SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Remark 1. S-CDRs are injective representations: if the support of two interval modules are different, then their corresponding S-CDRs (evaluated on a point that belongs to the support of one interval but not on the support of the other) will differ, provided that  $\delta$  is sufficiently small.
4 Numerical Experiments
In this section, we illustrate the efficiency of our S-CDRs with numerical experiments. First, we explore the stability theorem in Section 4.1 by studying the convergence rates, both theoretically and empirically, of S-CDRs on various data sets. Then, we showcase the efficiency of S-CDRs on classification tasks in Section 4.2, and we investigate their running times in Section 4.3. Our code for computing S-CDRs is based on the MMA [29] and Gudhi [36] libraries for computing candidate decompositions<sup>2</sup>. It is publicly available at https://github.com/DavidLapous/multipipers and will be merged as a module of the Gudhi library. We also provide pseudo-code in Appendix H.
4.1 Convergence rates
In this section, we study the convergence rate of S-CDRs with respect to the number of sampled points, when computed from specific bifiltrations. Similar to the single parameter persistence setting [16], these rates are derived from Theorem 1. Indeed, since concentration inequalities for multiparameter
2Several different approaches can be used for computing decompositions [1, 7]. In our experiments, we used MMA [29] (with a family of 1000 diagonal lines) because of its simplicity and rapidity.
The persistence modules have already been described in the literature, these concentration inequalities can transfer to our representations. Note that while Equations (7) and (8), which provide such rates, are stated for the S-CDR in (3), they also hold for the S-CDR in (2).
Measure bifiltration. Let  $\mu$  be a compactly supported probability measure of  $\mathbb{R}^D$ , and let  $\mu_n$  be the discrete measure associated to a sampling of  $n$  points from  $\mu$ . The measure bifiltration associated to  $\mu$  and  $\mu_n$  is defined as  $\mathcal{F}_{r,t}^\mu := \{x \in \mathbb{R}^D : \mu(B(x, r)) \leq t\}$ , where  $B(x, r)$  denotes the Euclidean ball centered on  $x$  with radius  $r$ . Now, let  $\mathbb{M}$  and  $\mathbb{M}_n$  be the multiparameter persistence modules obtained from applying the homology functor on top of the measure bifiltrations  $\mathcal{F}^\mu$  and  $\mathcal{F}^{\mu_n}$ . These modules are known to enjoy the following stability result [2, Theorem 3.1, Proposition 2.23 (i]):  $d_{\mathrm{I}}(\mathbb{M}, \mathbb{M}_n) \leq d_{\mathrm{Pr}}(\mu, \mu_n) \leq \min(d_W^p (\mu, \mu_n)^{\frac{1}{2}}, d_W^p (\mu, \mu_n)^{\frac{p}{p+1}})$ , where  $d_W^p$  and  $d_{\mathrm{Pr}}$  stand for the  $p$ -Wasserstein and Prokhorov distances between probability measures. Combining these inequalities with Theorem 1, then taking expectations and applying the concentration inequalities of the Wasserstein distance (see [26, Theorem 3.1] and [21, Theorem 1]) lead to:
$$
\delta \mathbb {E} \left[ \| V _ {\infty , \delta} (\mathbb {M}) - V _ {\infty , \delta} (\mathbb {M} _ {n}) \| _ {\infty} \right] \leq \left(c _ {p, q} \mathbb {E} \left(| X | ^ {q}\right) n ^ {- \left(\frac {1}{2 p \vee d}\right) \wedge \frac {1}{p} - \frac {1}{q}} \log^ {\alpha / q} n\right) ^ {\frac {p}{p + 1}}, \tag {7}
$$
where  $\vee$  stands for maximum,  $\alpha = 2$  if  $2p = q = d$ ,  $\alpha = 1$  if  $d \neq 2p$  and  $q = dp / (d - p) \wedge 2p$  or  $q > d = 2p$  and  $\alpha = 0$  otherwise,  $c_{p,q}$  is a constant that depends on  $p$  and  $q$ , and  $X$  is a random variable of law  $\mu$ .
