A generalized logistic-logit function and its application to multi-layer perceptron and neuron segmentation.

Logistic and logit functions play important roles in modern science, serving as foundational tools in various applications, such as artificial neural networks (ANN). While there are functions that could produce distinct logistic and logit curves, no single, unified framework has been developed to generate both logistic and logit curves. We introduce a Cannistraci-Muscoloni-Gu generalized logistic-logit function (CMG-GLLF) to fill this gap. CMG-GLLF provides four interpretable and trainable parameters that allow explicit control over: curve type and steepness, asymmetry, and upper and lower limits of x- and y-axes. CMG-GLLF's potential is explored in basic machine intelligence tasks. As a proof-of-concept on how this function can improve the performance of deep learning, we propose a trainable input feature modulator (IFM) that consists of learning the parameters of the CMG-GLLF for each input layer node during backpropagation for a multi-layer perceptron (MLP), which is a fundamental building block of many complex network architectures. Compared to various other learnable functions, across three different optimizers, CMG-GLLF allows superior MLP's accuracy and stable training behavior on CIFAR-10 and CIFAR-100 image classification, but at the cost of increased computational time. Hence, we identified limitations to address in future studies, notably the need to derive an explicit mathematical expression for the logit phase, which could: (i) mitigate numerical instability in more complex architectures (e.g., CNNs) while reducing computational overhead and (ii) enable a systematic evaluation of CMG as an activation function across all layers. Furthermore, CMG-GLLF, adopted as a data transformation function, enhances the accuracy of affinity-graph-based neuron segmentation. CMG-GLLF combines in a unique framework the ability of logistic and logit functions to modulate signals or variables, covering a full spectrum of attenuation or amplification transformations. CMG-GLLF is flexible and trainable, has the potential to advance machine learning models, and can inspire further applications in other data analysis challenges in different domains of science.