SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Abraham D. Smith
Geometric Data Analytics, Inc.
343 W. Main Street
Durham, NC 27701 USA
abraham.smith@geomdata.com
University of Wisconsin-Stout
Math, Stats, and CS Dept
Menomonie, WI 54751 USA
smithabr@uwstout.edu
Gabrielle Angeloro
gabrielle.angeloro@geomdata.com
Michael J. Catanzaro
michael.catanzaro@geomdata.com
Nirav Patel
nirav.patel@geomdata.com
Paul Bendich
paul.bendich@geomdata.com
Mathematics Dept
Durham, NC 27708 USA
bendich@math.duke.edu
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure.
Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between "overfitting" and "generalization" in applications of deep-learning.
In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset.
Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.
Suppose  $X$  is a finite subset of  $V = \mathbb{R}^n$  with the Euclidean metric. In data science—particularly in applications of DNNs—we often encounter the situation where  $X$  is a dataset, and some opaque algorithm has produced a trained model  $k: V \to \{0,1\}$ , where  $k(x) = 1$  for all  $x \in X$ . This defines the model as a set of accepted inputs  $K = \{x: k(x) = (1)\} \subset V$ , which has no available description beyond evaluation of the perception function  $k$  on samples.
Our main contribution in this paper is a method we call topological parallax to estimate the multiscale geometry of  $K$  from the persistent homology ([14, 37]) of  $X$ , in a situation where  $K$  does not have an explicit description. This method provides meaningful geometric information about  $K$  through a simple computational approach that can be applied to any perception model  $k$ . We prove that the resulting criterion of homological matching satisfies a stability property. We propose homological matching via parallax as a geometric specification that could be applied to many machine-learning systems. The measurement of homological matching also admits a back-propagation scheme, which could be used to improve the geometric similarity between the model  $K$  and the dataset  $X$ .
Because of the generality of Definition 1.1, it may be that  $V$  represents any layer of a neural network,  $X$  represents any dataset mapped into that layer, and  $K$  represents an activated region in that layer. For example, the "neural collapse" concept from Papyan et al. [29] can be seen as a special case of this specification, because the conception from [29] is that the dataset  $X$  becomes a tight blob in the penultimate layer, and the penultimate model is simply a Voronoi region  $K$  surrounding that blob.
1.1 Assumptions and Motivation
As discussed by Belkin [5], DNNs usually achieve high statistical accuracy, but some resulting models are better than others at capturing patterns in the dataset. Despite having perfect statistical accuracy on the dataset  $X$ , fundamental questions arise about the model  $K$ : "Is it safe to deploy? Is it trustworthy? Is it a good model?" To broadly paraphrase [5], it used to be good practice to tell data analysts not to overfit their data, because overfit models were "bad" due to poor generalization; however, essentially every DNN fits the training data perfectly, so it is not clear what distinguishes "good" models from "bad." We suggest that a model  $K$  is "good" if the geometry of  $K$  matches the geometry of  $X$ . Consider the two example models in Figure 1. Although both models achieve perfect statistical accuracy, only one of them appears to have learned the geometric structure of the dataset. This suggestion is likely intuitive to many ML engineers, but the subtlety lies in the implicit assumptions often made about the nature of the dataset and the available models. We assess this matching in a way that is independent of the architecture that produced  $k$ , and that makes very few assumptions about  $X$  and  $K$ , as encapsulated by Definition 1.1, which are assumed henceforth.
Definition 1.1 (Datasets and Models). Suppose that  $V$  is a geodesic space. We define a model  $K \subset V$  to be the closure of an open set,  $K = \overline{K^{\circ}} \subset V$ , colloquially known as a "solid." For any finite dataset  $X \subset V$ , we consider the collection of all models for which  $X$  is contained in the interior of the model,  $X \subset K^{\circ}$ ; let  $\mathcal{M}(X) := \{K \subset V : X \subset K^{\circ}, \overline{K^{\circ}} = K\}$ . With subset inclusions as morphisms,  $\mathcal{M}(X)$  is a small category. For any  $K \subset V$  with  $K = \overline{K^{\circ}}$ , let  $\mathcal{M}^{*}(K) := \{X \subset V : X \subset K^{\circ}, \# X < \infty\}$ . With functions of finite sets as morphisms,  $\mathcal{M}^{*}(K)$  is a small category. Of course,  $X \in \mathcal{M}^{*}(K) \Leftrightarrow K \in \mathcal{M}(X)$ . Note that  $V \in \mathcal{M}(X)$ .
