Generalized Persistence Landscapes

• Generalized persistence landscapes are vectorized summaries in topological data analysis that extend classical persistence landscapes to include continuous, multi-parameter, and spatiotemporal settings.
• They provide a stable and invertible mapping of persistent homological features into Banach or Hilbert spaces, facilitating rigorous statistical and machine learning applications.
• Incorporating kernel smoothing and weighted variants, these landscapes enhance discrimination in complex, noisy data environments, enabling efficient computation and practical integration into neural architectures.

Generalized persistence landscapes are a class of vectorized summaries in topological data analysis (TDA) that extend the classical persistence landscape construction to a broad family of contexts, including non-discrete, multi-parameter, measure-theoretic, geometric, labeled, and spatiotemporal settings. These constructions aim to provide interpretable, stable, and informative functional representations of persistent homological features, suitable for downstream statistical and machine learning tasks, beyond what is provided by the basic barcode or persistence diagram.

1. Classical Persistence Landscapes and Limitations

The classical persistence landscape, introduced by Bubenik, associates to a finite persistence diagram $D = \{(b_i,d_i)\}_{i=1}^N$ in a fixed homological degree a sequence of piecewise-linear functions $\lambda_k : \mathbb R \to \mathbb R$ defined as the $k$-th largest value among a family of tent functions: $f_{b_i,d_i}(t) = \max\{0, \min(t-b_i, d_i-t)\},$ and

$\lambda_k(t) = \text{%%%%3%%%%-th largest of } \{ f_{b_i,d_i}(t) \}.$

This construction embeds persistence diagrams into Banach or Hilbert spaces ($L^p$ of sequences of functions), enabling direct use of analytical and statistical machinery. The map is stable:

$$\|\lambda_k^D - \lambda_k^{D'}\|_\infty \leq d_B(D,D')$$

where $d_B$ is the bottleneck distance (Bubenik, 2018). The feature map is injective (invertible) under generic conditions. However, the classical landscape is limited to finite diagrams and single-parameter filtrations, and does not naturally incorporate measure-theoretic, multi-parameter, or geometric information (Zhao et al., 28 Nov 2025, Vipond, 2018).

2. Measure-Theoretic and Continuous Landscapes

Continuous persistence landscapes generalize $\lambda_k$ to the setting where diagrams are replaced by Borel measures $\mu$ on the open wedge $X = \{(b,d) \in \mathbb R^2 : b < d\}$. "q-tame" measures are those for which all relevant quadrant masses are finite: $$\mu((-\infty, t-h) \times (t + h, \infty)) < \infty, \quad \forall t \in \mathbb R, h \geq 0.$$ The continuous landscape is defined as

$$\lambda(a,t) = \sup\{h \geq 0 : \mu((-\infty,t-h]\times[t+h,\infty)) \geq a\}$$

and the classical layer is recovered by setting $a=k\in \mathbb N$. The construction is bijective: $\mu \mapsto \lambda$ is a bijection between q-tame measures and functions satisfying natural monotonicity, continuity, and additivity properties. Continuous landscapes are $L^1$-stable under an intrinsic Wasserstein-type cost metric and fully invertible, in contrast to the pointwise instability of classical landscapes under high multiplicity (Zhao et al., 28 Nov 2025).

3. Functional Generalizations and Kernel Smoothing

Generalized persistence landscapes include kernel-smoothing and weighted variants, expanding the family of summaries. For a kernel function $K$ (e.g. triangle, Epanechnikov, Gaussian)

$$\widetilde\Lambda_i(t;h) = \frac{y_i}{K_h(0)} K_h\left(\frac{t-x_i}{h}\right),\quad x_i = \frac{b_i+d_i}{2},\ y_i = \frac{d_i-b_i}{2}$$

and the generalized landscape is

$\tilde{\lambda}_k(t) = \text{%%%%13%%%%-th largest of } \{\widetilde\Lambda_i(t;h)\}_{i=1}^N.$

This family is stable under bottleneck distance, allows tuning the bandwidth $h$, and provides improved discrimination in high-noise or low-signal tasks (Berry et al., 2018).

Weighted landscape variants take the form

$\lambda^w_k(t) = w(k, t) \lambda_k(t),$

where $w$ is a user-chosen weight function. Specializing to Poisson weights $P_v(k)$ allows the construction of a characteristic, scale-sensitive kernel

$$K^v(D,D') = \sum_{k=1}^\infty P_v(k-1) \int_{\mathbb R} \lambda_k(D; t)\lambda_k(D'; t)\,dt,$$

which smoothly interpolates emphasis across landscape layers (Bubenik, 2018).

