## Connect-The-Dots: How many random points can a regular curve pass through? (2004)

### Cached

### Download Links

Citations: | 10 - 4 self |

### BibTeX

@MISC{Arias-Castro04connect-the-dots:how,

author = {Ery Arias-Castro and David L. Donoho and Xiaoming Huo and Craig Tovey},

title = {Connect-The-Dots: How many random points can a regular curve pass through?},

year = {2004}

}

### OpenURL

### Abstract

### Citations

628 | Constructive Approximation - Devore, Lorentz - 1993 |

382 |
Empirical Processes with Applications to Statistics
- Shorack, Wellner
- 1986
(Show Context)
Citation Context ...area not exceeding ( √ 2ε1 + 2ε)ε ≤ 11/(2n). Using this lemma, we obtain the bound, valid for all B > 0, max π∈Π n λ P � Y n (π) > B λ √ n � ≤ P � Bin(n, 6λ/ √ n) > B λ √ n � . Hoeffding’s inequality =-=[33]-=- gives us control over the tail of the Binomial distribution:sConnect-the-dots 13 Lemma 3. For C > 2, where P {Bin(n, p) > C np} ≤ ϕ(C) np , We get immediately that for B > 1 and n > n0(λ), Combining ... |

345 | On the distribution of the length of the longest increasing subsequence in a random permutation
- Baik, Deift, et al.
- 1999
(Show Context)
Citation Context ...alled Ulam’s Problem) attracted considerable attention in the 1990’s, with concentration of measure estimates [20], massive computational studies [30], and finally results on asymptotic distributions =-=[7]-=-. In the 1970’s, Vershik, Logan, and Shepp [37, 27] showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Groeneboom [21] gives a particular simple proof of this). The more delicate fluctuation dist... |

306 |
Wavelets and Operators
- Meyer
- 1992
(Show Context)
Citation Context ...tegral is equivalent to the usual Hölder class, and B α 2,2[0, 1] d is equivalent to the usual L 2 -Sobolev class of smoothness α. The bump algebra B 1 1,1[0, 1] is an interesting approximation to BV =-=[29]-=-, and several other interesting spaces can be obtained by proper choice of α, p, q. We use F k,d p,q (α, β) for classes of immersions built from Triebel function classes F α p,q[0, 1] d . Again α > 0 ... |

275 |
Contour integration by the human visual system: evidence for a local ”association field
- Field, Hayes, et al.
- 1993
(Show Context)
Citation Context ... to determine the asymptotic behavior of Nn(Γ), as we do in this paper. For more on such problems, see [4, 22]. • Vision Research. An interesting stream of vision research started with the two papers =-=[19, 25]-=-. Both experiments presented specially prepared images to human subjects who were asked to (quickly and reflexively) judge whether the images were ‘purely random’ or ‘contained a curve buried in clutt... |

267 | Concentration of measure and isoperimetric inequalities in product spaces
- TALAGRAND
- 1995
(Show Context)
Citation Context ...≥ 1}. Note that this applies to all kinds of CTD problems: curves, hypersurfaces, etc. The proof of this apparently powerful result is actually just a simple application of a framework of Talagrand’s =-=[35]-=-. First, we introduce Talagrand’s key abstract notion: Definition 2. (Talagrand 1995, Def. 7.1.7.) L : X n ↦→ R is a configuration function if given any x n = (xi) ∈ X n , there exists a subset J of {... |

190 | Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 - Federer - 1969 |

127 |
L.A.Shepp, A variational problem for random Young tableaux
- Logan
- 1977
(Show Context)
Citation Context ...ttention in the 1990’s, with concentration of measure estimates [20], massive computational studies [30], and finally results on asymptotic distributions [7]. In the 1970’s, Vershik, Logan, and Shepp =-=[37, 27]-=- showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Groeneboom [21] gives a particular simple proof of this). The more delicate fluctuation distributional properties have been determined by Baik,... |

126 |
Wedgelets: Nearly minimax estimation of edges
- Donoho
- 1999
(Show Context)
Citation Context ... (b) (c) Figure 6: Panel (a): A horizontal beamlet and its horizontal neighbors; Panel (b): a vertical beamlet and its vertical neighbors; Panel (c): a diagonal beamlet and its diagonal neighbors. in =-=[15, 5, 14]-=-. The set of edges in Gn, which is denoted by En, links any two line segments in Vn that are in “good continuation”, which here means their directions are close enough [4]. Formally, two horizontal be... |

