O N SUBSAMPLING AND INFERENCE FOR MULTIPARAMETER PERSISTENCE HOMOLOGY
A BSTRACT
In topological data analysis, multi-parameter persistence homology is a framework for extracting topological information for point cloud datasets equipped with multiple ﬁltrations.
However, existing algorithms become computationally expensive for large datasets, limiting their applicability.
In the single-parameter case, subsampling algorithms have been used to reduce the time complexity of computing persistence homology.
Convergence properties of the persistence barcodes have also been established in this setting.
We extend these results to the multiparameter persistence homology, and develop subsampling algorithms can be used to approximate the ﬁbered barcode in this setting.
We conduct experiments on the point cloud dataset ModelNet to demonstrate the efﬁciency of these algorithms.
I NTRODUCTION
Topological data analysis is an rapidly growing ﬁeld that producing statistical summaries of datasets by leveraging their underlying topological structure, inspired by the theory of homology from algebraic topology.
Notable applications include identifying biological datasets [21] and image segmentation [19].
One-parameter persistent homology ([13], [25]) extracts topological information from a dataset by constructing a family of simplicial complexes and studying how the homology varies.
The resulting statistical summary is presented as a persistence barcode.
In many real-world applications, there are multiple parameters attached to the dataset and it is important to incorporate this additional information.
However, the entire combinatorial structure of the multi-parameter persistence module cannot be encoded in a simple diagram analogous to the persistence barcode ([4]).
The ﬁbered barcode ([5]) is a statistical summary that can be associated to a dataset equipped with a multi-parameter ﬁltration.
While it is not an invariant, it does captures much of the interesting topological information.
These algorithms are computationally expensive for large datasets and it is desirable to have efﬁcient methods of approximating the statistical summaries.
An approach proposed by [8] in the one-parameter setting is to choose several subsamples from the dataset, compute their persistence landscapes and combine the information to produce an approximation.
These subsampling algorithms build on earlier work [9] establishing convergence properties of the persistence homology for samples drawn from a measure on a compact metric space, and use the language of persistence landscapes (following [3]).
The key technical step required is the stability of the persistence barcode with respect to the Haussdorff distance.
Our contributions.
In the present work, we study subsampling algorithms for multiparameter persistence homology and analyze their performance.
We focus on two common multiparameter ﬁltrations - the multi-level set ﬁltration, and the degree-RIPS ﬁltration.
First we study the convergence properties of the ﬁbered barcode for samples drawn from a measure µ on a compact metric space M , extending the main results of [9] to both of these ﬁltrations.
Outline.
Section 2 gives an overview of the background needed on multiparameter persistence homology, and the stability results.
Section 3 establishes convergence results for the ﬁbered barcode of ﬁnite samples drawn from a measure on a compact metric space, with respect to the matching distance.
Section 4 presents the subsampling algorithms, and analyzes their theoretical performance.
Section 5 provides experimental results on a synthetic dataset, and on the point cloud dataset
ModelNet.
All proofs are given in the Appendix.
B ACKGROUND MATERIAL
This section introduces the notations that will be used throughout the paper, and gives an expository overview of the results about persistence homology that will be needed.
None of the results in this section are new.
Let M be a ﬁnite metric space.
In this paper we focus on two-parameter persistence homology using the language of bi-ﬁltered simplicial complexes.
A bi-ﬁltered simplicial complex built on M is a family of simplicial complexes { K s,t } s,t ≥ 0 , equipped with inclusions K s,t
→ K s (cid:48) ,t (cid:48) if s ≤ s (cid:48) and t ≤ t (cid:48)
(these inclusions satisfy the natural compatibilities).
First we recall the deﬁnition of the ﬁbered barcode for multi-parameter ﬁltrations, following Section 1.5 of [20] (see Appendix A for an exposition of persistence barcodes in the single-parameter setting).
Deﬁnition 2.1.
Let L be the space of all afﬁne lines in R 2 with non-negative slope.
Given L ∈ L with equation y =
l 1 x + l 2 , deﬁne
K s,L := K s,l 1 s + l
Denote by β L ( K ) the persistence barcode of the ﬁltered simplicial complex { K s,L } s ≥ 0 .
The ﬁbered barcode β ( K ) is the function with domain L that sends L to β L ( K ) .