Cech complex and density. A limitation of the measure bifiltration is that it can be difficult to compute. Hence, we now focus on another, easier to compute bifiltration. Let  $X$  be a smooth compact  $d$ -submanifold of  $\mathbb{R}^D$  ( $d \leq D$ ), and  $\mu$  be a measure on  $X$  with density  $f$  with respect to the uniform measure on  $X$ . We now define the bifiltration  $\mathcal{F}^{C,f}$  with:
$$
\mathcal {F} _ {u, v} ^ {C, f} := \operatorname {C e c h} (u) \cap f ^ {- 1} ([ v, \infty)) = \left\{x \in \mathbb {R} ^ {D}: d (x, X) \leq u, f (x) \geq v \right\}.
$$
Moreover, given a set  $X_{n}$  of  $n$  points sampled from  $\mu$ , we also consider the approximate bifiltration  $\mathcal{F}^{C,f_n}$ , where  $f_{n}\colon X\to \mathbb{R}$  is an estimation of  $f$  (such as, e.g., a kernel density estimator). Let  $\mathbb{M}$  and  $\mathbb{M}_n$  be the multiparameter persistence modules associated to  $\mathcal{F}^{C,f}$  and  $\mathcal{F}^{C,f_n}$ . Then, the stability of the interleaving distance [27, Theorem 5.3] ensures:
$$
d _ {\mathrm {I}} (\mathbb {M}, \mathbb {M} _ {n}) \leq \| f - f _ {n} \| _ {\infty} \vee d _ {H} (X, X _ {n}),
$$
where  $d_H$  stands for the Hausdorff distance. Moreover, concentration inequalities for the Hausdorff distance and kernel density estimators are also available in the literature (see [16, Theorem 4] and [23, Corollary 15]). More precisely, when the density  $f$  is  $L$ -Lipschitz and bounded from above and from below, i.e., when  $0 < f_{\mathrm{min}} \leq f \leq f_{\mathrm{max}} < \infty$ , and when  $f_n$  is a kernel density estimator of  $f$  with associated kernel  $k$ , one has:
$$
\mathbb {E} \left(d _ {H} (X, X _ {n})\right) \lesssim \left(\frac {\log n}{n}\right) ^ {\frac {1}{d}} \text {a n d} \mathbb {E} (\| f - f _ {n} \| _ {\infty}) \lesssim L h _ {n} + \sqrt {\frac {\log \left(1 / h _ {n}\right)}{n h _ {n} ^ {d}}},
$$
where  $h_n$  is the (adaptive) bandwidth of the kernel  $k$ . In particular, if  $\mu$  is a measure comparable to the uniform measure of a  $d = 2$ -manifold, then for any stationary sequence  $h_n \coloneqq h > 0$ , and considering a Gaussian kernel  $k$ , one has:
$$
\delta \mathbb {E} \left[ \| V _ {\infty , \delta} (\mathbb {M}) - V _ {\infty , \delta} \left(\mathbb {M} _ {n}\right) \| _ {\infty} \right] \lesssim \sqrt {\frac {\log n}{n}} + L h. \tag {8}
$$
Empirical convergence rates. Now that we have established the theoretical convergence rates of S-CDRs, we estimate and validate them empirically on data sets. We will first study a synthetic data set and then a real data set of point clouds obtained with immunohistochemistry. We also illustrate how the stability of S-CDRs (stated in Theorem 1) is critical for obtaining such convergence in Appendix B, where we show that our main competitor, the multiparameter persistence image [11], is unstable and thus cannot achieve convergence, both theoretically and numerically.
Annulus with non-uniform density. In this synthetic example, we generate an annulus of 25,000 points in  $\mathbb{R}^2$  with a non-uniform density, displayed in Figure 3a. Then, we compute the bifiltration  $\mathcal{F}^{C,f_n}$  corresponding to the Alpha filtration and the sublevel set filtration of a kernel density estimator, with
bandwidth parameter  $h = 0.1$ , on the complete Alpha simplicial complex. Finally, we compute the candidate decompositions and associated S-CDRs of the associated multiparameter module (in homology dimension 1), and their normalized distances to the target representation, using either  $\| \cdot \|_2^2$  or  $\| \cdot \|_\infty$ . The corresponding distances for various numbers of sample points are displayed in log-log plots in Figure 3b. One can see that the empirical rate is roughly consistent with the theoretical one  $(-1/2$  for  $\| \cdot \|_\infty$  and  $-1$  for  $\| \cdot \|_2$ ) even when  $p \neq \infty$  (in which case our S-CDRs are stable for  $d_{\mathrm{B}}$  but theoretically not for  $d_{\mathrm{I}}$ ).