Note that we make no assumptions whatsoever about the architecture or training method that yielded  $K$ . With such a broad definition of  $X$  and  $K$ , we need a notion of "geometry" with very few assumptions. We use standard topological notation [23, 17, 14].
Definition 1.2 (Void). Given a set  $K \subset V$ , an interpolative void in  $K$  is a bounded open set in the complement of  $K$ ,  $\Omega \subset K^c$ , such that there exists a pair of points  $x, y \in K$  for which all  $V$ -geodesics pass through  $\Omega$ . If  $V = \mathbb{R}^n$ , this means  $\Omega$  intersects the convex hull of  $K$ .
<sup>1</sup>The theoretical results apply if  $V$  is any geodesic space—a metric space in which each distance is realized by a path—but our motivation and examples use  $V = \mathbb{R}^n$  to avoid distraction from our central theme of model assessment.
2Topological parallax was named by analogy to the method in astronomy, where an inaccessible object cannot be measured directly (in this case, the geometry of  $K$ ), so we must infer its location by comparing multiple observations from available vantage points (in this case, the dataset  $X$ ).
Figure 1: 1: A dataset  $X \subset \mathbb{R}^2$ . 2: The persistence diagram of the Rips complex of  $X$ , showing three components (two of which die at the dim-0 dots near 2.0) and two cycles (the dim-1 dots, representing the annulus and the overall arrangement) among the persistent features. 3: A tree model  $K_1$ . 4: A neural network model  $K_2$ . The tree model has many small voids that would forbid interpolation or perturbation in many locations. Without the luxury to visualize models  $K \subset V$  for  $\dim V > 3$ , Parallax can measure geometric similarity between  $X$  and  $K$  via the Rips complex  $R$  of  $X$ .
Figure 2: Five models for the same dataset, and the scales detected by parallax. The circle have radii 20 and 30, respectively. See Section 6 for general interpretation. The missing edge causing  $\lambda_{\mathrm{sup}}$  is highlighted in red at filtration radius 7.1, and the missing edges that cause  $\lambda_{\mathrm{hi}}$  are highlighted in blue. The dashed blue edge does not affect  $\lambda_{\mathrm{hi}}$  because the larger cycle determines that value.
Why focus on voids? As observed by Balestriero et al. [2], when  $V$  is high-dimensional, it is unlikely that any points in  $X$  lie in the convex hull of any others. However, it also seems unlikely that a "good" model  $K$  of  $X$  will be merely a convex solid. If convex models were sufficient for real-world problems, DNNs would be unnecessary, and the field would have concluded with PCA, Gaussian kernels, and convex polytopes. For example, many models implicitly or explicitly rely on the so-called manifold hypothesis, which is the hope that realistic datasets  $X$  will tend to be distributed near a union of lower-dimensional manifolds immersed in  $V$ , in which case a "good" model  $K$  would be a slight thickening of those manifolds to allow for measurement error. If there are multiple manifolds or there is nonzero curvature, such a model will have voids.
Any void in  $K$  indicates a region where  $K$  does not allow interpolation, which is where interesting geometry occurs. Voids occur if and only if some geodesic in  $K$  is strictly longer than the corresponding geodesic in  $V$ . Hence, for this article, we interpret "geometry of  $K$ " as the presence or absence of voids. We do not make assumptions about the geometric features of  $X$  or about the family from which  $K$  is chosen; rather, we ask that  $K$  respects the features of  $X$ , whatever they are. We propose that a "good" model  $K$  is one whose voids represent the highly persistent features of  $X$ , in the sense of Topological Data Analysis [TDA, [14, 27]]
A model  $K$  can be "bad" because it has too many or too few voids at various scales. For example, mismatch of voids of  $K$  and features of  $X$  would indicate that  $K$  is over-sensitive to small error or under-sensitive to large error, either of which could lead to adversarial attack. Also, numerous small-scale voids could make  $K$  incompatible with some forms of the manifold hypotheses, by obstructing the coverage of  $K$  by an atlas of local convex charts of moderate size.
