Difference between revisions of "Inner product"
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* TODO: Fourier representation | * 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. <ref>Theorem 9 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, [https://www.cs.toronto.edu/~toni/Papers/nips2013.pdf On the Representational Efficiency of Restricted Boltzmann Machines], NIPS 2013</ref> | * The inner product function requires either exponential weights or an exponential number of nodes to be represented with a restricted Boltzmann machine. <ref>Theorem 9 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, [https://www.cs.toronto.edu/~toni/Papers/nips2013.pdf On the Representational Efficiency of Restricted Boltzmann Machines], NIPS 2013</ref> | ||
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| + | == Open questions == | ||
| + | * Is it true that Inner Product requires super-polynomial sized [[Circuit_complexity#AC0 | AC<sup>0</sup>]] circuits with parity gates? | ||
== References == | == References == | ||
Revision as of 08:26, 1 October 2018
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
- ↑ Theorem 9 in James Martens, Arkadev Chattopadhyay, Toniann Pitassi, Richard Zemel, On the Representational Efficiency of Restricted Boltzmann Machines, NIPS 2013
