### Abstract

We consider the problem of estimating the surface area of an unknown n-dimensional set F given membership oracle access. In contrast to previous work, we do not assume that F is convex, and in fact make no assumptions at all about F. By necessity this means that we work in the property testing model; we seek an algorithm which, given parameters A and e, satisfies: if surf(F) ≤ A then the algorithm accepts (whp); if F is not e-close to some set G with surf(G) ≤ KA, then the algorithm rejects (whp). We call k ≥ 1 the "approximation factor" of the testing algorithm. The n - 1 case (in which "surf(F) = 2m" means F is a disjoint union of to intervals) was introduced by Kearns and Ron [KR98], who solved the problem with k = 1/ε and O(1/ε) oracle queries. Later, Balcan et al. [BBBY12] solved it with with k = 1 and 0(l/ε^{4}) queries. We give the first result for higher dimensions n. Perhaps surprisingly, our algorithm completely evades the "curse of dimensionality": for any n and any k > 4/π ≈ 1.27 we give a test that uses O( 1/ε) queries. For small n we have improved bounds. For n = 1 we can achieve k = 1 with O( 1/ε^{3.5}) queries (slightly improving [BBBY12]), or any K > 1 with 0( 1/ε) queries (improving [KR98]). For n = 2,3 we obtain K ≈ 1.08,1.125 respectively, with 0( 1/ε) queries. Getting an arbitrary k > 1 for n > 1 remains an open problem.Finally, motivated by the learning results from [KOSO8], we extend our techniques to obtain a similar tester for Gaussian surface area for any n, with query complexity O( 1/ε) and any approximation factor k > 4/π ≈ 1.27.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1204-1214 |

Number of pages | 11 |

ISBN (Print) | 9781611973389 |

DOIs | |

State | Published - Jan 1 2014 |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
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Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 1204-1214). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery. https://doi.org/10.1137/1.9781611973402.89