Section 2 introduces our key object, the bi-graded [7] parallax complex, in Definition 2.2, which measures geodesic distortion via the Rips complex. Section 3 provides a notion of dataset perturbation
and shows that the parallax complex and its homology remain stable under those perturbations. Section 4 defines local simplicial matching and shows how parallax detects small-scale changes in the Rips complex of  $X$  in  $K$  versus  $V$ , giving a clearly interpretable scale of locality,  $\lambda_{\mathrm{lo}}$ . Section 5 defines homological matching and shows how parallax detects large-scale voids in  $K$ , giving a scale  $\lambda_{\mathrm{hi}}$  above which homological features in  $X$  are respected by voids in  $K$ . Together, these results provide an overall interpretation as a specification in Section 6, which largely achieves the goals laid out in Section 1. Section 7 provides computational approaches to computing parallax, and links to our open-source software that has many practical improvements not detailed in this paper. Section 8 illustrates the effectiveness of parallax as a specification, as demonstrated on two models using the cyclo-octane dataset [22]. Additional proofs, details, and examples are provided in the Supplementary Material appendices.
To the best of our knowledge this is the first work to use TDA to express a desired geometric relationship that holds directly between datasets and models trained on them. There has been some work, for example the Manifold Topology Divergence of Barannikov et al. [3] or the Geometric Score of Khrulkov and Oseledets [20], which uses various TDA-based measures to quantify the difference between training data and new data generated by generative models. Quite a few other papers (see Fernandez et al. [15] for a very recent example) use TDA-based constructions to infer properties of underlying data manifolds, usually under very strict sampling assumptions.
More broadly, there has been a recent explosion of work (e.g. Hensel et al. [19]) connecting TDA to ML/DL. Several works (e.g. Adams et al. [1], Bendich et al. [6]) use TDA as a feature extraction method, pre-processing more complicated data objects before running standard ML pipelines. Later works (e.g. Chen et al. [10], Demir et al. [12], Solomon et al. [32], Nigmatov and Morozov [25]) use TDA to define novel losses within ML algorithms. Note that we comment below on ways in which our notion of homological matching can be used to define a TDA-based loss. TDA has also been used (e.g. Naitzat et al. [24], Wheeler et al. [36]) to analyze the behavior of data as it passes through the layers of a DNN. Some works (e.g. Guss and Salakhutdinov [16]) assess the capacity of a specific DNN to classify datasets with specific shapes, but do not provide tools to quantify shape mismatch between model and dataset. Finally, several works (e.g. Carriere et al. [8], Papillon et al. [28]) use TDA to define novel DNN architectures, including GNNs and other higher-order combinatorial structures.
There is also a recent stream (e.g. Liu et al. [21], Wang et al. [35]) of work that builds validation and verification/falsification techniques for desired properties of DNN-trained models; these mostly focus on the mechanics of how to verify/falsify such properties, rather than attempting to define them as we do. Perhaps the closest work in this stream to ours is Dola et al. [13], which uses a prior assumption on the underlying data distribution to verify/falsify DNN properties.
2 The Parallax Bi-Complex
Let  $B_{\alpha}(x)$  denotes the closed geodesic ball of radius  $\alpha$  around  $x \in V$ . For a formal edge  $e = (x_0, x_1)$  between points in  $X$ ,  $\rho_V(e)$  is the minimum radius for which  $B_{\rho_V(e)}(x_0)$  intersects  $B_{\rho_V(e)}(x_1)$ . Thus,  $2\rho_V(e)$  is the geodesic distance between  $x_0$  and  $x_1$ . The Rips complex  $R(X, V)$  is the simplicial complex generated by these edges, as filtered by  $\rho_V(e)$ . A chain is a formal sum of simplices in a complex [14]. More generally, for any  $K \in \mathcal{M}(X)$ , the Rips complex  $R(X, K)$  and its filtration by  $0 \leq \alpha < \infty$  is defined by
$$
(x _ {0}, \dots , x _ {d}) \in R _ {d} (X, K) _ {\alpha} \text {i f a n d o n l y i f} B _ {\alpha} \left(x _ {i}\right) \cap B _ {\alpha} \left(x _ {j}\right) \cap K \neq \emptyset , \forall 0 \leq i <   j \leq d. \tag {2.1}
$$
For any chain  $Y \in R(X, K)$ , let  $\rho_{K}(Y) = \min \{ \alpha : Y \in R(X, K)_{\alpha} \}$ . When  $X$  and  $V$  are understood in context, we abbreviate  $R = R(X, V)$  for the ambient case  $K = V$ .
