Stochastic Multiresolution Persistent Homology Kernel

Xiaojin Zhu, Ara Vartanian, Manish Bansal, Duy Nguyen, Luke BrandlDepartment of Computer Sciences, University of Wisconsin–Madison

1210 W. Dayton Street, Madison, Wisconsin, USA 53706

AbstractWe introduce a new topological feature representa-tion for point cloud objects. Specifically, we con-struct a Stochastic Multiresolution Persistent Ho-mology (SMURPH) kernel which represents an ob-ject’s persistent homology at different resolutions.Under the SMURPH kernel two objects are simi-lar if they have similar number and sizes of “holes”at these resolutions. Our multiresolution kernel cancapture both global topology and fine-grained topo-logical texture in the data. Importantly, on largepoint clouds the SMURPH kernel is more com-putationally tractable compared to existing topo-logical data analysis methods. We demonstrateSMURPH’s potential for clustering and classifica-tion on several applications, including eye diseaseclassification and human activity recognition.

1 Introduction and BackgroundThere has been growing interest in bringing topologicaldata analysis (TDA) into machine learning [Bubenik, 2015;Chazal et al., 2015b; Reininghaus et al., 2015; Ahmed et

al., 2014; Li et al., 2014; Chazal et al., 2015a; Pachauri et

al., 2011]. However, no method exists that is simultaneouslyrich in topological information, efficient in computation, andeasy to plug into a machine learning system. We take a stepin this direction by proposing a Stochastic MUltiResolutionPersistent Homology (SMURPH) kernel. Our kernel can beviewed as a topological feature extractor for machine learningthat compares the number and sizes of “holes” in two pointclouds. Unlike prior TDA methods, SMURPH captures boththe global topology and fine-grained “topological texture” inpoint cloud objects. Using SMURPH for machine learningrequires little background in topology, as it just produces akernel matrix.

Persistent homology is a TDA method for studying homol-ogy classes, or “topological holes.” A thorough tutorial isout of scope of this paper and can be found in e.g. [Edels-brunner and Harer, 2010; Nanda and Sazdanovic, 2014;Carlsson, 2009; Zhu, 2013]. We will focus on first-order ho-mology over Z

2

(binary) coefficients. For this purpose thefollowing background knowledge suffices. On a point cloudX in Rd we construct an ✏-hypergraph by creating edges

(known as 1-simplices) between all pair of points xi, xj 2 Xwithin distance ✏. We also create triangle faces (2-simplices)among all xi, xj , xk whose pairwise distances are within ✏.This hypergraph is known as a simplicial complex. Follow-ing the terminology of [Edelsbrunner and Harer, 2010] wedefine a first-order homology group whose linearly indepen-dent generators represents holes (cycles that are not filled inby triangles) in the graph.

Now imagine we repeat this hypergraph construction sep-arately for all thresholds ✏ > 0. This series of hypergraphsis called a Vietoris-Rips filtration. As ✏ increases more edgesand triangles will be created. The homology group changesby gaining or losing generators as holes are born (when a cy-cle is established) at some ✏

1

and die (the cycle is filled) atsome ✏

2

� ✏1

. Persistent homology tracks such birth anddeath events at critical ✏ values. This information is tradition-ally stored as a persistence diagram (PD) in the TDA litera-ture. Intuitively, a PD is a multiset of N points (bi, di) in 2Dfor the birth and death time (i.e. ✏ threshold) of the N holesencountered during Vietoris-Rips filtration.

The space of PDs is a metric space under the Wassersteindistance but not a vector space (hence not a Hilbert space).We instead build our kernels on the recently-proposed per-sistence landscape (PL) representation, which is a benignfunction space [Bubenik, 2015]. To obtain the PL from PD,one first rotates the PD so that the diagonal becomes the x-axis, and scale it by 1/

p2. In the new coordinate system, a

(birth, death) pair (bi, di) takes the coordinate (si, ti) wheresi = (bi + di)/2, ti = (di � bi)/2 for i = 1 . . . N .One defines a “tent function” ⇤i at each (si, ti): ⇤i(s) =

max(ti � |s � si|, 0), s 2 R. The PL is a collection ofpiecewise linear functions N ⇥ R 7! R indexed by level l:�(l, s) = lmax

Ni=1

⇤i(s) where lmax returns the lth largestvalue in the set, or 0 if the set has less than l items.

