Inner product

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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 AC0 circuits with parity gates?

References

  1. Theorem 9 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, On the Representational Efficiency of Restricted Boltzmann Machines, NIPS 2013