Proof of the Universal Approximation Theorem for EML Trees

Using the recently trending EML (Elementary Function Composition) function, we proved a theorem showing that EML-based tree structures can serve as a Universal Approximator.

The main points are as follows:

• Approximation Principle: This leverages the fact that composing EML functions can represent polynomials, and polynomials can densely represent continuous functions or functions within a specific Sobolev space.
• Construction Method: The approach takes a 'Lego block' style, explicitly constructing binary operations, polynomials, hyperbolic tangent, and approximations of partition of unity with EML to build complex functions.
• Technical Resolution: To solve the problem that the natural logarithm ($\text{ln}$) is undefined for negative numbers, sign-based decomposition and an appropriate affine map were introduced, enhancing the theoretical completeness.
• Extensibility: For theoretical and practical reasons, this paper generalizes and treats the topic in the form of EML(-type), which adds learnable parameters.