Local Homology Transfer and Stratification Learning

Transcription

1 Local Homology Transfer and Stratification Learning Paul Bendich Bei Wang Sayan Mukherjee Abstract The objective of this aer is to show that oint cloud data can under certain circumstances be clustered by strata in a lausible way. For our uroses, we consider a stratified sace to be a collection of manifolds of different dimensions which are glued together in a locally trivial manner inside some Euclidean sace. To adat this abstract definition to the world of noise, we first define a multi-scale notion of stratified saces, roviding a stratification at different scales which are indexed by a radius arameter. We then use methods derived from kernel and cokernel ersistent homology to cluster the data oints into different strata. We rove a correctness guarantee for this clustering method under certain toological conditions. We then rovide a robabilistic guarantee for the clustering for the oint samle setting we rovide bounds on the minimum number of samle oints reuired to state with high robability which oints belong to the same strata. Finally, we give an exlicit algorithm for the clustering. 1 Introduction. Manifold learning is a basic roblem in geometry, toology, and statistical inference that has received a great deal of recent attention. One formulation of the roblem is: given a oint cloud of data samled from a manifold in an ambient sace R N, infer the dimension and structure of the underlying manifold. A limitation of this roblem statement is that it does not aly to sets that are not manifolds. For examle, we may consider the more general class of stratified saces that can be decomosed into strata manifolds of varying dimension each of which fit together in some uniform way inside the higher dimensional sace. This aer is meant as a first ste towards the following roblem in stratification learning: given a oint cloud samled from a stratified sace, how do we cluster oints that belong to the same stratum iece together while keeing oints in different strata aart. Intuitively, a reasonable strategy would be to lace two oints in the same stratum iece if they look the same locally they have identical neighborhoods within the larger sace at some very small scale. However, the notion of local becomes unclear in the context of the uncertainty induced from samling, since everything becomes noisy at small scales (in a metahori- Duke University and IST Austria,bendich@math.duke.edu SCI Institute, University of Utah, beiwang@sci.utah.edu Duke University, sayan@stat.duke.edu cal sense, if one zooms in closely enough on a oint cloud, then one will always see what looks like an oen set of the to ambient dimension, no matter what the intrinsic dimension should have been). In resonse, we introduce a radius or scale arameter r and define a notion of local euivalence r at each scale r. The euivalences classes of r will then be taken as the connected comonents of our strata. We will use tools derived from algebraic toology. In articular, we define our euivalence relations r via mas that transfer information carried by local homology grous, and we then use ersistent homology [16] methods to infer the roerties of these mas. Prior work. Consistency in manifold learning has often been recast as homology inference as the number of oints in a oint cloud goes to infinity, the homology inferred from the oint cloud converges to the true homology of the underlying sace. Results of this nature have been given for manifolds [30, 31] and a large class of comact subsets of Euclidean sace [8]. Stronger results in homology inference for closed subsets of a metric sace are given in [13]. Geometric aroaches to stratification learning have also been develoed. These include inference of a mixture of linear subsaces [27], mixture models for general stratified saces [23], and generalized Princial Comonent Analysis (GPCA) [32] which was develoed for dimension reduction for mixtures of manifolds. The study of stratified saces has long been a focus of ure mathematics; see, for examle, [20, 33]. The roblem of inference for the local homology grous of a samled stratified sace in a deterministic setting has been addressed in [4]. Contributions. In this aer we roose an aroach to stratification learning based on local homology inference. The results in this aer are: 1. A toological definition of two oints belonging to the same stratum by assessing the multi-scale local structure of the oints through a local homology transfer ma ( 3 Definition 3.1); 2. Toological conditions on the relation between the oint samle and the underlying sace under which this characterization holds ( 3 Theorem 3.2); 3. Finite samle bounds for the minimum number of oints reuired in the samle to state with high robability which oints belong to the same stratum ( 4 Theorem 4.1);

2 4. An algorithm that comutes which oints belong to the same stratum ( 5). 2 Background. We first describe general ersistence modules [34, 7], focusing mainly on those that arise from mas between absolute or relative homology grous induced by inclusions of toological saces or airs of such saces. We then discuss stratifications and their connection to the local homology grous of a toological sace. Basics on homology itself are assumed; for a readable background, see [29] or [24], or [16] for a more comutationally oriented treatment. Persistence modules. For simlicity, our treatment of ersistence modules adated from [7] is restricted to Z/2Zvector saces (in general, ersistence can be defined for vector saces over any field, Z/2Z is used here for comutational simlicity). Let A be some subset of R. A ersistence module F A is a collection {F α } α A of Z/2Z-vector saces, together with a family {fα β : F α F β } α β A of linear mas such that α β γ imlies fα γ = f γ β f α β, and fα α = id Fα. We will assume that the index set A is either R or R 0 and not exlicitly state indices unless necessary. A real number α is said to be a regular value of the ersistence module F if there exists some ε > 0 such that the ma f α+δ α δ is an isomorhism for each δ < ε. Otherwise we say that α is a critical value of the ersistence module; if A = R 0, then α = 0 will always be considered to be a critical value. We say that F is tame if it has a finite number of critical values and if all the vector saces F α are of finite rank. Any tame R 0 -module F must have a smallest non-zero critical value ρ(f); we call this number the feature size of the ersistence module. Assume F is tame and so we have a finite ordered list of critical values 0 = c 0 < c 