Now recall the following of the matching distance between two ﬁbered barcodes, following [5] (see also Section 3 of [16]):
Deﬁnition 2.2.
Given two ﬁbered barcodes β 1 and β 2 , the matching distance d M ( β 1 , β 2 ) between them is deﬁned as follows (here L ∈ L as above, and w ( l ) is its “weight”):
[Equation]
We now deﬁne a multi-level set ﬁltration as follows.
Our input datum consists of a ﬁnite metric space
M with distance function d
M : M × M → R , and a continuous functional f : M → R .
Deﬁnition 2.3.
We deﬁne a two-parameter ﬁltration as follows.
Given a ﬁnite subset U ⊂ M , let the
Vietoris-Rips complex Rips fα,β ( U ) consist of all simplicies [ x 1 , · · · , x k ] such that d M ( x i , x j ) ≤ α and f ( x i ) ≤ β for all 1 ≤ i, j ≤ k .
Let β f ( M ) denote its ﬁbered barcode.
Given two subsets M 1 , M 2 of M , recall that the Haussdorff distance can be deﬁned as follows.
These deﬁnitions can be extended to compact metric spaces M ; see Appendix A for more details.
[Equation]
S UBSAMPLING ALGORITHMS FOR TWO - PARAMETER PERSISTENCE HOMOLOGY
In this section we present algorithms for approximating the ﬁbered barcode by subsampling.
We are primarily interested in the following scenario: suppose we have a dataset X consisting of N points, and we seek to compute its ﬁbered barcode.
Computing it precisely is often infeasible when
N is large, so instead we proceed by drawing smaller subsamples.
We present two algorithms for subsampling, extending the techniques from [8] in the single-parameter setting.
Keeping the notation from the previous section, we now describe two algorithms for approximating the ﬁbered barcode β f ( X ) obtained from a ﬁnite dataset X , and a function f : X → R .
Algorithm
Input for i = m − 1 do
Let X [ i ] be a randomly chosen sample of X with size n ; let d i =
d H ( X [ i ] ,f , X f ) end for j = arg min 1 ≤ i ≤ m d H ( X , X i ); X (cid:48) =
X j ; (cid:15) =
d H ( X , X (cid:48) )
1 is an extension of the closest subsample approach from [8]. The Haussdorff distance between ﬁnite metric spaces can be computed efﬁciently (for instance, using the SciPy “directed Haussdorff” package). Experimental results from Section 4.1 indicate that for some real-world datasets, Algorithm 2 is more effective. Algorithm 2 Haussdorff subsampling algorithm
>
Let M (cid:48)
= ∅ , V =
X while V (cid:54)
=
∅ do
Pick x ∈ V randomly.
Let M (cid:48)
=
M (cid:48) ∪ x for y ∈ V : do if d ( y, x ) < (cid:15) : then
V ← V − { y } end if end for end while
E XPERIMENTS
In this section, we present experiments that illustrate the practicality of the subsampling algorithm from the previous section on both synthetic and real datasets.
In both cases, the subsampling algorithm in the previous section can be used to quickly approximate the ﬁbered barcode, within a reasonable margin of error.
All experiments in this section were performed using the package Gudhi on Google
CoLab Pro with a V100 GPU and 25GB RAM.
4. 3D POINT CLOUD DATASETS
ModelNet10 is a 3D point cloud classiﬁcation datasets consists of 3991 training samples andtest instances belonging to 10 classes of objects (see [24]).
We select one poses from three different classes: “Toilet”, “Chair” and “Bed”.
For 0 ≤ i ≤ 2 , let X i consist of a N = 1000 points chosen from the training samples.
The following example illustrates the efﬁcacy of Algorithm 2 in this context (speciﬁcally in approximating the longest intervals of the persistent barcodes of β f ( X i ) restricted to the line y = x ).
It is unclear if the topological information obtained in this manner can discriminate between classes in ModelNet10.
Example 4.1.
First we rescale the data points so that they are centered at zero, and each of the three components has unit variance.
Let f : X i → R be the function deﬁned by f ( x ) =
|| x
|| 10 for each point x ∈ X i .
Consider the bi-ﬁltered simplicial complex on X i arising from the multi-level set construction, and let β f ( X i ) denote its ﬁbered barcode.
Our objective in this example is to use the algorithms in Section 3 to approximate β f ( X i ) , queried along a line L: y
= x .
