## Abstract

A sign-linear one-bit map from the d-dimensional sphere S^{d} to the N-dimensional Hamming cube H^{N} = {-1,+1}^{n} is given by × ↦ {sign(x · z_{j}): 1 ≤ j ≤ N}, where {z_{j}} C S^{d}. For 0 < δ < 1, we estimate N(d,δ), the smallest integer N so that there is a sign-linear map which has the δ-restricted isometric property, where we impose the normalized geodesic distance on S^{d} and the Hamming metric on H^{N}. Up to a polylogarithmic factor, N(d,δ) ≈ δ^{-2+2/(d+1)}, which has a dimensional correction in the power of δ. This is a question that arises from the one-bit sensing literature, and the method of proof follows from geometric discrepancy theory. We also obtain an analogue of the Stolarsky invariance principle for this situation, which implies that minimizing the L^{2}-average of the embedding error is equivalent to minimizing the discrete energy Σ_{i,j} (1/2 - d(z_{i}, z_{j})^{2}, where d is the normalized geodesic distance.

Original language | English (US) |
---|---|

Pages (from-to) | 744-763 |

Number of pages | 20 |

Journal | Sbornik Mathematics |

Volume | 208 |

Issue number | 6 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Funding Information:The research was partially supported by NSF (grants nos. DMS 1101519 and DMS 1265570). AMS 2010 Mathematics Subject Classification. Primary 11K38, 94A12, 94A20; Secondary 52C99.

Publisher Copyright:

© 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

Copyright:

Copyright 2019 Elsevier B.V., All rights reserved.

## Keywords

- Discrepancy
- One-bit sensing
- Restricted isometry property
- Stolarsky principle