Metrics on Persistence Modules and an Application to Neuroscience

Transcription

1 Metrics on Persistence Modules and an Application to Neuroscience Magnus Bakke Botnan NTNU November 28, 2014

2 Part 1: Basics

3 The Study of Sublevel Sets A pair (M, f ) where M is a topological space and f : M R such that H k (f 1 (, s]) is finitely generated for all s R. For every s t we get inclusions... and at the level of homology f 1 (, s] f 1 (, t] F s := H k (f 1 (, s]) H k (f 1 (, t]) =: F t Note: homology is computed with coefficients in a field k (usually Z/pZ for p a small prime).

4 Persistence Modules Definition A persistence module is a functor F : R vec (finite dimensional vector spaces). Example A sublevel filtration f : M R gives rise to a persistence module.

5 Persistence Modules Definition Let J R be an interval. The interval persistence module over J is the persistence module I J defined by { I J k if t J (t) = 0 otherwise with id : I J (t) I J (t ) whenever t t J and 0 otherwise. Note: I J is indecomposable and I is the 0 persistence module.

6 Persistence Modules Theorem (Krull Remak Schmidt Azumaya) If F = I J l F = I J k, l L then there is a bijection σ : L K such that J l = J σ(l) for all l L. So whenever F admits such a decomposition we can characterize it by the multiset Dgm(F ) = {J l : l L} of intervals. k K

7 Persistence Modules Theorem (Crawley-Boevey) A persistence module decomposes as a direct sum of interval persistence modules. Note: If we replace R with Z then the theorem follows from the structure theorem for finitely generated modules over a PID.

8 Persistence Modules Pointcloud Data Let P R n be a finite set of points. The distance function d : R n R, d(x) = min p P x p 2 defines a sublevel filtration of R n. We depict the sublevel sets as balls of radius r around the points of P. The subelevel set d 1 (0, r] is homotopy equivalent to the nerve of the cover formed by the union of radius r balls around each point of P.

9 Persistence Modules (Complex getting hard to draw by hand... ) H 1 ε H 0 ε

10 Interleavings Two persistence modules F and G are ɛ-interleaved if there exist an ɛ 0 and two families of homomorphisms {φ t : F (t) G(t + ɛ)} t R and {ψ t : G(t) F (t + ɛ)} t R such that the following four diagrams commute for all t t : F (t ɛ) F (t + ɛ) F (t + ɛ) F (t + ɛ) G(t) G(t) G(t ) F (t) F (t) F (t ) G(t ɛ) G(t + ɛ) G(t + ɛ) G(t + ɛ) Note: 0-interleaving means that F and G are naturally isomorphic.

11 Interleavings Definition The interleaving distance d I between two persistence modules F and G is d I (F, G) = inf[ɛ [0, ) F and G are ɛ-interleaved]

12 Interleavings Definition A matching between Dgm(F ) and Dgm(G) is a subset S (Dgm(F ) { }) (Dgm(G) { }) such that every element of Dgm(F ) and Dgm(G) appears in exactly one pair. Definition The bottleneck distance between F = J Dgm(F ) I J and G = J Dgm(G) I J is defined by d B (F, G) = inf S Match(F,G) max d I (I J1, I J2 ) (J 1,J 2) S Dgm(F ) Dgm(G)

13 Interleavings Algebraic Stability Theorem (AST) Theorem (Chazal et al., Bauer & Lesnick) The interleaving distance equals the bottleneck distance: d I = d B Thus, the interleaving distance can be computed by matching indecomposables.

14 Interleavings Application 1: Stability Let f, g : M R be such that f (x) g(x) ɛ for all x M. We get inclusions f 1 (, x] g 1 (, x + ɛ] f 1 (, x + 2ɛ]... and homomorphisms F x G x+ɛ F x+2ɛ. This defines an ɛ-interleaving between F and G. Thus, d I (F, G) ɛ and sublevel persistence is stable.

15 Interleavings Application 2: Approximations Need to approximate persistence computations as the memory needed for the standard constructions grows exponentially in the number of input points. A set of k close points introduces 2 k 1 simplices in the simplicial complex. With AST we can give provable error bounds in the bottleneck distance by defining an interleaving between the approximate persistence module and the original one. Joint work with Gard Spreemann.