Computing these persistence barcode β f,L ( X i ) exactly requires more than 310 seconds for each of the three point clouds; see Figure 1 for diagrams of the barcodes obtained.
When using Algorithm 1, with m =subsamples of n = 400 points, we found that the approximate barcodes obtained are too coarse to retain much useful information.
When we use the Haussdorff subsampling algorithm with (cid:15) = 0 .
2 , we found that the longest intervals of the barcodes β f,L ( X (cid:48) i ) closely resemble those of β f,L ( X i ) [Table] (speciﬁcally, the intervals whose length is at least 0 . 4 ).
The results are presented in Table 1.
Note however that most of the shorter intervals in β f,L ( X i ) cannot be reconstructed by subsampling in this fashion.
D ISCUSSION AND CONCLUSION
In this work, we developed statistical techniques for analyzing multiparameter persistence homology.
In particular, we establish convergence results the ﬁbered barcode for a dataset (equipped with a multi-level set ﬁltration) and analyze subsampling algorithms for efﬁciently approximating this quantity.
This extends earlier work of [9] and [8] in the one-parameter setting.
In future work, we will investigate improvements to the subsampling algorithms using multiparameter persistence landscapes.
A S INGLE - PARAMETER PERSISTENCE HOMOLOGY
Let M be a compact metric space with distance function d M : M × M → R .
In practice M will usually be a ﬁnite subset of Euclidean space R n but the results here are stated in more generality.
The deﬁnition of persistence homology uses the language of simplicial complexes.
Here we outline the key concepts and refer the reader to [7] for a detailed treatment and [18] for an introduction to simplicial homology.
See also [13] and [25] for the original papers in the setting where M is a ﬁnite set.
A simplicial complex C is a set of simplexes (i e. points, lines, triangles and their higher-dimensional counterparts) such that any face of a simplex in C is also in C , and the intersection of two simplices in C is either empty or a face of both simplices.
The simplicial complexes that are of interest to us are the Vietoris-Rips complexes Rips s ( M ) , deﬁned for a metric space M and s ≥ 0 .
A simplex [ m 0 , · · · , m k
] ∈
Rips s ( M ) if d M ( m i , m j ) ≤ s for 0 ≤ i ≤ k .
Note that if s ≤ s (cid:48) , then there is an inclusion from Rips s ( M ) to Rips s (cid:48)
( M ) ; we refer to the family { Rips s ( M ) } s ≥ 0 as a ﬁltered simplicial complex.
The persistent barcode β
i ( M ) of this ﬁltered simplicial complex is obtained by considering the homology groups of the simplicial complexes, H i
( Rips s ( M )) .
These are vector spaces equipped with linear maps H i
( Rips s ( M ))
→ H i ( Rips s (cid:48) ( M )) coming from the above inclusions.
The persistent barcode β
i ( M ) is a statistical summary consisting of intervals { [ b k , d k ] } 1 ≤ k ≤ n .
The bottleneck distance d b between two persistent barcodes is the smallest (cid:15) such that there is an (cid:15) -matching between the two barcodes (i.e. any interval [ b k , d k ] from one barcode of length more than (cid:15) must be matched to an interval [ b (cid:48)
j , d (cid:48)
j ] from the other barcode, with the Cartesian distance between the two points in
R 2 being less than (cid:15) ).
B C ONVERGENCE OF THE FIBERED BARCODE FOR TWO - PARAMETER PERSISTENCE HOMOLOGY B. S TABILITY RESULTS FOR PERSISTENCE HOMOLOGY
One of the most important properties of persistence barcodes is their stability with respect to the
Haussdorff distance, which was established in [11] and [7].
Given two subsets M 1 , M 2 of a compact metric space M , recall that the Haussdorff distance:
[Equation]
The stability property is the following:
[Equation]
This statement is often phrased using the Gromov-Haussdorff distance instead, but we will not need that level of generality here.
Now we state extensions of these results to the multi-parameter setting, speciﬁcally for the multi-level set ﬁltration and the Degree-Rips ﬁltration.
Deﬁnition B.1.
Let (cid:98) f
: M → M × R be the embedding deﬁned by (cid:98) f
( m ) =
( m, f ( m )) .