2 The SMURPH Kernel

We will define three kernels on n point cloud objectsX

1

. . . Xn. The first kernel K[ captures global persistent ho-mology but does not scale to large point clouds. The secondkernel K] introduces multi-resolution but also does not scale.The SMURPH kernel K solves the scalability issue of K]

with Monte Carlo sampling.

Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16)

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point cloud size10 1 10 2 10 3

time

(sec

onds

)

10 -2

10 -1

10 0

10 1

10 2

10 3JavaplexPerseusTDA:DionysusTDA:Gudhi

Figure 1: Time to compute a PD vs. point cloud size, log-logscale

2.1 A Global Persistent Homology Kernel K[

For any fixed level l, the lth landscape function �(l, s) 2 L2

and is 1-Lipschitz. We define the inner product between twoPL functions �,�0 by

h�,�0i =X

l

Z 1

�1�(l, s)�0

(l, s)ds. (1)

We will use |X| to denote the number of points in point cloudX . The summation over level l is finite, since the maximumnonzero level in a PL is upper bounded by max(|X|, |X 0|). Ithas been noted that this space of PL functions is a Hilbertspace [Bubenik, 2015]. Therefore, we can define a posi-tive definite topological kernel K[ between two point cloudsX,X 0 via the inner product of their PL functions �X ,�X0 :

K[(X,X 0

) = h�X ,�X0i. (2)

While K[ captures global topological information, there isa major roadblock in using it for data analysis. Figure 1 showsthe time to compute PD, the building block of PL and K[,versus the point cloud size |X|. We compared several popularTDA software: JavaPlex [Tausz et al., 2011], Perseus [Nanda,2013], the R TDA package [Fasy et al., 2014] with Diony-sus [Morozov, 2007] and GUDHI [Maria, 2014] solvers, re-spectively. The reported time is in seconds on a typical desk-top computer for the software to compute 1

st persistence di-agram. The slopes of these log-log time curves suggest thatcomputation time scales as |X|3 to |X|4 for 1st persistent ho-mology. Objects with more than a few hundred points arecomputationally prohibitive.

A naive solution to the scalability issue is to sampleb points from X to form a sparser point cloud b(X), assuggested in [Chazal et al., 2015b; Chazal et al., 2015a].Repeating the process s times, we obtain point cloudsb1

(X) . . . bs(X). A bootstrapping PL function can then becomputed as ¯�sb

=

1

s

Pj �bj(X)

, where the average appliesto each PL level separately.

Unfortunately, this bootstrap procedure introduces a lossof resolution. As an example, consider the two point cloudsin Figure 2. At the global level (a) they look the same, butzooming in (b) we notice that the top one is made of numer-ous small rings while the bottom one is simply an annulus.

(a) (b) (c) (d)

Figure 2: (a) top: the ring-of-rings dataset, bottom: the annu-lus dataset. (b) zoom. The two datasets look identical in theirglobal PL ¯�msb

r with r = diam(X) = 2 in (c), but can bedistinguished at a finer resolution r = 0.06 in (d).

Figure 3: A mug point cloud inside a ball, its persistence land-scape �B(c,r), and its average persistence landscape ¯�msb

r overm = 20, s = 1 samples.

Subsampling globally (c) ignores this difference – justifiablyso if one is only concerned with the overall shape of �X , sincethe finer resolution only contributes minimally to the globalPL. Nonetheless, this difference in “topological texture” maybe important for classifying or clustering. For better topolog-ical data analysis, we need the ability to identify differencesin localized topology.

