Computing Projective Clusters via Certificates Cecilia Procopiuc - PowerPoint PPT Presentation

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 Compression Image Processing Computing Projective Clusters via Certificates 1

Example Computing Projective Clusters via Certificates 2

Example Computing Projective Clusters via Certificates 3

✝ ✖ ✏ ☛ ✥ ☎ ✔ ✕ ✔ ✠ ✟ ✍ � ✣ ✜ ✟ ✢ ✣ ✤ ✢ ✎ ✌ ✂ ✂ ✁ ✥ ✠ ✂ ✦ ✂ ✝ ✟ ☞ ✟ ✟ ✟ ✟ ✝ ✟ ✦ ☛ � Definition - : 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 . Computing Projective Clusters via Certificates 4

☞ ✡ ✖ ✣ ✢ ☛ ✝ ✜ ✂ ✠ ✞ ✂ ✟ ✖ ✗ ✞ ☎ ✜ ✖ ✜ ✝ ✗ ✂ ✗ ☞ ✁ ✣ ✢ ✜ ☎ ✄ ✂ ✍ ✖ ✂ � ✗ ✂ ✜ ✌ ✂ ✁ ✂ ✥ ✗ ✝ ✂ ✤ ✂ ✓ ✖ ✗ � ✂ ✖ ✤ ✂ ✗ ✁ ✝ � ✤ ✖ ✕ ✏ ✥ ✗ ✗ ✂ ✜ ✂ ✤ ✥ ✢ ✣ ✂ ✝ ✖ ✤ ✥ ✖ ✁ ✜ ✎ ✂ ✞ Results 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. Computing Projective Clusters via Certificates 5

✁ � ✂ ✗ ✍ ✤ ✡ ✦ � ✝ ✡ ☎ ✦ ✜ ✁ ✁ ✥ ☎ ✖ ✜ � ✗ ✟ ✝✞ ✘ ✄ ✆ ✄☎ ✁ ☎ ✝ Core-Sets (Har-Peled & Varadarajan) 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). Computing Projective Clusters via Certificates 6

✗ � ✂ � ✍ ✤ ✡ � ✡ ✝ � ✖ ✜ ☎ ☎ ✁ � � � � ✂ � � ✝ Certificates (Agarwal, Procopiuc & Varadarajan) 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 . Computing Projective Clusters via Certificates 7

� ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 8

� ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 9

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 10

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 11

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 12

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 13

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 14

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 15

� � ✁ ✁ ✂ ✄ -Strip Certificate Computing Projective Clusters via Certificates 16

� -Strip Certificate Computing Projective Clusters via Certificates 17

� -Strip Certificate Computing Projective Clusters via Certificates 18

� -Strip Certificate Computing Projective Clusters via Certificates 19

� -Strip Certificate Computing Projective Clusters via Certificates 20

� ✂ ✂ ☎ � � ✂ � ✂ � ✁ � � Line Certificate : 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. Computing Projective Clusters via Certificates 21

� ☎ � ✁ � ✂ ✤ � Line Certificate Computing Projective Clusters via Certificates 22

� ☎ � ✁ � ✂ ✤ � Line Certificate Computing Projective Clusters via Certificates 23

� ☎ � ✁ � ✂ ✤ � Line Certificate Computing Projective Clusters via Certificates 24

✎ � ✏ ✎ ☎ ✁ ✗ ✠ ✏ � ✂ ✠ ✁ ✗ ✂ � ✗ ✠ ✏ ✎ ✜ ☎ ✁ ✂ ✗ ✁ Line Certificate 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 Computing Projective Clusters via Certificates 25

✥ Line Certificate Open Problems 1. Certificates of smaller size? 2. Constructive proof for certificates. 3. Extensions to -flats. Computing Projective Clusters via Certificates 26