Lemma 2.1. If  $K_{1}, K_{2} \in \mathcal{M}(X)$  with  $K_{1} \subset K_{2} \subset V$ , then there is a natural inclusion of filtered modules,  $R(X, K_{1})_{\alpha} \subset R(X, K_{2})_{\alpha} \subset R_{\alpha}$ . That is,  $\rho_{V}(Y) \leq \rho_{K_{2}}(Y) \leq \rho_{K_{1}}(Y)$ , with the convention  $\min \varnothing = \infty$ .
The previous lemma is simply because geodesic lengths in  $K_{1}$  are never shorter than geodesic lengths in  $K_{2}$ , and neither is shorter than geodesic lengths in  $V$ . Our approach to the question "does the geometry of  $K$  match the geometry of  $X$ ?" relies on detecting the inequality  $\rho_{V}(Y) < \rho_{K}(Y)$ .
Definition 2.2 (Parallax Complex). For  $K \in \mathcal{M}(X)$ , let  $P(X, K, V)$  denote the subcomplex of  $R$  defined for each real pair  $(\alpha, \varepsilon)$  by
$$
P (X, K, V) _ {\alpha , \varepsilon} = \{Y \in R: \rho_ {K} (Y) \leq \alpha , \rho_ {V} (Y) \leq \rho_ {K} (Y) \leq \rho_ {V} (Y) + \varepsilon \}. \tag {2.2}
$$
When  $X, K, V$  are understood in context, we abbreviate  $P = P(X, K, V)$ .
The parameter  $\varepsilon$  measures the distortion of geodesic length in  $K$  versus  $V$ . The next few lemmas are immediate consequences of the definition.
Lemma 2.3 (Parallax is Bi-Filtered). If  $\alpha \leq \alpha'$ , then  $P_{\alpha, \varepsilon} \subset P_{\alpha', \varepsilon}$ . If  $\varepsilon < \varepsilon'$ , then  $P_{\alpha, \varepsilon} \subset P_{\alpha, \varepsilon'}$ .
Lemma 2.4. For all  $\alpha, \varepsilon$ , we have  $P_{\alpha, \varepsilon} \subset R(X, K)_{\alpha} \subset R_{\alpha}$ .
Let  $\iota : P_{\alpha, \varepsilon} \to R_{\alpha}$  denote the inclusion of complexes, and let  $\iota_* : HP_{\alpha, \varepsilon} \to HR_{\alpha}$  denote the induced homomorphism on homology [17].
Corollary 2.5 (Homology Deaths are later in Parallax). Suppose that  $[Y]$  is a class in  $HP_{\alpha_1,\varepsilon_1}$  such that the bi-transition map  $HP_{\alpha_1,\varepsilon_1} \to HP_{\alpha_2,\varepsilon_2}$  takes  $[Y] \mapsto [0]$ . Then there exists  $t \leq \alpha_2$  such that the transition map  $HR_{\alpha_1} \to HR_t$  satisfies  $\iota_*([Y]) \mapsto [0]$ .
Lemma 2.6. For all  $\varepsilon \geq \alpha$ , we have  $P_{\alpha, \varepsilon} = P_{\alpha, \infty} = R(X, K)_{\alpha} \subset R_{\alpha}$ .
The proofs of 2.5 and 2.6 are given in the Supplemental Material.
It is sometimes useful to create single-parameter filtrations through  $P$ , parameterized by Rips radius, for the purpose of computing barcodes and persistence diagrams.
Definition 2.7 (Rips-like Path). A Rips-like path is a filtered module  $L_{\alpha} = P_{\alpha, \varepsilon(\alpha)}$  for  $0 \leq \alpha < \infty$  such that  $\varepsilon(\alpha)$  is a non-decreasing function satisfying  $\varepsilon(0) = 0$ .
A Rips-like path has homology  $HL$  and a barcode or persistence diagram. By Lemma 2.6, one Rips-like path is  $R(X, K)_\alpha = P_{\alpha, \alpha}$ . Another is the "inflexible" path  $L_\alpha = P_{\alpha, 0}$ .
