On the impossibility of dimension reduction in /spl lscr//sub 1/

Brinkman, Bo; Charikar, Moses

doi:10.1109/sfcs.2003.1238224

Cited by 10 publications

(19 citation statements)

References 22 publications

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“…This result is meaningful since, although ℓ 2 isometrically embeds into L p for every 1 ≤ p ≤ ∞, there is no known ℓ p analogue of the Johnson-Lindenstrauss dimension reduction lemma [31] (in fact, the Johnson-Lindenstrauss lemma is known to fail in ℓ 1 [19], [33]). These bounds are almost best possible.…”

Section: Theorem 15 (Near Tightness)mentioning

confidence: 99%

On metric ramsey-type phenomena

Bartal

Linial

Mendel

2003

Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing

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This paper deals with Ramsey-type theorems for metric spaces. Such a theorem states that every n point metric space contains a large subspace which can be embedded with some fixed distortion in a metric space from some special class.Our main theorem states that for any > 0, every n point metric space contains a subspace of size at least n 1− which is embeddable in an ultrametric with O( log(1/ ) ) distortion. This in particular provides a bound for embedding in Euclidean spaces. The bound on the distortion is tight up to the log(1/ ) factor even for embedding in arbitrary Euclidean spaces. This result can be viewed as a non-linear analog of Dvoretzky's theorem, a cornerstone of modern Banach space theory and convex geometry.Our main Ramsey-type theorem and techniques naturally extend to give theorems for classes of hierarchically wellseparated trees which have algorithmic implications, and can be viewed as the solution of a natural clustering problem.We further include a comprehensive study of various other aspects of the metric Ramsey problem. The full version appears mostly in a manuscript entitled the same. Some parts appear in the papers entitled "On some low distortion metric Ramsey problems", "On Fréchet embedding of metric spaces", and "On Lipschitz embedding of ultrametrics in low dimension" by the same authors. The manuscripts can be downloaded from

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Section: Theorem 15 (Near Tightness)mentioning

confidence: 99%

On metric ramsey-type phenomena

Bartal

Linial

Mendel

2003

Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing

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“…Nevertheless, it required substantial effort to even show that, say, one has k α n ( 1 ) n/2 for some universal constant α: This is achieved in the forthcoming work [4] which obtains the estimate k α n ( 1 ) n/α. The question whether 1 admits metric dimension reduction was open for many years, until it was resolved negatively in [17] by showing that there exists a universal constant 1 Formally, for the purpose of efficient approximate nearest neighbor search one cannot use a dimension reduction statement like (2) as a "black box" without additional information about the low-dimensional embedding itself rather than its mere existence. One would want the embedding to be fast to compute and "data oblivious," as in the classical Johnson-Lindenstrauss lemma [44].…”

Section: Introductionmentioning

confidence: 99%

“…Thus, S1 does not admit a dimension reduction result á la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with S1 replaced by the Banach space 1 of absolutely summable sequences via the work of Brinkman and Charikar (2003).In fact, the above set C can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in S1. The challenge is to demonstrate that C cannot be faithfully realized in an arbitrary low-dimensional subspace of S1, while Brinkman and Charikar obtained such an assertion only for subspaces of S1 that consist of diagonal operators (i.e., subspaces of 1).…”

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confidence: 98%

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confidence: 99%

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Impossibility of dimension reduction in the nuclear norm

Pisier

Scheclitman

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abstractvet S 1 @the htten!von xeumnn tre lssA denote the fnh spe of ll ompt liner opertors T : 2 → 2 whose nuler norm T S 1 = ∞ j=1 σ j (T ) is (niteD where {σ j (T )} ∞ j=1 re the singulr vlues of T F e prove tht for ritrrily lrge n ∈ N there exists suset C ⊆ S 1 with |C| = n tht nnot e emedded with iEvipshitz distortion O(1) into ny n o(1) Edimensionl liner suspe of S 1 F C is not even O(1)Evipshitz quotient of ny suset of ny n o(1) Edimensionl liner suspe of S 1 F husD S 1 does not dmit dimension redution result á l tohnson nd vindenstruss @IWVRAD whih omplements the work of rrrowD wontnro nd hort @PHIIA on the limittions of quntum dimension redution under the ssumption tht the emedding into low dimensions is quntum hnnelF uh sttement ws previously known with S 1 repled y the fnh spe 1 of solutely summle sequenes vi the work of frinkmn nd ghrikr @PHHQAF sn ftD the ove set C n e tken to e the sme set s the one tht frinkmn nd ghrikr onsideredD viewed s olletion of digonl mtries in S 1 F he hllenge is to demonstrte tht C nnot e fithfully relized in n ritrry lowEdimensionl suspe of S 1 D while frinkmn nd ghrikr otined suh n ssertion only for suspes of S 1 tht onsist of digonl opertors @iFeFD suspes of 1 AF e estlish this y proving tht the wrkov PEonvexity onstnt of ny (nite dimensionl liner suspe X of S 1 is t most universl onstnt multiple of log dim(X)F

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“…Cauchy C(0, 1) [9]. However, the recent impossibility result [5] has ruled out estimators that could be metrics for dimension reduction in l 1 .…”

Section: A Brief Introduction To Random Projectionsmentioning

confidence: 99%

Conditional Random Sampling: A Sketch-based Sampling Technique for Sparse Data

Li¹,

Church²,

Hastie³

2007

Advances in Neural Information Processing Systems 19

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We 1 develop Conditional Random Sampling (CRS), a technique particularly suitable for sparse data. In large-scale applications, the data are often highly sparse. CRS combines sketching and sampling in that it converts sketches of the data into conditional random samples online in the estimation stage, with the sample size determined retrospectively. This paper focuses on approximating pairwise l 2 and l 1 distances and comparing CRS with random projections. For boolean (0/1) data, CRS is provably better than random projections. We show using real-world data that CRS often outperforms random projections. This technique can be applied in learning, data mining, information retrieval, and database query optimizations.

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On the impossibility of dimension reduction in /spl lscr//sub 1/

Cited by 10 publications

References 22 publications

On metric ramsey-type phenomena

On metric ramsey-type phenomena

Impossibility of dimension reduction in the nuclear norm

Conditional Random Sampling: A Sketch-based Sampling Technique for Sparse Data

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