2.2 A Multiresolution Kernel K]

Inspired by multiresolution data analysis, especially the workon local relative homology in [Ahmed et al., 2014], we pro-pose a multiresolution persistent homology description of Xto compute localized topology like those in Figure 2(d). Tothis end consider balls of a particular radius r, which we callthe resolution. Let B(c, r) = {x 2 X : kx � ck r} be theset of points in the ball centered at c with radius r. Let �B(c,r)be the corresponding PL function computed by performing afiltration on B(c, r). As an example, Figure 3(left) shows a3D point cloud of a mug. The ball is shown in yellow, B(c, r)shown in red1, and �B(c,r) in Figure 3(mid).

We draw centers c from a probability distribution PC . Inthis paper, we assume PC is supported on X; that is, each ballmust center on one of the data points in X . A simple choiceof PC is the uniform distribution over points in X , thougha domain expert can design other PC to emphasize certaininteresting regions. Our quantity of interest is the expectedPL function at resolution r:

⇡r :

= Ec⇠PC [�B(c,r)]. (3)

1Actually a subset of B(c, r), see section 2.3

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When PC is the uniform distribution over X , this quan-tity is simply ⇡r =

1

|X|P

c2X �B(c,r). Let the diameterof X be diam(X) = maxi,j2X ki � jk. Noting that ifr � diam(X), B(c, r) = X for all c 2 X , we can estab-lish that 8r � diam(X), ⇡r = �X . Thus r � diam(X)

recovers the global PL function. To obtain topological infor-mation of X at finer resolutions, we consider L decreasingradii r

1

> . . . > rL and compute the corresponding expectedPL functions ⇡r1 , . . . ,⇡rL .Definition 1 The multiresolution persistent homology repre-

sentation of X at radii r1

. . . rL is (⇡r1 , . . . ,⇡rL).

We now have all the ingredients to define a multiresolu-tion persistent homology kernel K]. Specifically, we definehomology kernel K] between two point cloud objects X andX 0 as

K](X,X 0

) =

LX

i=1

wih⇡ri ,⇡0rii (4)

where w1

, . . . , wL are nonnegative weights to combine dif-ferent resolutions. A particularly useful weighting scheme isto set

wi = (r1

/ri)3

, i = 1 . . . L. (5)To understand this scheme, imagine two holes both born atb = 0 and die at d = r

1

. The inner product (1) betweentheir PLs is r3

1

/12. Now imagine two other holes both withb = 0, d = ri. Their inner product is r3i /12. The weightingscheme thus scales up the finer resolution so that all resolu-tions contribute equally to the kernel.

However, the kernel K] (4) is defined without regard tocomputation. In fact K] is more costly to compute than K[

due to two issues: 1. The expectation inside ⇡ requires enu-merating all possible centers c 2 X; 2. B(c, r) may still con-tain many points, making persistent homology software slow(recall Figure 1). We address both issues with sampling next.

2.3 The SMURPH Kernel KThe procedure below is carried out independently at each res-olution r. We sample m centers c

1

. . . cm ⇠ PC . For eachcenter c, we generate j = 1 . . . s bootstrap samples withinthe ball B(c, r). Each bootstrap sample bj(c, r) consists ofb points sampled with replacement from B(c, r). Instead ofcomputing �B(c,r), we compute �bj(c,r). The value of b ischosen with computation speed in mind, so that state-of-the-art persistent homology software can finish in a reasonableamount of time. We define average persistence landscape

(APL), an estimator of ⇡r, as follows:

¯�msbr =

1

ms

mX

i=1

sX

j=1

�bj (ci, r). (6)

The superscripts msb remind the reader that ¯�msbr is subject

to three kinds of randomness: m balls, s bootstraps per ball,b points per bootstrap.

To illustrate, we go back to Figure 2. Both the ring-of-rings point cloud and the annulus point cloud in (a) containone million points. This poses no computational difficulty for¯�msbr in (c) and (d) if we choose e.g. b = 300, which is easily

handled by state of the art TDA software, and m = 20, s = 1.

As another example, Figure 3(right) shows the average persis-tence landscape on the mug. The most significant persistenthomology class (due to the handle) survives the averaging,even though we do not expect every random ball to containthe handle.Definition 2 The stochastic multiresolution persistent

homology representation of X at radii r1

. . . rL is

(

¯�msbr1 , . . . , ¯�msb

rL ).

