Operator-Valued Kernel
An operator-valued kernel is a kernel function that returns an operator (matrix) in a vector-valued RKHS. It encodes correlations among the components of output vectors, and specific block structures can be designed so that functions in the hypothesis space automatically satisfy structural constraints, such as sum consistency in hierarchical time series forecasting.
Timeline (last 90 days)
The paper uniformly models hierarchical forecasting as a multivariate regression problem in a vector-valued reproducing kernel Hilbert space, embedding the consistency constraint into the hypothesis space through a structured operator-valued kernel.
Functions in a vector-valued RKHS take values in R^d, and their kernel function K(x,x') returns a d×d matrix (operator), encoding not only sample similarity but also the correlations among output components.
An operator-valued kernel must satisfy operator-valued positive definiteness in order to guarantee that the corresponding RKHS exists.
By choosing an operator-valued kernel K with a specific block structure, the output vector of any function in the hypothesis space can be made to lie in the column space of the aggregation matrix, thereby achieving 'consistency by construction'.
The idea of operator-valued kernels is in the same lineage as the operator-valued kernels used in multi-task learning and structured output regression, but it is systematically used by this paper for consistency guarantees in hierarchical forecasting for the first time.
All Facts (5)
The paper uniformly models hierarchical forecasting as a multivariate regression problem in a vector-valued reproducing kernel Hilbert space, embedding the consistency constraint into the hypothesis space through a structured operator-valued kernel.
50%UnverifiedFunctions in a vector-valued RKHS take values in R^d, and their kernel function K(x,x') returns a d×d matrix (operator), encoding not only sample similarity but also the correlations among output components.
50%UnverifiedAn operator-valued kernel must satisfy operator-valued positive definiteness in order to guarantee that the corresponding RKHS exists.
50%UnverifiedBy choosing an operator-valued kernel K with a specific block structure, the output vector of any function in the hypothesis space can be made to lie in the column space of the aggregation matrix, thereby achieving 'consistency by construction'.
50%UnverifiedThe idea of operator-valued kernels is in the same lineage as the operator-valued kernels used in multi-task learning and structured output regression, but it is systematically used by this paper for consistency guarantees in hierarchical forecasting for the first time.
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