[Paper Review] Sigmoid functions and exponential Riordan arrays

Paul Barry|arXiv (Cornell University)|Feb 13, 2017

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper introduces a novel framework linking sigmoid functions and their derivatives to exponential Riordan arrays, particularly focusing on the derivative subgroup [f′(x), f(x)]. It demonstrates that such arrays generate polynomial families, some of which are orthogonal, and establishes connections to combinatorial sequences like Stirling numbers of the second kind. A key contribution is the explicit link between the Gompertz sigmoid function and Stirling numbers via Riordan array composition.

ABSTRACT

Sigmoid functions play an important role in many areas of applied mathematics, including machine learning, population dynamics and probability. We place the study of sigmoid functions in the context of the derivative sub-group of the group of exponential Riordan arrays. Links to families of polynomials are drawn, and it is shown that in some cases these polynomials are orthogonal. In the non-orthogonal case, transformations are given that produce orthgonal systems. Alternative means of characterisation are given, based on the production (Stieltjes) matrix associated to the relevant Riordan array.

Motivation & Objective

• To establish a systematic framework for analyzing sigmoid functions using the derivative subgroup of exponential Riordan arrays.
• To investigate the polynomial families generated by the action of sigmoid-based Riordan arrays on the monomial basis.
• To determine when these polynomial families are orthogonal and, when not, to derive transformations that produce orthogonal systems.
• To clarify combinatorial connections, particularly between the Gompertz sigmoid and Stirling numbers of the second kind, using Riordan array composition.
• To provide parametric visualizations of the (f′(t), f(t)) pairs to illustrate the geometric and analytic behavior of the sigmoid-derivative pairs.

Proposed method

• Utilizes the group structure of exponential Riordan arrays [g(x), f(x)] with a focus on the derivative subgroup [f′(x), f(x)] for analytic sigmoid functions.
• Applies the bivariate generating function g(x)e^{yf(x)} to represent the array elements t_{n,k} = n! / k! [x^n] g(x)f(x)^k.
• Employs the production (Stieltjes) matrix to characterize the Riordan arrays and analyze their moment sequences.
• Derives explicit formulas for array elements using generating functions and complex exponential expansions, particularly for trigonometric and hyperbolic sigmoid pairs.
• Uses Riordan array multiplication and compositional inverses to relate the Gompertz sigmoid array to Stirling numbers of the second kind via [e^{1−x−e^{−x}}, 1−e^{−x}] and [1, e^x−1].
• Applies parametric plotting of (f′(t), f(t)) to visualize the geometric shape of the sigmoid-derivative pair, such as the unit circle for [cos(x), sin(x)].

Experimental results

Research questions

• RQ1 How can the derivative subgroup of exponential Riordan arrays be used to systematically study sigmoid functions and their derivatives?
• RQ2 Under what conditions do the polynomial families generated by sigmoid-based Riordan arrays become orthogonal?
• RQ3 What combinatorial structures, such as Stirling numbers of the second kind, underlie the coefficients of the Gompertz sigmoid function’s derivative?
• RQ4 How do the production matrices of these Riordan arrays reflect the moment sequences of the associated orthogonal polynomials?
• RQ5 What geometric insights can be gained from parametric plots of the (f′(t), f(t)) pairs for various sigmoid functions?

Key findings

• The exponential Riordan array [e^{1−x−e^{−x}}, 1−e^{−x}] is the moment matrix for a family of orthogonal polynomials defined by the recurrence P_n(x) = (x + (n−1))P_{n−1}(x) + (n−1)P_{n−2}(x), with initial conditions P_0(x)=1, P_1(x)=1.
• The moment sequence 1, 0, −1, 12, −9, 9, ... associated with this orthogonal polynomial family has an ordinary generating function given by a continued fraction involving x² and linear terms.
• The Gompertz sigmoid array [e^{1−t−e^{−t}}, e^{1−e^{−t}}−1] is shown to be equal to the product of [e^{−t}, 1−e^{−t}] and [e^t, e^t−1], linking it directly to Stirling numbers of the second kind.
• The general element g_{n,k} of the Gompertz sigmoid array is given by g_{n,k} = ∑_{j=0}^n S2(n+1,j+1)(−1)^{n−j} S2(j+1,k+1), where S2(n,k) denotes the Stirling numbers of the second kind.
• The derivative of the Gompertz sigmoid function has coefficients g_{n,0} = ∑_{j=0}^n S2(n+1,j+1)(−1)^{n−j}, which are the signed sums of Stirling numbers of the second kind.
• Parametric plots of (f′(t), f(t)) for various sigmoid functions, such as (sech²(t), tanh(t)) and (1/(1+t²), tan⁻¹(t)), reveal distinct geometric shapes, including closed curves and S-shaped trajectories.