Operator theory, kernels, and Feedforward Neural Networks
In this paper we show how specific families of positive definite kernels
serve as powerful tools in analyses of iteration algorithms for multiple layer
feedforward Neural Network models. Our focus is on particular kernels that
adapt well to learning algorithms for data-sets/features which display
intrinsic self-similarities at feedforward iterations of scaling.
Palle E. T. Jorgensen, Myung-Sin Song, James Tian
# Operator theory, kernels, and feedforward neural networks

###### Abstract.

In this paper we show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.

Key words and phrases:algorithms, multipliers, spectral resolutions, normal operators, iterated function systems, fractal measures, feedforward neural network, explicit kernels, ReLU, reproducing kernel Hilbert spaces, positive definite kernels, composition operators 2
believe that this then yields a more direct tool for kernel-based feedforward NN models. This advantage of our approach is based on two facts. First, we identify a direct notion of kernel iteration which accounts for traditional function theoretic feedforward NN steps. Secondly, our approach offers a more direct and natural choices of kernels which govern approximations involved in deep NN models, for example graph NN constructions.

While positive definite kernels and their associated reproducing kernel Hilbert spaces have found diverse applications in pure and applied mathematics, we shall focus here on a new role of kernels in feedforward network models. In more detail, the main purpose of our paper is a presentation of choices of particular families of positive definite kernels which serve as powerful tools in analyses of multiple layer feedforward Neural Networks.

In general, reproducing kernel constructions, and the corresponding RKHSs are powerful tools in diverse applications. In the present framework of kernel neural networks (KNNs), their role may be summarized as follows: Starting with the problem at hand, when we build our RKHS(\(\mu\)) via IFS iterations (e.g., via Cantor-like fractal limits), then the Cantor-like \(\mu\) activation functions arise as relative reproducing dipole functions for RKHS(\(\mu\)) as in Figure 3.1 below.

## 2. Neural networks (NN), and reproducing kernel Hilbert spaces (RKHS)

A main theme in our paper is a development of new tools for design of feedforward Neural Network constructs. For this purpose we point out the use of positive definite kernels, and associated generating function for the NN algorithms. These kernel based functions include the more familiar ReLu functions, see Theorems 3.3 and 3.4 below. We stress that the particular RKHS constructs will be relative in the sense of Theorems 3.3 and 3.4, i.e., the inner product reproduces differences of function values.

Our approach to the use of kernels and functions for feedforward Neural Network (NN) algorithms, is based on a systematic study of two classes of operators. They act as follows: (i) between prescribed kernel Hilbert spaces, and (ii) other operators acting at indexed levels in the network, i.e., operators acting at fixed levels, so within choices of kernel Hilbert spaces. Case (i) includes a systematic study of composition operators (see Corollaries 2.7 and 2.9) in the context of kernel Hilbert spaces; and case (ii), the study of multiplier operators and their adjoints, see e.g., Theorem 2.15. We emphasize that the two classes of operations discussed below depend on choices of kernels at each level in particular NN-network models. Together these families of operators allow for realizations of black box filter-entries in associated generalized multi-resolutions systems, including operators which consist of composition followed by multiplication. Specific 3D applications are presented in the subsequent sections, secs 3 and 4.

**Conventions.** Inside the paper we shall work with Hilbert spaces of functions, e.g., reproducing kernel Hilbert spaces (RKHSs), \(L^{2}\) spaces, and Sobolev spaces. It will be assumed that these are Hilbert spaces of real valued functions. Inner products will be written \(\langle\cdot,\cdot\rangle\), and we shall use subscripts on \(\langle\cdot,\cdot\rangle\) to indicate the Hilbert space under consideration. Moreover, in our use of differentiation, or differential operators, we shall mean weak derivatives, i.e., differentiation in the sense of distributions, or making use of the natural duality for the spaces under consideration.
Our restriction here to the real valued case is dictated by our present applications to feedforward Neural Networks. However, many of our general results in Section 2 below extend to complex RKHS theory. The latter in turn are important in the study of geometry and potential theory of complex domains, see e.g., [1].

The power of kernel machines derives in part from the following facts. First, kernel machines serve to map points in a low-dimensional data sets (typically non-linear) into higher dimensions. The dimensionality of this linear "hyperspace" may be infinite but is designed for optimization and efficient encoding of features. Hence the kernel method allows one to find coefficients of separating hyperplanes for the problem at hand via RKHS-inner products, one selected for each pair of high-dimensional features. While kernel machines of various types have been used for decades, it was with the invention of support vector machines (SVMs) that kernels have now taken center stage (see e.g., [1, 14, 15, 16, 17, 18]). By now, SVMs are used in diverse applications, including in bioinformatics (for finding similarities between different protein sequences), machine vision, and handwriting recognition. Deep neural networks (to be discussed in Sections 3 and 4 below) are made of layers of artificial neurons: input layer, an output layer, and multiple hidden layers in-between them. Deeper the networks have more hidden layers. The parameters of the network represent the strengths of the connections between layers. Training a network yield determination of values of parameters. Once trained, the ANN represents a model for turning an input (say, an image) into an output (a label or category).

The variety of uses of forward Neural Network algorithms, the recent literature is substantial and diverse, especially with regards to applications. See e.g., [1, 1, 2, 10, 11].

The following lemma is a basic result in the theory of RKHSs. For details, see e.g., [11, 12, 13, 14, 15], and also [16] and the papers cited therein.

**Lemma 2.1**.: _Fix a p.d. kernel \(X\times X\xrightarrow{K}\mathbb{R}\left(\text{or }\mathbb{C}\right)\), let \(\mathscr{H}_{K}\) denote the corresponding RKHS. Then a function \(F\) on \(X\) is in \(\mathscr{H}_{K}\) if and only if there exists a constant \(C_{F}<\infty\), such that the following estimate holds for all \(n\in\mathbb{N}\), all \(\left(\xi_{i}\right)_{i=1}^{n}\), \(\xi_{i}\in\mathbb{R}\left(\text{or }\mathbb{C}\right)\), and all \(\left(x_{i}\right)_{i=1}^{n}\), \(x_{i}\in X\):_

\[\left|\sum_{i=1}^{n}\xi_{i}F\left(x_{i}\right)\right|^{2}\leq C_{F}\sum_{i=1}^ {n}\sum_{j=1}^{n}\overline{\xi}_{i}\xi_{j}K\left(x_{i},x_{j}\right). \tag{2.1}\]

_Remark 2.2_.: With the construction \(K\mapsto\mathscr{H}_{K}\) (referring to a RKHS of a fixed p.d. kernel \(K\)), we arrive at the following two conclusions:

1. For all \(x\in X\), the function \(K_{x}:=K\left(\cdot,x\right)\) is in \(\mathscr{H}_{K}\); and
2. For all \(F\in\mathscr{H}_{K}\), and \(x\in X\), we have \[F\left(x\right)=\left\langle F,K\left(\cdot,x\right)\right\rangle_{\mathscr{H} _{K}},\] (2.2) i.e., the values of functions \(F\) in \(\mathscr{H}_{K}\) are _reproduced_ via the inner product \(\left\langle\cdot,\cdot\right\rangle_{\mathscr{H}_{K}}\), and the kernel functions.

In addition to (2.2), we shall also consider _relative reproducing kernels_, and _relative_ RKHSs. As noted in [1], the _relative_ reproducing property takes the following form

\[F\left(y\right)-F\left(x\right)=\left\langle F,v_{x,y}\left(\cdot\right) \right\rangle_{\mathscr{H}_{\text{rel}}}, \tag{2.3}\]
now valid for all pairs of points \(x,y\in X\). So this entails double-indexed kernel functions \(v_{x,y}\in\mathscr{H}_{rel}\).

