Symmetric Representation Topology Divergence

• SRTD is a topological divergence metric that measures discrepancies between paired representations by comparing their Vietoris–Rips filtrations.
• It constructs a single symmetric auxiliary complex that captures both union and intersection structures to derive persistent homology barcodes.
• SRTD-lite offers enhanced computational efficiency by leveraging Minimum Spanning Trees, making it practical for rapid diagnostic evaluation in representation learning.

Searching arXiv for the specified SRTD and related RTD papers. Symmetric Representation Topology Divergence (SRTD) is a topological divergence for comparing two paired representations of the same inputs by measuring discrepancies between the induced Vietoris–Rips filtrations. It was introduced as a symmetric completion of the Representation Topology Divergence (RTD) framework, addressing the directional asymmetry of earlier variants while retaining a barcode-based diagnostic interpretation for structural differences between representations (Wang et al., 4 Jun 2026). In the representation-analysis setting, SRTD operates on two point clouds with one-to-one correspondence, constructs a single auxiliary complex encoding both intersection and union structure, computes persistent homology on that complex, and aggregates persistence interval lengths into a nonnegative scalar score (Trofimov et al., 2023, Wang et al., 4 Jun 2026).

1. Origins within topology-preserving representation learning

RTD was introduced in work on learning topology-preserving data representations, where the objective was to preserve topological similarity between a data manifold and a latent representation by enforcing similarity in topological features and their localization (Trofimov et al., 2023). In that setting, the core construction compared pairwise distance structures in the original space and the latent space through a cross-barcode derived from an auxiliary graph, and the resulting RTD was used as a loss term for an autoencoder. The method "RTD-AE" was reported to better preserve the global structure and topology of the data manifold than state-of-the-art competitors as measured by linear correlation, triplet distance ranking accuracy, and Wasserstein distance between persistence barcodes (Trofimov et al., 2023).

The original RTD formulation was directional. For point clouds

$$X=\{x_i\}_{i=1}^N\subset\mathbb R^d,\qquad Z=\{z_i\}_{i=1}^N\subset\mathbb R^p,$$

with one-to-one correspondence $x_i\leftrightarrow z_i$, RTD was defined from persistent homology on a doubled-vertex construction involving the pairwise distance matrices $w_{ij}=\|x_i-x_j\|$ and $\tilde w_{ij}=\|z_i-z_j\|$ (Trofimov et al., 2023). A common symmetric loss in that work was

$\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$

and a fully symmetric divergence was stated as

$\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$

equivalently $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$, with an optional averaging factor $1/2$ (Trofimov et al., 2023).

Subsequent work reformulated this symmetrization more fundamentally. Rather than summing two directional computations, it introduced a single symmetric auxiliary graph whose filtration tracks the union and intersection of the two Vietoris–Rips complexes simultaneously, thereby defining SRTD as a one-pass symmetric cross-barcode divergence (Wang et al., 4 Jun 2026). This suggests a conceptual shift from “symmetrized RTD” as an arithmetic combination of directional terms to SRTD as its own homological construction.

2. Formal construction

In the unified topological framework for representation analysis, SRTD is defined for two representations

$$P=\{x_1,\dots,x_n\}\subset\mathbb R^d,\qquad P'=\{x'_1,\dots,x'_n\}\subset\mathbb R^{d'},$$

of the same $n$ inputs, with pairwise dissimilarity matrices

$x_i\leftrightarrow z_i$0

(Wang et al., 4 Jun 2026). The construction first forms the element-wise minimum and maximum,

$x_i\leftrightarrow z_i$1

A single symmetric auxiliary graph is then encoded by the $x_i\leftrightarrow z_i$2 distance matrix

$x_i\leftrightarrow z_i$3

where $x_i\leftrightarrow z_i$4 means “replace the strict upper triangle of $x_i\leftrightarrow z_i$5 by $x_i\leftrightarrow z_i$6” so that the lower-block and upper-block together encode precisely the intersection and union filtrations (Wang et al., 4 Jun 2026). Ordinary persistent homology is then computed on the Vietoris–Rips filtration induced by $x_i\leftrightarrow z_i$7.