Figure 3: Convergence rate of synthetic data set.
Immunohistochemistry data. In our second experiment, we consider a point cloud representing cells, taken from [40], see Figure 4a. These cells are given with biological markers, which are typically used to assess, e.g., cell types and functions. In this experiment, we first triangulate the point cloud by placing a  $100 \times 100$  grid on top of it. Then, we filter this grid using the sublevel set filtrations of kernel density estimators (with Gaussian kernel and bandwidth  $h = 1$ ) associated to the CD8 and CD68 biological markers for immune cells. Finally, we compute the associated candidate decompositions of the multiparameter modules in homology dimensions 0 and 1, and we compute and concatenate their corresponding S-CDRs. Similar to the previous experiment, the theoretical convergence rate of our representations is upper bounded by the one for kernel density estimators with the  $\infty$ -norm. The convergence rates are displayed in Figure 4b. Again, one can see that the observed and theoretical convergence rates are consistent.
Figure 4: Convergence rate of immunohistochemistry data set.
4.2 Classification
In this section, we illustrate the efficiency of S-CDRs by using them for classification purposes. We show that they perform comparably or better than existing topological representations as well as standard baselines on several UCR benchmark data sets, graph data sets, and on the immunohistochemistry data set. Concerning UCR, we work with point clouds obtained from time delay embedding applied on the UCR time series, following the procedure of [10], and we produce S-CDRs with bifiltrations coming from combining either the Rips filtration with sublevel sets of a kernel density estimator (as in Section 4.1), or the Alpha filtration with the sublevel sets of the distance-to-measure
with parameter  $m = 0.1$  (as in [10] and the baselines therein). Concerning graph datasets, we produce S-CDRs by filtering the graphs themselves directly using Heat Kernel Signature with parameter  $t = 10$ , Ricci curvature and node degree (similarly to what is used in the literature [11, 22, 45]). In all tasks, every point cloud or graph has a label (corresponding to the type of its cells in the immunohistochemistry data set, and to pre-defined labels in the UCR and graph data sets), and our goal is to check whether we can predict these labels by training classifiers on the corresponding S-CDRs.
For point clouds (immunohistochemistry and UCR), we compare the performances of our S-CDRs (evaluated on a  $50 \times 50$  grid) to the one of the multiparameter persistence landscape (MPL) [39], kernel (MPK) [17] and images (MPI) [10], as well as their single-parameter counterparts (P-L, P-I and PSS-K). We also compare to some non-topological baselines: we used the standard Ripley function evaluated on 100 evenly spaced samples in [0, 1] for the immunohistochemistry data set, and k-NN classifiers with three difference distances for the UCR time series (denoted by B1, B2, B3), as suggested in [19]. For graps, we compare S-CDRs to the Euler characteristic based multiparameter persistence methods ECP, RT, and HTn, introduced in [22]. In order to also include non topological baselines, we also compare against the state-of-the-art graph classification methods RetGK [44], FGSD [37], and GIN [43].
All scores on the immunohistochemistry data set were computed after cross-validating a few classifiers (random forests, support vector machines and xgboost, with their default Scikit-Learn parameters) with 5 folds. For the time series data, our accuracy scores were obtained after also cross-validating the following S-CDR parameters;  $p \in \{0,1\}$ , op ∈ {sum, mean},  $\delta \in \{0.01,0.1,0.5,1\}$ ,  $h \in \{0.1,0.5,1,1.5\}$  with homology dimensions 0 and 1, and the following bandwidth parameters for kernel density estimation:  $b \in \{0.1\%, 1\%, 10\%, 20\%, 30\% \}$ , which are percentages of the diameters of the point clouds. with 5 folds. Parameters and results on graph data sets were cross-validated and averaged over 10 folds, following the pipelines of [22]. All results can be found in Table 1 (immunohistochemistry and UCR—UCR acronyms are provided in Appendix I) and Table 2. Bold indicates best accuracy and underline indicates best accuracy among topological methods. Note that there are no variances for UCR data sets since pre-defined train/test splits were provided. One can see that S-CDR almost always outperform topological baselines and are comparable to the standard baselines on the UCR benchmarks. Most notably, S-CDRs radically outperform the standard baseline and competing topological measures on the immunohistochemistry data set. For graph data sets, results are competitive with both topological and non-topological baselines; S-CDRs perform even slightly better on COX2.