A particular class of \(\mathscr{H}_{rel}\) spaces are considered in Theorem 3.4 below. There the setting is \(X=\mathbb{R}\), and the relative kernel functions \(v_{a,b}\) take the form of activation functions for classes of feedforward-NN-algorithms, see e.g., Figure 3.1.

A systematic study of (2.3) is undertaken in [1] where it is shown that the setting of relative reproducing is characterized by _conditionally negative definite functions_.

We now recall the RKHS for the standard \(1\)-dimensional Brownian motion. (See e.g., [11, 12, 13, 14].)

**Lemma 2.3**.: _When \(K\) is the Brownian motion kernel on \(\mathbb{R}_{\geq}\times\mathbb{R}_{\geq}\), i.e.,_

\[K\left(x,y\right)=x\wedge y=\frac{\left|x\right|+\left|y\right|-\left|x-y \right|}{2},\quad x,y\in\mathbb{R}_{\geq}, \tag{2.4}\]

_the corresponding RKHS \(\mathscr{H}_{K}\) is the Hilbert space of absolutely continuous functions \(f\) on \(\mathbb{R}\) such that the derivative \(f^{\prime}=df/dx\) is in \(L^{2}\left(\mathbb{R}\right)\), and \(f\left(0\right)=0\). Moreover,_

\[\left\|f\right\|_{\mathscr{H}_{K}}^{2}=\int_{\mathbb{R}_{\geq}}\left|f^{ \prime}\left(x\right)\right|^{2}dx,\quad\text{for all }f\in\mathscr{H}_{K}. \tag{2.5}\]

Proof.: The key observation is that, if \(x>0\), the function

\[\mathbb{R}_{\geq}\ni y\longmapsto F_{x}\left(y\right):=K\left(y,x\right)= \begin{cases}y&\text{if }y\leq x\\ x&\text{if }y>x\end{cases} \tag{2.6}\]

has weak derivative. Indeed, we have

\[\frac{dF_{x}}{dy}=\chi_{\left[0,x\right]}, \tag{2.7}\]

i.e., the indicator function of the interval \(\left[0,x\right]\). Hence if \(f\) is a function with \(f^{\prime}\in L^{2}\left(\mathbb{R}\right)\) and \(f\left(0\right)=0\), then

\[f\left(x\right)=f\left(x\right)-f\left(0\right)=\int_{0}^{x}f^{\prime}\left( y\right)dy=\int_{\mathbb{R}}F_{x}^{\prime}\left(y\right)f^{\prime}\left(y \right)dy, \tag{2.8}\]

and the RHS in (2.8) is the inner product from the Hilbert space defined by the RHS in (2.5).

The corresponding implication follows from the general theory of RKHS. Recall that the RKHS of a kernel is a Hilbert space completion of the functions

\[y\longmapsto K\left(x,y\right) \tag{2.9}\]

as \(x\) varies over \(\mathbb{R}\). Moreover, for \(K\left(x,y\right)=x\wedge y\),

\[\left\langle K\left(\cdot,x_{1}\right),K\left(\cdot,x_{2}\right)\right\rangle _{\mathscr{H}_{K}}=K\left(x_{1},x_{2}\right)=x_{1}\wedge x_{2},\]

and we can compute as follows:

\[\int_{\mathbb{R}}\left(\frac{d}{dy}K\left(\cdot,x_{1}\right) \right)\left(\frac{d}{dy}K\left(\cdot,x_{2}\right)\right)dy =\int_{\mathbb{R}}\chi_{\left[0,x_{1}\right]}\left(y\right)\chi_ {\left[0,x_{2}\right]}\left(y\right)dy\] \[=\lambda\left(\left[0,x_{1}\right]\cap\left[0,x_{2}\right]\right)\] \[=x_{1}\wedge x_{2}=K\left(x_{1},x_{2}\right)\]

where \(\lambda=dy\) denotes the Lebesgue measure.
_Remark 2.4_.: Note that the functions \(v_{a,b}\) (called dipoles) in \(\mathscr{H}_{K}\) which satisfy

\[f\left(b\right)-f\left(a\right)=\left\langle f,v_{a,b}\right\rangle,\quad\text{ for all }f\in\mathscr{H}_{K}\]

(see (2.5) and (2.8)) are as follows:

\[v_{a,b}\left(x\right)=\begin{cases}0&\text{if }x<a\\ x-a&\text{if }a\leq x<b\\ b-a&\text{if }x>b,\end{cases}\]

as illustrated in Figure 2.1. Also compare with Theorem 3.4 and Figure 3.1, and the iterations in Section 4.

**Induced metrics**

For a general p.d. kernel \(K\) on \(X\times X\), there is an induced metric on \(X\),

\[d_{K}:X\times X\to\mathbb{R}_{+}\]

defined as (see e.g., [1])

\[d_{K}\left(x,y\right)=\left\|K\left(\cdot,x\right)-K\left(\cdot,y\right)\right\| _{\mathscr{H}_{K}}^{2}. \tag{2.10}\]

In particular,

\[d_{K}\left(x,y\right)=K\left(x,x\right)+K\left(y,y\right)-2\Re\left\{K\left(x,y\right)\right\}.\]

Note that \(d_{K}^{1/2}\) is also a metric on \(X\times X\).

**Example 2.5**.: For \(K\left(x,y\right)=x\wedge y\) on \(\mathbb{R}\times\mathbb{R}\) as in (2.4),

\[\left\|K\left(\cdot,s\right)-K\left(\cdot,t\right)\right\|_{ \mathscr{H}_{K}}^{2} =\left\|\left(\cdot\wedge s\right)^{\prime}-\left(\cdot\wedge t \right)^{\prime}\right\|_{L^{2}}^{2}\] \[=\left\|\chi_{[0,s]}-\chi_{[0,t]}\right\|_{L^{2}}^{2}\] \[=\left|s-t\right|.\]

The results below deal with a general framework of pairs of sets \(X\) and \(Y\), each equipped with a positive definite kernel, \(K\) resp., \(L\), \(K\) on \(X\), and \(L\) on \(Y\). With view to realization of feedforward Neural Network-functions, we will present an explicit framework (see (2.11) and (2.20)) which allows us to pass from (nonlinear) functions \(f:X\to Y\) to linear operators \(T_{f}\) acting between the respective RKHSs \(\mathscr{H}_{K}\) and \(\mathscr{H}_{L}\). This will be a representation in the sense that composition of functions will map into products of the corresponding linear operators. Some care must be exercised as the linear operators \(T_{f}\) will in general be unbounded. Nonetheless, we shall show that the operators still fall in a class where spectral resolutions are available, see Theorem 2.11 and Corollary 2.12.

Figure 2.1. The generating dipole function \(\left\{v_{a,b}\right\}\) indexed by pairs \(a,b\) such that \(a<b\). Compare with Figure 3.1 below.
**Theorem 2.6**.: _Consider p.d. kernels \(X\times X\overset{K}{\longrightarrow}\mathbb{R}\) and \(Y\times Y\overset{L}{\longrightarrow}\mathbb{R}\). Let \(f:X\to Y\) be Lipschitz continuous with respect to the induced metrics \(d_{K},d_{L}\), i.e.,_

\[d_{L}\left(f\left(x\right),f\left(y\right)\right)\leq c_{f}d_{K}\left(x,y \right),\quad x,y\in X,\]

_for some constant \(c_{f}\). Define the operator \(T_{f}:\mathscr{H}_{K}\rightarrow\mathscr{H}_{L}\) by_

\[T_{f}\left(K_{x}\right)\left(y\right)=L\left(f\left(x\right),y\right)\]

_and extend it by linearity and density._

_Then, for any fixed \(c\in Y\), the function_

\[F_{f}:X\rightarrow\mathbb{R},\quad F_{f}\left(x\right):=L\left(f\left(x \right),c\right) \tag{2.11}\]

