an orthonormal basis of. By the positivity of for all One can also show that maps continuously into the space of continuous functions and therefore we may choose continuous functions as the eigenvectors, that is, for all Then by Mercer's theorem may be written in terms of the eigenvalues and continuous eigenfunctions as
for all such that
This above series representation is referred to as a Mercer kernel or Mercer representation of.
Furthermore, it can be shown that the RKHS of is given by
where the inner product of given by
This representation of the RKHS has application in probability and statistics, for example to the Karhunen-Loève representation for stochastic processes and kernel PCA.

Feature maps

A feature map is a map, where is a Hilbert space which we will call the feature space. The first sections presented the connection between bounded/continuous evaluation functions, positive definite functions, and integral operators and in this section we provide another representation of the RKHS in terms of feature maps.
We first note that every feature map defines a kernel via
Clearly is symmetric and positive definiteness follows from the properties of inner product in. Conversely, every positive definite function and corresponding reproducing kernel Hilbert space has infinitely many associated feature maps such that holds.
For example, we can trivially take and for all. Then is satisfied by the reproducing property. Another classical example of a feature map relates to the previous section regarding integral operators by taking and.
This connection between kernels and feature maps provides us with a new way to understand positive definite functions and hence reproducing kernels as inner products in. Moreover, every feature map can naturally define a RKHS by means of the definition of a positive definite function.
Lastly, feature maps allow us to construct function spaces that reveal another perspective on the RKHS. Consider the linear space
We can define a norm on by
It can be shown that is a RKHS with kernel defined by. This representation implies that the elements of the RKHS are inner products of elements in the feature space and can accordingly be seen as hyperplanes. This view of the RKHS is related to the kernel trick in machine learning.

Properties

The following properties of RKHSs may be useful to readers.

Bilinear kernels

The RKHS corresponding to this kernel is the dual space, consisting of functions satisfying

Polynomial kernels

[Radial basis function kernel]s

These are another common class of kernels which satisfy Some examples include:
We also provide examples of Bergman kernels. Let X be finite and let H consist of all complex-valued functions on X. Then an element of H can be represented as an array of complex numbers. If the usual inner product is used, then Kx is the function whose value is 1 at x and 0 everywhere else, and can be thought of as an identity matrix since
In this case, H is isomorphic to
The case of is more sophisticated. Here the Bergman space H square| is the space of square-integrable holomorphic functions on It can be shown that the reproducing kernel for is
Lastly, the space of band limited functions in with bandwidth are a RKHS with reproducing kernel

Extension to vector-valued functions

In this section we extend the definition of the RKHS to spaces of vector-valued functions as this extension is particularly important in multi-task learning and manifold regularization. The main difference is that the reproducing kernel is a symmetric function that is now a positive semi-definite matrix for any in. More formally, we define a vector-valued RKHS as a Hilbert space of functions such that for all and
and
This second property parallels the reproducing property for the scalar-valued case. We note that this definition can also be connected to integral operators, bounded evaluation functions, and feature maps as we saw for the scalar-valued RKHS. We can equivalently define the vvRKHS as a vector-valued Hilbert space with a bounded evaluation functional and show that this implies the existence of a unique reproducing kernel by the Riesz Representation theorem. Mercer's theorem can also be extended to address the vector-valued setting and we can therefore obtain a feature map view of the vvRKHS. Lastly, it can also be shown that the closure of the span of coincides with, another property similar to the scalar-valued case.
We can gain intuition for the vvRKHS by taking a component-wise perspective on these spaces. In particular, we find that every vvRKHS is isometrically isomorphic to a scalar-valued RKHS on a particular input space. Let. Consider the space and the corresponding reproducing kernel
As noted above, the RKHS associated to this reproducing kernel is given by the closure of the span of where
for every set of pairs.
The connection to the scalar-valued RKHS can then be made by the fact that every matrix-valued kernel can be identified with a kernel of the form of via
Moreover, every kernel with the form of defines a matrix-valued kernel with the above expression. Now letting the map be defined as
where is the component of the canonical basis for, one can show that is bijective and an isometry between and.
While this view of the vvRKHS can be useful in multi-task learning, this isometry does not reduce the study of the vector-valued case to that of the scalar-valued case. In fact, this isometry procedure can make both the scalar-valued kernel and the input space too difficult to work with in practice as properties of the original kernels are often lost.
An important class of matrix-valued reproducing kernels are separable kernels which can factorized as the product of a scalar valued kernel and a -dimensional symmetric positive semi-definite matrix. In light of our previous discussion these kernels are of the form
for all in and in. As the scalar-valued kernel encodes dependencies between the inputs, we can observe that the matrix-valued kernel encodes dependencies among both the inputs and the outputs.
We lastly remark that the above theory can be further extended to spaces of functions with values in function spaces but obtaining kernels for these spaces is a more difficult task.

Connection between RKHS with ReLU function

The ReLU function is commonly defined as and is a mainstay in the architecture of neural networks where it is used as an activation function. One can construct a ReLU-like nonlinear function using the theory of reproducing kernel hilbert spaces. Below, we derive this construction and show how it implies the representation power of neural networks with ReLU activations.
We will work with the Hilbert space of absolutely continuous functions with inner product
Let and. We begin by constructing the reproducing kernel via the Fundamental Theorem of Calculus,
where
and
This implies reproduces, and we can write down its general form as
By taking the limit, we obtain the ReLU function,
Using this formulation, we can apply the Representer theorem to the RKHS, letting one prove the optimality of using ReLU activations in neural network settings.