Computing Projective Clusters via Certificates Cecilia Procopiuc - - PowerPoint PPT Presentation

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Computing Projective Clusters via Certificates Cecilia Procopiuc AT&T Labs (joint work with Pankaj Agarwal and Kasturi Varadarajan) Applications Shape Fitting Database Indexing Information Retrieval Data

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SLIDE 1

Computing Projective Clusters via Certificates Cecilia Procopiuc AT&T Labs (joint work with Pankaj Agarwal and Kasturi Varadarajan)

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SLIDE 2
  • Shape Fitting
  • Database Indexing
  • Information Retrieval
  • Data Compression
  • Image Processing

Applications

Computing Projective Clusters via Certificates 1

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SLIDE 3

Example

Computing Projective Clusters via Certificates 2

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SLIDE 4

Example

Computing Projective Clusters via Certificates 3

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SLIDE 5
  • : set of
✁

points

  • ✂

: integer

✂
  • Line-Center: Find
✂

lines

✄✆☎✞✝ ✟ ✟ ✟ ✝ ✄✡✠

that minimize

☛ ☞ ✌ ✍ ✎ ✏ ☛ ✑✓✒ ☎ ✔ ✕ ✔ ✠ ✖ ✗✙✘ ✝ ✄✛✚ ✜ ✟ ✢ ✣ ✤

minimum value so that

  • can be

covered by

✂

hyper-cylinders of diameter

✢ ✣ ✟

Projective Clustering: Find

✥
  • dimensional flats
✦ ☎✞✝ ✟ ✟ ✟ ✝ ✦ ✠

, for some integer

✥

.

Definition

Computing Projective Clusters via Certificates 4

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SLIDE 6
  • 1. Most variants of projective clustering problems are NP-Hard:

Meggido and Tamir ’82.

2.

✖ ✤
  • ✝
✁

,

✂ ✤ ✂ ✝
  • : Houle & Toussaint ’98, Agarwal & Sharir ’96,

Jaromczyk & Kowaluk ’95.

3.

✂ ✤ ✂

, general

✖

,

✗ ✂ ✄✆☎ ✜
  • approx.:
  • ✥
✤ ✖ ✝ ✂

(width): Duncan et al. ’97, Chan ’00.

  • ✥
✤ ✂

(enclosing cyl.):Har-Peled & Varadarajan ’01, B˘

adoiu et al. ’02.

  • general
✥

: Har-Peled & Varadarajan ’03.

  • 4. General
✂

and

✖

:

  • ✞
✗ ✖ ✂✟ ✠ ✡ ✂ ✜

hyper-cylinders of diameter

☛ ✢ ✣

in

☞ ✞ ✗ ✖ ✁ ✂ ✌ ✜

time:

Agarwal & Procopiuc ’00

  • ✂

hyper-cylinders of diameter

✗ ✂ ✄ ☎ ✜ ✢ ✣

in

☞ ✞ ✗ ✁ ✍ ✗ ✂ ✝ ✖ ✝ ☎ ✜ ✜

time:

Agarwal, Procopiuc & Varadarajan ’02.

  • ✂
✥
  • flats of diameter
✗ ✂ ✄✆☎ ✜ ✢ ✣

in

✖ ✁ ✎ ✏✒✑ ✏ ✠✔✓ ✕ ✓ ✖ ✗ ✗

time: Har-Peled &

Varadarajan ’02.

Results

Computing Projective Clusters via Certificates 5

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SLIDE 7

For each flat

✦

in optimal cover, there exists small subset

✂✁

s.t.

✄☎ ✆ ✄ ✘ ✝✞ ✟ ✗ ✠✁ ✜

contains

☎
  • approx. flat.
  • ✁

: core-set of

✦

.

✡ ☛☞✁
  • ✁
✡ ✤ ✍ ✗ ✂ ✝ ✥ ✝ ☎ ✜

: independent of

✁

and

✖

!

  • 1. Find core-sets
  • ✁

(brute force enumeration).

  • 2. Compute
☎
  • approx. solution (brute force).

Core-Sets (Har-Peled & Varadarajan)

Computing Projective Clusters via Certificates 6

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SLIDE 8

There exists small subset

  • s.t.
  • covered by
✂

congruent hyper-cylinders

  • covered by the
☎
  • expanded hyper-cylinders.
  • : certificate of
  • .
✡
  • ✡
✤ ✍ ✗ ✂ ✝ ☎ ✝ ✖ ✜

: independent of

✁

!

  • 1. Find certificate
  • (iterative sampling).
  • 2. Compute optimal solution on
  • (brute force).
  • 3. Expand to solution on
  • .

Certificates (Agarwal, Procopiuc & Varadarajan)

Computing Projective Clusters via Certificates 7

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SLIDE 9
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 8

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SLIDE 10
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 9

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SLIDE 11
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 10

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SLIDE 12
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 11

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SLIDE 13
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 12

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SLIDE 14
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 13

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SLIDE 15
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 14

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SLIDE 16
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 15

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SLIDE 17
  • ✁
✁ ✂ ✄
  • Strip Certificate

Computing Projective Clusters via Certificates 16

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SLIDE 18
  • Strip Certificate

Computing Projective Clusters via Certificates 17

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SLIDE 19
  • Strip Certificate

Computing Projective Clusters via Certificates 18

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SLIDE 20
  • Strip Certificate

Computing Projective Clusters via Certificates 19

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SLIDE 21
  • Strip Certificate

Computing Projective Clusters via Certificates 20

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SLIDE 22
  • : set of points on real line.
  • ✁
  • :
✂
  • certificate if any
✂

intervals that cover

  • can be
☎
  • expanded

to cover

  • .

Claim: A

✂
  • strip certificate can be obtained from the union of
✂
  • certificates of all grid lines.

Line Certificate

Computing Projective Clusters via Certificates 21

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SLIDE 23
  • ☎
  • ✁
  • ✂
✤
  • Line Certificate

Computing Projective Clusters via Certificates 22

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SLIDE 24
  • ☎
  • ✁
  • ✂
✤
  • Line Certificate

Computing Projective Clusters via Certificates 23

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SLIDE 25
  • ☎
  • ✁
  • ✂
✤
  • Line Certificate

Computing Projective Clusters via Certificates 24

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SLIDE 26

Lemma 1: For any set of points in

  • , there exists a line certificate of size
✗ ✂ ✁ ☎ ✜ ✎ ✏ ✠ ✗

. Lemma 2: For any set of points in

  • ✁

, there exists a certificate of size

✂ ✎ ✏ ✠ ✗ ✁ ☎ ✎ ✏ ✁ ✂ ✠ ✗

.

  • Iterative random sampling

Line Certificate

Computing Projective Clusters via Certificates 25

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SLIDE 27

Open Problems

  • 1. Certificates of smaller size?
  • 2. Constructive proof for certificates.
  • 3. Extensions to
✥
  • flats.

Line Certificate

Computing Projective Clusters via Certificates 26