1 <... < c m. We choose regular values {a i } m i=0 such that c i 1 < a i 1 < c i < a i for all 1 i m, and we adot the shorthand notation F i := F ai and f j i : F i F j, for 0 i j m. A vector v F i is said to be born at level i if v im fi 1 i, and such a vector dies at level j if f j i (v) im f j j 1 i 1 but fi (v) im f j 1 i 1. This is illustrated in Figure 1 (a). We then define P i,j to be the vector sace of vectors that are born at level i and then subseuently die at level j, and let β i,j denote its rank. We note that it is in general ossible to have vectors which are born at or which never die, and that some care is needed with these definitions; in our context, this will not haen, so we do not get into these technicalities. Persistence diagrams. Let R = R {, } denote the extended real line. The extended lane R 2 = R R is endowed with the l norm. That is, for any two oints u = (x, y) and u = (x, y ) in the extended lane, we define u u = max{ x x, y y }. The information contained within a tame module F can be comactly reresented by a ersistence diagram, Dgm(F), which is a multi-set of oints in this lane. It contains β i,j coies of the oints (c i, c j ), as well as infinitely many coies of each oint along the major diagonal y = x. In Figure 3 (a) the ersistence diagrams for a curve and a oint cloud samled from it are dislayed, see below for a full exlanation of this figure. We define the bottleneck distance between any two ersistence diagrams D and D to be: d B (D, D ) = inf su u Γ(u), Γ:D D u D where Γ ranges over all bijections from D to D. Under certain conditions which we now describe, ersistence diagrams will be stable under the bottleneck distance. Two ersistence modules F and G are said to be strongly ɛ-interleaved if, for some ositive ɛ, there exist two families {ξ α : F α G α+ɛ } α and {ψ α : G α F α+ɛ } of linear mas which commute with the module mas {f β α } and {g β α} in the aroriate manner. More recisely, we reuire that, for each α β, the four diagrams in Figure 2 all commute [7]. F α ɛ ξ α ɛ ψ α ɛ G α ɛ G α F α f β+ɛ α ɛ g β α f β α g β+ɛ α ɛ G β F β ξ β F β+ɛ ψ β G β+ɛ ψ α G α F α ξ α F α+ɛ g β α f β α G α+ɛ f β+ɛ α+ɛ G β F β g β+ɛ α+ɛ ξ β F β+ɛ ψ β G β+ɛ Figure 2: Commuting diagrams for strongly interleaving ersistence modules. We can now state the diagram stability result ([7], Theorem 4.4), that we will need below. THEOREM 2.1. (DIAGRAM STABILITY THEOREM) Let F and G be two tame ersistence modules and ɛ > 0. If F and G are strongly ɛ-interleaved, then d B (Dgm(F), Dgm(G)) ɛ. When we wish to comute the ersistence diagram associated to a module F, it is often convenient to substitute another module G, usually one defined in terms of simlicial comlexes or other comutable objects. The following theorem ([16],.159) gives a condition under which this is ossible. THEOREM 2.2. (PERSISTENCE EQUIVALENCE THEOREM) Given two ersistence modules F and G, suose there exist for each α isomorhisms F α = Gα which commute with the module mas, then Dgm(F) = Dgm(G).

3 F i 1 F i F j 1 F j v... F α f β α F β... φ α φ β f i i 1 imf i i 1 f j 1 i imf j 1 i 1 f j j 1... G α g β α G β... (a) (b) Figure 1: (a) The vector v is born at level i and then it dies at level j. (b) Commuting diagrams for (co)kernel modules. α 4 α 1 death death α 3 death (0, c 4 ) α 4 α 2 (c 2, c 3 ) (0, c 3 ) 0 birth 0 birth (a) α 2 α 3 Figure 3: (a) Illustration of a oint cloud and its ersistence diagram: left, X is the curve embedded as shown in the lane and U is the oint cloud; middle, the ersistence diagram Dgm 1 (d X ); right, the ersistence diagram Dgm 1 (d U ). The diagrams are generated by thickening X (or U) while tracking the birth and death of homology classes. (b) Illustration of relative homology and its ersistence diagram: left, the sace X is in solid line and the closed ball B has dotted boundary; right, the ersistence diagram for the module {H 1 (X α B, X α B)}. Here, α goes through four non-zero critical values c 1 < c 2 < c 3 < c 4 that corresond to the four colored level sets, where the oints in the ersistence diagram corresond to the birth and death of the four relative homology classes resectively. In articular, α 4 is created when the level set at value c 2 touches B. (b) α 1 (0, c 1 ) 0 birth That is, if all the vertical mas are isomorhisms and all suares commute in the following diagram, then Dgm(F) = Dgm(G).... F α F β... = =... G α G β... (Co)Kernel modules. Suose now that we have two ersistence modules F and G along with a family of mas {φ α : F α G α } which commute with the module mas for every air α β, we have gα β φ α = φ β fα β. In other words, every suare commutes in the diagram shown in Figure 1 (b). Then, for each air of real numbers α β, the restriction of fα β to ker φ α mas into ker φ β, giving rise to a new kernel ersistence module, with ersistence diagram denoted by Dgm(ker φ). Similarly, we obtain a cokernel ersistence module, with diagram Dgm(cok φ). For details on ersistent homology for kernels and cokernels, see [14]. Homology and distance functions. Consider a family of toological saces {X α }, along with inclusions X α X β for all α β. The inclusions induce mas H j (X α ) H j (X β ), for each homological dimension j 0, and hence we have ersistence modules for each j. Defining H(X α ) = j H j(x α ) and taking direct sums of mas in the obvious way, will also give one large direct-sum ersistence module {H(X α )}. Given a comact toological sace X embedded in some Euclidean sace R N, we define d X as the distance function which mas each oint in the ambient sace to the distance from its closest oint in X. We let X α denote the sublevel set [0, α]; each sublevel set should be thought of as a thickening of X within the ambient sace. Increasing the thickening arameter roduces a growing family of sublevel sets, giving rise to the ersistence module {H(X α )} α R 0 ; we denote the ersistence diagram of this module by Dgm(d X ) and use Dgm j (d X ) for the diagrams of the individual modules for each homological dimension j. In Figure 3(a), we see an examle of such an X embedded in the lane, along with the ersistence diagram Dgm 1 (d X ). We also have the ersistence diagram Dgm 1 (d U ), where U is a dense oint samle of X. Note that the two diagrams are uite close in bottleneck distance. Indeed, the bottleneck distance between the two diagrams will always be uer-bounded by the Hausdorff distance between the sace and its samle. d 1 X