3 Perturbation
This section establishes lemmas that ensure Parallax and its consequences (notably Theorem 5.4) are stable under certain types of perturbations, which means that the parallax is reasonable in the presence of noise.
Definition 3.1 (Pointwise Perturbation). Given  $X \in \mathcal{M}^*(V)$ , a pointwise  $\kappa$ -perturbation is  $X' \in \mathcal{M}^*(V)$  such that the sets  $X$  and  $X'$  admit a one-to-one correspondence  $f: X \to X'$  satisfying  $\| f(x) - x \| \leq \kappa$ . We write  $f: X \stackrel{\kappa}{\approx} X'$ .
Definition 3.2 (Pointwise  $K$ -perturbation). Suppose  $f: X \stackrel{\kappa}{\approx} X'$  such that each  $(x, f(x))$  pair is connected by a  $K$ -geodesic of length  $\leq \kappa$ . We write  $f: X \stackrel{\kappa}{\approx}_K X'$ .
Note that any  $f: X \stackrel{\kappa}{\approx} X'$  is an isomorphism in the category  $\mathcal{M}^*(V)$ , and any  $f: X \stackrel{\kappa}{\approx}_K X'$  is an isomorphism in the category  $\mathcal{M}^*(K)$ . The relation  $\stackrel{\kappa}{\approx}$  is reflexive and symmetric, but not transitive; hence, it is a way of describing proximity but does not provide an equivalence relation. Of course,  $X' \stackrel{\kappa}{\approx} X$  implies the Hausdorff distance satisfies  $d_H(X, X') \leq \kappa$ . A pointwise  $\kappa$ -perturbation can cause length distortions by  $2\kappa$ , as said formally in the following lemma.
Lemma 3.3 (Data Perturbation Lemma). If  $f: X \stackrel{\kappa}{\approx} X'$ , then the Rips complexes identified via  $f$  admit a  $\kappa$ -interleaving
$$
\dots \longrightarrow R (X ^ {\prime}, V) _ {\alpha - \kappa} \stackrel {f _ {\sharp} ^ {- 1}} {\longrightarrow} R (X, V) _ {\alpha} \stackrel {f _ {\sharp}} {\longrightarrow} R (X ^ {\prime}, V) _ {\alpha + \kappa} \stackrel {f _ {\sharp} ^ {- 1}} {\longrightarrow} R (X, V) _ {\alpha + 2 \kappa} \longrightarrow \dots
$$
Specifically, for any edge  $e = (x_i, x_j) \in R(X, V)_\alpha$  defined by the existence of a  $V$ -geodesic of length  $2\alpha$ , the edge  $f_{\sharp}(e) = (f(x_i), f(x_j))$  in  $R(X', V)$  has length  $2\alpha'$  satisfying  $2\alpha - 2\kappa \leq 2\alpha' \leq 2\alpha + 2\kappa$ .
Moreover, the same holds when replacing  $V$  with  $K$ , under the assumption  $X \stackrel{\kappa}{\approx}_K X'$ .
Proof. Note that a geodesic of length  $2\alpha$  corresponds with the intersection of two balls of radius  $\alpha$ ; hence, the factor of 2. The worst-case perturbation is to move each of  $x_{i}$  and  $x_{j}$  by  $\kappa$  in opposite directions, away from each other, along their geodesic.
Lemma 3.4 (Parallax Interleaving Lemma). Suppose  $f: X \stackrel{\kappa}{\approx}_K X'$ . Let  $P = P(X, K, V)$  and  $P' = P(X', K, V)$ . For any  $\alpha, \varepsilon$ , these parallax complexes admit a  $(\kappa, 2\kappa)$ -interleaving
$$
\dots \longrightarrow P _ {\alpha , \varepsilon} \xrightarrow {f _ {\sharp}} P _ {\alpha + \kappa , \varepsilon + 2 \kappa} ^ {\prime} \xrightarrow {f _ {\sharp} ^ {- 1}} P _ {\alpha + 2 \kappa , \varepsilon + 4 \kappa} \longrightarrow \dots
$$
Corollary 3.5 (Parallax Interleaving Lemma, Homology Version). Suppose  $f: X \stackrel{\kappa}{\approx}_K X'$ . Let  $HP = HP(X, K, V)$  and  $HP' = HP(X', K, V)$ . For any  $\alpha, \varepsilon$ , these homology groups admit a  $(\kappa, 2\kappa)$ -interleaving  $\dots \longrightarrow HP_{\alpha, \epsilon} \xrightarrow{f_*} HP_{\alpha + \kappa, \epsilon + 2\kappa}^{\prime} \xrightarrow{(f^{-1})_*} HP_{\alpha + 2\kappa, \epsilon + 4\kappa} \xrightarrow{f_*} \dots$
The proof of Lemma 3.4 is a worst-case distance estimate given in the Supplemental Material, and Corollary 3.5 follows functorially.