If

$x_i\leftrightarrow z_i$8

denotes the multiset of $x_i\leftrightarrow z_i$9-dimensional persistence intervals, then the dimension-$w_{ij}=\|x_i-x_j\|$0 SRTD is

$w_{ij}=\|x_i-x_j\|$1

and the total divergence is

$w_{ij}=\|x_i-x_j\|$2

In practice, only $w_{ij}=\|x_i-x_j\|$3 are often used (Wang et al., 4 Jun 2026).

Algorithmically, the procedure is concise: after optionally normalizing $w_{ij}=\|x_i-x_j\|$4 by their $w_{ij}=\|x_i-x_j\|$5-quantiles, one builds $w_{ij}=\|x_i-x_j\|$6, feeds it to a Vietoris–Rips persistent homology engine, and sums all interval lengths (Wang et al., 4 Jun 2026). In the earlier RTD-based formulation, the corresponding directional pipeline was batch-wise: compute two distance matrices, build the doubled-graph weight matrix, feed it to a Rips-persistence solver such as GPU-Ripser++, extract the first-degree cross-barcode, and return the sum of interval lengths (Trofimov et al., 2023).

3. Relation to RTD and the resolution of asymmetry

The central motivation for SRTD is that RTD is directional. In the 2026 formulation, directional RTD uses auxiliary matrices $w_{ij}=\|x_i-x_j\|$7 and $w_{ij}=\|x_i-x_j\|$8, producing two scalar divergences $w_{ij}=\|x_i-x_j\|$9 and $\tilde w_{ij}=\|z_i-z_j\|$0; in general,

$\tilde w_{ij}=\|z_i-z_j\|$1

(Wang et al., 4 Jun 2026). The stated source of this asymmetry is that new features can appear earlier in one filtration than in the other, producing “private” topological features unique to each direction (Wang et al., 4 Jun 2026).

SRTD resolves this by collapsing union-and-intersection into one mapping-cone complex $\tilde w_{ij}=\|z_i-z_j\|$2 (Wang et al., 4 Jun 2026). The precise homological relation is given as

$\tilde w_{ij}=\|z_i-z_j\|$3

showing that SRTD is the large, shared symmetric core, while directional RTD and Max-RTD add private components (Wang et al., 4 Jun 2026).

This relation is especially explicit in the $\tilde w_{ij}=\|z_i-z_j\|$4-dimensional lightweight setting. Corollary statements reported for the lite version are

$\tilde w_{ij}=\|z_i-z_j\|$5

together with

$\tilde w_{ij}=\|z_i-z_j\|$6

These inequalities place SRTD-lite between the directional extremes and remove directional asymmetry by construction (Wang et al., 4 Jun 2026).

A common misconception is to treat SRTD merely as the arithmetic average of two RTD directions. The literature supports two distinct usages. In the original topology-preserving representation-learning work, a “fully symmetric divergence” was defined as the sum of directional RTD terms, optionally averaged (Trofimov et al., 2023). In the later unified framework, SRTD is instead a single symmetric construction based on one auxiliary complex and one cross-barcode computation (Wang et al., 4 Jun 2026). The second formulation is the one specifically designed to “complete the RTD framework” and eliminate heuristic asymmetry.

4. Barcode interpretation and diagnostic role

SRTD is not only a scalar divergence but also a barcode-based diagnostic object. In the 2026 framework, each interval $\tilde w_{ij}=\|z_i-z_j\|$7 in the cross-barcode identifies one connected component that is born in the $\tilde w_{ij}=\|z_i-z_j\|$8-filtration at time $\tilde w_{ij}=\|z_i-z_j\|$9 but does not die until time $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$0 in the $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$1-filtration (Wang et al., 4 Jun 2026). By lifting the responsible edge back to the original point pair $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$2, one can identify which sample pair contributes most strongly to the divergence. In language-model experiments, this enabled tracing a few anomalous question–answer pairs that drove large SRTD-lite divergences (Wang et al., 4 Jun 2026).

