The Choquet Integral generalizes metric functions

This is a really cool integral that basically maps any metric function on sets like the MAXIMUM, MINIMUM, MEDIAN

Choquet Integral: The Math

Where $\pi(j)$ sorts the inputs based on their output values, $h_{\pi(j)}$, which is the output for input $x_j$. Little g is a measure such that $g(A) < g(B)$ if $A \subseteq B$, $g(\emptyset) = 0$, $g(\mathbf{X}) = 1$. ChI adheres to monotonicity, idempotency, subcontinuity, etc. This includes the Linear Order Statistic (LOS) the Order Weighted Average (OWA), Maximum, Minimum, etc. $$ \int h \odot g = C_g(h) = \sum_{j=1}^N h_{\pi(i)} \left( g(A_{\pi(j)}) - g(A_{\pi(j-1)}) \right) $$ $\pi$ is the function that sorts the inputs according to their values $h(x_j)$ $A_{\pi(j)}$ is the sorted vector = ${x_{\pi(1)}, x_{\pi(2)}, …}$ $g$ is the learned measure (subsumes LOS, OWA, etc.)

LOS/OWA

Also known as LOS or Ordered Weighted Average (OWA), LOS sorts the inputs based on their magnitude, and has a vector of weights the same dimension as the input. Inputs are sorted in descending order, and all the weights sum to 1. This is differentiable, so we can use Pytorch, and the integral basically acts as a Multilayer perceptron (another thing that can represent linear statistics on data).

Example with three inputs (sort them first in descending order, largest first) $w = [1,0,0]$ is the maximum operator $w = [0,0,1]$ is minimum $w = [0,1,0]$ is median $w=[\frac{1}{3},\frac{1}{3},\frac{1}{3}]$ is average $w = [0.7,0.3,0]$ is a soft-max