Finally, we define the Stochastic Multi-Resolution Persis-tent Homology (SMURPH) kernel as:

K(X,X 0) =

LX

i=1

wih¯�msbri (x), ¯�0msb

ri i. (7)

Note that K is stochastic due to sampling but given the sam-ples it is a positive semi-definite kernel matrix. During testtime when one needs to compute the kernel K(Xi, X

⇤) be-

tween a training object Xi and a new point cloud objectX⇤, it is important that one uses the same representation�¯�msbr1 , . . . , ¯�msb

rL

�for Xi during training to ensure that the

kernel matrix is well-defined.Algorithm 1 specifies the computation of SMURPH ker-

nel K. If we take the state-of-the-art persistent homologytime complexity to be O(b3) as indicated by Figure 1, thecomplexity of Algorithm 1 is O(nLmsb3 + n2

). We alsonote that the inner product (1) can be efficiently computedin closed-form since PL is piecewise linear as a consequenceof lmax over tent functions. In practice, computing each ob-ject’s stochastic multiresolution persistent homology repre-sentation (

¯�msbr1 , . . . , ¯�msb

rL ) takes a few seconds for modest bin the hundreds.

Algorithm 1 SMURPH KernelInput: n point cloud objects X

1

, . . . Xn

Parameters: radius scheme (r1

, . . . , rL), kernel weightscheme (w

1

, . . . , wL), center distribution PC , number ofcenters m, bootstrap sample size b, number of bootstrapss, filtration F .for object X 2 {X

1

. . . Xn} dofor resolution r 2 {r

1

. . . rL} dofor center i = 1 . . .m do

Sample ci ⇠ PC

for bootstrap j = 1 . . . s doSample b points within B(ci, r) to form bj(ci, r)Apply filtration F on bj(ci, r) to compute PL�bj (ci, r)

end forend for¯�msbr =

1

ms

Pmi=1

Psj=1

�bj (ci, r)end forRepresent object X by

�¯�msbr1 , . . . , ¯�msb

rL

�

end forDefine the n ⇥ n resolution-r kernel matrix asKr(Xi, Xj) = h¯�msb

r (Xi), ¯�msbr (Xj)i

Output: SMURPH kernel matrix K =

PLi=1

wiKri

For theoretical consideration, define the n ⇥ n SMURPHkernel matrix K = [Kij ] where Kij = K(Xi, Xj) for

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i, j = 1, . . . , n. Similarly, we define K]=

hK]

ij

i. Fol-

lowing the technique in [Chazal et al., 2015a; 2013], wecan show that K approximates K]: under mild conditions

E||⇡r � ¯�msbr ||1 O

⇣log bb

⌘1/�

�for some constant �.

This means that ¯�msbr is an asymptotically unbiased estima-

tor of ⇡r. Furthermore, we can bound E||K]�K||max

where||.||

max

is the max matrix norm. If s,m are of the order⇣

blog b

⌘2/�

, then E||K] �K||max

O⇣

log bb

⌘1/�

�.

3 ExperimentsWe present three applications in clustering and classificationto demonstrate the potential of the SMURPH kernel.

3.1 Pots and PansWe demonstrate SMURPH kernel’s ability to embed pointcloud objects in a meaningful way based on persistent homol-ogy features. The dataset consists of 41 point cloud kitchenutensils (e.g. pans, cups, bottles, knives) [Neumann et al.,2013]. We compute SMURPH kernel matrix using a radiusof r = 0.1, m = 20 centers per point cloud, s = 1 samplesper center, and a budget of b = 350 points per sample.