This diagnostic interpretation is continuous with the earlier RTD program, which emphasized not only topological features such as clusters, loops, and $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$3D voids, but also their localization (Trofimov et al., 2023). The 2026 paper states that SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature and allows precise localization of structural discrepancies (Wang et al., 4 Jun 2026). A plausible implication is that SRTD is particularly useful when scalar similarity scores alone are insufficient and one needs localized structural attribution.

The same work also positions SRTD as an optimization objective. Because SRTD and SRTD-lite are single symmetric functions of $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$4, they can be inserted directly into gradient-based training, for example as regularizers in an autoencoder (Wang et al., 4 Jun 2026). The paper reports that replacing the usual RTD-AE loss with SRTD yields the same quality but roughly half the runtime because there is no need to compute two directional RTDs (Wang et al., 4 Jun 2026).

5. Efficient variant: SRTD-lite

SRTD-lite is the $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$5-dimensional, computationally efficient variant. It exploits the classical equivalence between the $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$6D Vietoris–Rips barcode and the Minimum Spanning Tree (MST) (Wang et al., 4 Jun 2026). The procedure is summarized as follows: compute

$\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$7

sort both in nondecreasing order of edge weight, interpret each edge in $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$8 with weight $\RTD(X,Z):=\frac12\bigl[\RTD_1(X,Z)+\RTD_1(Z,X)\bigr],$9 as a birth event, then scan through $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$0 to find the smallest weight $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$1 that reconnects the same two components; the pair $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$2 defines a barcode interval, and the final score is the sum of all $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$3 (Wang et al., 4 Jun 2026).

The paper gives the following complexity characterizations for SRTD-lite (Wang et al., 4 Jun 2026):

Component	Complexity
Building pairwise distances	$\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$4
Two MSTs by Kruskal/Prim	$\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$5 or $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$6
Nested loop over birth and death edges	$\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$7

It is reported to run in under a second for $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$8 up to tens of thousands, making it highly scalable (Wang et al., 4 Jun 2026). The paper recommends deterministic tie-breaking in MST/Kruskal and mid-rank ties in Spearman’s ranks to ensure reproducibility (Wang et al., 4 Jun 2026).

The authors also recommend $\SRTD(X,Z):=\RTD(X,Z)+\RTD(Z,X),$9-quantile pre-normalization before computing RTD or SRTD, rescaling all distances so that the $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$0 quantile equals $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$1, in order to reduce extreme scale effects (Wang et al., 4 Jun 2026). They further recommend using SRTD-lite for quick diagnostics, local anomaly detection, and as a drop-in loss (Wang et al., 4 Jun 2026). Because the underlying score remains unbounded and scale-dependent, they suggest switching to Normalized Topological Similarity (NTS) for cross-scenario benchmarking (Wang et al., 4 Jun 2026).

6. Theoretical properties

Several formal properties are explicitly stated for SRTD and the surrounding RTD framework. First, symmetry is built in: $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$2 by construction (Wang et al., 4 Jun 2026). Second, SRTD is nonnegative, and if $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$3, then $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$4, the mapping cone is trivial, and $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$5 (Wang et al., 4 Jun 2026).

The key homological interpretation is that the chain complex on $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$6 is homotopy equivalent to the mapping cone of the inclusion

$\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$7

which underpins the exact sequence relating RTD, Max-RTD, and SRTD (Wang et al., 4 Jun 2026). This establishes SRTD as a genuinely topological comparison functional rather than a heuristic graph statistic.

