Normalized Persistence in Topological Data Analysis

• Normalized persistence is a family of normalization strategies that adjust persistence-based summaries to remove dependencies on scale, total point count, or persistence mass.
• These techniques facilitate consistent statistical estimation and scale-invariant comparisons, as seen in normalized bottleneck distances and survival fractions.
• By converting raw persistence measures into probability densities, survival ratios, or normalized scores, the approach enhances robustness and interpretability of topological features.

Normalized persistence is not a single standardized construction. In contemporary literature, the term denotes several normalization procedures applied to persistence-based objects in order to remove dependence on total point count, absolute scale, total persistence mass, or ambient feature multiplicity. In topological data analysis, these procedures include normalizing persistence diagrams into probability measures, scaling point clouds before applying bottleneck distance, converting persistent Betti counts into survival fractions, and normalizing vectorized summaries such as neural persistence or persistence-velocity descriptors. In adjacent areas, the same phrase also appears in stochastic-process persistence, where an exponentially decaying persistence probability is divided by its dominant decay rate, and in time-series analysis of cell migration, where a normalized Shannon entropy is identified as “normalized persistence” (Wu et al., 2023, May et al., 2023, Lowe et al., 3 Jul 2026, Rieck et al., 2018, Alsmeyer et al., 2021, Liu et al., 2020).

1. Statistical normalization of persistence diagrams

A central statistical meaning of normalized persistence arises when a persistence diagram is treated as a random discrete measure on a domain $\Omega \subset \mathbb{R}^2$. In this setting, a diagram $D$ is viewed as a counting measure $\mu$, and its normalized diagram is the probability measure

$\widetilde{\mu}(A)=\mu(A)/N(D),$

where $N(D)=\mu(\Omega)$. For random diagrams, the expected counting measure $E[\mu]$ and the expected probability measure $E[\widetilde{\mu}]$ admit Lebesgue densities under mild conditions: the persistence intensity $\lambda(\omega)$ and the persistence density $p(\omega)$, respectively. The normalization changes the object from an expected count density to the density of a “typical” point in a random normalized diagram, with $\int_\Omega p(\omega)\,d\omega=1$ (Wu et al., 2023).

This distinction has direct inferential consequences. Kernel estimators for $D$0 and $D$1 are constructed from i.i.d. diagrams by replacing the empirical sample in a standard KDE with the random discrete measures $D$2 or $D$3. Under bounded total $D$4-persistence, weighted boundedness, and Hölder smoothness of order $D$5, both estimators achieve the minimax-optimal uniform rate

$D$6

but the normalized estimator $D$7 enjoys uniform control on all of $D$8, whereas the estimator $D$9 is uniform only away from a $\mu$0-band near the diagonal. The stated reason is that $\mu$1 is a KDE of a probability measure, while $\mu$2 must exclude the diagonal region where infinitely many small-persistence points may accumulate. Accordingly, the normalized persistence density provides strictly stronger statistical guarantees (Wu et al., 2023).

The same paper makes this normalization operational for linear representations of persistence diagrams. For a linear functional

$\mu$3

its expectation is represented through either $\mu$4 or $\mu$5, and sup-norm control of the corresponding estimator yields uniform error control for bounded-integral functionals. Special cases explicitly listed include Betti curves and persistence surfaces; the summary statement extends this to silhouettes and related linear summaries. A plausible implication is that normalization is not merely a technical convenience for density estimation, but a way of upgrading uniform consistency for a broad class of persistence-based statistical functionals (Wu et al., 2023).

2. Scale invariance and normalized distances between persistence diagrams

A second major use of normalized persistence concerns comparison of persistence diagrams under global rescaling. The ordinary bottleneck distance $\mu$6 can become arbitrarily large for point clouds sampled from topologically similar manifolds when those clouds differ by a large scaling factor. To address this, a normalized bottleneck distance is defined by scaling each metric space to unit diameter before computing bottleneck distance: $\mu$7 This construction yields a scale-invariant pseudometric, since for any $\mu$8,

$\mu$9

The same work develops a metric decomposition framework, writing distance matrices as $\widetilde{\mu}(A)=\mu(A)/N(D),$0 for a bijection between equal-cardinality metric spaces and using the optimal scalar $\widetilde{\mu}(A)=\mu(A)/N(D),$1 to derive a stability bound

$\widetilde{\mu}(A)=\mu(A)/N(D),$2

This gives an explicit way to quantify topological distortion after normalization (May et al., 2023).

