## Bounded geometries, fractals, and low-distortion embeddings

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Citations: | 156 - 32 self |

### BibTeX

@MISC{Gupta_boundedgeometries,,

author = {Anupam Gupta and Robert Krauthgamer and James R. Lee},

title = {Bounded geometries, fractals, and low-distortion embeddings},

year = {}

}

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### Abstract

The doubling constant of a metric space (X; d) is thesmallest value * such that every ball in X can be covered by * balls of half the radius. The doubling dimension of X isthen defined as dim(X) = log2 *. A metric (or sequence ofmetrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaceswhich contains many families of metrics that occur in applied settings.We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider bothgeneral doubling metrics, as well as more restricted families such as those arising from trees, from graphs excludinga fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, andan analysis of a fractal in the plane due to Laakso [21]. Finally, we discuss some applications and point out a centralopen question regarding dimensionality reduction in L2.

### Citations

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(Show Context)
Citation Context ...are very natural and are thought to occur in real-world phenomena such as peer-to-peer networks (see e.g. [30]) and data analysis (e.g., when the input data resides on a low-dimensional manifold, cf. =-=[34]-=-). In fact, various algorithms can be tailored to run efficiently on certain classes of growth-restricted metrics, as demonstrated in [8, 31, 16, 17]. The metrics considered there are either equivalen... |

1717 | The Probabilistic Method
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Citation Context ...bility of the latter event is at most 1 8 according to the analysis of Theorem 3.2, and thus EY x # m 8 . Finally, let E m x be the event that Y x > m 2 = 4 EY x . A standard Chernoff bound (see e.g. =-=[1]-=-) shows that Pr[E m x ] # (9/10) m . 7 Claim 3.4. If N is an #r-net in X then Pr # # y#N E m y # > 0. Notice that the above claim suffices to prove the Theorem. Indeed, if #P (y) # 2#r for every y # N... |

557 | Predicting Internet Network Distance with Coordinates-based Approaches
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(Show Context)
Citation Context ...sting objects in their own right, but they are also of practical concern. Growth restrictions are very natural and are thought to occur in real-world phenomena such as peer-to-peer networks (see e.g. =-=[30]-=-) and data analysis (e.g., when the input data resides on a low-dimensional manifold, cf. [34]). In fact, various algorithms can be tailored to run efficiently on certain classes of growth-restricted ... |

517 | Accessing nearby copies of replicated objects in a distributed environment
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(Show Context)
Citation Context ...hen the input data resides on a low-dimensional manifold, cf. [34]). In fact, various algorithms can be tailored to run efficiently on certain classes of growth-restricted metrics, as demonstrated in =-=[8, 31, 16, 17]. The metr-=-ics considered there are either equivalent to or a subclass of those metrics which are doubling; see Section 1.3. 1.1 Results and techniques We are concerned with the broad roles of "volume"... |

455 | The Geometry of Graphs and some of Its Algorithmic Applications
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Citation Context ...undamental properties of the metric (X, d). The general case is well-understood. Bourgain [4] showed that every n-point metric embeds into L p with O(log n) distortion for any fixed p; it is shown in =-=[25]-=- that this bound is tight, for all p # 2, for the shortest path metric on constant-degree expander graphs. This was later extended in [26], showing a tight upper bound of O( log n p ) for any L p spac... |

425 |
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Citation Context ...tions, we studied the distortion of general metrics in terms of their doubling dimension. Here, let us consider an n-point set X # # n 2 with dim(X) = O(1). The Johnson-Lindenstrauss flattening lemma =-=[15]-=- tells us that there is a 1 + # embedding of X into # O(# -2 log n 2 ). We pose the following intriguing question which was asked independently in [21]. Question 1. Can every doubling submetric of # 2... |

278 |
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Citation Context ...arch and Computer Science Division, U.C. Berkeley. Email: jrl@cs.berkeley.edu distortion in terms of certain fundamental properties of the metric (X, d). The general case is well-understood. Bourgain =-=[4]-=- showed that every n-point metric embeds into L p with O(log n) distortion for any fixed p; it is shown in [25] that this bound is tight, for all p # 2, for the shortest path metric on constant-degree... |

