--- title: Normalized Topological Similarity (NTS) url: https://www.emergentmind.com/topics/normalized-topological-similarity-nts type: topic --- # Normalized Topological Similarity (NTS) Searching arXiv for recent and foundational papers on Normalized Topological Similarity and closely related constructs. Normalized Topological Similarity (NTS) is a normalized, topology-aware similarity measure for comparing two paired neural representations. In its explicit formulation, NTS measures the rank correlation of hierarchical merge orders derived from the \(0\)-dimensional Vietoris–Rips, single-linkage, and minimum spanning tree (MST) structure of two dissimilarity matrices, yielding a scale-invariant score in \([-1,1]\) intended for cross-scenario benchmarking [2606.06342]. The term is not fully standardized across the broader literature: several earlier works define closely related normalized topological comparisons under different names or within different pipelines rather than as a single standalone coefficient. ## 1. Concept and terminological scope In the strict sense, NTS denotes the measure introduced in "Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis" [2606.06342]. That work presents NTS as the normalized counterpart to RTD- and SRTD-style topological divergences. Its stated purpose is to replace unbounded, sample-dependent scalar summaries based on sums of barcode lengths with a bounded similarity based on the rank correlation of hierarchical merge orders. Across adjacent literatures, however, the same phrase is often only approximately applicable. "Inter-layer Information Similarity Assessment of Deep Neural Networks Via Topological Similarity and Persistence Analysis of Data Neighbour Dynamics" defines **Nearest Neighbour Topological Similarity (NNTS)** rather than NTS, using intersection-over-union of \(k\)-nearest-neighbour sets across layers [2012.03793]. "Topological Similarity Index and Loss Function for Blood Vessel Segmentation" introduces a path-feasibility-based **topological similarity index** \(m(\mathbf{P},\mathbf{Y})\), also normalized to \([0,1]\), but specialized to binary vessel masks [2107.14531]. "Topological Representational Similarity Analysis in Brains and Beyond" does not define a metric called NTS at all; instead, it introduces tRSA, geo-topological transforms, representational geo-topological matrices (RGTMs), and representational geodesic-distance matrices (RGDMs), which collectively function as a normalized, topology-sensitive representational comparison pipeline [2408.11948]. This suggests that NTS is best understood as a family of normalized topology-sensitive comparison strategies, with a particularly precise modern instantiation in [2606.06342]. ## 2. Formal definition Let \[ w,\tilde w \in \mathbb{R}^{n\times n} \] be pairwise dissimilarity matrices for two paired representations on the same \(n\) samples. In the NTS formulation, these matrices are assumed symmetric, nonnegative, and to have zero diagonal [2606.06342]. Let \(T_w\) and \(T_{\tilde w}\) be minimum spanning trees of the complete weighted graphs induced by \(w\) and \(\tilde w\), with edge sets \(E_w\) and \(E_{\tilde w}\). The **core pair set** is \[ \mathcal{C} := E_w \cup E_{\tilde w}. \] Since each MST has \(n-1\) edges, \(|\mathcal{C}| \le 2n-2\). NTS is defined through **Spearman’s rank correlation**. For vectors \(x,y\in\mathbb{R}^m\), with rank vectors \(R(x)\) and \(R(y)\), the paper gives \[ \rho_S(x,y) := \mathrm{corr}\!\big(R(x),R(y)\big) = \frac{\sum_{k=1}^{m}\big(R(x)_k-\overline{R(x)}\big)\big(R(y)_k-\overline{R(y)}\big)} {\sqrt{\sum_{k=1}^{m}\big(R(x)_k-\overline{R(x)}\big)^2}\; \sqrt{\sum_{k=1}^{m}\big(R(y)_k-\overline{R(y)}\big)^2}} \in [-1,1]. \] The construction has two variants. For **NTS-E**, one uses raw dissimilarities on the core pair set. For each core edge \(e=(i,j)\in \mathcal{C}\), define \[ x_e := w_{ij}, \qquad \tilde x_e := \tilde w_{ij}. \] Then \[ \mathrm{NTS\text{-}E}(w,\tilde w) := \rho_S\!