The normalized distance is then analyzed under several dimension-reduction schemes. For a Johnson–Lindenstrauss map preserving pairwise distances up to tolerance $\widetilde{\mu}(A)=\mu(A)/N(D),$3, the paper shows

$\widetilde{\mu}(A)=\mu(A)/N(D),$4

For metric multidimensional scaling, bounds are expressed in terms of the eigenvalues of the covariance matrix, and for $\widetilde{\mu}(A)=\mu(A)/N(D),$5-biLipschitz maps one obtains

$\widetilde{\mu}(A)=\mu(A)/N(D),$6

The computational examples are included to show that $\widetilde{\mu}(A)=\mu(A)/N(D),$7 can remain small when $\widetilde{\mu}(A)=\mu(A)/N(D),$8 is large due to global rescaling; in the reported clustering experiment on frogs, chairs and tori of varying overall scale, clustering via $\widetilde{\mu}(A)=\mu(A)/N(D),$9 perfectly separated the three shape classes, whereas ordinary bottleneck distance mis-clustered several shapes (May et al., 2023).

A related normalization appears in persistence-based change statistics for high-dimensional point clouds. There, persistence landscapes are truncated to the first $N(D)=\mu(\Omega)$0 layers, aggregated into $N(D)=\mu(\Omega)$1, regularized by adding a uniform floor, and converted into a density $N(D)=\mu(\Omega)$2 on $N(D)=\mu(\Omega)$3. The comparison statistic is then the Jensen–Shannon distance

$N(D)=\mu(\Omega)$4

The resulting “PL+JS” statistic is stated to be stable, scale- and shift-invariant, with $N(D)=\mu(\Omega)$5-Hölder continuity under point-cloud perturbations. Scale invariance is enforced in practice by rescaling all point clouds by max pairwise distance and fixing $N(D)=\mu(\Omega)$6 globally (Nakayama, 2 Nov 2025). This suggests that normalization by diameter or by total landscape mass is a recurrent strategy for separating geometric structure from trivial scale effects.

3. Fractional persistence and normalized persistence as a survival ratio

In computational and quantum-complexity treatments, normalized persistence is formulated as a ratio rather than a density or a rescaled metric. For a filtered pair of simplicial complexes $N(D)=\mu(\Omega)$7, with persistent Betti number $N(D)=\mu(\Omega)$8, the normalized persistence fraction is defined as

$N(D)=\mu(\Omega)$9

namely the fraction of $E[\mu]$0-dimensional holes in $E[\mu]$1 that survive into $E[\mu]$2. Equivalently, in terms of the projectors onto harmonic subspaces of the $E[\mu]$3-th combinatorial Laplacians, it is the rank of the restriction of the later kernel projector to the earlier kernel, divided by the dimension of the earlier kernel (Lowe et al., 3 Jul 2026).

This ratio has two roles in that work. First, it is presented as a practically motivated and easily interpretable version of persistent homology, because it counts a fraction rather than a raw count. Second, it is embedded into a broader family of low-energy spectral-overlap problems for local Hamiltonians, including normalized quasi-persistence, low-energy normalized subtrace, low-energy spectral density, and exact normalized persistence. The main complexity-theoretic conclusions are that a variant of normalized persistence is $E[\mu]$4-hard and contained in $E[\mu]$5, while exact normalized persistence for $E[\mu]$6-local Hamiltonians is $E[\mu]$7-hard (Lowe et al., 3 Jul 2026).

Algorithmically, the quantum procedures all share the same high-level structure: prepare the uniform maximally mixed state on the normalizing subspace, use Hamiltonian simulation and phase estimation to project onto a low-energy window or exact kernel, apply fixed-point amplitude amplification or oblivious amplitude amplification, and repeat $E[\mu]$8 times to obtain additive $E[\mu]$9 accuracy with confidence $E[\widetilde{\mu}]$0. The complexity-theoretic significance is that normalized persistence is treated as a subspace-normalized spectral estimation problem rather than as a purely geometric descriptor (Lowe et al., 3 Jul 2026).