4 We can also have ersistence modules of relative homology grous. For examle, referring to the left of Figure 3(b), we let X be the sace drawn in solid lines and B the closed ball whose boundary is drawn as a dotted circle. By restricting d X to B and also to B, we roduce airs of sublevel sets (X α B, X α B). Using the mas induced by the inclusions of airs, we obtain the ersistence module {H(X α B, X α B)} α R 0 of relative homology grous. The ersistence diagram, for homological dimension 1, aears in Figure 3 (b) right. Here, α goes through four nonzero critical values c 1 < c 2 < c 3 < c 4 that corresond to the four level sets, where the oints in the ersistence diagrams (Figure 3 (b) right) corresond to the birth and death of the four relative homology classes resectively. Stratified saces. We assume that we have a toological sace X embedded in some Euclidean sace R N. A d- dimensional stratification of X is a decreasing seuence of closed subsaces X = X d X d 1... X 0 X 1 =, such that for each i, the i-dimensional stratum S i = X i X i 1 is a (ossibly emty) i-manifold. The connected comonents of each stratum S i are called i-dimensional ieces. See Figure 4 (a) for an illustration. One usually also imoses a reuirement to ensure that the various ieces fit together uniformly. There are a number of different ways this can be done (see [25] for an extensive survey). For examle, one might assume that for each x S i, there exists a small enough neighborhood N(x) X and a (d i 1)-dimensional stratified sace L x (the link of x) such that N(x) is stratum-reserving homeomorhic to the roduct of an i-ball and the cone on L x ; one can then show that the sace L x deends only u to homeomorhism on the articular iece containing x. This definition, formally known as a cs-sace [22], is illustrated in Figure 4 (b). Since the toology on X is that inherited from the ambient sace, this neighborhood N(x) can take the form X B r (x), where B r (x) is a small enough ball around x in the ambient sace. We note that the above definition does not reuire all strata to be contained within the closure of the todimensional stratum. For examle, a two-dimensional lane that has been unctured by a line would be an examle of a two-dimensional cs-sace. Local homology. Recall ([29]) that the local homology grous of a sace X at a oint x X are the grous H i (X, X x) in each homological dimension i. If X haens to be a d-manifold, or if x is simly a oint in the todimensional stratum of a d-dimensional stratification, then these grous are rank one in dimension d and trivial in all other dimensions. On the other hand, the local homology grous for lower-stratum oints can be more interesting; for examle if x is the crossing oint in Figure 3 (b) (not the center of ball B), then H 1 (X, X x) has rank three. Suose that x and y are two oints from the same iece of a articular stratification of the cs-sace X. Then it is easy to show, by using the local roduct structure around each oint, that the local homology grous of x and y must have the same ranks in all dimensions. The converse is certainly false, as can be seen by examining Figure 4(b): the oints x and y must be laced into different strata, but their local homology grous are both rank two in dimension two, and zero otherwise. We note, however, that whenever z is a onestratum oint sufficiently close to the inch oint x, let = x and = z, the ma φ X (,, r) defined in (3.1) fails to be an isomorhism for all small enough radii r. This is discussed in detail in the following section. 3 Toological Inference Theorem. In this section we build a useful analytical tool based on the contraositive of the statement above: given two oints x and y samled from a sace X, we can hoe to state, based on some aroriate definition of their local homology grous, that the two oints should not be laced in the same iece of any stratification. To do this, we first adat the definition of these local homology grous, as well as certain mas between them which we shall define below, into a multiscale and robust framework. More secifically, we introduce a radius arameter r and a notion of local euivalence, r, which allows us to grou the oints of X, as well as oints of the ambient sace, into strata at this radius scale. We then give the main result of this section: toological conditions under which the oints in a oint cloud U samled from X can be clustered according to these newly-defined strata at different radius scales. 3.1 Local Euivalence. We begin by describing the local homology intersection ma that is crucial in our formulation. Local homology intersection ma. We assume that we are given a cs-sace X embedded in some Euclidean sace in R N. For each radius r 0, and for each air of oints, R N, we define the following relative homology ma φ X (,, r): (3.1) H(X B r (), X B r ()) H(X B r () B r (), X (B r () B r ())). Intuitively, this ma can be understood as taking a chain, throwing away the arts that lie outside the smaller range, and then modding out the new boundary. Alternatively, one may think of it as being induced by a combination of inclusion and excision. A third ossibility is to consider the ma (3.2) H(X, X B r ()) H(X, X (B r () B r ())), although the former is easier to comute combinatorially than the later. For a formal and technical definition, see Aendix A. For examle, consider the sace X drawn in the lane as shown in Figures 5 (a), (b), and (c). For each air

5 = x y (a) (b) Figure 4: (a) The coarsest stratification of a inched torus with a sanning disc stretched across the hole. (b) The sace in (a) is a cs-sace, where the oints x and y are resectively in the 0-stratum and the 1-stratum, their neighborhoods are highlighted. (a) (b) α 2 α 1 γ 1 β 1 α 3 α 1 γ 1 β 1 γ 2 β 2 α 1 α 2 α 3 kerf f g γ 1 β 1 f g α 1 γ 1 β 1 γ 2 β 2 (c) α 1 γ 1 β 1 cokf f g α 1 γ 1 β 1 Figure 5: Let f = φ X (,, r) and g = φ X (,, r). The local homology classes are labeled in their corresonding locations. (a) and do not have the same local structure at radius r since ker f 0. (b) and do not have the same local structure at radius r since cok f 0. (c) and have the same local structure at radius r since ker f = cok f = 0 and ker g = cok g = 0. (, ) of oints shown in the three arts of the figure, we let f = φ X (,, r) and g = φ X (,, r). Then the oints and are considered to have the same local structure if f and g are both isomorhisms; euivalently, if ker f = cok f = 0 and if ker g = cok g = 0. In art (a), ker (f) 0, since the classes α 2 and α 3 go to zero when assing to the intersection. In art (b), there is a class γ 2 cok f. The mas f and g in art (c) are both isomorhisms. Returning to the general case, we use these mas to imose an euivalence relation on R N. DEFINITION 3.1. (LOCAL