_is in the RKHS \(\mathscr{H}_{K}\) if, and only if_

\[L_{c}\in dom(T_{f}^{*}), \tag{2.12}\]

_the domain of the adjoint operator._

_Moreover,_

\[F_{f,y}\left(x\right):=L\left(f\left(x\right),y\right)\in\mathscr{H}_{K},\ \forall y\in Y\]

\[\Updownarrow\]

\[T_{f}\]

_is closable._

_(See also Theorem 2.8.)_

Proof.: Let the setting be as in the statement of the theorem, i.e., \(X\overset{f}{\longrightarrow}Y\) assumed continuous with respect to the two metrics, \(d_{K}\) on \(X\) and \(d_{L}\) on \(Y\); so in particular, for pairs of points \(x_{1},x_{2}\in X\), we have

\[d_{K}\left(x_{1},x_{2}\right) =\left\|K\left(\cdot,x_{1}\right)-K\left(\cdot,x_{2}\right)\right\| _{\mathscr{H}_{K}}^{2} \tag{2.13}\] \[=K\left(x_{1},x_{1}\right)+K\left(x_{2},x_{2}\right)-2K\left(x_{1 },x_{2}\right). \tag{2.14}\]

We further fix a point \(c\in Y\), and set \(F=F_{f,c}\), specified as follows:

\[F\left(x\right)=L\left(f\left(x\right),c\right),\text{ for all }x\in X,\]

so \(F:X\rightarrow\mathbb{R}_{+}\cup\{0\}\).

Now, for every \(N\), and every subset \(S_{N}=\left(x_{1},x_{2},\ldots,x_{N}\right)\subset X\), consider the following matrix operations (in \(N\) dimensions):

\[\underbrace{F\big{|}_{N}:=\begin{bmatrix}F\left(x_{1}\right)\\ \vdots\\ F\left(x_{N}\right)\end{bmatrix}}_{\text{column vectors}},\text{ and }\quad \underbrace{Q_{N}:=\left|F\big{|}_{N}\left\rangle\left\langle\,F\big{|}_{N} \right|\right|}_{\text{matrix of a rank-1 operator}}\,, \tag{2.15}\]

i.e., the rank-1 operator on \(\mathbb{R}^{N}\) written in Dirac's notation, defined as

\[Q_{N}\left(\xi\right)=\left\langle F_{N},\xi\right\rangle F_{N} \tag{2.16}\]

for all \(\xi\in\mathbb{R}^{N}\). Set

\[K_{N}:=\left(K\left(x_{i},x_{j}\right)\right)_{i,j=1}^{N}=\begin{bmatrix}K \left(x_{1},x_{1}\right)&\cdots&K\left(x_{1},x_{N}\right)\\ \vdots&\ddots&\vdots\\ K\left(x_{N},x_{1}\right)&\cdots&K\left(x_{N},x_{N}\right)\end{bmatrix}, \tag{2.17}\]

a sample matrix.
For the convex cone of all positive definite \(N\times N\) matrices, we introduce the following familiar ordering, \(K\ll_{C}K^{\prime}\) iff (Def.) \(\exists C<\infty\) such that

\[\xi^{T}K_{N}\xi\leq C\xi^{T}K_{N}^{\prime}\xi\text{ for all }\xi\in\mathbb{R}^{N}. \tag{2.18}\]

Now an application of Lemma 2.1 above shows that the assertion in the theorem is equivalent to the existence of a finite constant \(C\) (independent of \(S_{N}=\left(x_{i}\right)_{i=1}^{N}\)) satisfying \(Q_{N}\ll_{C}K_{N}\), i.e., the estimate

\[\left|\sum_{i}\xi L\left(f\left(x_{i}\right),c\right)\right|^{2}\leq C\sum_{i} \sum_{j}\xi_{i}K\left(x_{i},x_{j}\right)\xi_{j} \tag{2.19}\]

holds for all \(N\), all \(S_{N}=\left\{x_{i}\right\}_{i=1}^{N}\), and all \(\xi\in\mathbb{R}^{N}\). We get this from the assumption on \(f\) in the theorem. See details below. 

**Summary of Theorem 2.6:** Start with \(K\) p.d. on \(X\times X\), \(L\) p.d. on \(Y\times Y\), and \(f:X\to Y\). We introduce the metrics \(d_{K}\) on \(X\), and \(d_{L}\) on \(Y\), and we consider \(f\) continuous, or Lipchitz. To get the desired conclusion

\[\left(x\longmapsto L\left(f\left(x\right),y\right)\right)\in\mathscr{H}_{K},\]

we must introduce an operator \(T_{f}:\mathscr{H}_{K}\to\mathscr{H}_{L}\). The right choice is

\[L_{y}\in dom(T_{f}^{*}).\]

See details below:

Fixing two kernels \(K\) and \(L\), assumed p.d. on \(X\times X\), and on \(Y\times Y\). Pass to the corresponding RKHS \(\mathscr{H}_{K}\) and \(\mathscr{H}_{L}\).

**Problem**.: Find conditions on functions \(X\overset{f}{\longrightarrow}Y\) with the property that, for \(\forall y\in Y\), then the induced function

\[\underbrace{\left(X\ni x\longmapsto L\left(f\left(x\right),y\right)\right)}_{ F_{f},\left(\cdot\right)\text{ as a function on }X}\in\mathscr{H}_{K}. \tag{2.20}\]

The argument stressed below is via dual operators (bounded)

but the unbounded case is also interesting.

Some remarks on the definition of the operator \(T_{f}:\mathscr{H}_{K}\to\mathscr{H}_{L}\) in the case when no additional assumptions are placed on \(X\overset{f}{\longrightarrow}Y\).

We define

\[T_{f}\left(K_{x}\right)\left(y\right)=L\left(f\left(x\right),y\right);\]

and so we extend \(T_{f}\) to linear combinations:

\[\mathscr{D}_{K}:=\big{\{}\underbrace{\sum_{i}c_{i}K_{x_{i}}}_{\text{function on }X}\xrightarrow{T_{f}}\underbrace{\sum_{i}c_{i}L\left(f\left(x_{i} \right),\cdot\right)}_{\text{function on }Y}. \tag{2.21}\]

But to make sense of (2.21) so it is well defined, we must be careful with equivalence classes.
If \(f:X\to Y\) is a general function, the (generalized) operator (2.21) may be non-closable. However, we can still define the adjoint \(T_{f}^{*}\), but its domain might be "small".

Set

\[\mathscr{D}_{K}:=span\left\{K_{x}\right\}_{x\in X}, \tag{2.22}\]

then (Definition) a vector \(\psi\in\mathscr{H}_{L}\) is in \(dom(T_{f}^{*})\) iff \(\exists C_{\psi}<\infty\) s.t.

\[\left|\left\langle T_{f}\varphi,\psi\right\rangle_{\mathscr{H}_{L}}\right| \leq C_{\psi}\left\|\varphi\right\|_{\mathscr{H}_{K}},\quad\forall\varphi \in\mathscr{D}_{K}. \tag{2.23}\]

We then set \(T_{f}^{*}\psi=\) the solution to

\[\left\langle T_{f}\varphi,\psi\right\rangle_{\mathscr{H}_{L}}=\left\langle \varphi,T_{f}^{*}\psi\right\rangle_{\mathscr{H}_{K}},\quad\forall\varphi\in \mathscr{D}_{K}. \tag{2.24}\]

Let \(K,L,f\) be as specified, and assume for some \(y\in Y\), that we have \(L_{y}\in dom(T_{f}^{*})\), then (2.24) for \(L_{y}\in\mathscr{H}_{L}\overset{T_{f}^{*}}{\longrightarrow}\mathscr{H}_{K}\), \(\varphi=K_{x}\), \(\psi=L_{y}\) yields

\[x\longmapsto T_{f}^{*}\left(L_{y}\right)\left(x\right)=L\left(f\left(x\right),y\right)\in\mathscr{H}_{K}.\]

So the conclusion in Theorem 2.6 that

\[\left(x\longmapsto L\left(f\left(x\right),y\right)\right)\in\mathscr{H}_{K} \tag{2.25}\]

holds iff \(L_{y}\in dom(T_{f}^{*})\).

