Localized Topological Fidelity Score (LTFS)

• LTFS is a metric that integrates global topological similarity with local temporal overlap to evaluate forecasting quality.
• It extends the Topological Fidelity Score (TFS) by incorporating a Jaccard-based measure for assessing dominant cycle overlap.
• Utilizing Takens delay embedding and H1 persistent homology, LTFS detects oscillatory behavior and phase alignment missed by pointwise error metrics.

Searching arXiv for the LTFS paper and closely related work on topological fidelity, localized topology, and fidelity-based topology diagnostics. Localized Topological Fidelity Score (LTFS) is a topology-driven evaluation metric for time series forecasting that combines diagram-level structural fidelity with temporal localization of the dominant oscillatory feature. In the formulation introduced in “TopoCast: A Topological Fidelity Framework for Evaluating Transformer-Based Time Series Forecasting” (Weerasekara et al., 24 Jun 2026), LTFS extends the Topological Fidelity Score (TFS) by incorporating dominant cycle overlap, a Jaccard-based measure of whether the longest-lived $H_1$ feature in forecast and ground truth occurs at the same time points. The framework is motivated by the inadequacy of pointwise metrics such as Mean Squared Error (MSE) and Mean Absolute Error (MAE) for detecting over-smoothing, phase shift, frequency distortion, spurious oscillation injection, and loss of recurrence, all of which can preserve low numerical error while degrading the reconstructed dynamical structure (Weerasekara et al., 24 Jun 2026). In a broader research context, LTFS sits at the intersection of persistent homology, phase-space reconstruction, and fidelity-based topology diagnostics, with conceptual affinities to localized persistent homology for delineation (Oner et al., 2021), momentum-resolved fidelity susceptibility near topological transitions (Panahiyan et al., 2020), and local spectral-topological analysis via persistent local Laplacians (Liu et al., 8 Mar 2026).

1. Definition and conceptual scope

LTFS was introduced as part of the TopoCast framework for evaluating Transformer-based time series forecasting (Weerasekara et al., 24 Jun 2026). The framework reconstructs phase-space trajectories from forecast and ground-truth signals using Takens delay embedding, computes persistent homology on the resulting point clouds, derives four topological fidelity measures from the $H_1$ persistence diagrams, aggregates them into TFS, and then augments TFS with a temporal overlap measure for the dominant cycle. The stated purpose is to assess structural fidelity in settings where recurrent dynamics, oscillatory behavior, and phase alignment matter but are not captured by pointwise forecasting errors (Weerasekara et al., 24 Jun 2026).

The core distinction in the framework is between global topological similarity and localized temporal agreement. TFS measures whether the forecast preserves the topology of the reconstructed dynamics at the level of persistence summaries. LTFS adds a localization term that maps the dominant topological generator back into the temporal domain and measures whether the corresponding oscillatory pattern occurs at the correct time points. The intended formula, reconstructed from the paper’s prose because the displayed equation is malformed in the extracted text, is

$LTFS = TFS \times \text{Overlap}.$

This means LTFS is low whenever either the topology is mismatched or the dominant oscillation is temporally mislocalized (Weerasekara et al., 24 Jun 2026).

This construction is specific to time series forecasting and should be distinguished from other localized topology-aware methods in the literature. “Persistent Homology with Improved Locality Information for more Effective Delineation” introduces a persistent-homology-based loss with improved locality information for curvilinear segmentation, but it does not define LTFS (Oner et al., 2021). “Local Laplacian: theory and models for data analysis” develops persistent local Laplacians to capture local topological and geometric signatures around vertices, which suggests a different notion of locality based on links and local homology rather than temporal support (Liu et al., 8 Mar 2026). “Fidelity susceptibility near topological phase transitions in quantum walks” treats fidelity susceptibility as a local density in momentum space, again conceptually related but not identical to LTFS (Panahiyan et al., 2020).

2. Mathematical pipeline in TopoCast

The TopoCast pipeline begins with a forecast sequence $\hat{y}$ and aligned ground truth $y$, potentially multivariate with $C$ channels (Weerasekara et al., 24 Jun 2026). The authors state that a raw one-dimensional time series has trivial topology, so each channel $x_i(t)$ is lifted into a higher-dimensional phase space by Takens delay embedding with embedding dimension $m=3$ and delay $\tau=2$: $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$ with

$H_1$0

The $H_1$1 channel embeddings are then concatenated column-wise into a joint point cloud $H_1$2 (Weerasekara et al., 24 Jun 2026).

Persistent homology is computed on the embedded point clouds using a Vietoris–Rips filtration built from pairwise Euclidean distances. The framework focuses exclusively on $H_1$3, because periodic or seasonal dynamics become loop-like structures in phase space after delay embedding. Each $H_1$4 class yields a birth–death interval $H_1$5 with lifetime $H_1$6. Features with lifetime

$H_1$7

are discarded as numerical noise (Weerasekara et al., 24 Jun 2026).