−0.6 −0.4 −0.2 0.0 0.2 0.4

−0.4

−0.2

0.0

0.2

0.4

Figure 4: Kernel PCA embedding with SMURPH kernel ma-trix on the pots and pans dataset

We embed the 41 objects in 2D using kernel PCA on theSMURPH kernel matrix. For better visualization we draweach object centered on its embedding coordinates, see Fig-ure 4. All objects are drawn at the same scale. We identifythree salient groups with similar persistent homology in thisembedding: (1) mugs with a handle, pots with handles, andlong bottles near (�0.4, 0.2); (2) ladle, knifes, screw drivers

near (0.3, 0); (3) small cans near (�0.4,�0.4). We now ex-plain why the grouping is topologically meaningful.

Most objects in the group (1) have handles. We have shownone such mug in Figure 3. Another mug is shown in Fig-ure 5(left). Colors are solely for better visualization. Han-dles produce large holes in Vietoris-Rips filtration, whose ef-fect is preserved in the average persistence landscape as a talland long hump. Recall the SMURPH kernel is computed asinner products between these APL functions. The groupingin kPCA embedding space reflects the overall similarity be-tween APL functions. Therefore, objects with handles aregrouped close.

Figure 5: Three representative point clouds from the Pots andPans dataset and their average persistence landscape ¯�msb

r .

In contrast, objects in group (2), such as the ladle shown inFigure 5(middle), have no intrinsic holes. Their average per-sistence landscape is almost flat and distinct from group (1).Note their APL is not exactly the zero line – this is an artifactof sampling. To see why, imagine a regular 2D grid of pointsin a ball B(c, r). SMURPH samples b points from the ball.When b is small relative to the number of points in the ball,spurious holes will be created during Vietoris-Rips filtration.In practice, however, this artifact has limited effect and doesnot prevent kPCA from producing meaningful embedding.

Finally, group (3) reveals an interesting property ofSMURPH. First, the three cans are defective and missing thebottom in their point cloud (Figure 5 right), making themsimilar to the wine glass at (�0.6,�0.2). Second, whenSMURPH samples from a ball centered near the bottom, theball would contain the whole can except the cap. The pointsin the ball then form a cylinder, which has a large hole. Thishomology feature will enter APL. In other words, SMURPHreflects both the overall topological structure of the object andthat of its parts. We call the latter topological texture.2 Thetopological texture of the cans and the wine glass has its owndistinct APL signature, and is responsible for group (3).

2Such topological texture is also responsible for the long bottles,which has no intrinsic 1st order holes, to enter group (1).

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3.2 Drusen Detection in Fundus PhotographyDrusen are small yellowish hyaline deposits that develop be-neath the retinal pigment epithelium. While drusen are nat-ural phenomena that occur with aging, The presence of nu-merous, larger drusen is predictive of macular degeneration.Some drusen sites form small holes in skeletonized images(Figure 6 right). This suggests SMURPH kernel can help inclassifying drusen vs. non-drusen images.

Non-drusen image Drusen image

Figure 6: Fundus photographs and the point clouds afterskeletonizing.

Data. Our sample consists of 67 retinal images, collectedas part of the STARE [Hoover and Goldbaum, 2013] project.Of these, 35 have been labeled by experts as images with-out drusen and 32 as images with drusen. We pre-processthe images by first binarizing (setting all non-zero pixel val-ues to 1) and then skeletonizing with MATLAB’s bwmorph.Our point cloud is then the coordinates of the nonzero pix-els of this skeletonized image. To compute SMURPH ker-nel, we sample patch centers with a uniform distribution overthe point cloud. We choose a resolution level of r = 8 andm = 50, s = 1, b = 300.

Method. We compare SMURPH to two baselines: lin-ear kernel and radial basis function (RBF) kernel. Whilemore sophisticated computer vision processing can undoubt-edly enhance the baselines, our goal is to demonstrate the po-tential of SMURPH without heavy feature engineering. Wetune all parameters (regularization C for SVM and band-width � for RBF kernel) using an inner cross validation (CV)inside the outer CV training portion. The tuning grid isC 2 {2�7, 2�6, . . . , 27} and � 2 {100, 101, . . . , 103}. Wefeed the kernels to an SVM (kernlab [Karatzoglou et al.,2004]) for classification and report 5-fold CV accuracy.