In this case there are no difficulties with (2.21) and we get a dual pair \(T_{f}\) and \(T_{f}^{*}\),

\[\left\langle T_{f}\varphi,\psi\right\rangle_{\mathscr{H}_{L}}=\left\langle \varphi,T_{f}^{*}\psi\right\rangle_{\mathscr{H}_{K}} \tag{2.26}\]

for \(\forall\varphi\in\mathscr{D}_{K}\), and \(\psi\in\mathscr{D}_{L}\),.

Setting \(\varphi=K_{x}\), and \(\psi=L_{y}\), (2.26) implies

\[T_{f}^{*}\left(L_{y}\right)\left(x\right)=L\left(f\left(x\right),y\right)= \left(T_{f}\left(K_{x}\right)\right)\left(y\right). \tag{2.27}\]

But the previous condition \(L_{y}\in dom(T_{f}^{*})\) (compare (2.26)) amounts to the assertion that \(T_{f}^{*}\left(L_{y}\right)\in\mathscr{H}_{K}\), and by (2.27), this is then the conclusion for Theorem 2.6.

**Corollary 2.7** (composition operators).: _Let \(X\), \(Y\), \(K\) and \(L\) be as specified above; in particular, \(K\) is a fixed p.d. kernel on \(X\times X\), and the RKHS \(\mathscr{H}_{K}\) is a Hilbert space of scalar valued functions on \(X\). Similarly, \(\mathscr{H}_{L}\) is a Hilbert space of scalar valued functions on \(Y\). Both \(\mathscr{H}_{K}\) and \(\mathscr{H}_{L}\) satisfy the defining axioms for RKHSs; see Lemma 2.1 above. As noted, every function \(f\), \(X\overset{f}{\longrightarrow}Y\), induces a linear operator_

\[\mathscr{H}_{K}\overset{T_{f}}{\longrightarrow}\mathscr{H}_{L}, \tag{2.28}\]

_with dense domain \(\mathscr{D}_{K}\); see the statement of Theorem 2.6. For the adjoint operator \(T_{f}^{*}\),_

\[\mathscr{H}_{L}\overset{T_{f}^{*}}{\longrightarrow}\mathscr{H}_{K}, \tag{2.29}\]

_we have the following: For a function \(\psi\) in \(\mathscr{H}_{L}\), the two characterizations (2.30) and (2.31) hold:_

\[\psi\in dom(T_{f}^{*}) \tag{2.30}\]
\[\Updownarrow\] \[\psi\circ f\in\mathscr{H}_{K}. \tag{2.31}\]

_In the affirmative,_

\[T_{f}^{*}\left(\psi\right)=\psi\circ f:X\rightarrow\mathbb{R}, \tag{2.32}\]

_i.e., \(T_{f}^{*}\) is the composition operator._

Proof.: (2.30)\(\Rightarrow\)(2.31). Assume (2.30), we then apply (2.23), and get \(C_{\psi}<\infty\) with:

\[\left|\sum_{i}c_{i}\left(T_{f}^{*}\left(\psi\right)\right)\left(x_{i}\right) \right|^{2}\leq C_{\psi}\sum_{i}\sum_{j}c_{i}c_{j}K\left(x_{i},x_{j}\right). \tag{2.33}\]

But

\[T_{f}^{*}\left(\psi\right)\left(x_{i}\right) =\left\langle K_{x_{i}},T_{f}^{*}\left(\psi\right)\right\rangle_ {\mathscr{H}_{K}} \tag{2.34}\] \[=\left\langle T_{f}\left(K_{x_{i}}\right),\psi\right\rangle_{ \mathscr{H}_{L}}\] \[=\left\langle L\left(f\left(x_{i}\right),\cdot\right),\psi \right\rangle_{\mathscr{H}_{L}}\] \[=\psi\left(f\left(x_{i}\right)\right),\]

where we used the RKHS property for \(\mathscr{H}_{L}\) in the last step. Substitution into (2.33) yields

\[\left|\sum_{i}c_{i}\psi\left(f\left(x_{i}\right)\right)\right|^{2}\leq C_{\psi }\sum_{i}\sum_{j}c_{i}c_{j}K\left(x_{i},x_{j}\right), \tag{2.35}\]

and, so by Lemma 2.1 applied to \(F=\psi\circ f\), conclusion in (2.31) follows.

(2.31)\(\Rightarrow\)(2.30). Assume (2.31), we then reverse the above reasoning to get

\[\left|\left\langle T_{f}\underbrace{\left(\sum_{i}c_{i}K_{x_{i}}\right)}_{ \in\mathscr{D}_{K}},\psi\right\rangle_{\mathscr{H}_{L}}\right|\leq\sqrt{C_{ \psi}}\left\|\sum_{i}c_{i}K_{x_{i}}\right\|_{\mathscr{H}_{K}} \tag{2.36}\]

which states that \(\psi\in dom(T_{f}^{*})\), which is condition (2.30). Now combine this with (2.34), and we conclude that (2.32) is satisfied for \(\psi\), i.e., that \(T_{f}^{*}\psi=\psi\circ f\) holds. 

### Basis approach

Let \(X,Y,K,L,f\) be as usual, and define \(T_{f}:\mathscr{H}_{K}\rightarrow\mathscr{H}_{L}\). Since \(K\) is p.d. on \(X\times X\), the RKHS \(\mathscr{H}_{K}\) allows an ONB \(\left\{h_{i}\right\}_{i\in\mathbb{N}}\), \(h_{i}\in\mathscr{H}_{K}\); by general theory, we get the pointwise formula:

\[K\left(x_{1},x_{2}\right)=\sum_{i\in\mathbb{N}}h_{i}\left(x_{1}\right)h_{i} \left(x_{2}\right). \tag{2.37}\]

Then our condition in Theorem 2.6 is equivalent to the following:

\[\left(X\ni x\longmapsto L\left(f\left(x\right),y\right)\right) \in\mathscr{H}_{K}\] (a) \[\Updownarrow\] \[\sum_{i\in\mathbb{N}}\left|\left(T_{f}\left(h_{i}\right)\right) \left(y\right)\right|^{2}<\infty.\] (b)

Proof.: \(\left(a\right)\Rightarrow\left(b\right)\) is detailed below; but the converse implication will follow by the same argument. So by \(\left(a\right)\), \(L_{y}\in dom(T_{f}^{*})\) and therefore:

\[T_{f}^{*}\left(L_{y}\right)\left(\cdot\right)\in\mathscr{H}_{K}. \tag{2.38}\]
Since \(\left\{h_{i}\right\}\) is an ONB in \(\mathscr{H}_{K}\),

\[\sum_{i}\left|\left\langle h_{i},\underbrace{T_{f}^{*}\left(L_{y}\right)}_{ \in\mathscr{H}_{K}}\right\rangle\right|^{2}=\left\|T_{f}^{*}\left(L_{y}\right) \right\|_{\mathscr{H}_{K}}^{2}<\infty. \tag{2.39}\]

But from the LHS\({}_{\left(\ref{eq:2.39}\right)}\):

and \(\left(b\right)\) follows. 