The forecast and ground truth are processed separately but with identical embedding and persistence settings. This yields two $H_1$8 persistence diagrams from which the four TFS components are extracted. A plausible implication is that the method presupposes that the salient structure of the forecasting problem is oscillatory or recurrent enough to manifest through $H_1$9, rather than through higher-dimensional homology classes or purely transient morphology. The paper itself identifies support for higher homology dimensions as future work rather than part of the current metric (Weerasekara et al., 24 Jun 2026).

3. Topological Fidelity Score as the global component

TFS is defined as the geometric mean of four topological fidelity measures derived from the forecast and ground-truth $LTFS = TFS \times \text{Overlap}.$0 persistence diagrams (Weerasekara et al., 24 Jun 2026). Each component is a log-ratio score bounded in $LTFS = TFS \times \text{Overlap}.$1, with value $LTFS = TFS \times \text{Overlap}.$2 corresponding to exact agreement.

The four components are organized as follows:

Component	Underlying descriptor	Reported role
$LTFS = TFS \times \text{Overlap}.$3	number of significant $LTFS = TFS \times \text{Overlap}.$4 features	loop count preservation
$LTFS = TFS \times \text{Overlap}.$5	maximum lifetime $LTFS = TFS \times \text{Overlap}.$6	dominant cycle strength preservation
$LTFS = TFS \times \text{Overlap}.$7	total persistence $LTFS = TFS \times \text{Overlap}.$8	total topological energy preservation
$LTFS = TFS \times \text{Overlap}.$9	persistence entropy $\hat{y}$0	persistence diagram complexity preservation

The score formulas, reconstructed from the table because of typesetting corruption in the extracted text, are

$\hat{y}$1

$\hat{y}$2

$\hat{y}$3

$\hat{y}$4

with

$\hat{y}$5

The aggregate TFS is given, again reconstructed from the prose and table, by

$\hat{y}$6

The geometric mean is explicitly motivated as a conjunctive aggregation: collapse in any single component strongly lowers the final score (Weerasekara et al., 24 Jun 2026).

The four descriptors encode complementary aspects of loop structure. $\hat{y}$7 penalizes loop loss and loop injection. $\hat{y}$8 isolates the strongest recurrent feature. $\hat{y}$9 measures aggregate cyclic content. $y$0 captures how persistence is distributed across loops rather than merely its total magnitude. The paper does not specify how zero-valued descriptors are handled inside the logarithms, and this omission is a reproducibility caveat (Weerasekara et al., 24 Jun 2026).

A broader interpretation is suggested by related work on persistent homology and local topology. Localized topological simplification of scalar data distinguishes between combinatorial topological fidelity and numerical fidelity in a localized setting (Lukasczyk et al., 2020). TopoCast’s TFS similarly measures structural preservation independently of pointwise amplitude error, although in the specific context of persistence diagrams from Takens embeddings rather than PL scalar fields (Weerasekara et al., 24 Jun 2026).

4. Localization through dominant cycle overlap

The defining addition in LTFS is the localization mechanism. TFS remains a diagram-level measure: it compares persistence summaries but does not indicate whether the dominant cycle is realized at the correct time points. The paper therefore extracts a representative cocycle for the longest-lived $y$1 feature from the forecast and ground-truth diagrams using Ripser with cocycle extraction enabled (Weerasekara et al., 24 Jun 2026).

For the embedded point cloud $y$2, each row corresponds to a time step in the original signal. The representative cocycle of the dominant $y$3 feature is mapped back to temporal indices through this row–time correspondence, producing two temporal support sets: $y$4 The exact set-construction rule is not formalized in detail beyond this index correspondence, and the lack of pseudocode or a simplex-to-time expansion procedure is one of the main under-specifications in the method (Weerasekara et al., 24 Jun 2026).

Temporal co-localization is quantified using the Jaccard coefficient: $y$5 Two edge cases are specified explicitly. If the ground-truth diagram contains no valid $y$6 generator, the corresponding window is excluded. If the forecast contains no valid dominant generator, then

$y$7

which is interpreted as complete dominant-cycle loss (Weerasekara et al., 24 Jun 2026).

This localization mechanism is what distinguishes LTFS from global topological distances on persistence diagrams. A phase-shifted forecast can preserve loop count, dominant lifetime, total persistence, and persistence entropy, and thus achieve $y$8, while still having zero dominant cycle overlap because the relevant oscillation occurs at the wrong times. The paper treats this as the canonical use case for LTFS (Weerasekara et al., 24 Jun 2026).

