The "Softmax" Nonlinearity: Derivation Using Statistical Mechanics and Useful Properties as a Multiterminal Analog Circuit Element

In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit is passive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks.