Induced matchings and the a lge b r a i c st ab ilit y of persisten c - PowerPoint PPT Presentation

Stab ilit y of persisten c e ba r c odes for f u n c tions T heorem ( C ohen- S teiner, E dels b r u nner, Ha rer 200 5) If two functions f , g ∶ K → R have distance ∥ f − g ∥ ∞ ≤ δ then there exists a δ -matching of their barcodes. 2 δ δ ∣ B : b ije c tion of s u b sets A ′ ⊆ A , B ′ ⊆ B • matching A → • δ -matching of barcodes : • m a t c hed inter v a ls h a v e endpoints w ithin dist a n c e ≤ δ • u nm a t c hed inter v a ls h a v e length ≤ 2 δ 9 / 2 8

Stab ilit y for f u n c tions in the b ig pi c t u re Da t a point c lo u d dist a n c e G eometr y f u n c tion s ub le v el sets T opolog y topologi ca l sp ac es homolog y A lge b r a v e c tor sp ac es ba r c ode C om b in a tori c s inter va ls 1 0 / 2 8

Stab ilit y for f u n c tions in the b ig pi c t u re Da t a point c lo u d dist a n c e G eometr y f u n c tion s ub le v el sets T opolog y topologi ca l sp ac es homolog y A lge b r a v e c tor sp ac es ba r c ode C om b in a tori c s inter va ls 1 0 / 2 8

Stab ilit y for f u n c tions in the b ig pi c t u re Da t a point c lo u d dist a n c e G eometr y f u n c tion s ub le v el sets T opolog y topologi ca l sp ac es homolog y A lge b r a v e c tor sp ac es ba r c ode C om b in a tori c s inter va ls 1 0 / 2 8

Stab ilit y for f u n c tions in the b ig pi c t u re Da t a point c lo u d dist a n c e G eometr y f u n c tion s ub le v el sets T opolog y topologi ca l sp ac es homolog y A lge b r a v e c tor sp ac es ba r c ode C om b in a tori c s inter va ls 1 0 / 2 8

Inter le av ings of s ub le v el sets L et • F t = f − 1 (−∞ , t ] , • G t = g − 1 (−∞ , t ] . 11 / 2 8

Inter le av ings of s ub le v el sets L et • F t = f − 1 (−∞ , t ] , • G t = g − 1 (−∞ , t ] . I f ∥ f − g ∥ ∞ ≤ δ then F t ⊆ G t + δ a nd G t ⊆ F t + δ . 11 / 2 8

Inter le av ings of s ub le v el sets L et • F t = f − 1 (−∞ , t ] , • G t = g − 1 (−∞ , t ] . I f ∥ f − g ∥ ∞ ≤ δ then F t ⊆ G t + δ a nd G t ⊆ F t + δ . S o the s ub le v el sets a re δ -int erle av ed: F t F t + 2 δ G t + δ G t + 3 δ 11 / 2 8

Inter le av ings of s ub le v el sets L et • F t = f − 1 (−∞ , t ] , • G t = g − 1 (−∞ , t ] . I f ∥ f − g ∥ ∞ ≤ δ then F t ⊆ G t + δ a nd G t ⊆ F t + δ . S o the s ub le v el sets a re δ -int erle av ed: H ∗ ( F t ) H ∗ ( F t + 2 δ ) H ∗ ( G t + δ ) H ∗ ( G t + 3 δ ) H omolog y is a f u n c tor: homolog y gro u ps a re interle av ed too. 11 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) • a line a r m a p M s → M t for e ac h s ≤ t ( tr a nsition m a ps ) 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) • a line a r m a p M s → M t for e ac h s ≤ t ( tr a nsition m a ps ) • respe c ting identit y : ( M t → M t ) = id M t a nd c omposition: ( M s → M t ) ○ ( M r → M s ) = ( M r → M t ) 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) • a line a r m a p M s → M t for e ac h s ≤ t ( tr a nsition m a ps ) • respe c ting identit y : ( M t → M t ) = id M t a nd c omposition: ( M s → M t ) ○ ( M r → M s ) = ( M r → M t ) A morphism f ∶ M → N is a n a t u r a l tr a nsform a tion : 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) • a line a r m a p M s → M t for e ac h s ≤ t ( tr a nsition m a ps ) • respe c ting identit y : ( M t → M t ) = id M t a nd c omposition: ( M s → M t ) ○ ( M r → M s ) = ( M r → M t ) A morphism f ∶ M → N is a n a t u r a l tr a nsform a tion : • a line a r m a p f t ∶ M t → N t for e ac h t ∈ R 1 2 / 2 8

Persistence modu les A persistence modu le M is a di a gr a m (f u n c tor) R → Vect : • a v e c tor sp ac e M t for e ac h t ∈ R (in this t a lk: dim M t < ∞ ) • a line a r m a p M s → M t for e ac h s ≤ t ( tr a nsition m a ps ) • respe c ting identit y : ( M t → M t ) = id M t a nd c omposition: ( M s → M t ) ○ ( M r → M s ) = ( M r → M t ) A morphism f ∶ M → N is a n a t u r a l tr a nsform a tion : • a line a r m a p f t ∶ M t → N t for e ac h t ∈ R • morphism a nd tr a nsition m a ps c omm u te: M s M t f s f t N s N t 1 2 / 2 8

