Optimization in pattern analysis Optimization approaches in pattern

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Optimization in pattern analysis Optimization approaches in pattern analysis Patterns in vector spaces Tijl

Optimization in pattern analysis Optimization approaches in pattern analysis Patterns in vector spaces Tijl De Bie K. U. Leuven / Southampton University School on pattern analysis - Erice

Optimization in pattern analysis Pattern analysis Overview • Pattern analysis – Pattern functions –

Optimization in pattern analysis Pattern analysis Overview • Pattern analysis – Pattern functions – Pattern matching – Pattern discovery Patterns in discrete spaces • Patterns in discrete spaces – Patterns in sets – Patterns in strings – Patterns in graphs Patterns in vector spaces • Patterns in vector spaces – Least squares problems – Convex optimization – Eigenvalue problems 9/9/2020 School on pattern analysis - Erice 2

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? santic.

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? santic. vinym. thyi. thapna. mycym Pattern discovery halch. zethui. thivacl. cesasin. thumsa. cilva neri. canva. carsi. pytnam. thycal. atnam tei. lena. haustish. enac. eshi. catnis. heci spurta. sylsle. napti. thui. laiscla. hechz. neri une. mlach nthi acal vacl. vinum. shantishtsh. celi. pen. trytym thi. thapneshtsh. trytanasha. hanthin. celi Patterns in discrete spaces tyr. hetym. vinym. thic. vacl. hechz. etnam ich. matam. cnticn. thsepen. teshamitn murce. thi. nunthen. etnam. thi. tryth. etnam hanthin. etnam. celycn. etnam. athumitn pethereni. eslem. zathrum. mur. in. velthinesh cilthsh. vacl. ara. thui. useti. catneti. slapichyn slapinash. favin. yfli. spyrta. eisna. hinthy cla. thesns. eslem. cealchus. etnam. aisna. cesal tychlac. ethri. syntnam. cecha cntnam. thesan. fler. veivesh. thezeri etnam. aisna. esa. ich. huthish. zathrumish flerchve. trin. nethunshl. cn. thunt. eitylvar. thunem. cealchus. etnam. ich. eslem. cialchysh than etnam. thesan flanac. farsi tunt. enac. etnam. athymica. thlupcva ceshum. tei. lanti. ininc. eshi. tei. chimth streta. satrs. enash. ucy. hamphetish. rinush thui. arash. mucum. ania. chesh. rasna. hilar cateyth. hamphet. etnam Patterns in vector spaces aisna. ich. nac. reushce. aiserash. sheush thunchulem. muth. hilarthune. etertic cathre. chim. enach. unchva. methlumth. puts muth. hilarthuna. tecum. etrinthi. muth nafaucn. unchva. hetum. hilarthuna. thenth hyrsic. caplthy. cecham. enac. eisna. hinthy hetym. hilarthuma. etertic. cathra etnam. aisna. ich. matam /// vacltnam thunem. cialchush. masn. ynialti. ursmnal athre. acil. an. shacnicn. cilth. cecha. sal cus. cluce. caperi. zamtic. svem. thumsa matan. clyctrash. hilar. 9/9/2020 School on pattern analysis - Erice 3

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? Pattern

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? Pattern discovery Patterns in discrete spaces Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 4

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? santic.