EQUIVALENCE) Two oints x and y are said to have euivalent local structure at radius r, denoted x r y, iff there exists a chain of oints x = x 0, x 1,..., x m = y from X such that, for each 1 i m, the mas φ X (x i 1, x i, r) and φ X (x i, x i 1, r) are both isomorhisms. In other words, x and y have the same local structure at this radius iff they can be connected by a chain of oints which are airwise close enough and whose local homology grous at radius r transfer isomorhically into each other via the intersection mas. Different choices of r will of course lead to different euivalence classes. For examle, consider the sace X drawn in the lane as shown in the left half of Figure 6 (a). At the radius drawn, oint z is euivalent to the cross oint and is not euivalent to either the oint x or y. Note that some oints from the ambient sace will now be considered euivalent to x and y, and some others will be euivalent to z. On the other hand, a smaller choice of radius would result in all three of x, y, and z belonging to the same euivalence class. (Co)Kernel ersistence. In order to relate the oint cloud U to the euivalence relation r, we must first define a

6 α 1 α 2 death death x y z α 3 #2 0 birth 0 ɛ 2ɛ 3ɛ 4ɛ birth (a) (b) Figure 6: (a) Illustration of euivalence relation: left, x r y, y r z; right, the 1-dim ersistence diagram, for the kernel of the ma going from the z ball into its intersection with the y ball. A number, i.e., #2, labeling a oint in the ersistence diagram indicates its multilicity. (b) Regions in X-diagrams and U-diagrams. The oint in the X-diagrams lie either along the solid black line or in the darkly shaded region. Adding the lightly shaded regions, we get the region of ossible oints in the U-diagrams. 4ɛ 3ɛ 2ɛ ɛ multi-scale version of the mas φ X (,, r); we do so by gradually thickening the sace X using the sublevel sets of its distance function. For each, R N and r, α 0, we will consider the intersection ma φ X α(,, r), which is defined by substituting X α for X in (3.1). Note of course that φ X (,, r) = φ X 0 (,, r). For the moment, we fix a choice of,, and r, and we use the following shorthand, B X (α) = X α B r (), B X (α) = X α B r (), B(α) X = X α B r () B r (), B(α) X = X α (B r () B r ()), and we also often write B X = B X (0) and B X = B(0). X By relacing X with U in this shorthand, we also write B U (α) = U α B r (), and so forth. For any air of non-negative real values α β the inclusion X α X β gives rise to the following commutative diagram: (3.3) H(B X (α), B X (α)) φx α H(B(α), X B(α)) X H(B X (β), B X (β)) φx β H(B(β), X B(β)) X Hence there are mas ker φ X α ker φ X β and cok φx α cok φ X β. Allowing α to increase from 0 to gives rise to two ersistence modules, {ker φ X α} and {cok φ X α}, with diagrams Dgm(ker φ X ) and Dgm(cok φ X ). Recall that a homomorhism is an isomorhism iff its kernel and cokernel are both zero. In our context then, the ma φ X is an isomorhism iff neither Dgm(ker φ X ) nor Dgm(cok φ X ) contain any oints on the y-axis above 0. Examle. As shown in the left art of Figure 6 (a), x, y, and z are oints samled from a cross embedded in the lane. Taking r as drawn, the right art of Figure 6 (a) dislays Dgm 1 (ker φ X ), where φ X = φ X (z, y, r); we now exlain this diagram in some detail. The grou H 1 (Bz X, Bz X ) has rank three; as a ossible basis we might take the three local homology classes reresented by α 1, α 2, and α 3, which are airs of segments defining the northwest-facing right angle, the northeast-facing right angle, and the southeast-facing right angle. Under the intersection ma φ X = φ X 0, the first of these classes α 1 mas to the generator of H 1 (B X zy, B X zy), while the other two ma to zero. Hence ker φ X 0 has rank two. As X starts to thicken into the ambient sace, both classes in this kernel eventually die, one at the α value which fills in the northeast corner of the larger ball, and the other at the α value which fills in the southeast corner; these two values are the same here due to symmetry in the icture. At this value, the ma φ X α becomes an isomorhism and it remains so until the intersection of the two balls fills in comletely. This gives birth to a new kernel class which subseuently dies when the larger ball finally fills in. The diagram Dgm 1 (ker φ X ) thus contains three oints; the leftmost two show that the ma φ X is not an isomorhism, and thus that z and y do not have the same local structure at the chosen radius level. 3.2 Toological Inference Theorem. Given a oint cloud U samled from X, we consider the following uestion: for a radius r, how can we infer whether or not any given air of oints in U has the same local structure at this radius? In this subsection, we rove a theorem which describes the circumstances under which we can make the above inference. Naturally, any inference will reuire that we use U to judge whether or not the mas φ X (,, r) are isomorhisms. The basic idea is that if U is a dense enough samle of X, then the (co)kernel diagrams defined by U will be good enough aroximations of the diagrams defined by X. (Co)Kernel stability. Again we fix,, and r, and write φ X = φ X (,, r). For each α 0, we let U α = d 1 U [0, α]. We consider φ U α = φ U α(,, r), defined by relacing X with U α in (3.1), as (3.4) H(U α B r (), U α B r ()) H(U α B r () B r (), U α (B r () B r ())).

7 Running α from 0 to, we obtain two more ersistence modules, {ker φ U α} and {cok φ U α}, with diagrams Dgm(ker φ U ) and Dgm(cok φ U ). If U is a dense enough samle of X, then the (co)kernel diagrams defined by U will be good aroximations of the diagrams defined by X. More recisely, we have the following theorem, THEOREM 3.1. ((CO)KERNEL DIAGRAM STABILITY) The bottleneck distances between the (co)kernel diagrams of φ U and φ X are uer-bounded by the Hausdorff distance between U and X: d B (Dgm(ker φ U ), Dgm(ker φ X )) d H (U, X), d B (Dgm(cok φ U ), Dgm(cok φ X )) d H (U, X). Proof. We rove the first ineuality; the roof of the second is identical. Put ɛ = d H (U, X). Then, for each α 0, the inclusions U α X α+ɛ and X α U α+ɛ induce mas ker φ U α ker φ X α+ɛ and ker φ X α ker φ U α+ɛ. These mas clearly commute with the module mas in the needed way, and hence we have the reuired ɛ-interleaving and can thus aeal to Theorem 2.1. Main inference result. We now suose that we have a oint samle U of a sace X, where the Hausdorff distance between the two is no more than some ε. In this case, we call U an ε-aroximation of X. Given two oints, U and a fixed radius r, we set φ X = φ X (,, r), and we wish to determine whether or not