Inner product
Definition
A function [math]f:\{-1,1\}^{2k} \to \{-1,1\}[/math] is called an inner product function if it is equal to the sign of the inner product of the first half of the input and the second half. More formally,
[math]f(x_1,\ldots,x_k, y_1,\ldots,y_k) = (-1)^{\boldsymbol{x} \cdot \boldsymbol{y}}. [/math]
Properties
- TODO: Fourier representation
- The inner product function requires either exponential weights or an exponential number of nodes to be represented with a restricted Boltzmann machine. [1]
- The inner product function has nearest neighbor complexity bounded by [math]NN(f) \geq 2^{n/2}[/math]. [2]
Open questions
- Is it true that Inner Product requires super-polynomial sized AC0 circuits with parity gates?
References
- ↑ Theorem 9 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, On the Representational Efficiency of Restricted Boltzmann Machines, NIPS 2013
- ↑ Péter Hajnal, Zhihao Liu, György Turán, Nearest neighbor representations of Boolean functions, Theorem 10
