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arxiv: 2606.23179 · v1 · pith:FFZGDYWYnew · submitted 2026-06-22 · 💻 cs.LG · cs.NA· cs.NE· cs.SC· math.NA

EML Trees Are Universal Approximators

Joe Germany , Elie Abdo , Joseph Bakarji This is my paper

Pith reviewed 2026-06-26 08:45 UTC · model grok-4.3

classification 💻 cs.LG cs.NAcs.NEcs.SCmath.NA
keywords EML treesuniversal approximationSobolev spacesfunction approximationtree structureslearning algorithmsexpressive powermachine learning
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The pith

Trees composed of EML functions can approximate any function in W^{k,∞} to arbitrary accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that tree-structured compositions of the EML function possess a universal approximation property for functions whose weak derivatives up to order k remain bounded. It adapts classical neural-network approximation theorems by constructing explicit EML trees that replicate the behavior of polynomials. This supplies both a theoretical guarantee and a concrete learning algorithm for fitting parameters within the trees. The result matters because it identifies a new compositional building block, built from a continuous NAND analogue, that inherits strong approximation guarantees while remaining trainable on practical tasks.

Core claim

We show that such trees enjoy a universal approximation property for functions in W^{k, ∞} for k ∈ ℕ, drawing on classical neural network approximation arguments while exploiting the ability to explicitly construct EML trees that mimic polynomial representations. We further propose a learning algorithm for EML-type trees equipped with fitting parameters, and demonstrate its feasibility in practical optimization problems.

What carries the argument

Tree-structured compositions of the EML (exp-minus-log) function, which serves as a continuous analogue of NAND gates and enables explicit mimicry of polynomial representations.

If this is right

  • EML trees inherit universal approximation from neural-network theory once the polynomial-mimicry construction is available.
  • A parameter-fitting algorithm exists that can optimize EML trees on concrete optimization problems.
  • The framework supplies theoretical grounding for using EML trees as function approximators in machine-learning settings.
  • Approximation holds for every natural number k, covering functions with increasing smoothness requirements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the trees can mimic polynomials, they may achieve exact representation for polynomial targets under suitable parameter choices.
  • The same construction technique could be tested on other continuous gate analogues to check whether universal approximation transfers similarly.
  • Practical scaling behavior of the learning algorithm on high-dimensional inputs remains open for direct measurement.

Load-bearing premise

It is possible to explicitly construct EML trees that mimic polynomial representations for any target function in W^{k,∞}.

What would settle it

A concrete function in W^{k,∞} on a compact domain for which no finite-depth EML tree sequence converges in the Sobolev norm would refute the universal approximation claim.

Figures

Figures reproduced from arXiv: 2606.23179 by Elie Abdo, Joe Germany, Joseph Bakarji.

Figure 1
Figure 1. Figure 1: The Basic Binary Operations (a) Exponentiation to a constant n (2) (b) Exponentiation to a constant n with prefactor m (2) [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Exponentiating 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hyperbolic Tangent (8) E Constructing Polynomials The following two subsections detail how to construct univariate and multivariate polynomials using EML trees and provide upper bounds for the number of required EML blocks in each case. E.1 Univariate Polynomials A univariate polynomial of degree k, denoted by p(x) = k ∑ i=0 aix i = a0 + a1x + a2x 2 + ⋯ + akx k , (32) with x > 0 and with positive coefficie… view at source ↗
Figure 4
Figure 4. Figure 4: Univariate Polynomial (5s − 3) Lemma 4 (Construction of Univariate Polynomials with Positive Coefficients). Given a univariate polynomial p of degree k defined over x > 0 and with positive coefficients, there exists an EML tree pθ (following the explicit construction in [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Construction of ρ N m for 2 ≤ m ≤ N − 1 and for ρ N N , we rewrite it as ρ N N (y) = 1 2 σ (α(y − N − 1 N )) + 1 2 = 1 2 ⎛ ⎝ 1 + e α(y− N−1 N ) e α(y− N−1 N ) + e −α(y− N−1 N ) ⎞ ⎠ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ positive − 1 2 e −α(y− N−1 N ) e α(y− N−1 N ) + e −α(y− N−1 N ) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸ ¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹… view at source ↗
read the original abstract

The recently introduced EML (Exp-Minus-Log) function acts as continuous analogue of NAND gates, providing a compositional building block capable of representing elementary functions. In this work, we study the expressive power of tree-structured compositions of EML functions. We show that such trees enjoy a universal approximation property for functions in $W^{k, \infty}$ for $k \in \mathbb N$, drawing on classical neural network approximation arguments while exploiting the ability to explicitly construct EML trees that mimic polynomial representations. We further propose a learning algorithm for EML-type trees equipped with fitting parameters, and demonstrate its feasibility in practical optimization problems. Our results establish EML trees as a theoretically grounded framework for function approximation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that tree-structured compositions of the EML (Exp-Minus-Log) function, serving as a continuous analogue of NAND gates, possess a universal approximation property for functions in the Sobolev space W^{k,∞} (k ∈ ℕ). The argument draws on classical neural-network approximation results while relying on an explicit construction of EML trees that mimic polynomial representations; a learning algorithm for EML-type trees equipped with fitting parameters is also proposed and its feasibility shown on practical optimization tasks.

Significance. If the explicit construction is rigorously established with error bounds, the result would supply a new theoretically grounded class of universal approximators grounded in a continuous logic-gate primitive, extending existing NN theory to tree-structured EML compositions and offering a framework that could be directly useful for approximation and optimization.

major comments (2)
  1. [Abstract and main theorem] Abstract and the main existence argument: the transfer of classical NN universal-approximation guarantees to EML trees for the full class W^{k,∞} rests entirely on the asserted ability to explicitly construct EML trees that mimic arbitrary polynomials. No derivation from the EML definition or the tree composition rules is supplied showing how addition, multiplication, or monomials of degree >1 are obtained, nor are approximation error bounds or verification for the Sobolev norm provided; without these steps the central claim cannot be verified.
  2. [Learning algorithm] Learning-algorithm section: the proposed fitting procedure introduces free parameters inside the EML trees, yet it is not shown whether the optimization step preserves the polynomial-mimicking property or the resulting approximation rate for W^{k,∞}; this interaction is load-bearing for any claim that the learned trees inherit the universal-approximation guarantee.
minor comments (1)
  1. [Preliminaries] The notation for tree depth, composition rules, and the precise definition of the EML primitive would benefit from an additional displayed equation block for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We respond to each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and main theorem] Abstract and the main existence argument: the transfer of classical NN universal-approximation guarantees to EML trees for the full class W^{k,∞} rests entirely on the asserted ability to explicitly construct EML trees that mimic arbitrary polynomials. No derivation from the EML definition or the tree composition rules is supplied showing how addition, multiplication, or monomials of degree >1 are obtained, nor are approximation error bounds or verification for the Sobolev norm provided; without these steps the central claim cannot be verified.

    Authors: We agree that the explicit constructions and error bounds are not derived in sufficient detail in the current manuscript. In the revision we will insert a new subsection that starts from the EML definition and tree composition rules, shows how addition and multiplication are realized, extends the construction to monomials of arbitrary degree, and supplies the corresponding approximation error bounds together with verification in the W^{k,∞} norm. This will make the appeal to classical neural-network results fully rigorous and verifiable. revision: yes

  2. Referee: [Learning algorithm] Learning-algorithm section: the proposed fitting procedure introduces free parameters inside the EML trees, yet it is not shown whether the optimization step preserves the polynomial-mimicking property or the resulting approximation rate for W^{k,∞}; this interaction is load-bearing for any claim that the learned trees inherit the universal-approximation guarantee.

    Authors: The universal-approximation statement is a property of the entire class of EML trees; any concrete tree, including one whose parameters have been fitted by the proposed algorithm, remains an element of that class. We will revise the manuscript to state this distinction explicitly and to note that while membership in the class guarantees the existence result, the concrete approximation rate achieved by a learned tree is governed by the optimization outcome rather than by the specific polynomial-mimicking constructions used in the existence proof. A short discussion of this point will be added to the learning-algorithm section. revision: yes

Circularity Check

0 steps flagged

No circularity detected; relies on external NN theory

full rationale

The paper's central claim transfers universal approximation from classical neural network results in W^{k,∞} by asserting an explicit construction of EML trees that mimic polynomials. This is an existence argument grounded in prior external theory rather than any self-referential definition, fitted parameter renamed as prediction, or self-citation chain. No equations or steps in the abstract reduce the result to its own inputs by construction. The construction step, while undetailed here, does not constitute circularity under the specified patterns as it is not shown to be tautological or statistically forced within the paper's own framework.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven ability to construct EML trees that exactly mimic polynomials (invoked to transfer classical NN results) and on the background validity of those classical NN approximation theorems; the learning algorithm introduces fitting parameters whose number and initialization are not specified in the abstract.

free parameters (1)
  • fitting parameters inside EML trees
    The abstract states that the learning algorithm equips EML-type trees with fitting parameters; these are chosen to match data and therefore count as free parameters whose values are not derived from first principles.
axioms (1)
  • domain assumption Classical neural network approximation arguments apply once EML trees can mimic polynomials
    The abstract explicitly says the proof draws on these arguments after establishing the polynomial-mimicry construction.

pith-pipeline@v0.9.1-grok · 5653 in / 1433 out tokens · 13509 ms · 2026-06-26T08:45:45.174532+00:00 · methodology

discussion (0)

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