Supplement to: Subsampling Methods for Persistent Homology

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1 Suppleent to: Subsapling Methods for Persistent Hoology A. Technical results In this section, we present soe technical results that will be used to prove the ain theores. First, we expand the notation introduced in the body of the paper (Section 3). For any positive integer, let : X!D T be the diagra and : D T!L T the landscape, i.e. (X) = D X, for any X = {x,,x } Xand (D X )= D X = X, for any D X 2D T. Given µ 2P(X), we denote by µ the push-forward easure of µ by, that is µ =( ) µ. Siilarly, we denote by µ the push-forward (induced) easure of µ by, that is µ =( ) µ. For a fixed integer >0, consider the etric space (X, ) and the space X endowed with a etric. We ipose two conditions on : (C) Given a real nuber p, for any X = {x,...,x } X and Y = {y,...,y } X, (X, Y ) X i= Subsapling Methods for Persistent Hoology (x i,y i ) p! p, (9) (C2) For any X = {x,...,x } X and Y = {y,...,y } X, H(X, Y ) (X, Y ). (0) Two exaples of distance that satisfy conditions (C) and (C2) are the Hausdorff distance and the L p -distance (X, Y )=( P i= (x i,y i ) p ) p. Lea 5. For any probability easures µ, 2 P(X) and any etric : X X! R that satisfies (C), we have W,p(µ, ) p W,p (µ, ). Reark: The bound of the above lea is tight: it is an equality when µ is a Dirac easure and any other easure. Proof. Let 2P(X X) be a transport plan between µ and. Up to reordering the coponents of X 2, is a transport plan between µ and whose p-cost is given by (X, Y ) p d (X, Y ) X X X (x i,y i ) p d (x,y ) d (x,y ) = X X i= X X (x,y ) p d (x,y ). The lea follows by taking the iniu over all transport plans on both sides of this inequality. Lea 6. For any probability easures µ, 2 P(X) and any etric : X X! R that satisfies (C2), we have W db,p 2W,p(µ, ). Proof. This is a consequence of the stability theore for persistence diagras. Given X, Y X, define (X, Y )=(D X,D Y ). If 2P(X X ) is a transport plan between µ and then, is a transport plan between µ and. Its p-cost is given by d b (D X,D Y ) p d, (D X,D Y ) D T D T = d b ( (X), (Y )) p d (X, Y ) X X 2 H(X, Y ) p d (X, Y ) (stability theore ) X X 2 (X, Y ) p d (X, Y ). X X The lea follows by taking the iniu over all transport plans on both sides of this inequality. Lea 7. Let µ and be two probability easures on X. Let X µ and Y. E µ [ X] E W db,p. Proof. Let be a transport plan between µ and. For any t 2 R we have E µ [ X](t) E (t) p = E[ X(t) Y (t)] p E [ X(t) Y (t) p ] (Jensen inequality) E [d b (D X,D Y ) p ] (Stability of landscapes) = d b (D X,D Y ) p d (D X,D Y ) D T D T = C p ( ) p.

2 Subsapling Methods for Persistent Hoology The following lea is siilar to Theore 2 in Chazal et al. (20c). Lea 8. Let X be a saple of size fro a easure µ 2P(X) that satisfies the (a, b, r 0 )-standard assuption. /b. Let r =2 log a log E [H(X, X µ )] r 0 +2 a +2C (a, b) log a /b (r 0,)(r )+ /b (log ) 2, where C (a, b) is a constant depending on a and b. Proof. Let r > r 0. It can be proven that q := Cv (X r/2) b _, where Cv(X 2r) denotes the nuber of balls of radius r/2 that are necessary to cover X µ. Let C = {x,...,x p } be a set of centers such that B(x,r/2),...,B(x p,r/2) is a covering of X µ., P (H(X, X µ ) >r) P (H(X, C)+H(C, X µ ) >r) P (H(X, C) >r/2) P (9i 2{,,p} such that X \ B(x i,r/2) = ;) px P (X \ B(x i,r/2) = ;) i= b b inf b exp i=...p P(B(x i,r/2)) 2 b a 2 b rb. E [H(X, X µ )] = P (H(X, X µ ) >r) dr r 0 + P (H(X, X µ ) >r) dr. () If r r 0 then the last quantity in () is bounded by r 0 + P (H(X, X µ ) >r) dr, r>r otherwise () is bounded by r 0 + P (H(X, X µ ) >r) dr r 0 + r + P (H(X, X µ ) >r) dr. r>r In either case, we follow the strategy in Chazal et al. (20c) to obtain the following bound: P (H(X, X µ ) >r) dr r>r 2 C(a, b) which iplies that log a /b (log ) 2, E [H(X, X µ )] r 0 + r (r0,)(r )+ /b log +2C(a, b) a (log ) 2. B. Main Proofs Proof of Theore 5 It iediately follows fro the three following inequalities of Leas 5, 6 and 7: upper bound on the Wasserstein distance between the tensor product of easures: W,p(µ, ) p W,p (µ, ) fro easures on X to easures on D: W db,p 2W,p(µ, ) fro easures on D to difference of the expected landscapes: E µ [ X] E Proof of Theore 7 2W db,p ke µ ( X ) E ( Y )k = P µ (k X Y k >") d" ">0 = " 0 + P µ (k X ">" 0 Y k >") d". (2) The event {k X Y k >"} inside the integral iplies that " 0 2 " 2 <H(X, Y ) H(X, X µ)+h(x X )+H(Y,X ), (3) where X and Y are two saples of points fro µ and, respectively. Let " 0 =2H(X X ). By (3), it follows that at least one of the following conditions holds: H(X, X µ ) H(Y,X ),.