117 |
A closed curve is much more than an incomplete one: Effect of closure in figure-ground segmentation
- Kovacs, Julesz
- 1993
(Show Context)
Citation Context ... to determine the asymptotic behavior of Nn(Γ), as we do in this paper. For more on such problems, see [4, 22]. • Vision Research. An interesting stream of vision research started with the two papers =-=[19, 25]-=-. Both experiments presented specially prepared images to human subjects who were asked to (quickly and reflexively) judge whether the images were ‘purely random’ or ‘contained a curve buried in clutt... |

101 | Universality at the edge of the spectrum in Wigner random matrices
- Soshnikov
- 1999
(Show Context)
Citation Context ...ipschitz graphs. The fact that study of CTD extends the range of such limit phenomena seems per se interesting, especially in view of other universality results regarding the Tracy-Widom distribution =-=[34]-=-. 2. Traveling Salesman Problem. Probabilists and operations researchers have been interested for decades in the problem of determining the shortest path through every point in a cloud of n uniform ra... |

70 | Emerging Challenges: Mobile Networking for Smart Dust,” KICS
- Kahn, Katz, et al.
- 2000
(Show Context)
Citation Context ...ce problem models data given as the output of a spatially distributed array of detectors, such as particle detectors in high energy physics [3] or, more recently, sensor networks forming a Smart Dust =-=[24]-=-. Sensor alarms caused by ‘background’ are ‘false detections’ uniformly scattered in space; sensor alarms caused by something interesting (a particle or intruder) are scattered along the path of the i... |

63 |
Function Spaces, Entropy Numbers, Differential Operators
- Edmunds, Triebel
- 1996
(Show Context)
Citation Context ... ≤ v�ψ ⌈α⌉ �∞ · η −(1+{α}) · �t − s�. � ≤ v�ψ ⌊α⌋ �∞ · �t − s� {α} . In all cases, (A.3)-(A.4) guarantee gk+i ∈ H k,d (α, β). � A.5. Besov and Triebel Objects The following results are classical (see =-=[16]-=-, p. 105). 1. We have the inclusions: B α p,p∧q ⊂ F α p,q ⊂ B α p,p∨q, (A.6) where F ⊂ B, say, means �f�B ≤ C�f�F for some C > 0. 2. If α1 − α2 − k(p −1 1 α1 < ∞, then − p−1 2 )+ > 0, where p1, p2, q1... |

55 | Beamlets and multiscale image analysis
- Donoho, Huo
- 2002
(Show Context)
Citation Context ... (b) (c) Figure 6: Panel (a): A horizontal beamlet and its horizontal neighbors; Panel (b): a vertical beamlet and its vertical neighbors; Panel (c): a diagonal beamlet and its diagonal neighbors. in =-=[15, 5, 14]-=-. The set of edges in Gn, which is denoted by En, links any two line segments in Vn that are in “good continuation”, which here means their directions are close enough [4]. Formally, two horizontal be... |

51 | Asymptotic experimental analysis for the Held-Karp traveling salesman bound
- Johnson, McGeoch, et al.
- 1996
(Show Context)
Citation Context ...mining the shortest path through every point in a cloud of n uniform random points. This path length grows like .7124 √ n , where .7124 is an approximation to the BeardswoodHalton-Hammersley constant =-=[23]-=-. The CTD problem considers instead the maximum number of points on a curve of fixed length λ independent of n. While the two problems would thus be closely connected if λ were variable, λ = λn ≈ .712... |

51 | On the distribution of the lengths of the longest monotone subsequences in random words, preprint
- Tracy, Widom
- 1999
(Show Context)
Citation Context ... this). The more delicate fluctuation distributional properties have been determined by Baik, Deift, and Johanssen [7] who showed that the asymptotic distribution follows the Tracy-Widom distribution =-=[36]-=-. It turns out that similar asymptotic results hold for a different CTD problem where the class Γ is made ofs6 Arias-Castro, et al. Lipschitz graphs. The fact that study of CTD extends the range of su... |

31 |
Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux
- Vershik, M, et al.
(Show Context)
Citation Context ...ttention in the 1990’s, with concentration of measure estimates [20], massive computational studies [30], and finally results on asymptotic distributions [7]. In the 1970’s, Vershik, Logan, and Shepp =-=[37, 27]-=- showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Groeneboom [21] gives a particular simple proof of this). The more delicate fluctuation distributional properties have been determined by Baik,... |

30 | E.M.Rains, On longest increasing subsequences in random permutations, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis
- Odlyzko
- 1998
(Show Context)
Citation Context ...sing subsequence of n such random numbers (sometimes called Ulam’s Problem) attracted considerable attention in the 1990’s, with concentration of measure estimates [20], massive computational studies =-=[30]-=-, and finally results on asymptotic distributions [7]. In the 1970’s, Vershik, Logan, and Shepp [37, 27] showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Groeneboom [21] gives a particular simp... |