16 Interleavings 5000 points on RP 2 embedded in R 4 ((x, y, z) (xy, xz, y 2 z 2, 2yz)). Note: Z 2 coefficients H H H Death Death Death Birth Birth Birth Simplex count ε = 1 No collapse Net tree Scale

17 Part 2: An Application to Neuroscience

18 Place Cell Data

19 Place Cell Data Place Cells and Place Fields A place cell is a neuron within the hippocampus that becomes active when then animal enters a particular place in the environment. Such a region is a place field of the neuron. The collection of place cells is thought to act as cognitive representation of the environment.

20 Place Cell Data Spike Trains

21 Homology of Neural Network Homological Analysis of Place Cell Network The place fields cover the environment so the spatial topology should be detectable from place cell correlations. There are place cells with spatially separate place field centers that are highly-correlated as some cells only fire when the animals head points in a specific direction. Such firings also introduce non-trivial topology in the network. The cells also fire due other external factors (stress, fear, light conditions etc.). Joint work with the Roudi group at the Kavli Institute, Nils Baas and Gard Spreemann.

22 Homology of Neural Network Example Place Fields Figure: Place fields for two place cells recorded from a rat exploring a 1m x 1m box.

23 Homology of Neural Network Setup: k place cells {c 1,..., c k } in the hippocampus. Compute correlation corr(c i, c j ) between any pair of place cells. Let k 1 be the full simplicial complex having the place cells as 0-simplices. Define a filtering function f : k 1 R by f (c i ) = 0 f ([c i, c j ]) = 1 corr(c i, c j ) Inductively we define the filtering function on a higher-dimensional simplex to the maximum filtration value over all its faces.

24 Homology of Neural Network Simulated Data (a) Place Field 270 (b) Head Direction and Activity

25 Homology of Neural Network Simulated Data: Persistence Diagrams 2.0 H H H Death 0.8 Death 0.8 Death Birth Birth 0.0 (a) Persistence diagram for H 1 where the environment is a box with a cylinder in the middle. (b) Same as (a) but without head direction tuning Birth (c) Same as (a) but without spatial tuning.

26 Homology of Neural Network Future Plan We will fit a model to the real data and try to understand the points in the persistence diagram by removing the following in succession: Spatial tuning: we expect to see a persistent H 1 coming from preferences in head direction. Head direction tuning: we don t know what will come out. Random-like or do we find interesting structure?

27 Homology of Neural Network Real Data: Persistence Diagram 30 H Death Birth Figure: Persistence diagram for H 1 computed from real data recorded from a rat exploring a square.

28 Part 3: Generalizations Joint work with Mike Lesnick.

29 Multidimensional Persistence Generalizations Why only restrict the domain of our functors to R? We can study functors F : R n vec.... or more generally from any poset.... or even more generally between arbitrary categories C and D.

30 Multidimensional Persistence Carlsson and Memoli (2013) show that single linkage clustering (H 0 in persistence) is the unique hiearchical clustering method satisfying certain desirable properties ( functoriality ). Not so nice in practice due to the chaining phenomenon (illustrated). Idea: filter by both density and radius 2D persistence module.

31 Multidimensional Persistence... V 0,2 V 1,2 V 2,2 V 0,1 V 1,1 V 2,1 V 0,0 V 1,0 V 2,0 We get a persistence module of this form. These persistence modules are very complicated objects and there is no general decomposition into simple indecomposables.

32 Multidimensional Persistence But the interlaving distance generalizes readily: F, G : R n vec are ɛ-interleaved for ɛ = (ɛ,..., ɛ) R n if we have two families of homomorphisms {φ t : F (t) G(t + ɛ)} and {ψ t : G(t) F (t + ɛ} such that the interleaving diagrams commute for all t t : F (t ɛ) F (t + ɛ) F (t + ɛ) F (t + ɛ) G(t) G(t) G(t ) F (t) F (t) F (t ) G(t ɛ) G(t + ɛ) G(t + ɛ) G(t + ɛ)

33 Multidimensional Persistence Definition (Interlaving Distance) d I (F, G) = inf[ɛ [0, ) F and G are ɛ-interleaved]. But can we compute this? For ɛ = 0 this belongs to a certain module isomorphism problem which can be checked in polynomial time. There are also randomized algorithms since invertible maps are abundant. For ɛ > 0: open problem! Naïvely one can solve a system of multivariate quadratic equations but this is NP-hard. Being an ɛ-interleaving morphism for ɛ > 0 is not a generic property (in any dimension) no randomized algorithms.