Dataset	B1	B2	B3	PSS-K	P-I	P-L	MPK	MPL	MPI	S-CDR (Rips + KDE)	S-CDR (Alpha + DTM)
DPOAG	62.6	62.6	77.0	76.9	69.8	70.5	67.6	70.5	71.9	71.9	71.9
DPOC	71.7	72.5	71.7	47.5	67.4	66.3	74.6	69.6	71.7	73.8	74.6
PPOAG	78.5	78.5	80.5	75.9	82.0	78.0	78.0	78.5	81.0	81.9	84.9
PPOC	80.8	79.0	78.4	78.4	72.2	72.5	78.7	78.7	81.8	79.4	83.2
PPTW	70.7	75.6	75.6	61.4	72.2	73.7	79.5	73.2	76.1	75.6	75.1
IPD	95.5	95.5	95.0	-	64.7	61.1	80.7	78.6	71.9	81.2	77.2
GP	91.3	91.3	90.7	90.6	84.7	80.0	88.7	94.0	90.7	96.3	92.7
GPAS	89.9	96.5	91.8	-	84.5	87.0	93.0	85.1	90.5	88.0	93.7
GPMVF	97.5	97.5	99.7	-	88.3	87.3	96.8	88.3	95.9	95.3	95.9
PC	93.3	92.2	87.8	-	83.4	76.7	85.6	84.4	86.7	93.1	90.0
Ripley	P	MPL	S-CDR
Immuno	67.2(2.3)	60.7(4.2)	65.3(3.0)	91.4(1.6)
Table 1: Accuracy scores for UCR and immunohistochemistry data sets.
<sup>3</sup>Note that the sizes of the point clouds in the immunohistochemistry data set were too large for MPK and MPI using the code provided in https://github.com/MathieuCarriere/multipipers, and that all three single-parameter representations had roughly the same performance, so we only display one, denoted as P.
Dataset	RetGK	FGSD	GIN	ECP	RT	HT nD	S-CDR
COX2	81.4(0.6)	-	-	80.3(0.4)	79.7(0.4)	80.6(0.4)	82.0(0.2)
DHFR	81.5(0.9)	-	-	82.0(0.4)	81.3(0.4)	83.1(0.5)	81.6(0.2)
IMDB-B	71.9(1.0)	73.6	75.1(5.1)	73.3(0.4)	74.0(0.5)	74.7(0.5)	73.5(0.2)
IMDB-M	47.7(0.3)	52.4	52.3(2.8)	48.7(0.4)	50.2(0.4)	49.9(0.4)	49.5(0.2)
MUTAG	90.3(1.1)	92.1	90(8.8)	90.0(0.8)	87.3(0.6)	89.4(0.7)	88.4(0.3)
PROTEINS	78.0(0.3)	73.4	76.2(2.6)	75.0(0.3)	75.4(0.4)	75.4(0.4)	73.9(0.2)
4.3 Running time comparisons
In this section, we provide running time comparisons between S-CDRs and the MPI and MPL representations, in which we measured the time needed to compute all the train and test S-CDRs and baselines of the previous data sets, averaged over the folds (again, note that since UCR data sets already provide the train/test splits, there is no variance in the corresponding results). All representations are evaluated on grids of sizes  $50 \times 50$  and  $100 \times 100$ , and we provide the maximum running time over  $p \in \{0,1,\infty\}$ . All computations were done using a Ryzen 4800 laptop CPU, with 16GB of RAM. We provide results in Table 3, where it can be seen that S-CDRs (computed on the pinched annulus and immunohistochemistry data sets) can be computed much faster than the other representations, by a factor of at least 25. As for UCR data sets, which contain only small time series and corresponding point clouds, it can still be observed that S-CDRs can be computed faster than the baselines. Interestingly, this sparse and fast implementation based on corners can also be used to improve on the running time of the multiparameter persistence landscapes (MPL), as one can see from Algorithm 4 in Appendix H (which retrieves the persistence barcode of a multiparameter persistence module along a given line; this is enough to compute the MPL) and from Table 3.
