2/22/2017. Sept 20, 2013: Persistent homology.

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1 //7 MATH:745 (M:35) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept, 3: Persistent homology. Fall 3 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa H = < a, b, c, d : tc + td, tb + c, ta + tb> H = <z, z : t z, t 3 z + t z > z = ad + cd + t(bc) + t(ab), z = ac + t bc + t ab p persistent k th homology group: i, p i i+p i H k = Z k /(B k Z k ) y U 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C + C C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C

2 //7 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C a + b + bc + abc + acd 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C a + b + bc + abc + acd (a + b, bc, abc + acd) 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C + C 3 + : + C k + C k (, ab, ) = (a + b,, ) = 3 C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C + C 3 + : + C k + C k (, ab, ) = (a + b,, ) ( + ab + ) = a+b C = Z abc, acd] C = Z ab, bc, ac, ad, cd] C = Z a, b, c, d] + C k = C + C + C + C 3 + : + C k + C k p persistent k th homology group: i, p i i+p i H k = Z k /(B k Z k ) y U

3 // C j C j C j C j C j C j C C C C C C C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + (a, c + ta, tc + t b, t c, t 4 b, t 5 b, t 6 b, ) H = < a, b, c, d : tc + td, tb + c, ta + tb> C C C C C C + C i <a, b> + <c, d, ta, tb> + <tc, td, t a, t = b> + (a, c + ta, tc + t b, t c, t 4 b, t 5 b, t 6 b, ) t (a, c + ta, tc + t b, t c, t 4 b, t 5 b, t 6 b, ) = (, ta, tc + t a, t c + t 3 b, t 3 c, t 5 b, t 6 b, t 7 b, ) C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + a: Z t] b: Z t] c: Z t] d: Z t] 3

4 // C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + a: Z t] b: Z t] c: Z t] d: Z t] acd: Z t] C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + a: Z t] = {n + n t + n t + + n k t k : n i in Z, k in Z + } = { (n, n, n,, n k,,, ) : n i in Z, k in Z + } C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + b: Z t] = {n + n t + n t + + n k t k : n i in Z, k in Z + } = { (n, n, n,, n k,,, ) : n i in Z, k in Z + } H = < a, b, c, d : tc + td, tb + c, ta + tb> C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> C C C C C C + C i = <a, b> + <c, d, ta, tb> + <tc, td, t a, t b> + c: Z t] = {n t + n t + : n i in Z } = { (, n, n, ) : n i in Z } d: Z t] = {n t + n t + : n i in Z } = { (, n, n, ) : n i in Z } 4

5 //7 H = < a, b, c, d : tc + td, tb + c, ta + tb> a: Z t] = {n + n t + n t + + n k t k : n i in Z, k in Z + } = { (n, n, n,, n k,,, ) : n i in Z, k in Z + } H = < a, b, c, d : tc + td, tb + c, ta + tb> Z t] = {n + n t + n t + : n i in Z } b: Z t]/t = {n : n in Z }= Z = { (n,,,, ) : n i in Z } H = < a, b, c, d : tc + td, tb + c, ta + tb> Z t] = {n + n t + n t + : n i in Z } d: Z t]/t = {n : n in Z } = { (, n,,,, ) : n in Z } H = < a, b, c, d : tc + td, tb + c, ta + tb> H = < a, b, c, d : tc + td, tb + c, ta + tb> Z t] = {n + n t + n t + : n i in Z } c: Z t]/ = { (,,, ) } C C C C C C H = < a, b, c, d : tc + td, tb + c, ta + tb> a: Z t] = {n + n t + n t + n j t j : n i in Z } b: Z t]/t = {n : n in Z }= Z c: Z t]/ = empty set d: Z t]/t = {n : n in Z }={(, n,,,, )} = Z t] + Z t]/t + ( Z t])/t 5

6 //7 H = < a, b, c, d : tc + td, tb + c, ta + tb> H = <z, z : t z, t 3 z + t z > z = ad + cd + t(bc) + t(ab), z = ac + t bc + t ab H = < a, b, c, d : tc + td, tb + c, ta + tb> H = < z, z : t z, t 3 z + t z > ( Z t])/t 3 + ( Z t])/t z = ad + cd + t(bc) + t(ab), z = ac + t bc + t ab H = < z, z : t z, t 3 z + t z > Z t] = {n + n t + n t + n t 3 + n t 4 + : n i in Z } z : ( Z t])/t 3 = { (,, n, n, n,,,, ) : n i in Z } H = < z, z : t z, t 3 z + t z > Z t] = {n + n t + n t + n t 3 + n t 4 + : n i in Z } z : ( Z t])/t = { (,,, n,,,, ) : n i in Z } In general when calculating homology over the field F n H k = ( + i Ft] ) + ( + j Ft]/(t j ) ) i= n k i= MATH:745 (M:35) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 3, 3: Multidimensional Persistence. Fall 3 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa 6