φ X is an isomorhism. Since we only have access to the oint samle U, we instead comute the diagrams Dgm(ker φ U ) and Dgm(cok φ U ); we rovide an algorithm for doing this in 5. Given any ersistence diagram D, which we recall is a multi-set of oints in the extended lane, and two ositive real numbers a < b, we let D(a, b) denote the intersection of D with the ortion of the extended lane which lies above y = b and to the left of x = a; note that these oints corresond to classes which are born no later than a and die no earlier than b. For a fixed choice of,, r, we consider the following two ersistence modules: {H(B X (α), B X (α))} and {H(B(α), X B(α))}. X We let σ(, r) and σ(,, r) denote the resective feature sizes of these modules and then set ρ(,, r) to their minimum. Geometrically, ρ(,, r) is related to a local reach and the gradient of d X (as detailed in [5]). We now give the main theorem of this section, which states that we can use U to decide whether or not φ X (,, r) is an isomorhism as long as ρ(,, r) is large enough relative to the samling density. THEOREM 3.2. (TOPOLOGICAL INFERENCE THEOREM) Suose that we have an ε-aroximation U from X. Then for each air of oints, R N such that ρ = ρ(,, r) 4ε, the ma φ X = φ X (,, r) is an isomorhism iff Dgm(ker φ U )(ε, 3ε) Dgm(cok φ U )(ε, 3ε) =. Proof. To simlify exosition, we will refer to oints in Dgm(ker φ X ) Dgm(cok φ X ) and Dgm(ker φ U ) Dgm(cok φ U ) as X-oints and U-oints, resectively. Whenever 0 < α < β < 4ɛ < ρ, the two vertical mas in diagram (3.3) will by definition both be isomorhisms. This is evidently an immediate conseuence of the definition of the feature size. Hence the mas ker φ X α ker φ X β and cok φ X α cok φ X β must also be isomorhisms, and so, as α increases from 0 to, any element of the (co)kernel of φ X must live until at least 4ɛ, and any (co)kernel class which is born after 0 must in fact be born after 4ɛ. In other words, any X-oint must lie either to the right of the line x = 4ɛ, or along the y-axis and above the oint (0, 4ɛ); see Figure 6 (b). Recall that φ X is an isomorhism iff ker φ X = 0 = cok φ X. Thus φ X is an isomorhism iff the black line in Figure 6 (b) contains no X-oints. On the other hand, Theorem 3.1 reuires that every U- oint must lie within ɛ of an X-oint. That is, all U-oints are contained within the two lightly shaded regions drawn in Figure 6 (b). Since the rightmost such region is more than ɛ away from the thick black line, there will be a U-oint in the left region iff there is an X-oint on the thick black line. But the U-oints within the left region are exactly the members of Dgm(ker φ U )(ɛ, 3ɛ) Dgm(cok φ U )(ɛ, 3ɛ). Figure 6 (b) illustrates Theorem 3.2, that is, under certain toological conditions, φ X is an isomorhism if and only if certain regions in the U-diagrams are emty. For examle, suose X is the cross shown in Figure 7 (a), with,, r as drawn. and are locally different at this radius level, as shown by the resence of two black emty oints on the y-axis of the kernel ersistence X-diagram (Figure 7 (c)). Suose X is unknown and we are only given U, an ε-aroximation of X (Figure 7 (b)). From the kernel U- diagram, which has two oints in the relevant rectangle, we can infer that and do not have the same local structure at radius level r by alying Theorem Probabilistic Inference Theorem. The toological inference of 3 states conditions under which the oint samle U can be used to infer stratification roerties of the sace X. The basic condition is that the Hausdorff distance between the two must be small. In this section we describe a robabilistic model for generating the oint samle U, and we rovide an estimate of how large this oint samle should be to infer stratification roerties of the sace X with a uantified measure of confidence. More secifically, we rovide a local estimate, based on ρ(,, r) and ρ(,, r), of how many samle oints are needed to infer the local relationshi at radius level r between two fixed

8 death #2 (ɛ, 3ɛ) (a) (b) 0 (c) Figure 7: (a) Sace X is the black cross, with,, r as drawn. (b) Sace U is an ε-aroximation of X. (c) The kernel ersistence diagram of φ X (,, r) (X-diagram) is shown to contain black emty oints; the kernel diagram of φ U (,, r) (U-diagram) is shown to contain red filled oints. Suose only U is given, we can use U-diagram to infer that oints and are not locally euivalent. birth oints and ; this same theorem can be used to give a global estimate of the number of oints needed for inference between any air of oints whose ρ-values are above some fixed low threshold. Samling strategy. We assume X to be comact. Since the stratified sace X can contain singularities and maximal strata of varying dimensions, some care is reuired in the samling design. Consider for examle a sheet of area one, unctured by a line of length one. In this case, samling from a naively constructed uniform measure on this sace would result in no oints being drawn from the line. This same issue arose and was dealt with in [31], although in a slightly different aroach than we will develo. A samling strategy that will deal with the roblem of varying dimensions is to use a mixture model. In the examle of the sheet and line, a uniform measure would be laced on the sheet, while another uniform measure would be laced on the line, and a mixture robability is laced on the two measures; for examle, each measure could be drawn with robability 1/2. We now formalize this aroach. Consider each (non-emty) i-dimensional stratum S i = X i X i 1 of X. All strata that are included in the closure of some higher-dimensional strata, in other words all non-maximal strata, are not considered in the model. A uniform measure is assigned to the closure of each maximal stratum, µ i (S i ), this is ossible since each such closure is comact. We assume a finite number of maximal strata K and assign to the closure of each such stratum a robability i = 1/K. This imlies the following measure F (x) = 1 K K j=1 µ i(x = x), where µ i is the measure on the closure of the i-th maximal stratum. The oint samle is generated from the following model: U = {x 1,...,.x n } iid F (x). We call this model M. Lower bounds on the samle size of the oint cloud. Our main theorem is the robabilistic analogue of Theorem 3.2. An immediate conseuence of this theorem is that, for two oints, U, we can infer with robability at least 1 ξ whether and are locally euivalent, r. The confidence level 1 ξ will be a monotonic function of the size of the oint samle. The theorem involves a arameter v(ρ), for each ositive ρ, which is based on the volume of the