**Key Question: When is \(F_{f,y}\left(\cdot\right)\in\mathscr{H}_{K}\)?** The cleanest answer to the question of what functions \(X\overset{f}{\longrightarrow}Y\) have the property that

\[F_{f,y}\left(x\right)=L\left(f\left(x\right),y\right)\text{ is in }\mathscr{H}_{K} \tag{2.40}\]

is this:

**Theorem 2.8**.: _Let \(K,L\) and \(f\) be given, then_

\[F_{f,y}\text{ in \eqref{eq:2.40} is in }\mathscr{H}_{K}\Longleftrightarrow L_{y}\in\] _dom \[\left(T_{f}^{*}\right)\],_ (2.41)

_where the operator \(T_{f}:\mathscr{H}_{K}\rightarrow\mathscr{H}_{L}\) is given by_

\[T_{f}\left(K_{x}\right):=L\left(f\left(x\right),\cdot\right).\]

_Moreover, (2.41) holds for all \(y\in Y\Longleftrightarrow T_{f}\) is closable._

### Dual pairs of operators

Consider a symmetric pair of operators with dense domains:

(\(T=T_{f}\), since it will depend on \(f\)) where

\[span\left\{K_{x}\right\}_{x\in X}\text{ is dense in }\mathscr{H}_{K} \tag{2.42}\]

and

\[span\left\{L_{y}\right\}_{y\in Y}\text{ is dense in }\mathscr{H}_{L} \tag{2.43}\]

such that

\[K_{x}\in dom(T_{f}),\text{ and } \tag{2.44}\] \[L_{y}\in dom(T_{f}^{*}) \tag{2.45}\]

where "\(dom\)" denotes the respective operator domains.

_Note._ We note that

\[T_{f}\left(K_{x}\right)\left(\cdot\right)=L\left(f\left(x\right),\cdot\right) \in\mathscr{H}_{L}\]

is always well defined, with dense domain, but the secret is \(T_{f}^{*}\).

Also note that (2.45) is the condition in Theorem 2.6.
Let \(f:X\to Y\) be as before, and the two p.d. kernels \(K\) and \(L\) are fixed. We then introduce the corresponding (densely defined) operator \(T_{f}:\mathscr{H}_{K}\rightarrow\mathscr{H}_{L}\) by setting

\[T_{f}\left(K_{x}\right)=L\left(f\left(x\right),\cdot\right)\in\mathscr{H}_{L}. \tag{2.46}\]

**Notation and convention.**\(K_{x}\left(\cdot\right)\) is the kernel function in \(\mathscr{H}_{K}\) as usual:

\[K_{x}\left(t\right) =K\left(x,t\right),\;\forall t\in X\;\text{and similarly,} \tag{2.47}\] \[L_{y}\left(u\right) =L\left(y,u\right),\;\forall u\in Y. \tag{2.48}\]

**Lemma 2.9**.: _When \(L_{y}\in dom(T_{f}^{\ast})\) then_

\[\left(T_{f}^{\ast}\left(L_{y}\right)\right)\left(x\right)=L\left(f\left(x \right),y\right)\;\text{on $X$,} \tag{2.49}\]

_equivalently,_

\[T_{f}^{\ast}\left(L_{y}\right)\left(\cdot\right)=L\left(f\left(\cdot\right),y \right)\;\text{on $X$.} \tag{2.50}\]

Proof of (2.49).: The conclusion (2.49) is equivalent to the following assertion:

The conclusion (2.49) follows since the respective kernel functions span dense subspaces. 

Recall,

\[\text{the function $x\longmapsto F_{f,y}\left(x\right)=L\left(f \left(x\right),y\right)\in\mathscr{H}_{K}$}\] \[\Updownarrow\] \[L_{y}\in dom(T_{f}^{\ast}).\]

Assume that \(L_{y}\in dom(T_{f}^{\ast})\), then apply the condition for functions in \(\mathscr{H}_{K}\) to \(F_{f,y}\left(\cdot\right)\), so \(\forall n,\)\(\forall\left(x_{i}\right)_{1}^{n},\)\(\forall\left(c_{i}\right)_{1}^{n},\)\(c_{i}\in\mathbb{R}\):

\[\left|\sum_{i}c_{i}F_{f,y}\left(x_{i}\right)\right|^{2} = \left|\sum_{i}c_{i}L\left(f\left(x_{i}\right),y\right)\right|^{2}\] \[\leq \left|\left\langle\sum_{i}c_{i}K_{x_{i}},T_{f}^{\ast}\left(L_{y} \right)\right\rangle_{\mathscr{H}_{K}}\right|^{2}\] \[\leq \left|\sum_{i}c_{i}K_{x_{i}}\right|^{2}_{\mathscr{H}_{K}}\left\| T_{f}^{\ast}\left(L_{y}\right)\right\|^{2}_{\mathscr{H}_{K}}\] \[= \sum_{i}\sum_{j}c_{i}c_{j}K\left(x_{i},x_{j}\right)\left\|T_{f}^{ \ast}\left(L_{y}\right)\right\|^{2}_{\mathscr{H}_{K}}.\]
**Lemma 2.10**.: _The implication below is both directions:_

\[X\ni x\longmapsto L\left(f\left(x\right),y\right)\in\mathscr{H}_{K} \text{ for }\forall y\] \[\Updownarrow\] _the condition in (_2.6_) is satisfied._

_Even if we fix \(y\in Y\), then_

\[L_{y}\in dom(T_{f}^{\ast})\Longleftrightarrow\left(x\longmapsto L\left(f \left(x\right),y\right)\right)\in\mathscr{H}_{K}. \tag{2.51}\]

Proof sketch.: By definition, \(L_{y}\in dom(T_{f}^{\ast})\), \(\exists C_{y}<\infty\Longleftrightarrow\)

\[\left|\left\langle T_{f}\varphi,L_{y}\right\rangle_{\mathscr{H}_{L}}\right|= \left|\left(T_{f}\left(\varphi\right)\right)\left(y\right)\right|\leq C_{y} \left\|\varphi\right\|_{\mathscr{H}_{K}}\]

holds for \(\forall\varphi\in span\left\{K_{x}\right\}_{x\in X}.\) But

\[T_{f}\left(K_{x}\right)\left(y\right)=L\left(f\left(x\right),y\right),\text{ and } \tag{2.52}\]

\[\left|\sum_{i}c_{i}\underbrace{L\left(f\left(x_{i}\right),y\right)}_{F_{f,y} \left(x_{i}\right)}\right|^{2} =\left|\left\langle\sum_{i}c_{i}K_{x_{i}},T_{f}^{\ast}\left(L_{y} \right)\right\rangle_{\mathscr{H}_{K}}\right|^{2} \tag{2.53}\]

\[\leq\left\|T_{f}^{\ast}\left(L_{y}\right)\right\|_{\mathscr{H}_{K}}^{2} \underbrace{\sum_{i}\sum_{j}c_{i}c_{j}K\left(x_{i},x_{j}\right)}_{<\infty} \tag{2.54}\]

and so by the basic lemma for \(\mathscr{H}_{K}\) (see the proof of Theorem 2.6), we conclude that functions \(F_{f,y}\in\mathscr{H}_{K}\), i.e., \(\left(x\longmapsto L\left(f\left(x\right),y\right)\right)\in\mathscr{H}_{K}\).

**Conclusion:** the bi-implication \(\Longleftrightarrow\) in (2.51) is valid. 

### Functions and Operators

In general if \(T:\mathscr{H}_{1}\rightarrow\mathscr{H}_{2}\) is an operator with dense domain \(\mathscr{D}\subset\mathscr{H}_{1}\), where \(\mathscr{H}_{i}\), \(i=1,2\), are two Hilbert spaces, we know that \(T\) is closable \(\Longleftrightarrow T^{\ast}\) is densely defined, i.e., iff \(dom(T^{\ast})\) is dense in \(\mathscr{H}_{2}\) (see e.g., [11]). So we apply this to \(T=T_{f}\), \(\mathscr{H}_{1}=\mathscr{H}_{K}\), \(\mathscr{H}_{2}=\mathscr{H}_{L}\), and the condition in Theorem 2.6 holds \(\Longleftrightarrow L_{y}\in dom(T_{f}^{\ast})\)\(\forall y\in Y\). Since \(span\left\{L_{y}\right\}_{y\in Y}\) is dense in \(\mathscr{H}_{L}\), the condition in Theorem 2.6\(\Longrightarrow T_{f}\) is closable.