A closely related but more classical normalization is implicit in graph-based persistence descriptors for neural networks. In neural persistence, each layer of a fully connected feed-forward network is viewed as a weighted bipartite graph, and the filtration is induced by normalized edge weights

$E[\widetilde{\mu}]$1

Raw neural persistence is a $E[\widetilde{\mu}]$2-norm of persistence lifetimes, while normalized neural persistence divides this by a theoretical upper bound

$E[\widetilde{\mu}]$3

yielding a layer-wise measure in $E[\widetilde{\mu}]$4. Aggregating these normalized values across layers gives a single network-level scalar (Rieck et al., 2018). Here normalization makes persistence values comparable across architectures with different layer sizes and weight ranges.

4. Normalized summaries of persistence magnitude and temporal change

Several recent constructions normalize persistence-derived summaries by total persistence, total feature count, or barcode-level moments in order to produce comparable descriptors.

In dynamic-network anomaly detection, the Overlap-Weighted Hierarchical Normalized Persistence Velocity (OW-HNPV) converts a persistence diagram $E[\widetilde{\mu}]$5 into an $E[\widetilde{\mu}]$6-vector summarizing how topological features “flow” across filtration intervals. Its key weight is the overlap

$E[\widetilde{\mu}]$7

whose subinterval sum is turned into a velocity and then normalized by total persistence

$E[\widetilde{\mu}]$8

Earlier variants in the same framework normalize by the number of features $E[\widetilde{\mu}]$9 or by $\lambda(\omega)$0 after ordinary-persistence weighting; OW-HNPV instead uses actual overlap length, so short-lived features automatically contribute little overlap without thresholding. The paper proves a Lipschitz stability bound with respect to the $\lambda(\omega)$1-Wasserstein distance and reports up to $\lambda(\omega)$2 AUC gain over baseline models for 7-day price movement predictions on Ethereum transaction networks (Khormali, 16 Dec 2025).

A variance-based normalization is developed in the Topological Stability Index framework. Given barcode lifetimes $\lambda(\omega)$3 with mean $\lambda(\omega)$4, the unnormalized quantity is

$\lambda(\omega)$5

Its normalized version is

$\lambda(\omega)$6

The paper identifies $\lambda(\omega)$7 exactly with the usual sample coefficient of variation of the lifetimes and derives the affine identity

$\lambda(\omega)$8

Hence $\lambda(\omega)$9 is a monotone reparametrization of Rényi entropy of order two. The paper emphasizes that, unlike persistent entropy, TSI captures absolute variability, whereas the normalized $p(\omega)$0 is scale invariant (Kirchner et al., 28 May 2026).

These constructions differ in object and intent, but they share a common pattern: persistence is first converted into a mass distribution over bars, overlaps, or lifetimes, and then normalized by an intrinsic aggregate such as total persistence or mean lifetime. This suggests a general distinction between raw persistence summaries, which retain absolute magnitude, and normalized persistence summaries, which emphasize composition or relative survival.

5. Normalized spaces and persistence in geometric topology

Normalization also enters persistence through the geometry of the underlying object, rather than through the persistence summary itself. In the deformation-theoretic framework for knot types, a knot type $p(\omega)$1 is assigned a family of normalized spaces of representatives

$p(\omega)$2

consisting of $p(\omega)$3 embeddings of thickness at least $p(\omega)$4 and length at most $p(\omega)$5, modulo orientation-preserving reparametrization and rigid motions. This forms a normalized moduli-type space equipped with a ropelength sublevel filtration (Ozawa, 20 Apr 2026).

An extended pseudometric is then defined by the infimum of swept areas of admissible deformations between representatives, producing admissible components and a genuine $p(\omega)$6-dimensional persistence module

$p(\omega)$7

The first birth time of this persistence is exactly the ropelength $p(\omega)$8, and the layer

$p(\omega)$9

is called the ideal stratum. By construction, it is the minimizer locus of the ropelength functional on the normalized representative space (Ozawa, 20 Apr 2026).