4 Sampling Density and Local Simplicial Matching
The goal "the geometry of  $K$  should match the geometry of  $X$ " requires that  $X$  has sufficient sampling density throughout  $K$  to express a meaningful comparison. The ideal situation would require there is a (small) scale  $\lambda$  for which: (1)  $K^{\circ}$  is homeomorphic to  $\bigcup_{x\in X}B_{\lambda}(x)$ , so that these  $\lambda$  balls capture the topology of  $K$ ; (2) all "highly persistent" homological features of  $X$  are born before  $\lambda$ ; and (3)  $\bigcup_{x\in X}B_{\lambda}(x)\subset K$ , so that perturbations of size  $\lambda$  in the dataset  $X$  are allowed, and so that these balls can be used as local charts in  $K$ . These sampling properties may or may not be true for any particular pair  $(X,K)$ , but Definition 4.1 provides scales for comparison.
Definition 4.1 (Locally Simplicially Matched). We say that  $X$  and  $K \in \mathcal{M}(X)$  are  $\lambda$ -locally simplicially matched [LSM] if the subset  $P_{t,0} \subset R_t$  is an equality for all  $t \leq \lambda$ . For any  $K \in \mathcal{M}(X)$ , the first non-LSM scale realized by the Rips complex is
$$
\lambda_ {\sup , X} (K) := \sup  \{\lambda : X, K \text {a r e} \lambda - L S M \} = \min  \left\{\rho_ {V} (Y): \rho_ {K} (Y) > \rho_ {V} (Y) \exists Y \in R \right\}.
$$
The last LSM scale realized by the Rips complex is
$$
\lambda_ {\operatorname {l o}, X} (K) := \max  \left\{\lambda <   \lambda_ {\sup , X} (K): \rho_ {V} (Y) = \lambda \exists Y \in R \right\}.
$$
Another LSM scale is  $\lambda_{\mathrm{ball},X}(K)\coloneqq \max \{\lambda :\bigcup_{x\in X}B_{\lambda}(x)\subset K^{\circ}\}$
When  $X$  and  $K$  are  $\lambda$ -LSM, we will identify  $P_{t,0}$  with  $R_{t}$  so that  $H_{*}P_{t,0} = H_{*}R_{t}$  whenever  $t \leq \lambda$ . Furthermore, locally simplicially matched implies  $P_{t,\epsilon} = R_{t}$  for any  $\epsilon > 0$  and  $t < \lambda_{\sup,X}(K)$ , following directly from the definition of  $P_{a,\epsilon}$ .
Lemma 4.2. If  $K\in \mathcal{M}(X)$  , then  $0 <   \lambda_{\mathrm{ball},X}(K) <   \lambda_{\sup ,X}(K).$
However, it may be that  $\lambda_{\mathrm{lo},X}(K) = 0$  , if the shortest edge  $e\in R$  has  $\rho_V(e) <   \rho_K(e)$
Corollary 4.3. For any  $K, K' \in \mathcal{M}(X)$ , there is some  $0 < \lambda$  such that each pair  $(X, K)$  and  $(X, K')$  is  $\lambda$ -locally simplicially matched.
Corollary 4.4. For any  $X, X' \in \mathcal{M}^*(K)$ , there is some  $0 < \lambda$  such that each pair  $(X, K)$  and  $(X', K)$  is  $\lambda$ -locally simplicially matched.
Lemma 4.5. If  $K_{1}, K_{2} \in \mathcal{M}(X)$  and  $K_{1} \subset K_{2}$ , then  $\lambda_{\sup, X}(K) \leq \lambda_{\sup, X}(K')$ .
The next lemma provides a bound on  $\lambda_{\mathrm{sup}}$  under small data perturbations, which is a form of stability.