intersection of ρ-balls with X. First we note that each maximal stratum of X comes with its own notion of volume: in the lane unctured by a line examle, we measure volume in the lane and in the line as area and length, resectively. The volume vol (Y) of any subsace Y of X is the sum of the volumes of the intersections of Y with each maximal stratum. For ρ > 0, we define v(ρ) = vol (B inf ρ/32 (x) X) x X vol (X). We then have our main theorem, THEOREM 4.1. (LOCAL PROBABILISTIC SAMPLING) Let {x 1, x 2,..., x n } be drawn from model M. Fix a air of oints, R N and a ositive radius r, and ( ut ρ = ) min{ρ(,, r), ρ(,, r)}. If n 1 v(ρ) log 1 v(ρ) + log 1 ξ, then, with robability greater than 1 ξ we can correctly infer whether or not φ X (,, r) and φ X (,, r) are both isomorhisms. Proof. A finite collection U = {x 1, x 2,..., x n } of oints in R N is ε-dense with resect to X if X U ε ; euivalently, U is an ε-cover of X. Let C(ε) be the ε-covering number of X, the minimum number of sets B ε X that cover X. Let P (ε) be the ε-acking number of X, the maximum number of sets B ε X that can be acked into X without overla. We consider a cover of X with balls of radius ρ/16. If there is a samle oint in each ρ/16-ball, then U will be an ε-aroximation of X, with ε 4(ρ/16) = ρ/4. This satisfies the condition of the toological inference theorem, and therefore we can infer the local structure between and. The following two results from [30] will be useful in comuting the number of samle oints n needed to obtain, with confidence, such an ε-aroximation. LEMMA 4.1. (LEMMA 5.1 IN [30]) Let {A 1, A 2,..., A l } be a finite collection of measurable sets with robability

9 measure µ on l i=1 A i, such that for all A i, µ(a i ) > α. Let U = {x 1, x 2,..., x n } be drawn iid according to µ. If n 1 α (log l + log 1 ξ ), then, with robability 1 ξ, i, U A i. LEMMA 4.2. (LEMMA 5.2 IN [30]) Let C(ε) be the covering number of an ε-cover of X and P (ε) be the acking number of an ε-acking, then P (2ε) C(2ε) P (ε). Again, we consider a cover of X by balls of radius ρ/16. Let {y i } l i=1 X be the centers of the balls contained in a minimal sub-cover. Put A i = B ρ/16 (y i ) X. Alying Lemma 4.1, we obtain the estimate n 1 ( log l + log 1 ), α ξ where l is the ρ/16-covering number, and α = min i vol (A i) vol (X). Alying Lemma 4.2, l = C(ρ/16) P (ρ/32) 1 On the other hand, α 1 v(ρ) follows. vol (X) vol (B ρ/32 X) 1 v(ρ). by definition, and the result To extend the above theorem to a more global result, one can ick a ositive ρ and radius r, and consider the set of all airs of oints (, ) such that ρ min{ρ(,, r), ρ(,, r}. Alying Theorem 4.1 uniformly to all airs of oints will give the minimum number of samle oints needed to settle the isomorhism uestion for all of the intersection mas between all airs. 5 Algorithm. The theorems in the revious sections give conditions under which a oint cloud U, samled from a stratified sace X, can be used to infer the local euivalences between oints on X. We now switch gears slightly, and imagine clustering the U-oints into strata. The basic strategy is to build a grah on the oint set, with edges corresonding to ositive isomorhism judgments. The connected comonents of this grah will then be our roosed strata. More recisely, we build a grah where each node in the grah corresonds uniuely to a oint from U. Two oints, U (where 2r) are connected by an edge iff both φ X (,, r) and φ X (,, r) are isomorhisms, euivalently iff Dgm(ker φ U )(ɛ, 3ɛ) and Dgm(cok φ U )(ɛ, 3ɛ) are emty. The connected comonents of the resulting grah are our clusters. A more detailed statement of this rocedure is giving in seudo-code, see Algorithm 5.1. Note that the connectivity of the grah is encoded by a weight matrix, and our clustering strategy is based on a 0/1-weight assignment. We discuss the robustness of our algorithm in a subseuent section. A crucial subroutine in the clustering algorithm is the comutation of the diagrams Dgm(ker φ U ) and Dgm(cok φ U ), for φ U = φ U (,, r) between all airs (, ) U U. We will focus our attention in this section on the comutation of the (co)kernel diagrams. To comute the diagrams Dgm(ker φ U ) and Dgm(cok φ U ) we reuire for each α 0 a simlicial analogue of the ma, φ U α : H(B U (α), B U (α)) H(B U (α), B U (α)). We define, for each α 0 (a) two airs of simlicial comlexes L 0 (α) L(α) and K 0 (α) K(α), and (b) a relative homology ma between them ψ α : H(L(α), L 0 (α)) H(K(α), K 0 (α)). Later, we give a correctness roof that Dgm(ker φ U ) = Dgm(ker ψ) and Dgm(cok φ U ) = Dgm(cok ψ). 5.1 Robustness of Clustering. Two tyes of errors in the clustering can occur: false ositives where the algorithm connects oints that should not be connected and false negatives where oints that should be connected are not. The current algorithm we state is somewhat brittle with resect to both false ositives as well as false negatives. We will suggest a very simle adatation of our current algorithm that should be more stable with resect to both false ositives and false negatives. The false ositives are driven by the condition in Theorem 3.2 that ρ < 4ɛ, so if the oint cloud is not samled fine enough we can get incorrect ositive isomorhisms and therefore incorrect edges in the grah. If we use transitive closure to define the connected comonents this can be very damaging in ractice since a false edge can collase disjoint comonents into one large cluster. The false negatives occur because our oint samle U is not fine enough to cature chains of oints that connect airs in U through isomorhisms, there may be other oints in X which if we had samled then the chain would be observed. The robability of these events in theory decays exonentially as the samle size increases and the confidence arameter ξ controls these errors. We now state a simle adatation of the algorithm that will make it more robust. It is natural to think of the 0/1- weight assignment on airs of oints, U as an association matrix W. A classic aroach for robust artitioning is via sectral grah theory [28, 26, 11]. This aroach is based an eigen-decomosition of the the grah Lalacian, L = D W with the diagonal matrix D ii = j W ij. The

