sigmoidWf

sigmoidWf is a term used to describe a class of sigmoid-transformed weighting functions, typically constructed by applying a sigmoid (logistic) function to a base weighting function f. The resulting weight Wf maps inputs to the range 0 to 1 and provides a smooth, differentiable transition that can be tuned through f and the sigmoid parameters.

Mathematically, Wf(x) can be written as σ(f(x)) with σ(z) = 1/(1 + exp(-z)). A common variant is Wf(x) = 1/(1 + exp(-α (f(x) - β))), where α controls the slope and β sets the midpoint. The derivative with respect to x is dWf/dx = α f'(x) σ(f(x)) (1 − σ(f(x))). If f is vector-valued, the application is typically element-wise, producing a weight vector.

Variants of the concept may use different sigmoids (such as tanh-based forms) or incorporate additional parameters to modulate curvature and asymmetry. The approach remains characterized by converting an underlying score or function into a bounded, smooth weight.

Applications of sigmoidWf appear across domains including machine learning, statistics, signal processing, and computer graphics. In neural networks, it serves as a soft gate or attention-like mechanism to emphasize or suppress features gradually. In robust statistics and optimization, it can implement soft constraints or soft-thresholding. In image and signal processing, sigmoidWf can enable smooth blending or region-of-interest weighting.

Terminology and definition vary by source; sigmoidWf is not a universally standardized object. Users should consult domain-specific definitions to understand the exact form and usage in a given context.

See also: logistic function, sigmoid function, weighting function, soft thresholding, attention mechanism, soft gating.