257 |
Geometry of Cuts and Metrics
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(Show Context)
Citation Context ...open question regarding the roles of volume and structure in dimensionality reduction in Euclidean spaces. 1.2 Preliminaries Here are some definitions used in the paper; the books by Deza and Laurent =-=[9]-=- and by Heinonen [13] give more de2 tails on metric spaces. Let (X, dX ) and (Y, d Y ) be two metric spaces, 1 and consider an injective map f : X # Y . We define contraction(f) = sup a,b#X dX (a, b) ... |

161 |
Lectures on Analysis on Metric Spaces
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(Show Context)
Citation Context ...ly, we exam-ine how the "volume growth" of a metric affects its embeddability into Lp spaces. The notion of growth that we use iswell-studied, and is very similar to a notion of Assouad [2], see also =-=[14]-=-. Our definition is technically slightly differentfrom Assouad's, but the flavor is left unaltered; in particular, the notion of bounded growth is equivalent under eitherframework. For a metric (X, d)... |

154 | Finding nearest neighbors in growth-restricted metrics
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(Show Context)
Citation Context ...hen the input data resides on a low-dimensional manifold, cf. [34]). In fact, various algorithms can be tailored to run efficiently on certain classes of growth-restricted metrics, as demonstrated in =-=[8, 31, 16, 17]. The metr-=-ics considered there are either equivalent to or a subclass of those metrics which are doubling; see Section 1.3. 1.1 Results and techniques We are concerned with the broad roles of "volume"... |

127 | Navigating nets: simple algorithms for proximity search
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- 2004
(Show Context)
Citation Context ...hen the input data resides on a low-dimensional manifold, cf. [34]). In fact, various algorithms can be tailored to run efficiently on certain classes of growth-restricted metrics, as demonstrated in =-=[8, 31, 16, 17]. The metr-=-ics considered there are either equivalent to or a subclass of those metrics which are doubling; see Section 1.3. 1.1 Results and techniques We are concerned with the broad roles of "volume"... |

124 | An O(log k) approximate min-cut max-flow theorem and approximation algorithm - Aumann, Rabani - 1998 |

118 | Algorithmic applications of low-distortion geometric embeddings
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(Show Context)
Citation Context ...ects of study lying at the intersection of analysis, combinatorics, and geometry, the ideas and techniques generated in this field have led to a number of powerful algorithmic applications (see, e.g. =-=[14, 24, 28]-=-). We consider embeddings of finite metric spaces into L p spaces. Given a metric (X, d), the goal is to find a map f : X # L p such that ||f (x)-f(y)|| p is close to d(x, y) for all x, y # X . The wo... |

112 | Nearest neighbor queries in metric spaces
- Clarkson
- 1997
(Show Context)
Citation Context |

111 |
Lectures on analysis on metric spaces, Universitext
- Heinonen
- 2001
(Show Context)
Citation Context ...y, we examine how the "volume growth" of a metric affects its embeddability into L p spaces. The notion of growth that we use is well-studied, and is very similar to a notion of Assouad [2],=-= see also [13]-=-. Our definition is technically slightly different from Assouad's, but the flavor is left unaltered; in particular, the notion of bounded growth is equivalent under either framework. For a metric (X, ... |

95 | Approximating the Bandwidth via Volume Respecting Embeddings
- Feige
(Show Context)
Citation Context ...erty is not maintained when even one point (the origin) is added. Local density. Finally, there is another natural notion of volume, which has been used widely in the study of the bandwidth of graphs =-=[10]-=-. Given an unweighted connected graph G = (V, E), the local density of G, denoted #(G), 3 is the smallest value # such that |B(v, r)| # # r for all v # V, r > 0. It is easy to see that since G is unwe... |

79 |
Plongements lipschitziens dans
- Assouad
- 1983
(Show Context)
Citation Context ...re specifically, we examine how the "volume growth" of a metric affects its embeddability into L p spaces. The notion of growth that we use is well-studied, and is very similar to a notion o=-=f Assouad [2]-=-, see also [13]. Our definition is technically slightly different from Assouad's, but the flavor is left unaltered; in particular, the notion of bounded growth is equivalent under either framework. Fo... |

67 |
The metrical interpretation of superreflexivity in Banach spaces
- Bourgain
- 1986
(Show Context)
Citation Context ...ibits a very natural class of metric spaces which embed into # 2 with O(1) distortion, but not isometrically. As discussed before, some tree metrics require# # log log n) distortion to embed into # 2 =-=[5]-=-, while we prove that some doubling metrics require distortion # # log n) (see Section 5). Thus it is precisely the synthesis of these two properties that yields an enormous improvement in embeddabili... |

