TY  - JOUR
ID  - Jorgensen2019
LB  - Jorgensen2019
AU  - Jorgensen, Palle
AU  - Tian, Feng
TI  - Decomposition of Gaussian processes, and factorization of positive definite kernels
ST  - Decomposition of Gaussian processes, and factorization of positive definite kernels
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2019
VL  - 39
IS  - 4
SP  - 497
EP  - 541
DO  - https://doi.org/10.7494/OpMath.2019.39.4.497
UR  - https://doi.org/10.7494/OpMath.2019.39.4.497
AB  - We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels \(K\), and the other for Gaussian processes, say \(V\). The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel \(K\), presented as a covariance kernel for a Gaussian process \(V\). We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel \(K\), vs for Gaussian process \(V\). Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for \(K\) is the exact same as that which yield factorizations for \(V\). Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
KW  - reproducing kernel Hilbert space
KW  - frames
KW  - generalized Ito-integration
KW  - the measurable category
KW  - analysis/synthesis
KW  - interpolation
KW  - Gaussian free fields
KW  - non-uniform sampling
KW  - optimization
KW  - transform
KW  - covariance
KW  - feature space
SN  - 1232-9274
ER  -