From the RTD literature inherited by SRTD, there are also continuity and barcode-matching guarantees. For any two weight-pairs $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$8, $\SRTD=\RTD_1(X,Z)+\RTD_1(Z,X)$9 on the same vertex set,

$1/2$0

so $1/2$1 is Lipschitz-continuous in the maximum norm on edge weights, and by chaining through pairwise distances, continuous in coordinates (Trofimov et al., 2023). The earlier work also states a zero-measure identity: if $1/2$2 for all $1/2$3, then the persistent barcodes of $1/2$4 and $1/2$5 coincide in every degree, and the corresponding classes are matched at identical filtration values (Trofimov et al., 2023). This suggests that the symmetrized divergence inherits a strong “zero implies topological coincidence” interpretation, although the exact zero-iff-identical statement for the 2026 SRTD is phrased at the level of equality of pairwise dissimilarities (Wang et al., 4 Jun 2026).

An important limitation is also explicit: like RTD, SRTD is not intrinsically bounded or scale-invariant and grows with $1/2$6 and the distance scale (Wang et al., 4 Jun 2026). The 2026 framework introduces NTS precisely to overcome this scale and sample dependence for benchmarking across heterogeneous settings (Wang et al., 4 Jun 2026).

7. Empirical behavior, usage, and scope

The 2026 study reports that SRTD and SRTD-lite increase monotonically in a synthetic task comparing a single Gaussian cluster to $1/2$7 subclusters on a circle, whereas RTD-lite inverts the trend and CKA is nearly flat (Wang et al., 4 Jun 2026). In UMAP embedding experiments, the SRTD family and NTS are reported to track smooth structural evolution as $1/2$8 varies, while CKA does not (Wang et al., 4 Jun 2026). In autoencoder experiments on F-MNIST and COIL-20, training a $1/2$9-dimensional autoencoder with SRTD loss is reported to match or beat RTD while reducing runtime by approximately $$P=\{x_1,\dots,x_n\}\subset\mathbb R^d,\qquad P'=\{x'_1,\dots,x'_n\}\subset\mathbb R^{d'},$$0 (Wang et al., 4 Jun 2026).

For CNN layer analysis, SRTD-lite heatmaps on an $$P=\{x_1,\dots,x_n\}\subset\mathbb R^d,\qquad P'=\{x'_1,\dots,x'_n\}\subset\mathbb R^{d'},$$1-layer TinyCNN are reported to reveal a strong structural break at the final pooling layer and a clear near-diagonal decay, while RTD and RTD-lite show counter-intuitive inversions under standard $$P=\{x_1,\dots,x_n\}\subset\mathbb R^d,\qquad P'=\{x'_1,\dots,x'_n\}\subset\mathbb R^{d'},$$2-quantile normalization (Wang et al., 4 Jun 2026). For LLM genealogy mapping, CKA is reported to saturate with most pairwise scores greater than $$P=\{x_1,\dots,x_n\}\subset\mathbb R^d,\qquad P'=\{x'_1,\dots,x'_n\}\subset\mathbb R^{d'},$$3, whereas NTS and, to a lesser extent, SRTD-lite barcodes uncover family-wise hierarchical fingerprints; the study further reports that NTS correctly identifies Qwen2.5-Math and its distilled DeepSeek child as highly similar where CKA fails (Wang et al., 4 Jun 2026). SRTD-lite barcodes also allow query-level diagnosis of representation shifts, though the final sum is sensitive to a few outlier intervals, which motivated NTS for robustness (Wang et al., 4 Jun 2026).

These results place SRTD within a broader TDA-based representation-analysis toolkit rather than as a universal replacement for all similarity measures. The paper explicitly frames SRTD as serving fine-grained structural diagnosis, while NTS is recommended for robust, standardized evaluation across layers, models, and datasets (Wang et al., 4 Jun 2026). A plausible implication is that SRTD is best suited to paired, topology-aware inspection where localizable discrepancies matter, whereas normalized statistics are preferable for large-scale comparative studies.

The acronym “SRTD” also appears in an unrelated finite-element literature as “Selective Replacement of Tensor Divergence” for Oldroyd viscoelastic flow models (Austin et al., 21 Jan 2025). That usage concerns a three-stage variational reformulation of a constitutive PDE system and is not related to Symmetric Representation Topology Divergence in representation analysis. Distinguishing these two expansions is important because both are current arXiv terms but belong to entirely different technical domains (Austin et al., 21 Jan 2025).