Given \(K\) and \(L\) as above, introduce

\[\mathscr{F}_{ub}\left(K,L\right) =\left\{f:T_{f}\text{ is closable}\right\},\text{ and } \tag{2.55}\] \[\mathscr{F}_{b}\left(K,L\right) =\left\{f:T_{f}\text{ is bounded from }\mathscr{H}_{K}\text{ into } \mathscr{H}_{L}\right\}. \tag{2.56}\]

In both cases, the operators \(T=T_{f}\) depends on the choice of function \(X\xrightarrow{f}Y\), but the two conditions (2.55) and (2.56) are different:

\[\left(T_{f}\left(K_{x}\right)\right)\left(y\right)=L\left(f\left(x\right),y \right)=\left(\left(T_{f}\right)^{\ast}\left(L_{y}\right)\right)\left(x\right), \tag{2.57}\]

for all \(x\in X\), and \(y\in Y\). See details below:

Some general comments about the operator \(T_{f}:\mathscr{H}_{K}\rightarrow\mathscr{H}_{L}\). As before, \(K\) and \(L\) are fixed p.d. kernels, and \(f:X\to Y\) is a function. We need to understand the conclusion from Theorem 2.6, i..e, when is

\[\left(X\ni x\longmapsto L\left(f\left(x\right),y\right)\right)\in\mathscr{H}_{ K}\text{ for all }y\in Y? \tag{2.58}\]

Answer: (2.58) \(\Longleftrightarrow\)\(L_{y}\in dom(T_{f}^{\ast})\).
Note that then the function in (2.58) is \(T_{f}^{*}\left(L_{y}\right)\); see (2.57). But note that, starting with a function \(X\xrightarrow{f}Y\), there are requirements for having (2.57) yield a well defined linear operator \(T_{f}\) with dense domain in \(\mathscr{H}_{K}\), s.t.

\[T_{f}\left(K_{x}\right)\left(\cdot\right)=L\left(f\left(x\right),\cdot\right). \tag{2.59}\]

The case when \(T_{f}\) is bounded is easy since then \(dom(T_{f}^{*})=\mathscr{H}_{L}\). Notationally, \(L\left(f\left(x\right),\cdot\right)\in L_{f\left(x\right)}\in\mathscr{H}_{L}\), but we must also verify the implicit kernel function for all finite sums:

\[\sum_{i}\sum_{j}c_{i}c_{j}K\left(x_{i},x_{j}\right)=0\Longrightarrow\sum_{i} \sum_{j}c_{i}c_{j}L\left(f\left(x_{i}\right),f\left(x_{j}\right)\right)=0.\]

### The case when \(K=l\)

As demonstrated in Section 3 below, for applications to multi-level NNs, the recursive constructions simplify when the same p.d. kernel \(K\) is used at each level. Hence below, we specialize to the case when \(X=Y\), and \(K=L\); see the setting in Theorems 2.6 and 2.8.

**Theorem 2.11**.: _Consider a positive definite kernel \(K\) on \(X\times X\), and the corresponding RKHS \(\mathscr{H}_{K}\), i.e., the Hilbert completion of \(\left\{K_{x}\right\}_{x\in X}\) where \(K_{x}:=K\left(\cdot,x\right)\). Fix a function \(X\xrightarrow{f}X\) with the property (see Theorem 2.6) that_

\[\left(X\ni x\longmapsto K\left(f\left(x\right),y\right)\right)\in\mathscr{H}_ {K}\,\text{for all }y\in X\,. \tag{2.60}\]

_Hence, the operator \(T_{f}:\mathscr{H}_{K}\to\mathscr{H}_{K}\) defined by_

\[T_{f}\left(K\left(\cdot,y\right)\right):=K\left(f\left(\cdot\right),y\right) \tag{2.61}\]

_is a densely defined operator from \(\mathscr{H}_{K}\) into \(\mathscr{H}_{K}\), with domain_

\[\mathscr{D}_{K}:=span\left\{K_{x}\right\}_{x\in X}. \tag{2.62}\]

1. _Then the closure of_ \(T_{f}\) _(also denoted_ \(T_{f}\)_) is well defined and normal, i.e., the two operators_ \(T_{f}\) _and_ \(T_{f}^{*}\) _commute._
2. _In particular,_ \(T_{f}\) _has a projection-valued spectral resolution, i.e., there is a projection-valued measure_ \(Q\left(\cdot\right)\) _on_ \(\mathscr{B}_{\mathbb{C}}\left(=\text{the Borel subsets in }\mathbb{C}\right)\) _such that_ \[T_{f}=\int_{spect\left(T_{f}\right)}\lambda\,Q\left(\lambda\right):\mathscr{D} _{K}\to\mathscr{H}_{K}.\] (2.63)

Proof.: Note that part (2) follows from (1) and the Spectral Theorem for normal operators (in the Hilbert space \(\mathscr{H}_{K}\).)

Part (1). When the operator \(T_{f}^{*}\) is introduced, we get the following commutativity:

which is the desired conclusion (1).

Figure 2.2. Commutativity of \(T_{f}\) and \(T_{f}^{*}\).
Given a function \(f:X\to X\) as in Theorem 2.11. Below we make use of the corresponding projection valued measure \(Q^{\left(f\right)}\) from Theorem 2.11 in order to establish an assignment from pairs \(\left(x,y\right)\) of points in \(X\), into systems of complex measures \(\mu_{\left(x,y\right)}\) on the spectrum of \(T_{f}\). In this assignment, \(n\)-fold composition-iteration of the function \(f\) yields the \(n\)th moment of each of the measures \(\mu_{\left(x,y\right)}\).

**Corollary 2.12**.: _Let \(K,X\) and \(f\) be as specified in Theorem 2.11, and let \(Q=Q^{\left(K,f\right)}\left(\cdot\right)\) be the corresponding projection valued measure in (2.63). Then for every pair \(x,y\in X\), we get a corresponding Borel measure_

\[\mu_{x,y}^{\left(f\right)}\left(B\right)=\left\langle K_{x},Q\left(B\right)K _{y}\right\rangle_{\mathscr{H}_{K}}=\left(Q\left(B\right)\left(K_{y}\right) \right)\left(x\right), \tag{2.64}\]

_for all \(B\in\mathscr{B}_{\mathbb{C}}\). Inductively, setting_

\[f^{\circ n}=\underbrace{f\circ\cdots\circ f}_{n\text{ fold}}\]

_we arrive at the following moment formula for the respective complex measures:_

\[\mu_{f^{\circ n}\left(x\right),y}^{\left(f\right)}\left(B\right)=\int_{B} \lambda^{n}\,\mu_{x,y}^{\left(f\right)}\left(d\lambda\right). \tag{2.65}\]

We now turn to the role of _multipliers_ in the RKHS \(\mathscr{H}_{K}\).

**Definition 2.13**.: A scalar valued function \(\varphi\) on \(X\) is said to be a _multiplier_ for \(\mathscr{H}_{K}\) iff one of the two equivalent conditions hold:

1. The multiplication operator \(M_{\varphi}\) acting on \(\mathscr{H}_{K}\) via \(M_{\varphi}F=\varphi F\) (via pointwise product) leaves \(\mathscr{H}_{K}\) invariant.
2. We have the following identity for the adjoint operator: \[M_{\varphi}^{\ast}\left(K_{x}\right)=\varphi\left(x\right)K_{x}\text{ for all }x\in X\] (2.66) where \(K_{x}\) denotes the kernel function \(K_{x}=K\left(\cdot,x\right)\).