34 Multidimensional Persistence Indecomposables can be computed but the interleaving distance can not be computed by a matching of indecomposables as the following example of 2-interleaved persistence modules illustrates: Two indecomposables One indecomposable k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 k 2 ( 0 1 ) k ( 0 1 ) k 2 k 2 k 2 ( 1 1 ) 0 k ( 1 1 ) k 2 k 2 ( 1 0 ) 0 0 k ( 1 0 ) k 2

35 Multidimensional Persistence Zigzag Diagrams Maybe we cannot understand the full decomposition but we can choose 1D paths V i,j+2 V i+1,j+2 V i+2,j+2 V i,j+1 V i+1,j+1 V i+2,j+1 V i,j V i+1,j V i+2,j... V i,j+2 V i+1,j+2 V i+1,j+1 V i+2,j+1 V i+2,j

36 Multidimensional Persistence Zigzag diagrams decompose nicely Theorem (Gabriel s Theorem) Diagrams of the form V i V i+1 V i+2 decompose into simple pieces like in 1D persistence. For a zigzag module V we have a well-defined diagram Dgm(V ). For zigzag modules V, W of the same type we have a distance d B between their diagrams. But how do we get an algebraic description of this metric? Not clear...

37 Levelset Zigzag Persistence Levelset Zigzag Persistence We return to the situation of a real-valued function f : M R. Let M t = f 1 (t) be the levelset at t and M I = f 1 (I ) for intervals I R. We assume that f is of morse type: There is a finite set of real-valued indices a 1 < a 2 < < a n called critical values, such that over each open interval I = (, a 1 ), (a 1, a 2 ),..., (a n 1, a n ), (a n, ) the slice M I is homeomorphic to a product of the form M I, with f being the projection onto the factor I. Each homeomorphism M I M I extends to a continuous function M Ī MĪ, where Ī is the closure of I R. We assume that each M t has finitely generated homology.

38 Levelset Zigzag Persistence Given (M, f ) of Morse type we select a set of indices s i such that < s 0 < a 1 < s 1 < a 2 < < s n 1 < a n < s n < and define an associated zigzag diagram M 0 0 M 1 0 M 1 1 M 2 1 M n n 1 M n n where M j i = M [si,s j ] = f 1 [s i, s j ]. Upon taking homology we get the levelset zigzag persistence f. We denote its persistence diagram by DgmZZ(f ).

39 Levelset Zigzag Persistence Example from Carlsson, de Silva, Morozov SoCG 2009

40 Levelset Zigzag Persistence Theorem (Diamond Principle, Carlsson et al. SoCG 2009) Any two monotone zigzags in the pyramid diagram carry the same information in their persistent homology. In particular, levelset zigzag carries more information than sublevel persistent homology.

41 Levelset Zigzag Persistence Theorem (Stability, Carlsson et al.) d B (DgmZZ(f), DgmZZ(g)) f g As in the 1D case we would like an algebraic description of the Bottleneck distance.

42 Levelset Zigzag Persistence Define a persistence module F : closed(r) vec by F [i, j] = H k (f 1 [i, j]) Using the diamond principle one can show that such a functor decomposes into a direct sum of four types of simple indecomposables. (, ) (, ) (, ) (, ) a b a b a b a b (, ) (, ) (, ) (, ) (a) (b) (c) (d)

43 Levelset Zigzag Persistence Observation If f g ɛ then we get inclusions f 1 [i, j] g 1 [i ɛ, j + ɛ] f 1 [i 2ɛ, j + 2ɛ] and thus an ɛ-interleaving between F and G. Conjecture The relation from 1D persistence generalizes to levelset zigzag persistence, i.e. d I = d B Partial result: we have reduced the problem to case of persistence modules built from pieces of type (a) on the previous slide. Approaches from 1D persistence don t work.

44 Cosheaves Other Aspects The previous discussion works equally if we study open sets of M. This defines a functor F : open(r) vec known as the Leray pre-cosheaf. Or, if we dualize, the Leray pre-sheaf. This has spawned a lot of interesting research into generalizations of persistent homology (Curry, Ghrist, MacPherson, Patel et al.).

45 Cosheaves Thank you!