Stability of 2-Parameter Persistent Homology

Abstract

The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the data. A natural solution is to consider 2-parameter persistence. This paper studies the stability of 2-parameter persistent homology: we show that several related density-sensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, Sheehy’s subdivision bifiltrations, and the degree bifiltrations. For the multicover and subdivision bifiltrations, we get 1-Lipschitz stability results closely analogous to the standard stability results for 1-parameter persistent homology. Our results for the degree bifiltrations are weaker, but they are tight, in a sense. As an application of our theory, we prove a law of large numbers for subdivision bifiltrations of random data.

Appendix A. A Computational Example of the Stability of Degree-Rips Bifiltrations

In this section, we explore the stability of the degree-Rips bifiltration in an example, using the 2-parameter persistence software RIVET [50, 51, 64]. We consider three point clouds X, Y, and Z, shown in Fig. 1:

• X consists of 475 points sampled uniformly from an annulus in ${\mathbb {R}}^2$ with outer radius .5 and inner radius .4.

• $Y=X\cup N$, where N consists of 25 points sampled uniformly from a disc of radius .4.

• Z consists of 500 points sampled uniformly from a disc of radius .5.

We would like to consider, for each $W\in \{X,Y,Z\}$ the homology module $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W))$ with coefficients in ${\mathbb {Z}}/2{\mathbb {Z}}$. However, working directly with $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W))$ is computationally expensive, so we instead work with an approximation H(W) of $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W))$; this is explained in Appendix A.1. In Appendix A.2, we illustrate the stability of degree bifiltrations in practice by using RIVET to visualize invariants of H(X), H(Y), and H(Z). Then, in Appendix A.3, we apply the stability result Proposition 3.9 (ii) to explain a small part of the similarity between H(X) and H(Y) observed in our visualizations. Recall that Proposition 3.9 (ii) applies to a nested pair of data sets; in Remark A.2, at the end of this section, we observe that Theorem 1.7 (ii), which does not assume that the data is nested, does not constrain the similarity between H(X) and H(Y).

We warn the reader that the remainder of this section is somewhat technical, in part because of the approximations involved. We invite the reader to skim the section on a first reading, focusing on understanding the figures.

Fig. 1

The point clouds X, Y, Z

1.1 A.1 Approximations to the Degree-Rips Bifiltrations

For discussing RIVET computations, it is convenient to introduce a variant of the normalized Degree-Rips bifiltration. First, for W a finite metric space, let $\bar{{\mathcal {R}}}(W):[0,\infty )\rightarrow \mathbf {Top}$ be the filtration defined by taking

$$\begin{aligned} \bar{{\mathcal {R}}}(W)_r=\lim _{r<s} \mathcal {R}(W)_s. \end{aligned}$$

This is precisely the variant of the Rips construction mentioned in Remark 2.3.

Now define a bifiltration

$$\begin{aligned} {{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W):{\mathbb {R}}^{\mathrm {op}}\times [0,\infty )\rightarrow {{\,\mathrm{\mathbf {Simp}}\,}}\end{aligned}$$

by taking ${{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)_{(k,r)}$ to be the maximal subcomplex of $\bar{{\mathcal {R}}}(W)_r$ whose vertices have degree at least $|W|(r-1)$. This bifiltration is slightly different from the normalized degree-Rips bifiltration ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W)$ defined in Sect. 2.3—for one thing, they are indexed by different posets—but it’s not hard to see that these two bifiltrations are equivalent, in the sense that each determines the other in a simple way. Moreover, the restriction of ${{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)$ to the poset $J$ is $\epsilon$-interleaved with ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W)$ for any $\epsilon >0$.

The largest simplicial complex in ${{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)$ is the simplex with vertices W; denote this as S. For any simplex $\sigma \in S$, we define the set of bigrades of appearance of $\sigma$ to be the set of minimal elements $(k,r)\in {\mathbb {R}}^{\mathrm {op}}\times [0,\infty )$ such that $\sigma \in {{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)_{(k,r)}$. This is a finite and nonempty subset of $J$.