Results. Table 1 summarizes our results. SMURPH kerneloutperforms the linear and RBF kernel baselines, demonstrat-ing that SMURPH captures topological features that help todetect drusen. To assess the significance of our result, we per-form a t-test over the accuracy of our classifiers across the CVfolds, testing the accuracy of the SMURPH kernel against theRBF kernel and the linear kernel. As reported in Table 1, foreither baseline we can reject the null hypothesis that accuracydoes not differ at the 0.05 level.

Kernel Parameters Accuracy p-valueSMURPH C = 32 70.2% -

RBF C = 2, � = 10 52.2% 0.040Linear C = 2 43.3% 0.007

Table 1: 5-fold cross validation accuracy of SVM classifierson Fundus data set

3.3 Human Activity RecognitionWe now demonstrate SMURPH kernel’s ability to handlepoint clouds from a state space. In an unsupervised learningsetting we show how kernel PCA embeds the n objects; in asupervised learning setting we will use activity as the classlabel and perform classification.

Data. The “daily and sports activities” data set [Altun et

al., 2010] contains sensor data of several everyday activi-ties, among which we choose five representative ones: sitting(A1), walking (A9), running (A12), jumping (A18), and play-ing basketball (A19). Each activity is performed by 8 peoplein their own style for five minutes. Each person’s data is atime series measured at 25 Hz, for a total of 5⇤60⇤25 = 7500

measurements. Each measurement is 45-dimensional: 5 sen-sor units placed on torso, right arm, left arm, right leg, andleft leg; each sensor unit contains x,y,z accelerometers, x,y,zgyroscopes, and x,y,z magnetometers.

We treat each activity-person combination as an object fora total of n = 5 ⇤ 8 = 40 objects. Each object contains 7500points. Each point has a time stamp and a 45-dimensionalmeasurement. No processing is performed on the measure-ments.

Filtration with Timeline. Since each object here is a timeseries, we present a special filtration design which is of inde-pendent interest. A similar filtration has been used to modelsequences of natural language paragraphs [Zhu, 2013]. Letthe point cloud object be X = {x = (t, z)} where t is thetime stamp and z is the measurement vector at time t. Ourfiltration has two unique steps:

(1) The ball is defined by t, not by z. That is, p is theuniform distribution over time stamps in X . Once we samplea center in time: c ⇠ Pt, we define the ball B(c, r) = {x =

(t, z) 2 X : kt� ck r}, which is in fact a time interval.(2) We add a “timeline” (a set of edges) before Vietoris-

Rips filtration begins. Within B(c, r), we randomly sam-ple b points x

1

. . . xb. Sort these points by time so thatt1

< . . . < tb. We create the timeline by connecting points(xi, xi+1

) adjacent in time for i = 1 . . . b � 1. The Vietoris-Rips filtration then proceeds as usual but uses the distancemetric on z. The final complex is the union of the timelineand the filtration. All in all, the timeline acts as a pre-existingtemporal skeleton to help enhance the filtration. We computea b ⇥ b distance matrix D where Di,j = kzi � zjk and setDi,i+1

= 0 for i = 1 . . . b � 1. Even though D no longersatisfies the triangle inequality it can still serve as a filtrationfunction.

Unsupervised kernel PCA Results. We use two reso-lution levels: r

1

= 125 and r2

= 25 which correspondto 10-second and 2-second intervals. We set the parame-ters m = 10, s = 1, b = 100. We compute the 40 ⇥ 40

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kernel PCA 1-300 -200 -100 0

kern

el P

CA

2

-100

-50

0

50

100

150 sittingwalkingrunningjumpingbasketball

Figure 7: Kernel PCA with the SMURPH kernel on eightpeople ⇥ five activities

SMURPH kernel matrix with weights w1

= w2

= 1 in (7).We then perform kernel PCA on this kernel matrix to embedthe 40 objects in 2D. Figure 7 shows the embedding with an-notated activities. Overall the activities form clear clusters.All eight participants’ sitting activities are overlapping at theorigin. This is expected because each sitting PL is nearlyempty. The participants’ walking activities are similar andform a tight cluster next to sitting. The cluster of playingbasketball is more spread out than walking but tighter thanrunning. We speculate that although basketball is more phys-ical, it also lacks clear periodicity, hence the in-between em-bedding. Running as a cluster occupies an extreme region inkernel PCA due to both strong activities and clear periodic-ity. Finally, jumping is the only activity that spread out in theembedding across the eight people. The original data collec-tion in [Altun et al., 2010] instructed people to perform eachactivity in free style. We suspect that the cluster spread maybe attributed to a large variation in how consistently peoplejumped for five minutes, given that it is a physically demand-ing activity.