_Remark 2.14_.: The equivalence of (1) and (2) follows from the standard reference on RKHSs; see e.g., [17].

**Theorem 2.15**.: _Let \(K\) be a fixed p.d. kernel on \(X\times X\), and let \(\mathscr{H}_{K}\) be the corresponding RKHS. Let \(X\xrightarrow{f}X\) be a function such that (2.60) holds, i.e., \(\left(X\ni x\longmapsto K\left(f\left(x\right),y\right)\right)\in\mathscr{H}_ {K}\) for all \(y\in X\)._

_Then for every multiplier \(\varphi\) for \(\mathscr{H}_{K}\), we have:_

\[M_{\left(\varphi\circ f\right)}T_{f}^{\ast}=T_{f}^{\ast}M_{\varphi}. \tag{2.67}\]

Proof.: It is clear that the conclusion (2.67) has the following equivalent form:

\[T_{f}M_{\left(\varphi\circ f\right)}^{\ast}=M_{\varphi}^{\ast}T_{f}; \tag{2.68}\]

and below we shall prove (2.68).

Let \(f\) and \(\varphi\) be as specified in the theorem. We then have the following commutative diagram:
In the verification of the assertions in Figure 2.3, we used the conclusions in Theorems 2.6 and 2.8 above. 

## 3. Neural Network-activation functions from p.d. kernels

In the previous section we introduced the use of positive definite kernels, and associated generating function for the NN algorithms. Below we make use of the kernel analysis in design of the generating NN functions.

The next definition makes use of the iterative generation of feedforward functions as in the literature, e.g., [1, 10, 11]. The recursive steps used here in the definition and Lemma 3.2 below serve as applications of our general framework from Theorems 2.6 and 2.8 above.

**Definition 3.1**.: Let \(K\) be a positive definite kernel on \(\mathbb{R}\). An \(l\)-layer feedforward network with kernel \(K\) is a function of the form

\[x\mapsto y_{1}=K\left(A_{1}x+b_{1},c_{1}\right)\mapsto y_{2}=K \left(A_{2}y_{1}+b_{2},c_{2}\right)\mapsto\cdots\\ \cdots\mapsto y_{l}=K\left(A_{l}y_{l-1}+b_{l},c_{l}\right)\mapsto y _{out}=K\left(\left\langle a_{l+1},y_{l}\right\rangle+b_{l+1},c_{l+1}\right)\]

where

* \(x\in\mathbb{R}^{n_{0}}\);
* \(A_{j}\in\mathbb{R}^{n_{j}\times n_{j-1}}\), \(b_{j},c_{j}\in\mathbb{R}^{n_{j}}\) for \(j=1,\cdots,l\);
* \(a_{l+1}\in\mathbb{R}^{n_{l}}\), \(b_{l+1},c_{l+1}\in\mathbb{R}\);

and for vectors \(x,y\in\mathbb{R}^{m}\),

\[K\left(x,y\right):=\left[K\left(x_{1},y_{1}\right),\cdots,K\left(x_{m},y_{m} \right)\right].\]

**Lemma 3.2**.: _Let \(K\left(x,y\right)=\min\left(x,y\right)\), and \(a,b,c,d\) be nonzero constants. Then_

1. \(K\left(ax+b,c\right)=aK\left(x,a^{-1}\left(c-b\right)\right)+b\)_;_
2. \(K\left(K\left(x,a\right),b\right)=K\left(x,K\left(a,b\right)\right)\)_;_
3. \(K\left(dK\left(ax+b,c\right)+e,f\right)=daK\left(x,K\left(a^{-1}\left(c-b \right),a^{-1}\left(d^{-1}\left(f-e\right)-b\right)\right)\right)+db+e\)_._

Proof.:
1. \(K\left(ax+b,c\right)=\begin{cases}ax+b&x<a^{-1}\left(c-b\right)\\ c&x>a^{-1}\left(c-b\right)\end{cases}\)
2. Assume \(a<b\), then \[K\left(K\left(x,a\right),b\right)=\begin{cases}x&x<a\\ a&x\geq a\end{cases}=K\left(x,a\right).\] The case \(a>b\) is similar.

Figure 2.3. Commutative diagram corresponding to (2.68).
3. This follows from (1)-(2): \[K\left(dK\left(ax+b,c\right)+e,f\right)\] \[= dK\left(K\left(ax+b,c\right),d^{-1}\left(f-e\right)\right)+e\] \[= dK\left(aK\left(x,a^{-1}\left(c-b\right)\right)+b,d^{-1}\left(f-e \right)\right)+e\] \[= d\left\{aK\left(K\left(x,a^{-1}\left(c-b\right)\right),a^{-1} \left(d^{-1}\left(f-e\right)-b\right)\right)+b\right\}+e\] \[= daK\left(K\left(x,a^{-1}\left(c-b\right),a^{-1}\left(d^{-1}\left( f-e\right)-b\right)\right)+db+e\] \[= daK\left(x,K\left(a^{-1}\left(c-b\right),a^{-1}\left(d^{-1}\left( f-e\right)-b\right)\right)\right)+db+e\]

In what follows, all the networks are restricted to be defined on compact subsets \(\Omega\) in \(\mathbb{R}^{d}\), e.g., \(\Omega=\left[0,1\right]^{d}\) (hypercubes). This is in consideration of standard normalizations in training neural networks.

In Theorems 3.3 and 3.4, we present in detail the particular _relative Reproducing Kernel Hilbert Spaces_ which have as their respective dipole system (see (3.5)) the generalized ReLu functions illustrated here in Figures 2.1 and 3.1.

Here we specify the kernel \(K_{1}\) for Brownian motion \(W\) indexed by \(\mathbb{R}\). As a result, the corresponding p.d. kernel on \(\mathbb{R}\times\mathbb{R}\) is as follows:

\[K_{1}\left(x,y\right)=\begin{cases}\left|x\right|\wedge\left|y\right|=\min \left(\left|x\right|,\left|y\right|\right)&\text{if $xy\geq 0$ (so same sign)}\\ 0&\text{if $xy<0$, so opposite sign.}\end{cases} \tag{3.1}\]

Proof.: The connection between the kernel \(K_{1}\) and the Brownian motion \(\left\{W_{x}\right\}_{x\in\mathbb{R}}\) is as follows:

\[K_{1}\left(x,y\right)=\mathbb{E}\left(\left(W_{x}-W_{0}\right)\left(W_{y}-W_{ 0}\right)\right) \tag{3.2}\]

for all \(x,y\in\mathbb{R}\). The asserted formula (3.1) follows from this, combined with the independence of increments for Brownian motion. 

