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Distance scales, embeddings, and efficient relaxations of the cut cone

Distance scales, embeddings, and efficient relaxations of the cut cone. James R. Lee. University of California, Berkeley. In general, we will consider euclidean embeddings and the geometry of high dimensional euclidean spaces. euclidean embeddings and geometry. Euclidean embedding:

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Distance scales, embeddings, and efficient relaxations of the cut cone

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  1. Distance scales, embeddings, and efficient relaxations of the cut cone James R. Lee University of California, Berkeley

  2. In general, we will consider euclidean embeddings and the geometry of high dimensional euclidean spaces. euclidean embeddings and geometry Euclidean embedding: A map f : X !Rk for some metric space (X,d). Why study this kind of geometry (in CS)? Distortion: Smallest number C such that… • Applicability of low-distortion Euclidean embeddings • Understanding semi-definite programs • Optimization, harmonic analysis, hardness of approximation, • cuts and flows, Markov chains, expansion, randomness…

  3. Offers some improvements on two nice ideas… this paper I.Measured descent Gluing Lemma [Krauthgamer-L-Mendel-Naor 04] II.Arora-Rao-Vazirani “Big Core” Theorem

  4. Our initial motivation was the conjecture: Every n-point metric of negative type embeds into a Euclidean space (L2 norm) with distortion , and this is tight. goals and results (known to imply a similar approximation for the general SparsestCut problem) We give improved embeddings of…

  5. Our initial motivation was the conjecture: Every n-point metric of negative type embeds into a Euclidean space (L2 norm) with distortion , and this is tight. goals and results Applications… Bandwidth, Euclidean volume-resp. embeddings (GL) Min-UnCut, 2CNF-deletion (BCT) [Agarwal, Charikar, Makarychev, Makarychev 04] vertex cover vanishing term (BCT) [Karakostas 04] general SparsestCut O(log n)3/4(BCT) [Chawla, Gupta, Racke 04] general SparsestCut O(log n log log n)1/2(BCT, GL) [Arora, L, Naor 04]

  6. Every metric space (X,d) is composed of many scales (resolutions)… the gluing lemma Often easy to construct an embedding for one scale:

  7. Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion. the gluing lemma Problem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddingsapart and puts them back together in complicated ways. Often easy to construct an embedding for one scale:

  8. Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion. the gluing lemma Problem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddingsapart and puts them back together in complicated ways. GLUING LEMMA:We can construct an embedding of the same quality using the component maps as a black box. Quantitatively, the distortion of the final map is . PROOF:

  9. Metrics of negative type arise as follows: metrics of negative type Let X µRk be endowed with the distance function If (X,d) is a metric space, then X is a space of negative type. x · 90o y z The family of negative type metrics is much bigger than the family of Euclidean metrics!

  10. THE ARV THEOREM the “big core” theorem Let X µ Sn-1 be an n-point subset of the unit sphere endowed with the squared Euclidean metricd(x,y) = ||x-y||2, and suppose that A Then there exist two subsets A, B µ X such that |A|,|B| ¸(n) and B Good enough for lots of applications… but not others. Improved version:Possible to choose the sets A and B “at random.”

  11. Randomized version: Choosing A and B “at random.” the “big core” theorem 1. Choose a random hyperplane. A 2. Prune the “exceptions.” B Pruning)d(A,B) is large. The hard part is showing that |A|,|B| = (n) whp. ARV yields . We obtain the optimal bound: .

  12. The proof is a modification of the ARV geometric chaining argument… the “big core” theorem Idea:If the matchings are large in most directions, then a large subset of points must be matching against each other in most directions. THE CORE CONTRADICTION! Theorem: Proof uses region growing to construct longer chains. (with param )

  13. A B open questions Does a better gluing lemma exist for L1? Can we pass from “deterministic” to “random” separators in a black-box manner? QUESTIONS?

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