28 |
The shortest path and the shortest road through n points in a region
- Few
- 1955
(Show Context)
Citation Context ...the Travelling Salesman Problem (TSP). Nearly half century ago, L. Few proved that given n points on a unit square, there is a curve of length not exceeding √ 2n + 7/4 that traverses all the n points =-=[18]-=-. Divide such a path into m = ⌈ √ 2n+7/4 λ ⌉ consecutive pieces γ1, . . . , γm, with length ≤ λ. By definition, Nn(Cλ) is larger than each X n (γi), and so larger than their average. Hence, Nn(Cλ) ≥ X... |

26 | Scheduling Dial-a-Ride Transportation Systems - STEIN - 1978 |

22 | On the length of the longest monotone subsequence in a random permutation
- Frieze
- 1991
(Show Context)
Citation Context ... of the length of the longest increasing subsequence of n such random numbers (sometimes called Ulam’s Problem) attracted considerable attention in the 1990’s, with concentration of measure estimates =-=[20]-=-, massive computational studies [30], and finally results on asymptotic distributions [7]. In the 1970’s, Vershik, Logan, and Shepp [37, 27] showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Gro... |

15 |
Adaptive multiscale detection of filamentary structures in a background of uniform random
- Arias-Castro, Donoho, et al.
- 2006
(Show Context)
Citation Context ...ection is possible. A sketch of the idea is illustrated in Figure 1. Hence it is of some interest to determine the asymptotic behavior of Nn(Γ), as we do in this paper. For more on such problems, see =-=[4, 22]-=-. • Vision Research. An interesting stream of vision research started with the two papers [19, 25]. Both experiments presented specially prepared images to human subjects who were asked to (quickly an... |

11 | Ulam’s problem and Hammersley’s process
- Groeneboom
(Show Context)
Citation Context ...e computational studies [30], and finally results on asymptotic distributions [7]. In the 1970’s, Vershik, Logan, and Shepp [37, 27] showed that asymptotic behavior of Nn(IncrGr) ∼ 2 √ n. (Groeneboom =-=[21]-=- gives a particular simple proof of this). The more delicate fluctuation distributional properties have been determined by Baik, Deift, and Johanssen [7] who showed that the asymptotic distribution fo... |

9 |
Geometric Measure Theory, Vol. 153 of Die Grundlehren der mathematischen Wissenschaften
- Federer
- 1969
(Show Context)
Citation Context ...ov and Tikhomirov, 1958.) Hε(H k,d (α, β), | · |∞) ≤ ck,d,α (β/ε) k/α , ε ∈ (0, 1). Finally we estimate the key volumic quantity M(ε). We note the connection with the notion of Minkowski content (see =-=[17]-=-). Indeed, fix S ∈ S k,d (α, β); then, µ((S)ε)/(vd−k ε d−k ) → vol(S), ε → 0, where vd−k denotes the volume of the (d − k)-dimensional unit ball, and vol(S) is the k-dimensional Hausdorff measure of S... |

7 | Limiting distribution of last passage percolation models
- Baik
- 2005
(Show Context)
Citation Context ...IncrGr denote the class of increasing curves, i.e., of sets {(x, f(x)) : x ∈ [0, 1]} where f : [0, 1] ↦→ [0, 1] is monotone increasing. Then Nn(IncrGr) measures the result of last-passage percolation =-=[6]-=-. Also, if we write Xi = (xi, yi), let π denote the sorting permutation x π(1) ≤ x π(2) ≤ . . . , and define wi = y π(i), i = 1, . . . , n, then Nn(IncrGr) is the length of the longest increasing subs... |

6 |
An orientation-selective neural network for pattern identification in particle detectors
- ABRAMOWICZ, HORN, et al.
- 1997
(Show Context)
Citation Context ...sampled at random along an unknown curve γ ∈ Γ. This inference problem models data given as the output of a spatially distributed array of detectors, such as particle detectors in high energy physics =-=[3]-=- or, more recently, sensor networks forming a Smart Dust [24]. Sensor alarms caused by ‘background’ are ‘false detections’ uniformly scattered in space; sensor alarms caused by something interesting (... |