\big((x_e)_{e\in\mathcal{C}},(\tilde x_e)_{e\in\mathcal{C}}\big). \] For **NTS-M**, one uses **merge times** induced by the hierarchy. For any pair \((i,j)\), the merge time under \(w\) is \[ m_w(i,j)\;:=\;\min\{\alpha:\ i \text{ and } j \text{ are connected at threshold } \alpha\}. \] The MST characterization is \[ m_w(i,j) = \max_{e\in \mathrm{path}_{T_w}(i,j)} w_e. \] For each core edge \(e=(i,j)\in\mathcal{C}\), define \[ x_e := m_w(i,j), \qquad \tilde x_e := m_{\tilde w}(i,j). \] Then \[ \mathrm{NTS\text{-}M}(w,\tilde w) := \rho_S\!\big((x_e)_{e\in\mathcal{C}},(\tilde x_e)_{e\in\mathcal{C}}\big). \] The paper explicitly states that NTS compares the **hierarchical connectivity structure** induced by the two representations, namely their \(0\)-dimensional Vietoris–Rips merge structure [2606.06342]. ## 3. Topological basis and computational pipeline NTS is built on the equivalence among \(0\)-dimensional Vietoris–Rips filtrations, single-linkage clustering, Kruskal-style merge sequences, and MSTs [2606.06342]. At threshold \(\alpha\), points \(i\) and \(j\) are connected when \(w_{ij}\le \alpha\). As \(\alpha\) increases, connected components merge. In dimension \(0\), the relevant topological information is therefore the sequence and ordering of those merges. The computation pipeline is concrete. Starting from paired representations \(X=\{x_i\}_{i=1}^n\) and \(X'=\{x'_i\}_{i=1}^n\), one computes dissimilarity matrices \[ w_{ij} = \mathrm{dissim}(x_i,x_j), \qquad \tilde w_{ij} = \mathrm{dissim}(x'_i,x'_j), \] with Euclidean distance or cosine dissimilarity given as examples. One then computes \(T_w\) and \(T_{\tilde w}\), forms \(\mathcal C=E_w\cup E_{\tilde w}\), constructs aligned vectors either of raw edge weights (NTS-E) or merge times (NTS-M), and returns their Spearman correlation. This design is intentionally sparse. The paper contrasts NTS with representational similarity analysis based on full vectorization of all pairwise distances: rather than using all \(\binom{n}{2}\) pairs, NTS restricts comparison to the topology-aware core set \(\mathcal C\) induced by the two MSTs [2606.06342]. In this sense, it is not simply “RSA on distances,” but a rank-correlation comparator applied to a summary of the \(0\)-dimensional topological hierarchy. The two variants differ in what they preserve. NTS-E is described as stricter because it matches the rank structure of the underlying edge weights on \(\mathcal C\), whereas NTS-M requires agreement only in the induced merge-event ordering. The paper provides pseudocode for both variants and specifies deterministic tie breaking for MST computation together with deterministic rank handling, such as mid-ranks, for Spearman’s \(\rho_S\) [2606.06342]. For complexity, the paper gives an explicit bound only for NTS-E: \[ O\!\big(n^2(\alpha_{\mathrm{uf}(n)}+d)\big), \] where \(\alpha_{\mathrm{uf}(n)}\) is the inverse Ackermann factor arising from union-find operations. The stated dominant costs are pairwise distance calculation and MST construction [2606.06342]. ## 4. Properties and interpretation The central normalization property is immediate from the definition: because NTS is a Spearman correlation, it is bounded in \[ [-1,1]. \] The measure is symmetric, since both Spearman correlation and the core set \(\mathcal C=E_w\cup E_{\tilde w}\) are symmetric in the two inputs [2606.06342]. The paper states a precise characterization of perfect **NTS-M**. Assuming nonzero rank variance, \[ \mathrm{NTS\text{-}M}(w,\tilde w)=1 \] if and only if the values \[ \{m_w(i,j)\}_{(i,j)\in\mathcal C} \quad\text{and}\quad \{m_{\tilde w}(i,j)\}_{(i,j)\in\mathcal C} \] induce the same **weak order** on \(\mathcal C\). The notion of weak order includes ties: for all \(e_1,e_2\in\mathcal C\), \[ V_{e_1}