Optimization in pattern analysis Pattern functions Pattern matching A framework for pattern analysis? santic. vinym. thyi. thapna. mycym Pattern discovery halch. zethui. thivacl. cesasin. thumsa. cilva neri. canva. carsi. pytnam. thycal. atnam tei. lena. haustish. enac. eshi. catnis. heci spurta. sylsle. napti. thui. laiscla. hechz. neri une. mlach nthi acal vacl. vinum. shantishtsh. celi. pen. trytym thi. thapneshtsh. trytanasha. hanthin. celi Patterns in discrete spaces tyr. hetym. vinym. thic. vacl. hechz. etnam ich. matam. cnticn. thsepen. teshamitn murce. thi. nunthen. etnam. thi. tryth. etnam hanthin. etnam. celycn. etnam. athumitn pethereni. eslem. zathrum. mur. in. velthinesh cilthsh. vacl. ara. thui. useti. catneti. slapichyn slapinash. favin. yfli. spyrta. eisna. hinthy cla. thesns. eslem. cealchus. etnam. aisna. cesal tychlac. ethri. syntnam. cecha cntnam. thesan. fler. veivesh. thezeri etnam. aisna. esa. ich. huthish. zathrumish flerchve. trin. nethunshl. cn. thunt. eitylvar. thunem. cealchus. etnam. ich. eslem. cialchysh than etnam. thesan flanac. farsi tunt. enac. etnam. athymica. thlupcva ceshum. tei. lanti. ininc. eshi. tei. chimth streta. satrs. enash. ucy. hamphetish. rinush thui. arash. mucum. ania. chesh. rasna. hilar cateyth. hamphet. etnam Patterns in vector spaces aisna. ich. nac. reushce. aiserash. sheush thunchulem. muth. hilarthune. etertic cathre. chim. enach. unchva. methlumth. puts muth. hilarthuna. tecum. etrinthi. muth nafaucn. unchva. hetum. hilarthuna. thenth hyrsic. caplthy. cecham. enac. eisna. hinthy hetym. hilarthuma. etertic. cathra etnam. aisna. ich. matam /// vacltnam thunem. cialchush. masn. ynialti. ursmnal athre. acil. an. shacnicn. cilth. cecha. sal cus. cluce. caperi. zamtic. svem. thumsa matan. clyctrash. hilar. 9/9/2020 School on pattern analysis - Erice 5

Optimization in pattern analysis Pattern functions Pattern matching Pattern discovery Patterns in discrete spaces

Optimization in pattern analysis Pattern functions Pattern matching Pattern discovery Patterns in discrete spaces Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 6

Optimization in pattern analysis Pattern functions Pattern discovery? Pattern matching santic. vinym. thyi. thapna.

Optimization in pattern analysis Pattern functions Pattern discovery? Pattern matching santic. vinym. thyi. thapna. mycym Pattern discovery halch. zethui. thivacl. cesasin. thumsa. cilva neri. canva. carsi. pytnam. thycal. atnam tei. lena. haustish. enac. eshi. catnis. heci spurta. sylsle. napti. thui. laiscla. hechz. neri une. mlach nthi acal vacl. vinum. shantishtsh. celi. pen. trytym thi. thapneshtsh. trytanasha. hanthin. celi Patterns in discrete spaces tyr. hetym. vinym. thic. vacl. hechz. etnam ich. matam. cnticn. thsepen. teshamitn murce. thi. nunthen. etnam. thi. tryth. etnam hanthin. etnam. celycn. etnam. athumitn pethereni. eslem. zathrum. mur. in. velthinesh cilthsh. vacl. ara. thui. useti. catneti. slapichyn slapinash. favin. yfli. spyrta. eisna. hinthy cla. thesns. eslem. cealchus. etnam. aisna. cesal tychlac. ethri. syntnam. cecha cntnam. thesan. fler. veivesh. thezeri etnam. aisna. esa. ich. huthish. zathrumish flerchve. trin. nethunshl. cn. thunt. eitylvar. thunem. cealchus. etnam. ich. eslem. cialchysh than etnam. thesan flanac. farsi tunt. enac. etnam. athymica. thlupcva ceshum. tei. lanti. ininc. eshi. tei. chimth streta. satrs. enash. ucy. hamphetish. rinush thui. arash. mucum. ania. chesh. rasna. hilar cateyth. hamphet. etnam Patterns in vector spaces aisna. ich. nac. reushce. aiserash. sheush thunchulem. muth. hilarthune. etertic cathre. chim. enach. unchva. methlumth. puts muth. hilarthuna. tecum. etrinthi. muth nafaucn. unchva. hetum. hilarthuna. thenth hyrsic. caplthy. cecham. enac. eisna. hinthy hetym. hilarthuma. etertic. cathra etnam. aisna. ich. matam /// vacltnam thunem. cialchush. masn. ynialti. ursmnal athre. acil. an. shacnicn. cilth. cecha. sal cus. cluce. caperi. zamtic. svem. thumsa matan. clyctrash. hilar. 9/9/2020 School on pattern analysis - Erice 7