10 Algorithm 5.1 Strata-Inference(U, r, ε) for all, U do if > 2r then W (, ) = 0 else Comute Dgm(ker φ U (,, r)) and Dgm(cok φ U (,, r)) Comute Dgm(ker φ U (,, r)) and Dgm(cok φ U (,, r)) if Dgm(ker φ U (,, r))(ɛ, 3ɛ) Dgm(cok φ U (,, r))(ɛ, 3ɛ) then W (, ) = 0 else if Dgm(ker φ U (,, r))(ɛ, 3ɛ) Dgm(cok φ U (,, r))(ɛ, 3ɛ) then W (, ) = 0 else W (, ) = 1 end if end if end for Comute connected comonents based on W. smallest nontrivial eigenvalue λ 1 of W is called the Fiedler constant and estimates of how well the vertex set can be artitioned [18]. The corresonding eigenvector v 1 is used to artition the vertex set. There are strong connections between sectral clustering and diffusions or random walks on grahs [11]. The roblems of sectral clustering and lower dimensional embeddings have been examined from a manifold learning ersective [2, 3, 19]. The idea central to these analyses is given a oint samle from a manifold construct an aroriate grah Lalacian and use its eigenvectors to embed the oint cloud in a lower dimensional sace. A theoretical analysis of this idea involves roving convergence of the grah Lalacian to the Lalace-Beltrami oerator on the manifold and the convergence of the eigenvectors of the grah Lalacian to the eigenvalues of the Lalace-Beltrami oerator. A key uantity in this analysis is the Cheeger constant which is the first nontrivial eigenvalue of the Lalace- Beltrami oerator [10]. An intriguing uestion is whether the association matrix we construct from the oint cloud can be related to the Lalacian on high forms. 5.2 Preliminaries. To construct the simlicial comlexes in our algorithm, we will comute Voronoi diagrams and nerves of sets of collections derived from these Voronoi diagrams. Voronoi diagram. Given a finite collection U of oints in R N and u i U, then the Voronoi cell of u i is defined to be: V i = V (u i ) = {x R N x u i x u j, u j U}. The union of cells V i covers the entire sace and forms the Voronoi diagram of R N, denoted as Vor (U R N ). If we restrict each V i to some subset X R N, then the set of cells V i X forms a restricted Voronoi diagram, denoted as Vor (U X). For a simlex σ with vertices in U, we set V σ = ui σv i. Nerves. The nerve N(C) of a finite collection of sets C is defined to be the abstract simlicial comlex with vertices corresonding to the sets in C and with simlices corresonding to all non-emty intersections among these sets, N(C) = {S C S }. Every abstract simlicial comlex can be geometrically realized, and therefore the concet of homotoy tye makes sense. Under certain conditions, for examle whenever the sets in C are all closed and convex subsets of Euclidean sace ([16],.59), the nerve of C has the same homotoy tye, and thus the same homology grous, as the union of sets in C. This imlies we can comute H(U α ), the absolute homology of the thickened oint cloud, by comuting the nerve of the collection of sets V i U α. The nerve of the restricted Voronoi diagram Vor (U X) is called the restricted Delaunay triangulation, denoted as Del (U X). It contains the set of simlices σ for which V σ X. Power cells, lunes, and moons. We need to comute the relative homology grous H(B U (α), B U (α)) and H(B(α), U B(α)), U for a fixed air of oints and. The direct argument used to comute absolute homology based on the nerve does not aly to comuting relative homology grous since the collection of the sets V i B U (α) and V i B(α) U need not be convex. To get around this roblem, we first define the ower cell with resect to B r (), P (α), as P (α) = {x R N x 2 r 2 x u 2 α 2, u U}, and we set P 0 (α) = B r () int P (α). Relacing with in this formula gives Q(α), the ower cell with resect to B r (). Finally, we set Z(α) = P (α) Q(α), and Z 0 (α) = (B r () B r ()) int Z(α). These definitions are illustrated in Figure

11 Z(α) (a) M L M P L P (α) Q(α) (b) Figure 8: (a) Illustration of intersection ower cell Z(α), as the grey shaded region. The unshaded convex regions are P (α) and Q(α) resectively. The dark ink and black shaded regions (ointed by single and double arrows) corresond to P 0 (α) and Q 0 (α) resectively. (b) Illustration of the lune and the moon. The shaded regions are the resective moons. The white regions within solid circles are the resective lunes. 8 (a). Note that P 0 (α) and Z 0 (α) are both contained in U α. It turns out that relacing B U (α) with P 0 (α) and B(α) U with Z 0 (α) has no effect on the relative homology grous in uestion. That is, the saces (B U (α), B U (α)) and (B U (α), P 0 (α)) are homotoy euivalent, so are the saces (B(α), U B(α)) U and (B(α), U Z 0 (α)). Conseuently, their homology grous are isomorhic. The first art of this statement was roven in [4], and a roof of the second aears in [5]. The sets V i P 0 (α) are convex, for all oints u i U [4]. Unfortunately, it is still ossible for V i Z 0 (α) to be non-convex, which reuires a further subdivision of the Voronoi cells by bisection. Consider the hyerlane P of oints in R N which are euidistant from and. This will divide R N into two half-saces with P and P denoting the half-saces containing and. Given P we define the -lune, L, and -moon, M, as follows (see Figure 8 (b)): L = P B r (), M = P B r (). (Note that here lune is defined differently from sherical lune or sherical wedge.) The -lune, L, and -moon, M, are defined similarly. Notice that L deends not only on but also on. For a fixed air of oints and, the lune and the moon divide each Voronoi cell into two arts, Vi L = V i L and Vi M = V i M, for all oints u i U. These sets are obviously convex, assuming they are non-emty, since they are each the intersection of two convex sets. It also turns out that the non-emty sets among Vi L Z 0(α) and Vi M Z 0 (α) are convex; see [5] for a roof. 5.3 Algorithm to Comute Simlicial Analogues. Our algorithm to comute simlicial analogues contains two stes: (a) defining the simlicial comlexes via the nerves of non-emty sets, and (b) defining the corresonding chain mas between relative simlicial chain grous. We first define the airs of simlicial comlexes L 0 (α) L(α) and K 0 (α) K(α). Set A to be the collection of the non-emty sets among Vi L B U (α) and Vi M B U (α). Define A 0 as the collection of the nonemty sets among Vi L P 0 (α) and Vi M P 0 (α). Note that A = B U (α) and A 0 = P 0 (α). Taking the nerve of both collections, we define the simlicial comlexes L(α) = N(A) and L 0 (α) = N(A 0 ). Similarly, we define C and C 0 to be the collections of the non-emty sets among, resectively, V L i and V L i Z 0(α) and V M i B(α) U and Vi M B(α), U Z 0 (α). We define K(α) = N(C) and K 0 (α) = N(C 0 ). See Figure 9 for an examle of the simlicial comlexes constructed in R 2 for a given U. To define the ma ψ α : H(L(α), L 0 (α)) H(K(α), K 0 (α)), we need the following technical lemma: LEMMA 5.1. (CONTAINMENT LEMMA) Assume that a simlex σ is in L 0 (α). If σ is also in K(α), then σ is in K 0 (α), as well. Proof. Recall that the lune and the moon divide each Voronoi cell into two arts, Vi L = V i L and Vi M = V i M.

