Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 383–403 | Cite as

Boolean autoencoders and hypercube clustering complexity

Open Access


We introduce and study the properties of Boolean autoencoder circuits. In particular, we show that the Boolean autoencoder circuit problem is equivalent to a clustering problem on the hypercube. We show that clustering m binary vectors on the n-dimensional hypercube into k clusters is NP-hard, as soon as the number of clusters scales like \({m^\epsilon (\epsilon >0 )}\) , and thus the general Boolean autoencoder problem is also NP-hard. We prove that the linear Boolean autoencoder circuit problem is also NP-hard, and so are several related problems such as: subspace identification over finite fields, linear regression over finite fields, even/odd set intersections, and parity circuits. The emerging picture is that autoencoder optimization is NP-hard in the general case, with a few notable exceptions including the linear cases over infinite fields or the Boolean case with fixed size hidden layer. However learning can be tackled by approximate algorithms, including alternate optimization, suggesting a new class of learning algorithms for deep networks, including deep networks of threshold gates or artificial neurons.


Autoencoders Clustering Boolean circuits Computational complexity 

Mathematics Subject Classification (2010)




Work supported in part by grants NSF IIS-0513376, NIH LM010235, and NIH NLM T15 LM07443 to PB.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

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