To control the cost of the computations, for each $W\in \{X,Y,Z\}$ we in fact work with a “coarsening”

$$\begin{aligned} F(W):{\mathbb {R}}^{\mathrm {op}}\times [0,\infty )\rightarrow \mathbf {Top}\end{aligned}$$

of the bifiltration ${{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)$, where the bigrades of appearance of all simplices are rounded upwards so as to lie on a uniform $100\times 100$ grid; the precise definition of this coarsening is given in [50]. The grid is chosen in a way that ensures that ${{\,\mathrm{\overline{\mathcal{DR}\mathcal{}}}\,}}(W)$ and F(W) are $(\tau ^{\frac{1}{100}},\mathrm {Id})$-interleaved. Let $F'(W)$ denote the restriction of F(W) to $J$. By the generalized triangle inequality for interleavings (Remark 2.40), ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(W)$ and $F'(W)$ are $(\tau ^{\frac{1}{100}+\epsilon },\tau ^\epsilon )$-interleaved for all $\epsilon >0$.

For $W\in \{X,Y,Z\}$, let H(W) denote the homology module $H_1(F(W))$ with coefficients in ${\mathbb {Z}}/2{\mathbb {Z}}$. Observe that H(W) is finitely presented.

1.2 A.2 Visualization

RIVET computes and visualizes three invariants of a finitely presented biperistence module $M:{\mathbb {R}}^{\mathrm {op}}\times [0,\infty )\rightarrow \mathbf {Top}$:

• The Hilbert function of M, i.e., the dimension of each vector space of M.

• The bigraded Betti numbers of M; these are functions $\beta _i^M:{\mathbb {R}}^2\rightarrow {\mathbb {N}}$, $i\in \{0,1,2\}$ which, roughly speaking, record the birth indices of generators, relations, and relations among the relations.

• The fibered barcode of M [16], i.e., the map sending each affine line $\ell \subset {\mathbb {R}}^{\mathrm {op}}\times [0,\infty )$ of non-positive slope to the barcode ${\mathcal {B}}_{M^{\ell }}$, where $M^{\ell }$ is the restriction of M to $\ell$. We regard ${\mathcal {B}}_{M^{\ell }}$ as a collection of intervals on the line $\ell$.

See [50, 51] for more details about these invariants and their computation in RIVET.

Figure 2 shows RIVET’s visualization of H(X), H(Y), and H(Z), and of the barcodes ${\mathcal {B}}_{H(X)^{\ell }}$, ${\mathcal {B}}_{H(Y)^{\ell }}$, and ${\mathcal {B}}_{H(Z)^{\ell }}$ for one choice of $\ell$. To explain the figure, first note that the x-axis is mirrored in each subfigure, relative to the usual convention, so that values decrease from left to right. In each figure, the Hilbert function is represented by grayscale shading, where the darkness is proportional to the homology dimension. The lightest non-white shade of gray shown in each figure corresponds to a value of 1. The bigraded Betti numbers are represented by translucent colored dots whose area is proportional to the value. The $0^{\mathrm {th}}$, $1^{\mathrm {st}}$, and $2^{\mathrm {nd}}$ bigraded Betti numbers are shown in green, red, and yellow, respectively. In each figure, the line $\ell$ is shown in blue, and the corresponding barcode is plotted in purple, with each interval offset perpendicularly from the line.

Note that the Hilbert function of H(X) takes value 1 on a large connected region parameter space, whose restriction to the left half-plane $k> 0$ looks roughly like a triangle. The Hilbert function of H(Y) also takes value 1 on a substantial region in parameter space, albeit one smaller than for H(X). In contrast, the Hilbert function of H(Z) does not take the value 1 in a large region of the parameter space, and in fact almost all of the support of H(Z) lies very near (0, 0).

Fig. 2

RIVET’s visualization of H(X), H(Y), H(Z) and the barcodes ${\mathcal {B}}_{H(X)^{\ell }}$, ${\mathcal {B}}_{H(Y)^{\ell }}$, ${\mathcal {B}}_{H(Z)^{\ell }}$, for one choice of line $\ell$

1.3 A.3 Stability Analysis

By appealing to Proposition 3.9 (ii), we can explain a small part of the similarity between the structures of H(X) and H(Y) observed in Fig. 2. More specifically, we will show that given H(X), Proposition 3.9 (ii) implies that the Hilbert function of H(Y) has non-trivial support on a small region of parameter space. The only property of Y we use in our analysis is that Y is a metric space of cardinality 500 containing X as a subspace.