A plausible implication is that locality in LTFS is not spatial or metric locality in the embedding space itself, but temporal locality after inverse mapping from persistent generators to signal indices. This differs fundamentally from locality notions in persistent local homology or persistent local Laplacian theory, where locality is defined relative to a vertex neighborhood or link complex (Liu et al., 8 Mar 2026).

5. Interpretation, behavior, and empirical signatures

The interpretive structure of LTFS is multiplicative: $y$9 Since

$C$0

the score is high only when the forecast preserves both the topology of the reconstructed dynamics and the temporal support of the dominant oscillation (Weerasekara et al., 24 Jun 2026).

The reported synthetic ECG experiments illustrate several canonical behaviors:

Scenario	MSE	TFS	Overlap	LTFS
Smooth forecast	0.0110	0.1172	0.0000	0.0000
Attenuated peaks	0.0081	0.4990	1.0000	0.4990
Missing beat	0.0020	0.9364	0.8806	0.8246
Phase shift	0.0321	0.9990	0.0000	0.0000

These cases establish the intended operational meaning of LTFS (Weerasekara et al., 24 Jun 2026). The smooth forecast exhibits modest numerical error but destroys recurrent geometry and loses localized cycle support, driving LTFS to zero. Attenuated peaks preserve timing but weaken the dominant cycle, resulting in intermediate LTFS. Missing beat yields low MSE yet measurable structural degradation. The phase-shift example is decisive: almost perfect TFS but zero overlap, so LTFS collapses to zero.

The paper extends this analysis to three benchmark datasets—ETTm2, Exchange, and ILI—and five Transformer architectures: Transformer, Informer, Autoformer, FEDformer, and PatchTST. The main claim is that models with similar MSE or MAE can exhibit substantially different LTFS values, revealing structural failure modes that conventional forecasting metrics do not detect (Weerasekara et al., 24 Jun 2026). On Exchange, for example, PatchTST often attains strong MSE but poor LTFS, which the authors interpret as spurious loop injection and poor temporal localization under weak periodicity.

This emphasis on localization resonates with other fidelity-based topology diagnostics, though with a different notion of locality. In disordered topological superconductors, fidelity susceptibility develops multiple peaks, some associated with the true topological transition and others with disorder-induced Majorana pinning (Tian et al., 2014). That work suggests that global fidelity can conflate bulk criticality with local rearrangements. TopoCast’s LTFS addresses an analogous issue for time series: TFS captures global topological structure, while overlap separates correct structure from misplaced structure (Weerasekara et al., 24 Jun 2026).

6. Relation to neighboring research and limitations

LTFS is specific to the TopoCast framework and should not be conflated with several superficially related ideas. Momentum-resolved fidelity susceptibility in Dirac models can be viewed as a local density on momentum space and is closely tied to the curvature function underlying a topological invariant (Panahiyan et al., 2020). Boundary fidelity susceptibility in the SSH model isolates a boundary contribution that changes sign across topological and trivial phases (Sirker et al., 2014). Persistent local Laplacians recover persistent local homology through the kernel of a local spectral operator and support localized topological analysis on simplicial complexes, point clouds, and graphs (Liu et al., 8 Mar 2026). These works all supply alternative notions of “localized topological fidelity,” but none defines the temporal-domain LTFS of TopoCast.

Within time series forecasting, the novelty claim of LTFS is precisely that it reconnects persistence-based topology to temporal support. Diagram distances such as Wasserstein distance remain diagram-level and do not indicate whether a generator occurs at the correct time points (Weerasekara et al., 24 Jun 2026). Dynamic Time Warping and related alignment measures quantify timing distortion but do not evaluate whether loop structure in reconstructed phase space is preserved. LTFS is intended to bridge these gaps.

Several limitations are explicit or strongly implied in the TopoCast formulation. The metric uses only $C$1, so non-cyclic structure is outside scope (Weerasekara et al., 24 Jun 2026). It is best suited to signals with meaningful oscillatory or recurrent dynamics; windows with no valid ground-truth $C$2 generator are excluded (Weerasekara et al., 24 Jun 2026). The inverse mapping from representative cocycles to original time indices is under-specified, which complicates exact reproduction. The handling of zero descriptors in the log-ratio TFS terms is not stated. Embedding hyperparameters are fixed at $C$3 and $C$4 without sensitivity analysis. These omissions do not negate the conceptual framework, but they constrain precise implementation fidelity (Weerasekara et al., 24 Jun 2026).

A plausible implication is that LTFS is best understood as a structured diagnostic rather than a universal metric. Its strongest use case is comparative evaluation when preserving recurrent dynamics and phase localization matters. In that regime, it functions as a two-factor score whose value lies precisely in separating topological plausibility from temporal correctness.