# 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]}

## Open questions

- Is it true that Inner Product requires super-polynomial sized AC
^{0}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