Table 2: Accuracy scores on graph datasets.
Annulus	Immuno	PPTW	GP
Ours (S-CDR)	250ms(2ms)	275ms(9.8ms)	33.0ms(3.99ms)	45.6ms(5.74ms)
Ours (MPL)	36.9ms(0.8ms)	65.9ms(0.9ms)	22.4ms(2.15ms)	31.8ms(2.95ms)
MPI (50)	6.43s(25ms)	5.67s(23.3ms)	65.2ms(12.9ms)	208ms(16.3ms)
MPL (50)	17s(39ms)	15.6s(14ms)	154ms(27.9ms)	630ms(30.0ms)
MPI (100)	13.1s(125ms)	11.65s(7.9ms)	289ms(75.0ms)	1.69s(77.7ms)
MPL (100)	35s(193ms)	31.3s(23.3ms)	843ms(200ms)	4.43s(186ms)
Table 3: Running times for S-CDRs and competitors.
In this article, we study the general question of representing decompositions of multiparameter persistence modules in Topological Data Analysis. We first introduce T-CDR: a general template framework including specific representations (called S-CDR) that are provably stable. Our experiments show that S-CDR is superior to the state of the art.
Limitations. (1) Our current T-CDR parameter selection is currently done through cross-validation, which can be very time consuming and limits the number of parameters to choose from. (2) Our classification experiments were mostly illustrative. In particular, it would be useful to investigate more thoroughly on the influence of the T-CDR and S-CDR parameters, as well as the number of filtrations, on the classification scores. (3) In order to generate finite-dimensional vectors, we evaluated T-CDR and S-CDR on finite grids, which limited their discriminative powers when fine grids were too costly to compute.
Future work. (1) Since T-CDR is similar to the PersLay framework of single parameter persistence [11] and since, in this work, each of the framework parameter was optimized by a neural network, it is thus natural to investigate whether one can optimize T-CDR parameters in a data-driven way as well, so as to be able to avoid cross-validation. (2) In our numerical applications, we focused on representations computed off of MMA decompositions [29]. In the future, we plan to investigate whether working with other decomposition methods [1, 7] lead to better numerical performance when combined with our representation framework.
Acknowledgments. The authors would like to thank the area chair and anonymous reviewers for their insightful comments and constructive suggestions. The authors would also like to thank Hannah Schreiber for her great help with the implementation of our method. The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support. DL was supported by ANR grant 3IA Côte d'Azur (ANR-19-P3IA-0002). MC was supported by ANR grant TopModel (ANR-23-CE23-0014). AJB was partially supported by ONR grant N00014-22-1-2679 and NSF grant DMS-2311338.
[1] H. Asashiba, E. Escolar, K. Nakashima, and M. Yoshiwaki. On approximation of 2d persistence modules by interval-decomposables. Journal of Computational Algebra, 6-7:100007, 2023.
[2] A. Blumberg and M. Lesnick. Stability of 2-parameter persistent homology. Foundations of Computational Mathematics, 2022.
[3] M. Botnan and M. Lesnick. An introduction to multiparameter persistence. In CoRR. arXiv:2203.14289, 2022.
[4] M. Botnan, S. Oppermann, and S. Oudot. Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions. In 38th International Symposium on Computational Geometry (SoCG 2022), volume 224, pages 19:1–19:18. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2022.
[5] M. B. Botnan and M. Lesnick. Algebraic stability of zigzag persistence modules. *Algebraic & Geometric Topology*, 18(6):3133-3204, Oct. 2018.
[6] M. Buchet, Y. Hiraoka, and I. Obayashi. Persistent homology and materials informatics. In Nanoinformatics, pages 75–95. Springer-Verlag, 2018.
[7] C. Cai, W. Kim, F. Mémoli, and Y. Wang. Elder-rule-stairstcodes for augmented metric spaces. In 36th International Symposium on Computational Geometry (SoCG 2020), pages 26:1-26:17. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2020.
[8] G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009.
[9] G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete & Computational Geometry, 42(1):71-93, 2009.
