An Explicit Neural Network Construction for Piecewise Constant Function Approximation
This provides a more practical explicit neural network construction for piecewise constant function approximation, which is incremental as it builds on existing explicit FNN work but avoids tensor structures.
The paper tackles the problem of approximating multivariate functions with piecewise constant approximations by presenting an explicit construction for a feedforward neural network with two hidden layers, where weights and thresholds are defined without numerical optimization, and it automatically creates Voronoi tessellations based on data, demonstrating properties through theoretical analysis and numerical examples.
We present an explicit construction for feedforward neural network (FNN), which provides a piecewise constant approximation for multivariate functions. The proposed FNN has two hidden layers, where the weights and thresholds are explicitly defined and do not require numerical optimization for training. Unlike most of the existing work on explicit FNN construction, the proposed FNN does not rely on tensor structure in multiple dimensions. Instead, it automatically creates Voronoi tessellation of the domain, based on the given data of the target function, and piecewise constant approximation of the function. This makes the construction more practical for applications. We present both theoretical analysis and numerical examples to demonstrate its properties.