64 | Approximation algorithms for the 0-extension problem
- Calinescu, Karloff, et al.
(Show Context)
Citation Context ...oubling dimension. Such decompositions are the main tool in many embedding results, as well as a number of other applications. To construct these decompositions, we adapt a probabilistic technique of =-=[7]-=-. For applications of the decomposition later in the paper, it is important that the probability space that we sample from be very compactly defined (e.g., of size O(1) for the case of doubling metric... |

63 | Embedding the diamond graph in Lp and dimension reduction in L1
- Lee, Naor
(Show Context)
Citation Context ...ing (see Section 5). The fact that these results can be extended to constant dimension is even more surprising, especially since recent results of Brinkman and Charikar [6] (see also a short proof of =-=[23]-=-, which even generalizes to the metrics exhibited in Section 5 [22]) show extremely strong lower bounds on the dimension required to embed simple series-parallel graphs into # 1 . On a high level, our... |

63 | Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics - Matouˇsek - 2002 |

50 |
distortion and volume preserving embeddings for planar and Euclidean metrics
- Small
- 1999
(Show Context)
Citation Context ...very tree metric embeds isometrically into L 1 . Matou sek [27] showed that every tree embeds into L p with distortion O((log log n) min( 1 2 , 1 p ) ) and that this bound is tight for all p > 1. Rao =-=[32]-=- showed that every planar graph embeds into L 2 with distortion O( # log n), and this in fact holds for any family which excludes a fixed minor. A matching lower bound, yielded by a family of series-p... |

49 |
Ahlfors Q-regular spaces with arbitrary Q > 1 admiting weak Poincare
- Laakso
- 2000
(Show Context)
Citation Context ...L 2 with distortion O( # log n), and this in fact holds for any family which excludes a fixed minor. A matching lower bound, yielded by a family of series-parallel metrics was given in [29] (see also =-=[19, 21]-=-). Gupta et al. [12] show that K 4 -free (series-parallel) and K 2,3 -free (outerplanar) graphs embed into L 1 with constant distortion. Here, we consider restrictions not on the topology of the metri... |

46 | Finite metric spaces - combinatorics, geometry and algorithms
- Linial
- 2002
(Show Context)
Citation Context ...ects of study lying at the intersection of analysis, combinatorics, and geometry, the ideas and techniques generated in this field have led to a number of powerful algorithmic applications (see, e.g. =-=[14, 24, 28]-=-). We consider embeddings of finite metric spaces into L p spaces. Given a metric (X, d), the goal is to find a map f : X # L p such that ||f (x)-f(y)|| p is close to d(x, y) for all x, y # X . The wo... |

36 | A lower bound on the distortion of embedding planar metrics into Euclidean space, Discrete Comput
- Newman, Rabinovich
(Show Context)
Citation Context ...ph embeds into L 2 with distortion O( # log n), and this in fact holds for any family which excludes a fixed minor. A matching lower bound, yielded by a family of series-parallel metrics was given in =-=[29]-=- (see also [19, 21]). Gupta et al. [12] show that K 4 -free (series-parallel) and K 2,3 -free (outerplanar) graphs embed into L 1 with constant distortion. Here, we consider restrictions not on the to... |

36 |
On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞-weights
- Semmes
- 1996
(Show Context)
Citation Context ... give an algorithmic version of Assouad's proof, and drastically improve the dependence of k and D to near-linear Assouad also conjectured that the above result holds even when # = 1. Although Semmes =-=[33]-=- disproved this conjecture, we have shown that it holds whenever (X, d) is a doubling tree metric. In Section 5, we exhibit a family of series-parallel doubling metrics which requires# # log n) distor... |

33 | On embedding trees into uniformly convex Banach spaces
- Matoušek
- 1999
(Show Context)
Citation Context ...ate connection with multicommodity flows and approximations for the sparsest cut, seee.g. [26, 3, 13, 33]. It is not too difficult to see that every tree metric embeds isometrically into L1. Matou^sek=-=[28]-=- showed that every tree embeds into Lp with distortion O((log log n)min( 12 , 1 p )) and that this bound is tight for all p > 1. Rao [34] showed that every planar graph embedsinto L2 with distortion O... |