Optimization in pattern analysis Pattern functions Pattern discovery Pattern matching Pattern discovery Patterns in

Optimization in pattern analysis Pattern functions Pattern discovery Pattern matching Pattern discovery Patterns in discrete spaces Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 8

Optimization in pattern analysis Pattern functions But… Pattern matching Pattern discovery Patterns in discrete

Optimization in pattern analysis Pattern functions But… Pattern matching Pattern discovery Patterns in discrete spaces Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 9

Optimization in pattern analysis Patterns in discrete spaces • Patterns in sets – Apriori

Optimization in pattern analysis Patterns in discrete spaces • Patterns in sets – Apriori algorithm Patterns in discrete spaces • Patterns in strings – Common/frequent substrings – More complex grammars – Probabilistic grammars Patterns in vector spaces • Patterns in graphs 9/9/2020 School on pattern analysis - Erice 10

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 11

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 12

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 13

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings Space of itemsets with items A, B, C, D Graphs AB A B C D AC AD BC BD ABC ABD ACD BCD CD Patterns in vector spaces ABCD 9/9/2020 School on pattern analysis - Erice 14

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings Space of itemsets with items A, B, C, D Graphs AB A B C D AC AD BC BD ABC ABD ACD BCD CD Patterns in vector spaces ABCD 9/9/2020 School on pattern analysis - Erice 15

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings

Optimization in pattern analysis Patterns in sets Patterns in discrete spaces Sets empty Strings Space of itemsets with items A, B, C, D Graphs A AB B D AD BD Patterns in vector spaces ABD 9/9/2020 School on pattern analysis - Erice 16

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 17

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 18

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs A Patterns in vector spaces AA B AB … AAA 9/9/2020 AAB School on pattern analysis - Erice 19

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs

Optimization in pattern analysis Patterns in strings Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 20

Optimization in pattern analysis Patterns in graphs, … Patterns in discrete spaces Sets Strings

Optimization in pattern analysis Patterns in graphs, … Patterns in discrete spaces Sets Strings Graphs Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 21

Optimization in pattern analysis Patterns in vector spaces • Least squares problems – Linear

Optimization in pattern analysis Patterns in vector spaces • Least squares problems – Linear regression (LSR/RR) – Fisher’s discriminant analysis (FDA) Patterns in discrete spaces • Convex optimization – Support vector machines (SVM) Patterns in vector spaces • Eigenvalue problems – Principal component analysis (PCA) 9/9/2020 School on pattern analysis - Erice 22

Optimization in pattern analysis Pattern analysis Least squares: LSR and FDA Patterns in discrete

Optimization in pattern analysis Pattern analysis Least squares: LSR and FDA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 23

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 24

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 25

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 26

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: LSR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 27

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 28

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 29

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: RR Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 30

Optimization in pattern analysis Pattern analysis Least squares: FDA Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: FDA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 31

Optimization in pattern analysis Pattern analysis Least squares: FDA Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Least squares: FDA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 32

Optimization in pattern analysis Pattern analysis Convex optimization: SVM Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Convex optimization: SVM Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 33

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 34

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 35

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns

Optimization in pattern analysis Pattern analysis Eigenvalue problems: PCA Patterns in discrete spaces Patterns in vector spaces Least squares Convex Opt. Eigenproblems 9/9/2020 School on pattern analysis - Erice 36

Optimization in pattern analysis Pattern analysis Summary Patterns in discrete spaces Patterns in vector

Optimization in pattern analysis Pattern analysis Summary Patterns in discrete spaces Patterns in vector spaces 9/9/2020 School on pattern analysis - Erice 37