By Proposition 3.9 (ii), ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X)$ and ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y)$ are $(\kappa ^{|X|/|Y|},\gamma ^{\delta })$-homotopy interleaved for any $\delta >\tfrac{25}{500}=\tfrac{1}{20}$. Note that $|X|/|Y|=\tfrac{475}{500}=\tfrac{19}{20}$. By the discussion of Appendix A.1, ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X)$, $F'(X)$ are $(\tau ^{\frac{1}{100}+\epsilon },\tau ^\epsilon )$-interleaved for all $\epsilon >0$, and the same is true for ${{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y)$, $F'(Y)$. For $\epsilon \ge 0$, let $\zeta ^\epsilon$ be the forward shift given by

$$\begin{aligned} \zeta ^{\epsilon }(x,y)=\left( \frac{19x}{20}-\frac{1}{100}- \epsilon ,y+\frac{1}{100}+\epsilon \right) , \end{aligned}$$

and let $\zeta =\zeta ^0$. A strict interleaving is also a homotopy interleaving, so by Proposition 2.38, $F'(X)$ and $F'(Y)$ are $(\zeta ^{\epsilon },\gamma ^{\frac{6}{100}+\epsilon })$-homotopy interleaved for all $\epsilon >0$. From this, one can show that in fact, $F'(X)$ and $F'(Y)$ are $(\zeta ,\gamma ^{\frac{6}{100}})$-homotopy interleaved, using an argument similar to the proof of [49, Theorem 6.1].

Letting $H'(X)$ and $H'(Y)$ denote the respective restrictions of H(X) and H(Y) to $J$ (i.e., $H'(X)=H_1(F'(X))$ and $H'(Y)=H_1(F'(Y))$), we then have that $H'(X)$ and $H'(Y)$ are $(\zeta ,\gamma ^{\frac{6}{100}})$-interleaved by Proposition 2.41. In what follows, we will show that this constrains certain vector spaces in $H'(Y)$ to have dimension at least one.

Let $\Delta$ denote the large triangle-like connected region in $J$ where the Hilbert function of $H'(X)$ takes value 1. The boundary of $\Delta$ intersects the vertical line $k=0$ and the horizontal line

$$\begin{aligned} r=\tfrac{7891389}{20000000}\approx .396. \end{aligned}$$

By inspecting the bigraded Betti numbers and fibered barcode of H(X), as shown in Figs. 2 and 3, it can be seen that $\Delta$ is in fact is contained in the support of a thin indecomposable summand of $H'(X)$. Thus, if $a\le b\in \Delta$ (with respect to the partial order on ${\mathbb {R}}^{\mathrm {op}}\times {\mathbb {R}}$), then ${{\,\mathrm{rank}\,}}({H'(X)}_{a,b})=1$.

Fig. 3

RIVET’s visualization of H(X), zoomed in near (0, 0)

In particular, if $(k,r)\in \Delta$ and also

$$\begin{aligned} \gamma ^{\frac{6}{100}}\circ \zeta (k,r)=\left( \frac{19k}{20}-\frac{7}{100},3r+\frac{9}{100}\right) \in \Delta , \end{aligned}$$

then

$$\begin{aligned} {{\,\mathrm{rank}\,}}H'(X)_{(k,r),\gamma ^{\frac{6}{100}}\circ \zeta (k,r)}=1. \end{aligned}$$

Now if $(k,r)\in \Delta$, then $(\frac{19k}{20}-\frac{7}{100},3r+\frac{9}{100})\in \Delta$ if and only if $k>c_x$ and $r<c_y$, where

$$\begin{aligned} c_x=\tfrac{7}{95}\approx .074,\quad c_y=\left( \tfrac{2030463}{20000000}\right) \approx .102. \end{aligned}$$

From the output of RIVET, it can be seen that the set

$$\begin{aligned} \Omega =\{(k,r)\in \Delta \mid k>c_x,\ r<c_y\} \end{aligned}$$

is a small but non-empty connected subregion of $\Delta$, containing the grades of three elements in any minimal set of generators for $H'(X)$. $\Omega$ is shown in Fig. 4A. Explicitly,

$$\begin{aligned} \Omega ={{\,\mathrm{Rect}\,}}(v_1)\cup {{\,\mathrm{Rect}\,}}(v_2)\cup {{\,\mathrm{Rect}\,}}(v_3), \end{aligned}$$

where

$$\begin{aligned} {{\,\mathrm{Rect}\,}}(a)&=\{(x,y)\mid a_1\ge x > c_x,\ a_2\le y < c_y\},\\ v_1&=\left( \tfrac{2657}{23750},\tfrac{1897929}{20000000}\right) \approx (.112,.095),\\ v_2&=\left( \tfrac{2183}{23750},\tfrac{1698147}{20000000}\right) \approx (.092,.085),\\ v_3&=\left( \tfrac{973}{11875},\tfrac{299673}{4000000}\right) \approx (.082,.075). \end{aligned}$$

Fig. 4

The regions $\Omega$ and $\zeta (\Omega )$, shown in solid black. [Note: These regions were drawn by hand in software, and so are not as precise as if they had been drawn algorithmically. However, the imprecisions are miniscule.]