Given any subset S ⊂ M , let S f ⊂ M ×
R be its image under the map (cid:98) f .
Proposition B.2.
Let M be a compact metric space equipped with a function f : M → R .
Given subsets S, S (cid:48) of M:
[Equation]
See Appendix A for a more detailed statement and proof.
B. M ULTIPARAMETER SUBLEVEL - SET FILTRATIONS
Let M be a compact metric space, and f : M → R is a continuous function.
Denote by β f ( M ) its ﬁbered barcode, as deﬁned in Section ?? .
Suppose that we observe a sample X
= { X 1 , · · · , X n } drawn from an unknown measure µ on M , and the values f ( X i ) for 1 ≤ i ≤ n .
In this section we analyze the performance of the quantity β f ( X ) as an estimator of β f ( M ) .
First we extend Theorem 3.1 from [15] to this setting.
Theorem B.3.
Let X 1 , X 2 , · · · be i.i.d valued random variables chosen from the measure µ on M , which is supported on a compact subset X µ .
Let X n = { X 1 , · · · , X n } .
The following holds almost surely:
[Equation]
Next we study the convergence rate for the above result with respect to the matching distance on ﬁbered barcodes, extending Corollary 3 of [9].
Let M f ⊂ M ×
R be as deﬁned in Deﬁnition B.1, and i f : M → M f be the natural map.
Let µ f = i f ∗ µ be the pushforward measure on M f .
Assumption 1.
Suppose that the induced measure µ f satisﬁes the ( a, b ) -standard assumption, for some ﬁxed constants a, b > 0 : given any x ∈ M × R and r > 0 , µ ( B ( x, r )) ≥ min { ar b , 1 } .
Theorem B.4.
Suppose that we have a sample of n points X n = { X 1 , · · · , X n } drawn from an unknown measure µ .
[Equation]
The above theorem can be used to construct conﬁdence sets for the ﬁbered barcode, following Section 3.4 in [9].
Proposition B.5.
Let α ∈ (0 , 1) be chosen arbitrarily.
Suppose c satisﬁes e − c c < αn 2 b .
The following is a conﬁdence interval for β f ( X µ ) with level 1 − α :
[Equation]
R ISK ANALYSIS OF THE CLOSEST SUBSAMPLE FOR MULTI - LEVEL SET FILTRATIONS
We return to the setting from Sections B.2.
Assume that M = X is a ﬁnite metric space, equipped with a function f : X → R and let µ be the discrete uniform distribution centered on X .
In particular, we assume that the induced measure µ f satisﬁes the ( a, b ) -standard assumption.
Let X [1] , · · · , X [ m ] be m independent subsamples of size n chosen from µ .
Deﬁnition B.6.
Deﬁne the closest subsample as follows (here H denotes the Haussdorff distance between the ﬁnite metric spaces X and X i , and the notation X
[ i ] ,f was introduced in Deﬁnition B.1):
[Equation]
The following proposition follows from Theorem B.4.
Proposition B.7.
[Equation]
[Table]
m C P ROOFS C. S TABILITY OF MATCHING DISTANCE FOR MULTI - PARAMETER PERSISTENCE FILTRATIONS
In this subsection we prove Proposition 2.6, and establish that the ﬁbered code β f ( M ) is well-deﬁned for any compact metric space M (in Section 2.2.1, β f ( M ) was deﬁned for ﬁnite metric spaces M ).
First we establish Proposition 2.6 in the setting where M is a ﬁnite metric space, keeping the notation from Section 2.2.1.
In particular, recall that d : M × M → R is the distance function, L is the space of all afﬁne lines in R 2 with non-negative slope, and L ∈ L is a line y
= l 1 x + l 2 .
Proposition 2.
[Equation]
Proof of Proposition 2.6, when M is ﬁnite.
By deﬁnition of the multiparameter interleaving distance, the above statement is equivalent to the below inequality.
Here Rips fL ( S ) =
{ Rips fL,a ( S ) } a ∈ R is the ﬁltered simplicial complex deﬁned via Rips fL,a ( S ) =
Rips ft 1 ( a ) ,t 2 ( a ) ( S ) where t 1 ( a ) =
[Equation]
To prove this, we follow the approach used in Lemma 4.3 of [7] to prove the analogous result in the the one-parameter setting, and start by recalling the deﬁnitions.
