I'm looking for any theoretical work/paper/development where the notion of Reproducing kernel Hilbert space (RKHS), defined via positive definite symmetric kernel $k(x,y) = k(y,x)$, is extended towards non-symmetrical kernels.
Basically, I'm looking for a similar notion to kernel learning/regression, which is where RKHS is mainly used, but that is based on non-symmetric similarity function $k(x,y) \neq k(y,x)$. Does such mathematical structure exist? Can anyone share a link for any relevant material?
Thanks
RKHS have been generalized to Reproducing Kernel Banach spaces (RKBS). In RKBS the inner product is replaced by a semi-inner product which is not necessarly symmetric anymore or more recently by Bilinearform on a pair of Banach spaces:
Definition Reproducing Kernels for RKBS (Lin et al, 2019, On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions, p. 3):
Let $B_1$ be an RKBS on a set $Ω_1$. If there exists a Banach space $B_2$ of functions on another set $Ω_2$, a continuous bilinear form $\langle ·, ·\rangle_{B1×B2}$, and a function K on $Ω_1 \times Ω_2$ such that K(x, ·) ∈ $B_2$ for all x ∈ $Ω_1$ and $f(x) = \langle f, K(x, ·)\rangle_{B_1×B_2}$ for all x ∈ $Ω_1$ and all f ∈ $B_1$, then we call K a reproducing kernel for $B_1$.
If, in addition, $B_2$ is also an RKBS on $Ω_2$ and it holds K(·, y) ∈ B1 for all y ∈ $Ω_2$ and g(y) = $\langle K(·,y), g\rangle_{B_1×B_2}$ for all y ∈ $Ω_2$ and all g ∈ $B_2$, then we call $B_2$ an adjoint RKBS of $B_1$ and call $B_1$ and $B_2$ a pair of RKBSs.
In this case, $\tilde{K}(x, y) := K(y, x)$, x ∈ $Ω_2$, y ∈ $Ω_1$, is a reproducing kernel for $B_2$.
The construction with a semi inner product has been applied to machine learning (see Zhang et al, Reproducing Kernel Banach Spaces for Machine Learning). There exist also constructions of this kind where the representer theorem can be applied: G. Song, H. Zhang, and F. J. Hickernell, Reproducing kernel Banach spaces with the ℓ1 norm.