  In this paper, we will introduce a novel deep model named Reconciled
Polynomial Network (RPN) for deep function learning. RPN has a very general
architecture and can be used to build models with various complexities,
capacities, and levels of completeness, which all contribute to the correctness
of these models. As indicated in the subtitle, RPN can also serve as the
backbone to unify different base models into one canonical representation. This
includes non-deep models, like probabilistic graphical models (PGMs) - such as
Bayesian network and Markov network - and kernel support vector machines
(kernel SVMs), as well as deep models like the classic multi-layer perceptron
(MLP) and the recent Kolmogorov-Arnold network (KAN).
  Technically, RPN proposes to disentangle the underlying function to be
inferred into the inner product of a data expansion function and a parameter
reconciliation function. Together with the remainder function, RPN accurately
approximates the underlying functions that governs data distributions. The data
expansion functions in RPN project data vectors from the input space to a
high-dimensional intermediate space, specified by the expansion functions in
definition. Meanwhile, RPN also introduces the parameter reconciliation
functions to fabricate a small number of parameters into a higher-order
parameter matrix to address the ``curse of dimensionality'' problem caused by
the data expansions. Moreover, the remainder functions provide RPN with
additional complementary information to reduce potential approximation errors.
We conducted extensive empirical experiments on numerous benchmark datasets
across multiple modalities, including continuous function datasets, discrete
vision and language datasets, and classic tabular datasets, to investigate the
effectiveness of RPN.