Deﬁnition C.1.
Let S =
( S a ) a ∈ R and T =
( T a ) a ∈ R be ﬁltered simplicial complexes with vertex sets
X and Y .
A multivalued map C : X (cid:16) Y is (cid:15) -simplicial
if for any σ ∈ S a and a ∈ R , every ﬁnite subset of C ( σ ) is a simplex of T a + (cid:15) .
Proposition 4.2 of [7] states that if C : X (cid:16) Y is a multi-valued map with an inverse C T , then if C and C T are both (cid:15) -simplicial, then the persistence modules H ( S ) and H ( T ) are (cid:15) -interleaved.
Now it sufﬁces to show that the ﬁltered complexes Rips fL ( S ) and Rips fL ( S (cid:48) ) are (cid:15)
-interleaved, when (cid:15) >
w ( l ) d H ( S f , S (cid:48) f ) .
Consider the subset X ⊂ S × S (cid:48) of points ( u, u (cid:48) ) such that d ( u, u (cid:48) ) ≤ d H ( S f , S (cid:48) f ) , and let C be the corresponding multi-valued map.
Given a simplex σ ∈ Rips fL ( S ) a , we must show that any ﬁnite subset τ ∈ C ( σ ) lies in Rips fL ( S (cid:48) )
a + (cid:15) .
Given any two points u (cid:48)
=
( m (cid:48) 1 , f (cid:48) 1 ) , u (cid:48)
2 =
( m (cid:48) 2 , f (cid:48) 2 ) ∈ C ( σ ) , suppose u (cid:48) 1 ∈ C ( u 1 ) , u (cid:48) 2 ∈ C ( u 2 ) for u 1 = ( m 1 , f 1 ) , u 2 = ( m 2 , f 2 ) ∈
σ .
Then for i = 1 , 2 :
[Equation]
It now follows that τ ∈ C ( σ ) lies in Rips l ( U ) a + (cid:15) , completing the proof.
For the rest of this section, M will be a compact metric space.
Now we will show that β f ( M ) can be deﬁned in this level of generality, and that Proposition 2.6 holds in this setting.
Both of these follow from Proposition C.4 below, which we will establish following the approach in Section 2.4 of [15].
Deﬁnition C.2.
Let F ( M ) denote the set of ﬁnite non-empty subsets of the metric space M .
Let
K ( M ) denote the set of compact subsets of the metric space M .
Deﬁnition C.3.
Denote by B f ∞ the space of all ﬁbered barcodes, viewed as a metric space with the distance given by the matching distance between two ﬁbered barcodes.
Let (cid:98) B f ∞ be the completion of this metric space.
Proposition C.4.
There is a unique continuous extension K ( M ) → (cid:98) B f ∞ of the map β f : F ( M ) →
B f ∞ , which we denote by the same symbol.
This extended map is also Lipschitz continuous with
Lipschitz constant 2 .
Proof.
Given a compact subset K ⊂ M , let X =
{ X 1 , X 2 , · · · } be i.i.d random variables chosen from the uniform distribution on K , and for each positive integer n let X n = { X 1 , X 2 , · · · , X n } .
Given (cid:15) > 0 , it is well-known that the following statements hold almost surely: for n (cid:29) 0 , d H ( K, X n ) < (cid:15) (see Lemma 5.1 above for the statement and Lemma 3.2 of [15] for the proof).
It then follows from that the sequence { β f ( X n ) } n ∈ Z converges for a suitable choice of X , so β f ( K ) can be deﬁned as its limit.
It is easy to check that it does not depend on the choice of X , and that
Lipschitz continuity is satisﬁed.
C. P ROOF OF KEY RESULTS
Now we prove the key results in Sections 3 and 4.
Proof of Theorem B.3.
This follows immediately from Proposition B.2 combined with Lemma 3.2 in [15].
Proof of Theorem B.4.
This follows from Proposition B.2, combined with Theorem 2 of [9].
Recall that the notation X µ,f was introduced in Deﬁnition B.1.
[Equation]
Proof of Theorem B.5.
This follows immediately from Theorem B.4.
If (cid:15) =
b (cid:112) cna , then na(cid:15)
b = c and the conclusion follows:
[Equation]
Proof of Theorem B.7.
This follows from Proposition B.4, combined with Proposition B.2 above:
[Equation]