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INCREMENTAL STABILITY OF PERSISTENCE MODULES
ABSTRACT INCREMENTAL STABILITY OF PERSISTENCE MODULES Petar Pavešić, University of Ljubljana We associate to every persistent module M a diagram of increments that provides a group-valued measure of the variation of M in the neighbourhood of its critical points. We then formulate and prove general stability theorems for increments of interleaved persistence modules. Castro Urdiales 1
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INCREMENTAL STABILITY OF PERSISTENCE MODULES
TOPOLOGICAL COMPLEXITY OUTLINE Persistent homology: study of scale-sensitive geometric/topological properties homology of filtered complexes, homology of sub-level sets labelled diagrams, bar-codes defined more generally for persistence modules, i.e. functors M: ( R,≤ ) Plan: - define -valued measure of the variation of M around critical points - achieve functoriality, sensibility to torsion, eliminate finiteness assumptions - investigate stability phenomena - more with less: simpler proofs of more general results 2
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SOME HOMOLOGICAL ALGEBRA
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY SOME HOMOLOGICAL ALGEBRA f g f g h q Im f Im g Im gf Im hgf Ker q i i’ Im gf q’ Ker q’ Im g Im hg Incr(f,g,h) Coker i Coker i’ Increment of composable morphisms f,g,h f epi or g mono Incr(f,g,h) = 0 3
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f g h Incr(f,g,h) =(,,,) Incr(f’,g’,h’) f’ g’ h’
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY f g h Incr(f,g,h) =(,,,) Incr(f’,g’,h’) f’ g’ h’ epi, mono mono epi, mono epi Im gf Im g Im hgf Im hg i’ i Coker i Coker i’ Incr(f,g,h) Im gf Im g Im hgf Im hg i’ i Coker i Coker i’ Incr(f,g,h) 4
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INCREMENTAL STABILITY OF PERSISTENCE MODULES
TOPOLOGICAL COMPLEXITY PERSISTENCE MODULES persistence module is a functor M: ( R,≤ ) (R-modules or any abelian category) xR is a regular point of M if exists s.t. M(a≤ b) is iso for every x- < a < b < x+ non-regular points are critical Crit(M), assume it is a discrete subspace of R Crit2(M) = { (x,y)| x,y Crit(M), x < y }, critical pairs (regular) rectangle K=[a,b][c,d], a < b < c < d regular points for M M(K) := Incr(M(a ≤ b), M(b ≤ c), M(c ≤ d)) increment of M over K 5
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(KK’-KK’) Crit2(M)= M(K) M(K’)
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY (KK’-KK’) Crit2(M)= M(K) M(K’) i.e. M(K) depends only on K Crit2(M) a a’ b’ b c c’ d’ d First assume K’ K and (K-K’)Crit2(M)=: If Crit (M) intersects [a,a’] then it cannot intersect [c,d], so K Crit2 (M) = and M(K)= M(K’)=0. Assume all points in [a,a’], [b’,b], [c,c’], [d’,d] are regular: Ma’ Mb’ Mc’ Md’ Mb Md Ma Mc M(K’) M(K) General case follows immediately. 6
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INCREMENTAL STABILITY OF PERSISTENCE MODULES
TOPOLOGICAL COMPLEXITY For (x,y)Crit2(M) define M(x,y) := M(K), where K Crit2(M) ={(x,y)}. Critical points are -1,0,1. 1 Alternating sum of Betti numbers at (0,1) Crit2(M) is 0. Z With Zp -coefficients labels are 1 if p divides n and 0 otherwise. n M(0,1)=Incr(ZZZ 0)=Zn n Z nZ nZ -1 Z Z Zn Zn 7
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STABILITY What happens with M when M is slightly perturbed?
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY STABILITY What happens with M when M is slightly perturbed? Persistence modules M,N are -interleaved if for every a < b there are natural transformation , and commutative interleaving diagrams M(a-) M(b+) Na Nb a- b N(a-) N(b+) Ma Mb a- b K=[a,b][c,d], c-b>2 K:=[a-,b+][c-,d+] 8
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generalized Hausdorff stability theorem
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY M,N -interleaved for every K the module N(K) is a subquotient of M(K). generalized Hausdorff stability theorem M(a-) M(b+) M(c-) M(d+) M(K) Na M(b+) Nc M(d+) Na Nb Nc Nd N(K) In particular, if N(x,y)0 and y-x > 2 then M(K(x,y)) 0. D(M) := {(x,y)|M(x,y)0} Crit2(M) M,N -interleaved Hausdorff distance between D(M) and D(N) is at most . 9
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M(K(x,y)) N(K(x,y)) for every (x,y)Crit2(M)
INCREMENTAL STABILITY OF PERSISTENCE MODULES TOPOLOGICAL COMPLEXITY Assume M,N -interleaved and any two critical points for M are at least 4 apart M(K(x,y)) N(K(x,y)) for every (x,y)Crit2(M) M(x-2) M(x+2) M(y-2) M(y+2) M(K2(x,y)) N(x-) M(x+2) N(y-) M(y+2) A Mx M(x+2) My M(y+2) M(K2(x,y)) A Similarly, N(K(x,y)) A. Finally, M(K(x,y)) M(K2 (x,y)) as M is -interleaved with itself. Define (bottleneck) distance between M and N using chains of interleavings (as in Bubenik-Scott or Chazal et al). M,N -interleaved bottleneck distance between M and N is at most . 10
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Loose ends… INCREMENTAL STABILITY OF PERSISTENCE MODULES
TOPOLOGICAL COMPLEXITY Loose ends… 11
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QUESTIONS? SUGGESTIONS? 12
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