32 |
Bilipschitz embeddings of metric spaces into space forms
- Lang, Plaut
- 2001
(Show Context)
Citation Context ...L 2 with distortion O( # log n), and this in fact holds for any family which excludes a fixed minor. A matching lower bound, yielded by a family of series-parallel metrics was given in [29] (see also =-=[19, 21]-=-). Gupta et al. [12] show that K 4 -free (series-parallel) and K 2,3 -free (outerplanar) graphs embed into L 1 with constant distortion. Here, we consider restrictions not on the topology of the metri... |

30 |
Plongements lipschitziens dans Rn
- Assouad
- 1983
(Show Context)
Citation Context ...ore specifically, we examine how the “volume growth” of a metric affects its embeddability into Lp spaces. The notion of growth that we use is well-studied, and is very similar to a notion of Assouad =-=[2]-=-, see also [14]. Our definition is technically slightly different from Assouad’s, but the flavor is left unaltered; in particular, the notion of bounded growth is equivalent under either framework. Fo... |

29 |
Plane with A∞-weighted metric not bi-Lipschitz embeddable to RN
- Laakso
(Show Context)
Citation Context ...om trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane due to Laakso =-=[21]-=-. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L2. 1 Introduction A basic goal in the study of finite metric spaces is to approxima... |

28 | Metric structures in L1: Dimension, snowflakes, and average distortion
- Lee, Mendel, et al.
- 2003
(Show Context)
Citation Context ...to constantdimension is even more surprising, especially since recent results of Brinkman and Charikar [6] (see also a short proofof [24], which even generalizes to the metrics exhibited in Section 5 =-=[23]-=-) show extremely strong lower bounds on thedimension required to embed simple series-parallel graphs into `1.On a high level, our embeddings consist of two steps. Without loss of generality, we can as... |

28 | On the impossibility of dimension reduction in ℓ1 - Brinkman, Charikar - 2003 |

25 |
Lectures on Discrete Geometry, volume 212 of Graduate Texts in Mathematics
- Matoušek
- 2002
(Show Context)
Citation Context ...jects of study lying at the intersection of analysis, combinatorics, and geometry, the ideas and techniques generated in this field have led to a number of powerful algorithmic applications (see e.g. =-=[15, 25, 29]-=-). We consider embeddings of finite metric spaces into Lp spaces. Given a metric (X, d), the goal is to find a map f : X → Lp such that ||f(x)−f(y)||p is close to d(x, y) for all x, y ∈ X . The worst-... |

24 | The intrinsic dimensionality of graphs
- Krauthgamer, Lee
- 2003
(Show Context)
Citation Context ... in the paper, it is important that the probability space that we sample from be very compactly defined (e.g., of size O(1) for the case of doubling metrics). For this purpose, we use some ideas from =-=[18]-=-, in conjunction with Lov asz Local Lemma, in order to exploit certain locality properties of our decomposition. Our use of the local lemma here, and elsewhere in the paper, can be made algorithmic us... |

24 | On embedding expanders into lp spaces - Matouˇsek - 1997 |

23 | On embedding trees into uniformly convex Banach spaces - Matouˇsek - 1999 |

22 |
Cuts, trees and l1-embeddings of graphs
- Gupta, Newman, et al.
- 1999
(Show Context)
Citation Context ... the restrictions considered have been mostly topological. For L1 embeddings, this is due partly to the intimate connection with multicommodity flows and approximations for the sparsest cut, see e.g. =-=[26, 3, 13, 33]-=-. It is not too difficult to see that every tree metric embeds isometrically into L1. Matoušek [28] showed that every tree embeds into Lp with distortion O((log logn)min( 1 2 , 1 p )) and that this b... |

18 | Cuts, trees and `1-embeddings of graphs
- Gupta, Newman, et al.
(Show Context)
Citation Context ...the restrictions considered have been mostly topological. For L 1 embeddings, this is due partly to the intimate connection with multicommodity flows and approximations for the sparsest cut, see e.g. =-=[25, 3, 12]-=-.It is not too difficult to see that every tree metric embeds isometrically into L 1 . Matou sek [27] showed that every tree embeds into L p with distortion O((log log n) min( 1 2 , 1 p ) ) and that t... |