6 |
2003),“Asymptotically Optimal Detection of Geometric Objects by Fast Multiscale Methods
- Arias, Donoho, et al.
(Show Context)
Citation Context ... (b) (c) Figure 6: Panel (a): A horizontal beamlet and its horizontal neighbors; Panel (b): a vertical beamlet and its vertical neighbors; Panel (c): a diagonal beamlet and its diagonal neighbors. in =-=[15, 5, 14]-=-. The set of edges in Gn, which is denoted by En, links any two line segments in Vn that are in “good continuation”, which here means their directions are close enough [4]. Formally, two horizontal be... |

6 | A note on Landau’s problem for bounded intervals - Chui, Smith - 1975 |

3 | Epsilon-entropy of convex sets and functions - Bronstein - 1976 |

3 |
Irregularities of distribution iii
- Schmidt
- 1969
(Show Context)
Citation Context ...at C is the class of all convex sets in [0, 1] 2 . We are given a set Xi of points in [0, 1] 2 and are interested in the discrepancy ∆(C) = sup |Nn(C) − nArea(C)|. C∈C It is known, by work of Schmidt =-=[31]-=-, [8, Theorem 15], that for any collection of n points, ∆(C) ≥ cn 1/3 . CTD leads to the same conclusion for random point sets. Suppose that the point set (Xi) is uniform random. Consider the class Co... |

2 | Entropies of sets of functions of bounded variation - Clements - 1963 |

2 |
Dynamic programming methods for “connecting the dots
- Huo, Donoho, et al.
(Show Context)
Citation Context ...ection is possible. A sketch of the idea is illustrated in Figure 1. Hence it is of some interest to determine the asymptotic behavior of Nn(Γ), as we do in this paper. For more on such problems, see =-=[4, 22]-=-. • Vision Research. An interesting stream of vision research started with the two papers [19, 25]. Both experiments presented specially prepared images to human subjects who were asked to (quickly an... |

2 | Dynamic programming methods for “connecting the dots” in scattered point sets - HUO, DONOHO, et al. - 2004 |

2 | Pister (2000). Emerging challenges: mobile networking for smart dust - Kahn, Katz, et al. |

1 |
Irregularities of distribution vol
- Beck, Chen
- 1987
(Show Context)
Citation Context ...repancy from uniform by comparing the fraction of points in a set with the fraction of volume in that set, and one maximizes the discrepancy over a class of sets (rectangles, disks, convex sets, ...) =-=[8, 28]-=-. As a referee has pointed out, CTD could be considered a variant of this approach, maximizing discrepancy over classes of curves. Since classes of curves involve objects of zero volume, discrepancy b... |

1 |
Administration, agencies failed to connect the dots
- Diamond, Kiely
- 2002
(Show Context)
Citation Context ...whose task is to connect every dot in as4 Arias-Castro, et al. particular sequence and see a picture emerge. We are not thinking of this usage. We think instead of recent usage in political discourse =-=[32, 13, 26]-=- where CTD refers to identifying a small subset of facts among many random conflicting ones, thereby detecting a subtle pattern. Thus, journalists writing in [32, 13, 26] all used the CTD phraseology ... |

1 |
Board not ready to connect the dots. Probe into Shuttle tragedy still working out the details. NewYork Newsday
- Lane
- 2002
(Show Context)
Citation Context ...whose task is to connect every dot in as4 Arias-Castro, et al. particular sequence and see a picture emerge. We are not thinking of this usage. We think instead of recent usage in political discourse =-=[32, 13, 26]-=- where CTD refers to identifying a small subset of facts among many random conflicting ones, thereby detecting a subtle pattern. Thus, journalists writing in [32, 13, 26] all used the CTD phraseology ... |

1 |
Geometric Discrepancy. An Illustrated Guide vol. 18 of Algorithms and Combinatorics
- Matousek
- 1999
(Show Context)
Citation Context ...repancy from uniform by comparing the fraction of points in a set with the fraction of volume in that set, and one maximizes the discrepancy over a class of sets (rectangles, disks, convex sets, ...) =-=[8, 28]-=-. As a referee has pointed out, CTD could be considered a variant of this approach, maximizing discrepancy over classes of curves. Since classes of curves involve objects of zero volume, discrepancy b... |

1 |
Connecting the dots. CNN.com
- Schneider
(Show Context)
Citation Context ...whose task is to connect every dot in as4 Arias-Castro, et al. particular sequence and see a picture emerge. We are not thinking of this usage. We think instead of recent usage in political discourse =-=[32, 13, 26]-=- where CTD refers to identifying a small subset of facts among many random conflicting ones, thereby detecting a subtle pattern. Thus, journalists writing in [32, 13, 26] all used the CTD phraseology ... |

1 | Connecting the Dots. CNN.com Inside Politics - Schneider - 2002 |