**Theorem 3.3**.: _Let \(K_{1}\) be the p.d. kernel (3.1) on \(\mathbb{R}\times\mathbb{R}\), with the corresponding RKHS_

\[\mathscr{H}_{K_{1}}=\left\{f:f^{\prime}\in L^{2}\right\},\quad\left\|f\right\| _{\mathscr{H}_{K_{1}}}^{2}=\int\left|f^{\prime}\right|^{2}d\lambda_{1}.\]

_On \(\Omega=\left[0,1\right]^{d}\), consider the p.d. kernel_

\[K_{d}\left(x,y\right)=K_{1}\left(x_{1},y_{1}\right)\cdots K_{1}\left(x_{d},y_ {d}\right),\]

_so that_

\[\mathscr{H}_{K_{d}}=\left\{f:\nabla f\in L^{2}\right\},\quad\left\|f\right\| _{\mathscr{H}_{K_{d}}}^{2}=\int\left|\nabla f\right|^{2}d\lambda_{d},\]

_where \(\lambda_{d}\) denotes the \(d\)-dimensional Lebesgue measure._

_Given \(f:\Omega\to\mathbb{R}\), and a fixed \(c\in\mathbb{R}\), set_

\[F:\mathbb{R}^{d}\to\mathbb{R}^{1},\quad F\left(x\right)=K_{1}\left(f\left(x \right),c\right).\]

_Then,_

\[F\in\mathscr{H}_{K_{d}}\Longleftrightarrow\iint_{f^{-1}\left(\left[0,c\right] \right)}\left|\nabla f\right|^{2}d\lambda_{d}<\infty.\]
**Theorem 3.4**.: _Let \(\mu\) be a non-atomic \(\sigma\)-finite measure on \(\left(\mathbb{R},\mathscr{B}_{\mathbb{R}}\right)\), and consider Stieltjes measures \(dF\) on \(\left(\mathbb{R},\mathscr{B}_{\mathbb{R}}\right)\) such that_

\[dF\ll\mu \tag{3.3}\]

(absolutely continuous). _Then the relative RKHS \(\mathscr{H}_{\mu}\) for the p.d. kernel_

\[K_{\mu}\left(A,B\right)=\mu\left(A\cap B\right) \tag{3.4}\]

_consists functions \(F\) such that_

\[F\left(b\right)-F\left(a\right)=\left\langle v_{a,b}^{\left(\mu\right)},F \right\rangle_{\mathscr{H}_{\mu}} \tag{3.5}\]

\[\int_{\mathbb{R}}\left|\frac{dF}{d\mu}\right|^{2}d\mu<\infty \tag{3.6}\]

_where the relative kernels \(v_{a,b}^{\left(\mu\right)}\) are as follows:_

\[\frac{dv_{a,b}^{\left(\mu\right)}}{d\mu}\left(x\right)=\chi_{\left[a,b\right] \left(x\right)},\]

_see Figure 3.1._

Proof.: See [1, 2] and the details in the proof of Theorem 3.3. 

_Remark 3.5_.: The positive definite kernel \(K_{\mu}\) which is "responsible" for the relative RKHS \(\mathscr{H}_{\mu}\) is defined on \(\mathscr{B}\times\mathscr{B}\), where \(\mathscr{B}\) denotes the Borel \(\sigma\)-algebra of subsets of \(\mathbb{R}\). Using [2], one checks that

\[K_{\mu}\left(A,B\right):=\mu\left(A\cap B\right)\text{ for all }A,B\in\mathscr{B}. \tag{3.7}\]

We further note that \(K_{\mu}\) is the covariance for the generalized \(\mu\)-Brownian motion \(\{W_{A}^{\left(\mu\right)}\}_{A\in\mathscr{B}}\), i.e., subject to

\[\mathbb{E}\left(W_{A}^{\left(\mu\right)}W_{B}^{\left(\mu\right)}\right)=\mu \left(A\cap B\right)\text{ for all }A,B\in\mathscr{B}. \tag{3.8}\]

The corresponding Ito-lemma for \(W^{\left(\mu\right)}\) is defined for differentiable functions \(f\) on \(\mathbb{R}\) via

\[f\left(W_{A}^{\left(\mu\right)}\right)-f\left(0\right)=\int_{A}f^{\prime} \left(W_{t}^{\left(\mu\right)}\right)dW_{t}^{\left(\mu\right)}+\frac{1}{2} \int_{A}f^{\prime\prime}\left(W_{t}^{\left(\mu\right)}\right)\mu\left(dt \right). \tag{3.9}\]

In particular, the measure \(\mu\) is the quadratic variation of \(W_{t}^{\left(\mu\right)}\).

Figure 3.1. Illustration dipole functions that reproduce differences of values of functions in the space \(\mathscr{H}_{\mu}\). Compare with Figure 2.1 above.
## 4. Applications to fractal images

In recent decades, it has become evident that fractal features arise in diverse datasets, in time series and in image analysis, to mention two. (See e.g., [11, 12].) Perhaps the best known examples of fractal features include precise symmetries of scales. Via a prescribed system of affine maps, they take the form of self-similarity. And a special case, includes iterated function systems (IFS), and maximal-entropy measures, also called IFS measures. The more familiar Cantor constructions, e.g., scaling by \(3\) or scaling by \(4\), are examples of IFS measures. For each of these cases, the RKHS framework we present in Section 3, serve as ideal tools for such adapted NN algorithms. In particular, this may be illustrated with large numbers of images, say \(5000\) generated images, each one is a fractal, either \(2\)D or \(3\)D, with random rotation, with zooming, and coloring; half of them have scaling \(3\), the other half have scaling \(4\). This leads to training of a network serving to classify the images by scaling factors.

In particular, the Cantor-type activation functions, or the cumulative functions of Cantor-like measures (Figure 3.1), have vanishing derivatives over structured subintervals of \([0,1]\). This feature may lead to several benefits in neural networks. For example, such functions can introduce sparsity and regularization into the network, which improves its generalization performance and reduces the risk of overfitting. Additionally, these functions can make the network more robust to noise and other perturbations in the input data, which improves its performance on unseen data. Furthermore, activation functions whose derivative is zero over subintervals allow the network to learn more complex and non-linear patterns in the data. This can improve the expressiveness and flexibility of the network, making it more accurate and effective for a wider range of tasks. Additionally, these functions can make the network easier to optimize and train, since the gradient of the activation function is well-structured, thus reduce the computational complexity and improve the convergence rate of the training algorithm.

More generally, a neural network with a custom activation function (see e.g. the dipoles in Figure 2.1) uses a non-standard activation function with adjustable parameters that can be trained and optimized during the learning process. This allows the network to learn more complex and non-linear relationships between the input and output data, which can improve the accuracy of the network's predictions.

The use of a custom activation function with trainable parameters can be useful in a variety of applications, such as image recognition, natural language processing, and time series forecasting. It can also be used to improve the performance of other machine learning algorithms, such as decision trees and support vector machines (see e.g., [1, 15, 16, 17, 18, 19, 20]).

Below we apply a custom activation function in a ConvNet to classify fractal images. In this setting, the activation function should be designed to capture the complex, self-similar patterns that are characteristic of the fractal images. The network is trained on a dataset of fractal images with corresponding labels. It is optimized using a gradient-based algorithm, such as stochastic gradient descent. Once trained, the network will be used to classify new fractal images and predict their classes with high accuracy.
In the example below, a dataset1 of 15,000 Cantor-like 3D images is generated in Mathematica. Parameters of each image, such as zoom factor, viewing angle, and scaling factor, are uniformly distributed. A sample of the images is shown in Figure 4.1.

Footnote 1: Available at [https://www.kaggle.com/dsv/4791103](https://www.kaggle.com/dsv/4791103).

The images are split into three categories according to their scaling factors, labeled as class "1", "2" and "3", respectively. The entire dataset is divided into a training set (size = 10,000), validation set (size = 2,500) and test set (size = 2,500). The task is to train a ternary classifier using the training set, along with the validation set (for model selection), whose performance is then tested on the test images.

In the experiment, a small ConvNet is implemented in Keras. Its architecture is shown in Figure 4.2. The loss and accuracy of the model are recorded for 20 epochs (Figure 4.3).

In comparison with a standard Relu network (Figure 4.3a) of the same architecture, the use of Cantor-like activation is better at reducing overfitting (Figure 4.3b); it is expected that with a systematic hyperparameter tuning, such a network has the potential to outperform Relu networks in certain applications.

Figure 4.1. A random sample of the dataset of 3D Cantor images.
Figure 4.2. A ConvNet for fractal image classification.

Figure 4.3. Training loss and validation loss, illustrated with use of a ConvNet.