17 | Graphs with small bandwidth and cutwidth
- Chung, Seymour
- 1989
(Show Context)
Citation Context ...ons of Computer Science (FOCS’03)s0272-5428/03 $17.00 © 2003 IEEEsLocal density. Finally, there is another natural notion of volume, which has been used widely in the study of the bandwidth of graphs =-=[8, 11]-=-. Given an unweighted connected graph G = (V,E), the local density of G, denoted β(G), is the smallest value β such that |B(v, r)| ≤ β r for all v ∈ V, r > 0. It is easy to see that since G is unweigh... |

15 |
On the impossibility of dimension reduction in 1
- Brinkman, Charikar
- 2003
(Show Context)
Citation Context ...tortion in any Euclidean embedding (see Section 5). The fact that these results can be extended to constant dimension is even more surprising, especially since recent results of Brinkman and Charikar =-=[6]-=- (see also a short proof of [24], which even generalizes to the metrics exhibited in Section 5 [23]) show extremely strong lower bounds on the dimension required to embed simple series-parallel graphs... |

12 |
Improved bandwidth approximation for trees and chordal graphs
- Gupta
(Show Context)
Citation Context ...trees and O(log 3 n) for general doubling graphs, improving over the general case by # log n and # log log n factors respectively. Furthermore, our tree coloring in conjunction with a modification of =-=[11]-=- show that the bandwidth of a graph with local density # is at most O(# 1.5 log 2 n). Distance-labelings of graphs assign labels to vertices so that the distance between two vertices can be computed (... |

12 |
On average distortion of embedding metrics into the line and into l1
- Rabinovich
- 2003
(Show Context)
Citation Context ... the restrictions considered have been mostly topological. For L1 embeddings, this is due partly to the intimate connection with multicommodity flows and approximations for the sparsest cut, see e.g. =-=[26, 3, 13, 33]-=-. It is not too difficult to see that every tree metric embeds isometrically into L1. Matoušek [28] showed that every tree embeds into Lp with distortion O((log logn)min( 1 2 , 1 p )) and that this b... |

11 |
On embedding expanders into lp spaces
- Matoušek
- 1997
(Show Context)
Citation Context ...o Lp with O(log n) distortion for any fixed p; it is shown in [26] that this bound is tight, for all p ≤ 2, for the shortest path metric on constant-degree expander graphs. This was later extended in =-=[27]-=-, showing a tight upper bound of O( log np ) for any Lp space. In light of this, a significant amount of effort has been made to understand the distortion achievable for restricted classes of metric s... |

9 |
On the impossibility of dimension reduction
- Brinkman, Charikar
(Show Context)
Citation Context ...tortion in any Euclidean embedding (see Section 5). The fact that these results can be extended to constant dimension is even more surprising, especially since recent results of Brinkman and Charikar =-=[6]-=- (see also a short proof of [23], which even generalizes to the metrics exhibited in Section 5 [22]) show extremely strong lower bounds on the dimension required to embed simple series-parallel graphs... |

3 |
Plane with A∞ -weighted metric not bi-Lipschitz embeddable to R
- Laakso
- 2002
(Show Context)
Citation Context ...om trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane due to Laakso =-=[20]-=-. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L 2 . 1 Introduction A basic goal in the study of finite metric spaces is to approxi... |

3 | Embedding the diamond graph in and dimension reduction in - Lee, Naor - 2003 |

2 |
Metric structures in L 1 : Dimension, snowflakes, and average distortion
- Lee, Mendel, et al.
- 2003
(Show Context)
Citation Context ... constant dimension is even more surprising, especially since recent results of Brinkman and Charikar [6] (see also a short proof of [23], which even generalizes to the metrics exhibited in Section 5 =-=[22]-=-) show extremely strong lower bounds on the dimension required to embed simple series-parallel graphs into # 1 . On a high level, our embeddings consist of two steps. Without loss of generality, we ca... |

2 |
Accessing NearbyCopies of Replicated Objects in a Distributed Environment
- Plaxton, Rajaraman, et al.
- 1997
(Show Context)
Citation Context ...n the input data re-sides on a low-dimensional manifold, cf. [36]). In fact, various algorithms can be tailored to run efficiently on cer-tain classes of growth-restricted metrics, as demonstrated in =-=[9, 32, 17, 19]-=-. The metrics considered there are eitherequivalent to or a subclass of those metrics which are doubling; see Section 1.3. 1.1 Results and techniques We are concerned with the broad roles of "volume" ... |

2 | trees and l -embeddings of graphs - Gupta, Newman, et al. - 2004 |

2 | On embedding expanders into spaces - Matouˇsek - 1997 |