7 //7 D persistence V = H V = H D persistence V = H V = H D persistence a b ta tb V = H V = H c Rank L(, ) = ; Rank L(, ) = ; Rank L(, ) = Note Rank determines persistence bars and vice versa Rank L(, ) = ; Rank L(, ) = ; Rank L(, ) = H, H, H o, H k i, p = Z ki /(B k i+p Z ki )= L(i, i+p)( H ki ) U ab a t b t c (ab) = t(b a) H Persistence module: Ft] + Ft]/(t) + Σ Ft] ( Σ f n t n, f, t Σ f n t n ) The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbner bases, Gunnar Carlsson Computing Multidimensional Persistence, Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian 7

8 //7 8

9 //7 (, ) Z x, x ]/x 9

10 //7 Z x, x ]/x Z x, x ]/<x, x >

11 //7 Z x, x ]/x (, ) Z Z x, x ]/x x, x ]/x + Z x, x ]/x + + Z x, x ]/<x, x > + Z x, x ]/x + Z x, x ] Z x, x ]

12 //7 (, ) Z Z x, x ]/x x, x ]/x + Z x, x ]/x + + Z x, x ]/<x, x > + Z x, x ]/x + Z x, x ] In general, use Buchberger s/ algorithm for constructing a Grobner basis. Algorithm described in Computing Multidimensional Persistence, by Carlsson, Singh, Zomorodian, can be computed in polynomial time. (, ) Z Z x, x ]/x x, x ]/x + Z x, x ]/x + + Z x, x ]/<x, x > + Z x, x ]/x + Z x, x ] In general, use Dionysus software, /dionysus/ See next lecture, week from today. MATH:745 (M:35) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Oct 6, 3: Zigzag Persistence and installing Dionysus part I. Fall 3 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa /ND /activities/Carlsson Gunnar/lecture4.pdf workshop_files/slides/desilva_tgda.pdf Lee Mumford Pedersen LMP] study only high contrast patches. Collection: 4.5 x 6 high contrast patches from a collection of images obtained by van Hateren and van der Schaaf Recall from Sept lecture

13 //7 M(, ) U Q where Q = 3 On the Local Behavior of Spaces of Natural Images, Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian, International Journal of Computer Vision 8, pp. The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbner bases, Gunnar Carlsson Computing Multidimensional Persistence, Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian 3

14 //7 (, ) Z x, x ]/x 4

15 //7 Witness Complexes Witness Complexes Time varying data Xt, t ] = data points existing at time t for t in t, t ] Xt, t ] Xt, t 3 ] Xt, t ] Xt, t 3 ] Xt, t 4 ] Time varying data Xt, t ] = data points existing at time t for t in t, t ] Xt, t ] Xt, t 3 ] Xt, t ] Xt, t 3 ] Xt, t 4 ] Time varying data Xt, t ] = data points existing at time t for t in t, t ] Xt, t ] Xt, t 3 ] Xt, t ] Xt, t 3 ] Xt, t 4 ] VR(Xt, t ], ε) VR(Xt, t 3 ], ε) VR(Xt, t ], ε) VR(Xt, t 3 ], ε) VR(Xt, t 4 ], ε) VR(Xt, t ], ε) VR(Xt, t 3 ], ε) VR(Xt, t ], ε) VR(Xt, t 3 ], ε) VR(Xt, t 4 ], ε) C C C C 3 C 4 5

16 //7 C C 3 C C C 4 H H 3 H H H 4 Persistent Homology: C C C C 3 C 4 H H H H 3 H 4 H k i, p = Z ki /(B k i+p Z ki )= L(i, i+p)( H ki ) U Zigzag Homology: C C C C 3 C 4 H H H H 3 H 4 H H H H 3 H 4 H H H H 3 H 4 Z Z H H H H 3 H 4 Z Z Z Z Z H H H H 3 H 4 Z Z x Z Z Z Z Z Z Z Z 6

17 //7 Gabriel (97) For Dynkin Coxeter graphs: F F H H H H 3 H 4 F F x F F F F F F F F F F F x F F F F F I(, ) I(, 4) The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbner bases, Gunnar Carlsson From Gunnar Carlsson, Lecture 7: Persistent Homology, 9/ND /activities/Carlsson G /i f h d t4 df 7

18 //7 From Gunnar Carlsson, Lecture 7: Persistent Homology, 9/ND /activities/Carlsson G /i f h d t4 df From Gunnar Carlsson, Lecture 7: Persistent Homology, 9/ND /activities/Carlsson G /i f h d t4 df From Gunnar Carlsson, Lecture 7: Persistent Homology, 9/ND /activities/Carlsson G /i f h d t4 df From Gunnar Carlsson, Lecture 7: Persistent Homology, 9/ND /activities/Carlsson G /i f h d t4 df Computing Multidimensional Persistence, Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian 8