This usage is adjacent to, but distinct from, the normalized persistence fraction or normalized persistence density. The normalization is imposed on the space of geometric representatives before persistence is constructed. A plausible implication is that normalization can operate at three different levels in persistence theory: on the input space, on the persistence object, or on the final summary statistic.

6. Uses of “normalized persistence” beyond persistent homology

Outside standard persistent homology, the phrase also appears in two distinct areas where “persistence” has its stochastic meaning.

In cell-migration analysis, the normalized Shannon entropy of a velocity power spectrum is explicitly identified as “normalized persistence” $\int_\Omega p(\omega)\,d\omega=1$0: $\int_\Omega p(\omega)\,d\omega=1$1 For an Ornstein–Uhlenbeck model of persistent random-walk velocities, the power spectral density is Lorentzian, and $\int_\Omega p(\omega)\,d\omega=1$2 is a monotonic function of the persistence time $\int_\Omega p(\omega)\,d\omega=1$3, allowing one to define $\int_\Omega p(\omega)\,d\omega=1$4. A time-varying version based on the wavelet power spectrum yields $\int_\Omega p(\omega)\,d\omega=1$5 for nonstationary motility (Liu et al., 2020). Although this is not topological persistence, it shares the same formal motive as TDA normalizations: convert a raw persistence-related signal into a bounded, scale-independent descriptor.

In autoregressive persistence probabilities, an AR(1) sequence with symmetric uniform innovations has $\int_\Omega p(\omega)\,d\omega=1$6-step persistence probability

$\int_\Omega p(\omega)\,d\omega=1$7

Because $\int_\Omega p(\omega)\,d\omega=1$8 decays exponentially, the normalized sequence

$\int_\Omega p(\omega)\,d\omega=1$9

is introduced so that $D$00 converges to a positive constant. Alsmeyer–Bostan–Raschel–Simon derive exact formulae for $D$01 in terms of Mallows–Riordan polynomials for $D$02 and show asymptotically that

$D$03

hence $D$04 (Alsmeyer et al., 2021). Here the normalization removes the dominant exponential decay and isolates the nontrivial limiting prefactor.

These usages are terminologically related but mathematically separate from persistence diagrams. They show that “normalized persistence” is a cross-disciplinary phrase whose meaning is determined by the object being normalized: entropy of a spectrum, exponentially decaying survival probability, density of diagram points, bottleneck distance under unit-diameter scaling, or fraction of surviving homology classes.

7. Conceptual distinctions and recurring themes

Across the cited literature, normalized persistence can be grouped by what is being normalized.

First, some constructions normalize the input geometry. The normalized bottleneck distance scales point clouds by diameter before computing $D$05, and the knot-theoretic framework fixes thickness at least $D$06 and bounds length by $D$07 in a normalized representative space (May et al., 2023, Ozawa, 20 Apr 2026).

Second, some normalize the persistence object itself. The persistence density replaces a counting measure by a probability measure on diagram points, and the normalized persistence fraction replaces a persistent Betti count by a survival ratio relative to the initial Betti number (Wu et al., 2023, Lowe et al., 3 Jul 2026).

Third, some normalize derived summaries. Neural persistence divides a lifetime norm by a layer-dependent upper bound; PL+JS regularizes and normalizes persistence landscapes into densities; OW-HNPV divides overlap-based velocity by total persistence; and $D$08 divides variance by squared mean lifetime (Rieck et al., 2018, Nakayama, 2 Nov 2025, Khormali, 16 Dec 2025, Kirchner et al., 28 May 2026).

A common misconception is to treat these as interchangeable. They are not. A normalized persistence density $D$09 is a probability density on diagram points; normalized bottleneck distance $D$10 is a scale-invariant pseudometric; $D$11 is a survival fraction of homology classes; normalized neural persistence is a complexity score in $D$12; and $D$13 is a normalized lifetime-dispersion statistic. The literature therefore supports a precise but plural usage: normalized persistence is best understood as a family of normalization principles applied to persistence-based constructions to improve comparability, stability, or statistical interpretability under varying scale, multiplicity, or total persistence mass (Wu et al., 2023, May et al., 2023, Lowe et al., 3 Jul 2026).