3 Subsapling Methods for Persistent Hoology We assue that the first condition holds (the other case follows siilarly). the last quantity in equation (2) can be bounded by " 0 + P ">" 0 =2H(X X )+ H(X, X µ ) u>0 d" P (H(X, X µ ) =2H(X X )+E[H(X, X µ )] /b log 2H(X X )+r 0 +8 a /b log +8C (a, b) a (log ) 2, u) du where the last inequality follows fro Lea 8. Proof of Theore 9 Lea 8. (r 0,)(r )+ It follows directly fro (8) and Proof of Theore 0 E hk Xµ i c n k h i 2E H(X C d n ) 2 P H(X C d n ) >r dr 2r 0 +2 [P (H(X S ) >r)] n dr 2r 0 +2 exp a n 2 b rb dr, where the last inequality follows fro Lea 8. If r r 0 then the last ter is upper bounded by 2r 0 +2 r>r exp a n 2 b rb dr, otherwise it is bounded by 2r 0 +2r +2 r>r b exp a 2 b rb n dr. In either case, r>r exp a n 2 b rb dr =2 2bn b (a) /b n u /b n exp( nu)du u>log log(2 b /b ) 2C 2 (a, b) a n [ log(2 b )] n+, where in the last inequality we applied the sae strategy used to prove Theore 2 in Chazal et al. (20c). C. About the (a, b, r 0 )-standard assuption The ai of this section is to explain why the (a, b, r 0 )- standard assuption is relevant, in particular when µ is a discrete easure. Our arguent is related to weighted epirical processes, which have been studied in details by Alexander; see Alexander (985; 987b;a). A new look on this proble has been proposed ore recently in Giné & Koltchinskii (2006); Giné et al. (2003) by using Talagrand concentration inequalities. The following result fro Alexander (985) will be sufficient here. Let (X,, ) be a easure etric space and let N be the epirical counterpart of. Proposition 9. Let C be a VC class of easurable sets of index v of X. for every, ">0there exists K such that h n o i sup N (C) (C) (C) : (C) Kv log N N,C2C >" = O(N (+ )v ). () Assue that µ is the discrete unifor easure on a point cloud X N = {x,...,x N } which has been sapled fro, thus µ = N. Assue oreover that satisfies an (a 0,b)-standard assuption (r 0 =0). Let r 0 be a positive function of N chosen further. For any (N) and any y 2 X µ : inf µ(b(y, r)) = inf N (B(y, r)) = inf (B(y, r)) ( ^ a 0 r b ) inf ( ^ a 0 r b ) inf ( sup x2x sup x2x,r 0 r 0(N) (B(y, r)) N (B(y, r)) (B(y, r)) (B(x, r)) N (B(x, r)) (B(x, r)) (B(x, r 0 )) N (B(x, r 0 )) (B(x, r 0 )) (5) Assue that the set of balls in (X, ) has a finite VCdiension v. For instance, in R d, the VC-diension of balls is d+. Under this assuption we apply Alexander s Proposition with (for instance) =and " =/2. Let K>0such that () is satisfied., by setting Kv r 0 (N) := a 0 /b log N, N we finally obtain using () and (5) that inf µ(b(y, r)) ^ a0,r r N 2 rb = O(N 2v ). )