12 u 1 u 2 Figure 9: Illustration of the simlicial comlexes constructed around two oints and. The underlying Voronoi decomosition of the sace is shown in thin dotted lines. u 1 and u 2 in U are the oints whose restricted Voronoi regions intersect with the lune at non-convex regions. These are defined as the artial Voronoi cells. For simlicity, for a simlex σ L(α) (similarly for a simlex in L 0, K and K 0 ), we define V σ as the intersection of the artial Voronoi cells that corresond to the vertices of σ. That is, σ L(α) iff V σ B U (α). By definition, σ L 0 (α) iff there exists some oint x V σ P 0 (α). We must show that the set V σ Z 0 (α) is non-emty. Note that x P 0 (α) imlies that x B r (), while x int P (α) imlies that x int Z(α). If x B r (), then we are done, since Z 0 (α) = B r () B r () int Z(α). Otherwise, choose some oint y V σ U α B r () B r (), which is ossible since σ K(α). Since both x and y belong to the same convex set V σ U α B r (), there exists a directed line segment γ from x to y within this set connecting them. We imagine moving along γ and first we suose that γ intersects B r () before it intersects int Q(α). Let z be the first oint of intersection. Then z B r () B r (), z / int Q(α). Therefore z V σ Z 0 (α). On the other hand, we may rove by contradiction that it is imossible for γ to intersect Q(α) before it intersects B r (). Let z be the first oint of such an intersection. Since z Q(α), by definition z 2 r 2 z u i 2 α 2, u i U. Since z U α, u i σ, z u i 2 α 2 0. Therefore z 2 r 2 z u i 2 α 2 0, u i σ. Since z is outside B r (), z 2 r 2 > 0. This is a contradiction. To define ψ α, we first construct a chain ma g = g α : C(L(α)) C(K(α)) as follows. Given a simlex σ L(α), we define g(σ) = σ if σ K(α), and g(σ) = 0 otherwise; we then extend g to a chain ma by linearity. Using the Containment Lemma, we see that g(c(l 0 (α))) C(K 0 (α)), and thus g descends to a relative chain ma f = f α : C(L(α), L 0 (α)) C(K(α), K 0 (α)). Since f clearly commutes with all boundary oerators, it induces a ma on relative homology, this is our ψ = ψ α. To comute the diagrams involving ψ, we reduce various boundary matrices via (co)kernel ersistence algorithm described in [14], in time at most cubic in the size of the simlicial comlexes reresenting the data. Correctness. We show that our algorithm is correct by roving the following theorem. A sketch of the roof is given here, with the details deferred to [5]. THEOREM 5.1. (CORRECTNESS THEOREM) The ersistence diagrams involving simlicial comlexes are eual to the ersistence diagrams involving the oint cloud, that is, Dgm(ker φ U ) = Dgm(ker ψ) and Dgm(cok φ U ) = Dgm(cok ψ). Proof sketch. To rove Theorem 5.1, we will rove, for each α β, that the following diagram (as well as a similar diagram involving cokernels) commutes, with the vertical

13 mas being isomorhisms. (5.5)... ker φ U α ker φ U β... = =... ker ψ α ker ψ β.... Alying Theorem 2.2 then finishes the claim. Therefore Dgm(ker φ U ) = Dgm(ker ψ) and Dgm(cok φ U ) = Dgm(cok ψ). Simle simulations. We use a simulation on simle synthetic data with oints samled from grids to illustrate how the algorithm works. We assume we know ε and we run our algorithm for 0 α 3ε. As shown in Figure 10 (a), if two oints x and y (also, z and w) are locally euivalent, their corresonding kernel and cokernel ersistence diagrams shown in Figure 10 (b) contain the emty uadrant redicted by our theorems. On the other hand, if two oints x and z are not euivalent, then the kernel ersistence diagrams shown in Figure 10 (c) do not contain such emty uadrants. 6 Discussion. We have resented a first ste towards learning stratified saces. There are several oen issues of interest including: algorithmic efficiency and scaling with dimension using Ris or Witness comlexes [15] instead of Delaunay triangulation, robustness of the algorithm and weighting local euivalence, and extensions to the noisy setting [30] when the mixture is concentrated around the stratified sace. Secifically, the algorithm to comute the (co)kernel diagrams from the thickened oint cloud is based on an adation of Delaunay triangulation and the ower-cell construction. This algorithm should be uite slow when the dimensionality of the ambient sace is high due to the runtime comlexity of Delaunay triangulation. One idea to address this bottleneck is to use Ris or Witness comlexes [15]. Another aroach is to use dimension reduction techniues such as rincial comonents analysis (PCA) or random rojection that aroximately reserve distance [12] as a rerocessing ste. Another idea that may work if the ambient dimension is not too high is using faster algorithms to construct Delaunay triangulations [6]. Here, comuting the mas into the intersection reuires a nice and dense samling, which is only a first ste towards samling conditions that give ractical results. Recent work in reconstructing metric grah (1D stratified sace) with guarantees [1] has shown some romise towards this direction, but still has trouble with noise. Using distance to a measure [9] might coe with noise and outliers to a certain extent. However deriving a sufficient samling condition in ractice remains a challenge, esecially in high dimensions. Acknowledgments All the authors would like to thank Herbert Edelsbrunner and John Harer for useful discussions and suggestions. PB would like to thank David Cohen-Steiner and Dmitriy Morozov for helful discussion, and SM would like to thank Shmuel Weinberger for useful comments. SM and BW would like to acknowledge the suort of NIH Grants R01 CA A1 and P50 GM , and SM would like to acknowledge the suort of NSF Grants DMS , DMS , and CCF and AFOSR FA The authors would like to thank the reviewers for their comments that hel imrove the manuscrit. Aendices. A Definition of the Intersection Ma. We give a more recise descrition of the ma φ = φ U α : H(B U (α), B U (α)) H(B U (α), B U (α)). The definition will be made on the chain level and will be given in terms of singular chains. A.1 Background. We give here some necessary background as well as some material from algebraic toology and homological algebra. Most of the descritions are adated from [24] and [29]. Chain homotoies. For our uroses, a chain comlex C is a seuence of Z/2Z- vector saces C, one for each integer, connected by boundary homomorhisms C : C C 1 such that 1 = 0 for all. The -th homology grou of such a chain comlex is defined by H = ker /im +1. A chain ma η : C D between two chain comlexes is a seuence of homomorhisms η : C D which commute with the boundary homomorhisms: D η = η 1 C. Every chain ma induces a ma η between the homology grous of the two comlexes. A chain homotoy F between two chain mas η, η : C D is a seuence of homomorhisms F : C D +1 which satisfy the following formula for each : η η = +1 D F F 1 C. We say that η and η are chain homotoic and note that they must then induce the same mas on homology: η = η. Finally, η is called a chain homotoy euivalence if there exists a chain ma ρ : D C such that η ρ and ρ η are both chain homotoic to the identity. In this case η and ρ will both induce homology isomorhisms. Singular homology. The standard -simlex is the subset of R +1 given by = {(t 0,..., t ) R +1 t i = 1, i, t i 0}. i=0