Supervised activity classification. We perform a classi-fication task to predict the five activities. The input is the40 ⇥ 40 SMURPH kernel matrix. We measure 8-fold CVerror, where in each fold we use 7 people (total of 35 activ-ities) as the training data, and leave all 5 activities from oneperson out as test data. We use SVM light [Joachims, 2008]with our SMURPH kernel. This SVM problem has a sin-gle regularization parameter C. We tune C by an inner CVwithin the first training fold of 7 people. On a parameter gridC 2 {10�5, 10�4, . . . , 105}, this inner CV selects C = 100.We fix this C for all other outer cross validation folds. TheSVM CV accuracy with the SMURPH kernel is 95.0%. Incontrast, with a linear kernel the SVM CV accuracy is 62.5%.

4 Related WorkThis paper is related to but differs from several recent workin topological data analysis:

multi-resolution single-resolutionkernel ours KHNLB15, RHBK15, PHCJS11

non-kernel AFW14 Bubenik15, LOC14

The closest work AFW14 is the local persistent homology

proposed by Ahmed, Fasy, and Wenk [2014]. They computedlocal relative homology based on PD at various resolutions tocompare street maps. SMURPH differs in that we build onPL; we address scalability issue with Monte Carlo; and weconstruct kernels to enable a much broader range of machinelearning applications.

Also closely related are KHNLB15 [Kwitt et al., 2015] andRHBK15 [Reininghaus et al., 2015]’s multi-scale kernel. Animportant difference is that the “scale” in their multi-scalekernel means different amount of heat diffusion. Therefore,their kernels always describe global topology but with vary-ing amount of smoothing. In contrast SMURPH kernels aredefined over varying spatial scales, which allows us to seeboth global topology and fine-grained “topological texture.”In addition our PL-based kernel avoids the need to performheat diffusion on PDs as they do, and enjoys both conceptualsimplicity and computational savings. Both kernels exhibitsstability [Bubenik, 2015; Reininghaus et al., 2015] and canbe used for machine learning.

Bubenik [2015] pointed out that PL is a Hilbert space.LOC14 [Li et al., 2014] used both PL and PD to measurethe global topological distance between point clouds. Theirmethods did not aim to define kernels, nor did they studymulti-resolution. PHCJS11 [Pachauri et al., 2011] proposeda topological kernel for studying Alzheimer’s disease. Thatkernel is based on kernel density estimation within a PD.However, this is a heuristic because a PD is not a 2D Eu-clidean space. Our paper inherits Chazal et al.’s bootstrapapproach and approximates the persistent homology on a fullpoint cloud with that on a random sample [Chazal et al.,2015b; Chazal et al., 2015a]. Their bootstrap approach losesresolution with sampling; our multi-resolution approach ame-liorates this issue.

5 ConclusionWe have presented a novel kernel that compares the topologyof point cloud objects. Our SMURPH kernel is multiresolu-tional and can handle large point clouds. It serves as a newbridge between topology and machine learning to encouragefurther research that can benefit both communities.

There are many ways one can extend SMURPH. The ex-periments in this paper are only proof-of-concept. One im-portant task is to identify applications where SMURPH, inconjunction with standard kernels, can significantly improvethe performance of machine learning applications. Anotherimportant task is to go beyond kernel methods and bring thesame topological information into probabilistic models.

Acknowledgments The authors thank Frederic Chazal andBrittany Fasy for helpful discussions. Support for this re-search was provided in part by NSF grant IIS 0953219.

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