Interva l P ersisten c e M od u les L et K b e a field. F or a n a r b itr a r y inter va l I ⊆ R , define the interva l persisten c e mod u le C ( I ) by ⎧ ⎪ ⎪ if t ∈ I , K C ( I ) t = ⎨ ⎪ ⎪ 0 other w ise ; ⎩ 1 3 / 2 8

Interva l P ersisten c e M od u les L et K b e a field. F or a n a r b itr a r y inter va l I ⊆ R , define the interva l persisten c e mod u le C ( I ) by ⎧ ⎪ ⎪ if t ∈ I , K C ( I ) t = ⎨ ⎪ ⎪ 0 other w ise ; ⎩ ⎧ ⎪ ⎪ id K if s , t ∈ I , C ( I ) s → C ( I ) t = ⎨ ⎪ ⎪ 0 other w ise . ⎩ 1 3 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . 14 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . T hen M is inter v a l-de c ompos ab le: 14 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . T hen M is inter v a l-de c ompos ab le: there e x ists a u niq u e c olle c tion of inter va ls B ( M ) 14 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . T hen M is inter v a l-de c ompos ab le: there e x ists a u niq u e c olle c tion of inter va ls B ( M ) s uc h th a t M ≅ ⊕ C ( I ) . I ∈ B ( M ) 14 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . T hen M is inter v a l-de c ompos ab le: there e x ists a u niq u e c olle c tion of inter va ls B ( M ) s uc h th a t M ≅ ⊕ C ( I ) . I ∈ B ( M ) B ( M ) is ca lled the ba r c ode of M . 14 / 2 8

The structure of persistence modu les T heorem ( C r aw le y - B oe w e y 20 1 2 ) Let M be a persistence modu le w ith dim M t < ∞ for a ll t . T hen M is inter v a l-de c ompos ab le: there e x ists a u niq u e c olle c tion of inter va ls B ( M ) s uc h th a t M ≅ ⊕ C ( I ) . I ∈ B ( M ) B ( M ) is ca lled the ba r c ode of M . • M oti v a tes u se of homolog y w ith field c oeffi c ients 14 / 2 8

Inter le av ings of persisten c e mod u les D efinition Tw o persisten c e mod u les M a nd N a re δ -int erle av ed 15 / 2 8

Inter le av ings of persisten c e mod u les D efinition Tw o persisten c e mod u les M a nd N a re δ -int erle av ed if there a re morphisms f ∶ M → N ( δ ) , g ∶ N → M ( δ ) 15 / 2 8

Inter le av ings of persisten c e mod u les D efinition Tw o persisten c e mod u les M a nd N a re δ -int erle av ed if there a re morphisms f ∶ M → N ( δ ) , g ∶ N → M ( δ ) s uc h th a t this di a gr a ms c omm u tes for a ll t : M t M t + 2 δ f t f t + 2 δ g t + δ N t + δ N t + 3 δ 15 / 2 8

Inter le av ings of persisten c e mod u les D efinition Tw o persisten c e mod u les M a nd N a re δ -int erle av ed if there a re morphisms f ∶ M → N ( δ ) , g ∶ N → M ( δ ) s uc h th a t this di a gr a ms c omm u tes for a ll t : M t M t + 2 δ f t f t + 2 δ g t + δ N t + δ N t + 3 δ • define M ( δ ) by M ( δ ) t = M t + δ 15 / 2 8

Inter le av ings of persisten c e mod u les D efinition Tw o persisten c e mod u les M a nd N a re δ -int erle av ed if there a re morphisms f ∶ M → N ( δ ) , g ∶ N → M ( δ ) s uc h th a t this di a gr a ms c omm u tes for a ll t : M t M t + 2 δ f t f t + 2 δ g t + δ N t + δ N t + 3 δ • define M ( δ ) by M ( δ ) t = M t + δ B ( M ) B ( M (δ)) (shift ba r c ode to the left by δ ) δ 15 / 2 8

A lge b r a i c st ab ilit y of persisten c e ba r c odes T heorem ( C h aza l et a l. 200 9, 20 1 2 ) If two persistence modu les a re δ -interle av ed, then there e x ists a δ -m a t c hing of their ba r c odes. 16 / 2 8

A lge b r a i c st ab ilit y of persisten c e ba r c odes T heorem ( C h aza l et a l. 200 9, 20 1 2 ) If two persistence modu les a re δ -interle av ed, then there e x ists a δ -m a t c hing of their ba r c odes. 2 δ δ 16 / 2 8

A lge b r a i c st ab ilit y of persisten c e ba r c odes T heorem ( C h aza l et a l. 200 9, 20 1 2 ) If two persistence modu les a re δ -interle av ed, then there e x ists a δ -m a t c hing of their ba r c odes. 2 δ δ • c on v erse st a tement a lso holds (isometr y theorem) 16 / 2 8

A lge b r a i c st ab ilit y of persisten c e ba r c odes T heorem ( C h aza l et a l. 200 9, 20 1 2 ) If two persistence modu les a re δ -interle av ed, then there e x ists a δ -m a t c hing of their ba r c odes. 2 δ δ • c on v erse st a tement a lso holds (isometr y theorem) • indire c t proof, 8 0 p a ge p a per ( C h a z a l et a l. 20 1 2 ) 16 / 2 8

Our approach Our proof takes a different approach: • direct proof (no inter pol a tion, m a t c hing immedi a tel y from interle av ing) 17 / 2 8