[10] M. Carrière and A. Blumberg. Multiparameter persistence image for topological machine learning. In Advances in Neural Information Processing Systems 33 (NeurIPS 2020), pages 22432-22444. Curran Associates, Inc., 2020.
[11] M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer, and Y. Umeda. PersLay: a neural network layer for persistence diagrams and new graph topological signatures. In 23rd International Conference on Artificial Intelligence and Statistics (AISTATS 2020), pages 2786-2796. PMLR, 2020.
[12] M. Carrière, S. Oudot, and M. Ovsjanikov. Local signatures using persistence diagrams. In HAL. HAL:01159297, 2015.
[13] M. Carrière, S. Oudot, and M. Ovsjanikov. Stable topological signatures for points on 3D shapes. Computer Graphics Forum, 34(5):1-12, 2015.
[14] O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee. Choosing Multiple Parameters for Support Vector Machines. Machine Learning, 46(1):131-159, Jan. 2002.
[15] F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules. SpringerBriefs in Mathematics. Springer-Verlag, 2016.
[16] F. Chazal, M. Glisse, C. Labruère, and B. Michel. Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16(110):3603-3635, 2015.
[38] O. Vipond. Local equivalence of metrics for multiparameter persistence modules. In CoRR. arXiv:2004.11926, 2020.
[39] O. Vipond. Multiparameter persistence landscapes. Journal of Machine Learning Research, 21(61):1-38, 2020.
[40] O. Vipond, J. Bull, P. Macklin, U. Tillmann, C. Pugh, H. Byrne, and H. Harrington. Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors. Proceedings of the National Academy of Sciences of the United States of America, 118(41), 2021.
[41] A. P. Witkin. Scale-space filtering. In Readings in Computer Vision: Issues, Problems, Principles, and Paradigms, pages 329-332. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, Jan. 1987.
[42] C. Xin, S. Mukherjee, S. Samaga, and T. Dey. GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning. In 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning. OpenReviews.net, 2023.
[43] K. Xu, W. Hu, J. Leskovec, and S. Jegelka. How powerful are graph neural networks? In 7th International Conference on Learning Representations (ICLR 2019). OpenReviews.net, 2019.
[44] Z. Zhang, M. Wang, Y. Xiang, Y. Huang, and A. Nehorai. RetGK: graph kernels based on return probabilities of random walks. In Advances in Neural Information Processing Systems 32 (NeurIPS 2018), pages 3964-3974. Curran Associates, Inc., 2018.
[45] Q. Zhao and Y. Wang. Learning metrics for persistence-based summaries and applications for graph classification. In Advances in Neural Information Processing Systems 33 (NeurIPS 2019), pages 9855–9866. Curran Associates, Inc., 2019.
A Limitation of single-parameter persistent homology
A standard example of single-parameter filtration for point clouds, called the Čech filtration, is to consider  $f: x \mapsto d_P(x) \coloneqq \min_{p \in P} \|x - p\|$ , where  $P$  is a point cloud. The sublevel sets of this function are balls centered on the points in  $P$  with growing radii, and the corresponding persistence barcode contains the topological features formed by  $P$ . See Figure 5a. When the radius of the balls is small, they form three connected components (upper left), identified as the three long red bars in the barcode (lower). When this radius is moderate, a cycle is formed (upper middle), identified as the long blue bar in the barcode (lower). Finally, when the radius is large, the balls cover the whole Euclidean plane, and no topological features are left, except for one connected component, that never dies (upper right).
(a) Persistence barcode obtained from growing balls on a clean point cloud.
(b) Persistence barcode obtained from growing balls on a noisy point cloud.
Figure 5: Example of persistence barcode construction for point clouds using Čech filtration. In the middle sublevel sets, we highlight the topological cycles in blue.
The limitation of the Čech filtration alone for noisy point cloud is illustrated in Figure 5b, where only feature scale is used, and the presence of three outlier points messes up the persistence barcode entirely, since they induce the appearance of two cycles instead of one. In order to remedy this, one could remove points whose density is smaller than a given threshold, and then process the trimmed point cloud as before, but this requires using an arbitrary threshold choice to decide which points should be considered outliers or not.
B Unstability of MPI