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In standard rounding, we want to map each value X in a large continuous space (e.g., R) to a nearby point P from a discrete subset (e.g., Z). This process seems to be inherently discontinuous in the sense that two consecutive noisy measurements X_1 and X_2 of the same value may be extremely close to each other and yet they can be rounded to different points P_1 P_2, which is undesirable in many applications. In this paper we show how to make the rounding process perfectly continuous in thearXiv:2008.03675v1 fatcat:dvidtydbfnhmrmdtuggez4d5v4
more »... that it maps any pair of sufficiently close measurements to the same point. We call such a process consistent rounding, and make it possible by allowing a small amount of information about the first measurement X_1 to be unidirectionally communicated to and used by the rounding process of X_2. The fault tolerance of a consistent rounding scheme is defined by the maximum distance between pairs of measurements which guarantees that they are always rounded to the same point, and our goal is to study the possible tradeoffs between the amount of information provided and the achievable fault tolerance for various types of spaces. When the measurements X_i are arbitrary vectors in R^d, we show that communicating log_2(d+1) bits of information is both sufficient and necessary (in the worst case) in order to achieve consistent rounding for some positive fault tolerance, and when d=3 we obtain a tight upper and lower asymptotic bound of (0.561+o(1))k^1/3 on the achievable fault tolerance when we reveal log_2(k) bits of information about how X_1 was rounded. We analyze the problem by considering the possible colored tilings of the space with k available colors, and obtain our upper and lower bounds with a variety of mathematical techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, and Čech cohomology.
In this paper we consider a new type of space partitioning which bridges the gap between continuous and discrete spaces in an error resilient way. It is motivated by the problem of rounding noisy measurements from some continuous space such as ℝ^d to a discrete subset of representative values, in which each tile in the partition is defined as the preimage of one of the output points. Standard rounding schemes seem to be inherently discontinuous across tile boundaries, but in this paper we showdoi:10.4230/lipics.icalp.2021.4 fatcat:op3plilqjvd5bajcvxnekmde5m
more »... ow to make it perfectly consistent (with error resilience ε) by guaranteeing that any pair of consecutive measurements X₁ and X₂ whose L₂ distance is bounded by ε will be rounded to the same nearby representative point in the discrete output space. We achieve this resilience by allowing a few bits of information about the first measurement X₁ to be unidirectionally communicated to and used by the rounding process of the second measurement X₂. Minimizing this revealed information can be particularly important in privacy-sensitive applications such as COVID-19 contact tracing, in which we want to find out all the cases in which two persons were at roughly the same place at roughly the same time, by comparing cryptographically hashed versions of their itineraries in an error resilient way. The main problem we study in this paper is characterizing the achievable tradeoffs between the amount of information provided and the error resilience for various dimensions. We analyze the problem by considering the possible colored tilings of the space with k available colors, and use the color of the tile in which X₁ resides as the side information. We obtain our upper and lower bounds with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Sperner's lemma, and Čech cohomology. In particular, we show that when X_i ∈ ℝ^d, communicating log₂(d+1) bits of information is both sufficient and necessary (in the worst case) to achieve positive resilience, and when d=3 we obtain a ti [...]
We also want to thank Karl Schlechta and Zeev Geyzel for their remarks that lead to definite improvements. ...arXiv:cs/0202031v1 fatcat:div4ti5ueza5pcwf5ltz2pgdmm