For any $(k,r)\in \Omega$, a $(\zeta ,\gamma ^{\frac{6}{100}})$-interleaving between $H'(X)$ and $H'(Y)$ provides a factorization of the non-zero linear map $H'(X)_{(k,r),(\frac{19k}{20}-\frac{7}{100},3r+\frac{9}{100})}$ through

$$\begin{aligned} H'(Y)_{\zeta (k,r)}=H'(Y)_{(\frac{19k}{20}-\frac{1}{100},r+\frac{1}{100})}. \end{aligned}$$

Therefore, the support of the Hilbert function of $H'(Y)$ must contain $\zeta (\Omega )$. $\Omega$ is shown in Fig. 4B. Letting

$$\begin{aligned} d_x=\tfrac{6}{100},\quad d_y=\tfrac{2230463}{20000000}\approx .112, \end{aligned}$$

$\zeta (\Omega )$ can be written explicitly as

$$\begin{aligned} \zeta (\Omega )={{\,\mathrm{Rect}\,}}(w_1)\cup {{\,\mathrm{Rect}\,}}(w_2)\cup {{\,\mathrm{Rect}\,}}(w_3), \end{aligned}$$

where

$$\begin{aligned} {{\,\mathrm{Rect}\,}}(a)&=\left\{ (x,y)\mid a_1\ge x > d_x ,\ a_2\le y < d_y\right\} ,\\ w_1&=\left( \tfrac{2407}{25000},\tfrac{2097929}{20000000}\right) \approx (.096,.105),\\ w_2&=\left( \tfrac{1933}{23750},\tfrac{1898147}{20000000}\right) \approx (.077,.095),\\ w_3&=\left( \tfrac{212}{3125},\tfrac{339673}{4000000}\right) \approx (.068,.085). \end{aligned}$$

Figure 4B indicates that the support of the Hilbert function of $H'(Y)$ does indeed contain $\zeta (\Omega )$, and in fact is much larger.

Remark A.1

We have shown that given H(X), Proposition 3.9 (ii) constrains the structure of H(Y). Using a similar argument, one can show that given $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X))$, Proposition 3.9 (ii) constrains the structure of $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y))$. However, a similar argument also shows that given H(Y), Proposition 3.9 (ii) provides no constraint on H(X); and similarly, given $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y))$, Proposition 3.9 (ii) provides no constraint on $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X))$.

Remark A.2

By Remark 2.16, $d_{GPr}(X,Y)\le d_P(X,Y)\le \frac{25}{500}=\frac{1}{20}$. Starting from this observation, one can perform a stability analysis similar to the one done above, but using the weaker interleaving provided by Theorem 1.7 (ii) in place of the one provided by Proposition 3.9 (ii). It is not difficult to check that the interleaving between $H'(X)$ and $H'(Y)$ provided by such an analysis can be taken to be trivial. Further, using a similar argument, one can show that the interleavings between $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X))$ and $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y))$ guaranteed to exist by Theorem 1.7 (ii) can also be taken to be trivial. Thus, the existence of this interleaving does not constrain the relationship between $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(X))$ and $H_1({{\,\mathrm{\mathcal{DR}\mathcal{}}\,}}(Y))$ at all. Because the shared topological signal present in X and Y is especially strong, this indicates that relative to the needs of applications, Theorem 1.7 is a rather weak result.

Remark A.3

Figure 2 makes clear that while H(X) and H(Y) have rather different global structure, they do share substantial qualitative similarities that neither module shares with H(Z). While our analysis demonstrates that Proposition 3.9 (ii) non-trivially constrains the relationship between H(X) and H(Y), most of the similarity between the two modules seen in Fig. 2 is not explained by the proposition (or, to the best of our knowledge, by any other known result). It would be valuable to develop a refinement of our stability theory which more fully explains the observed similarity.