4 Subsapling Methods for Persistent Hoology In this quite general context, we see that by taking r 0 /b, of the order of for large values of N the log N N (a, b, r 0 )-standard assuption is satisfied with high probability (in ). D. Robustness to Outliers The average landscape ethod is insensitive to outliers, as can be seen by the stability result of Theore 5. The probability ass of an outlier gives a inial contribution to the Wasserstein distance on the right hand side of the inequality. For exaple, suppose that X N = {x,...,x N } is a rando saple fro the unit circle S 2, and let Y N = X N \{x }[{(0, 0)}. See Figure 7. The landscapes X N and YN are very different because of the presence of the outlier (0, 0). On the other hand, the average landscapes constructed by ultiple subsaples of < N points fro X N and Y N are close to each other. Forally, let µ be the discrete unifor easure that put ass /N on each point of X N and siilarly let be the discrete unifor easure on Y N. The st Wasserstein distance between the two easure is /N and, according to Theore 5, the difference between the average landscapes is E µ [ X] E 2 N. More forally, we can show that the average landscape n can be ore accurate than the closest subsaple ethod when there are outliers. In fact, n can even be ore accurate than the landscape corresponding to a large saple of N points. Suppose that the large, given point cloud X N = {x,...,x N } has a sall fraction of outliers. Specifically, X N = G S B where G = {x,...,x G } are the good observations and B = {y,...,y B } are the outliers (bad observations). Let G = G, B = B so that N = G + B and let = B/N which we assue is sall but positive. Our target is the landscape based on the non-outliers, naely, G. The presence of the outliers eans that X N 6= G. Let =inf S S G >,for soe >0, where the infiu is over all subsets that contain at least one outlier. Thus, denotes the inial bias due to the outliers. We consider three estiators: X N : landscape fro full given saple X N ; n : average landscape fro n subsaples of size ; c n : landscape of closest subsaple, fro n subsaples of size. The last two estiators are defined in Section 3, and are constructed using n independent saples of size fro the discrete unifor easure that puts ass /N on each point of X N. Proposition 20. If = o(/n), then, for large enough n and, Ek n Gk < Ek XN Gk. (6) In addition, if n!then i P hek n Gk < k c n Gk!. (7) Proof. We say that a subsaple is clean if it contains no outliers and that it is infected if it contains at least one outlier. Let S,...,S n be the subsaples of X N of size. Let I = {i : S i is infected} and C = {i : S i is clean}. n = n 0 n 0 + n n where n 0 is the nuber of clean subsaples, n = n n 0 is the nuber of infected subsaples, 0 = (/n 0 ) P i2c S i and =(/n ) P i2i S i. Hence, k n Gk n 0 n k 0 Gk + n n k Gk n 0 n k 0 Gk + Tn 2n. A subsaple is clean with probability ( ). Thus, n 0 Binoial(n, ( ) ) and n Binoial(n, ( ) ). Let = ( ). By Hoeffding s inequality Tn P 2n > Tn T = P 2 2n 2 > T 2 2! 2 exp 2n. T Since = o(/n), we eventually have that = ( ) < T r log n 2n, Tn which iplies that P 2n > 2 < /n. So, except on a set of probability tending to 0, k n Gk n 0 n k 0 Gk + 2 k 0 G k + 2 and thus, as soon as n, and N are large enough, Ek n Gk = k XN Gk. This proves the first clai. To prove the second clai, note that the probability that at least one subsaple is infected is ( ) n e n!. So with probability tending to one, there will be an infected subsaple. This subsaple will iniize H(X, S j ) and the landscape based on this selected subsaple will have a bias of order.

5 Subsapling Methods for Persistent Hoology X N + (0,0) (Death+Birth)/2 (Death Birth)/2 Diagras D XN and D YN (di ) D XN D YN st Landscape (di ) landscape of X N landscape of Y N E Ψµ (λ) E Ψν (λ) Figure 7. Left: X N is the set of N =500points on the unit circle; Y N = X N \{x }[{(0, 0)}. Middle: persistence diagras (di ) of the VR filtrations on X N and Y N, in the sae plot, with different sybols. Right: landscapes of X N, Y N and the corresponding average landscapes constructed by subsapling =00points fro the two sets, for n =30ties. In practice, we can increase the robustness further, by using filtered subsapling. This can be done using the distance to the k-th nearest neighbor or using a kernel density estiator. For exaple, let bp h (x) = N NX j= K x Xi h be a kernel density estiator with bandwidth h and kernel K. Suppose that all subsaples are chosen fro the filtered set F = {X i : bp h (X i ) >t}. Suppose that the good observations G are sapled fro a distribution on a set A [0, ] d satisfying the (a, b)-standard condition with b<d, a>0 and that B consists of B observations sapled fro a unifor on [0, ] d. For any x 2 A, E[bp h (x)] ahb h d and for any outlier X i we have (for h sall enough) that bp h (X i )=/(nh d ). Hence, if we choose t such that nh d <t< a h d b then F = G with high probability.