4. Multiparameter, Labeled, Geometric, and Spatiotemporal Landscapes

Multiparameter landscapes extend the landscape formalism to persistence modules indexed by $(\mathbb R^d, \leq)$. For module $M$ and norm $\|\cdot\|_p$, the uniform multiparameter landscape is

$$\lambda^M_p(k, \mathbf x) = \sup\{ \epsilon \geq 0 : \operatorname{rank}_M(\mathbf x - \mathbf h, \mathbf x + \mathbf h) \geq k\ \forall \mathbf h, \|\mathbf h\|_p \leq \epsilon \}$$

and is 1-Lipschitz in $\mathbf x$, stable under multiparameter interleaving distances, and reconstructs the module's rank invariant on axis-aligned hypercubes (Vipond, 2018, García-Redondo et al., 1 Apr 2025).

Labeled generalized landscapes define, for a labeled metric space $\mathcal X = (X, d_X, L_X)$ with subsets $X_1,\dots,X_k$, a multi-parameter landscape

$$\lambda(n, r, p) = \sup\{\epsilon \geq 0 : \operatorname{rank}(V(r-\epsilon,p)\rightarrow V(r+\epsilon ,p)) \geq n\}$$

which is Lipschitz stable under a Gromov–Hausdorff-type distance respecting label structure (Fu et al., 9 Dec 2025).

Geometric generalizations use cycle representatives associated to barcode intervals via merge forests and Alexander duality, applying functionals (arc-length, enclosed area, curvature) to the time-evolving cycles and obtaining function-valued features more sensitive to geometric nuances than classical summaries (Lenzen et al., 10 Dec 2025).

Spatiotemporal landscapes, defined for extended zigzag modules indexed by $ZZ \times \mathbb Z$, use the generalized rank invariant over spatiotemporal boxes: $$\lambda_k(x) = \sup\{ \delta \geq 0 : \operatorname{rk}(M; R_x^\delta ) \geq k \}$$ and live in Banach (Lebesgue) spaces with stability with respect to a suitable interleaving distance, allowing principled statistical analysis for dynamic data (Flammer et al., 2024).

5. Theoretical Properties: Stability, Invertibility, and Expressiveness

The essential properties of generalized persistence landscapes across these settings include:

• Stability: Each construction is Lipschitz or contractive with respect to an appropriate diagram/module metric: 1-Wasserstein, bottleneck, interleaving, or Gromov–Hausdorff-type (Zhao et al., 28 Nov 2025, Vipond, 2018, Fu et al., 9 Dec 2025, Flammer et al., 2024). Graded persistence landscapes further refine stability constants and offer more discriminative (layerwise) metrics (Betthauser et al., 2019).
• Invertibility: Several constructions—continuous, classical, and graded—are mathematically invertible, allowing recovery of (a measure version of) the original data from its landscape map (Zhao et al., 28 Nov 2025, Bubenik, 2018, Betthauser et al., 2019).
• Banach/Hilbert structure: Landscapes live in Banach or Hilbert spaces, permitting application of the law of large numbers, central limit theorems, and use as features in kernel methods (SVMs, GP, kernel-PCA) (Bubenik, 2018, Vipond, 2018, Flammer et al., 2024).
• Functional data analysis: The landscape, or its generalizations, is subject to averaging, bootstrap confidence bands, and hypothesis testing directly in function space without reparametrizing to diagrams (Berry et al., 2018, García-Redondo et al., 1 Apr 2025).

6. Practical Computation and Machine Learning Applications

Generalized persistence landscapes facilitate fast, vectorized computation and integration into machine learning and deep learning pipelines. Efficient algorithms exist for kernel-smoothing, multiparameter slices, and geometric functionals, with complexities typically $O(NM)$ (diagram size × function grid points) for single-parameter versions and polynomial in the worst case for multiparameter or geometric variants (Berry et al., 2018, Vipond, 2018, Lenzen et al., 10 Dec 2025).

Neural architectures such as PLLay embed differentiable landscape layers into networks. These layers are fully compatible with gradient-based learning, exhibit stability to input perturbation, and empirically yield improved robustness and classification accuracy under several modalities (images, point clouds, dynamical systems), outperforming prior topological layers (Kim et al., 2020).

Empirical results show that generalized and kernel-averaged landscapes provide improved discrimination in high-dimensional, noisy, or geometrically ambiguous scenarios and outperform classical landscapes in classification, hypothesis testing, and shape recognition tasks (Berry et al., 2018, Lenzen et al., 10 Dec 2025).

7. Connections with Other Summaries and Future Directions

Generalized persistence landscapes form part of a broader unifying framework for persistence-based functional summaries. Persistence curves (PCs) subsume landscapes, silhouettes, Betti curves, and others, illustrating the flexibility of "diagram-to-function" vectorizations (Chung et al., 2019). Graded diagrams align precisely with the kink structure of landscapes and refine layerwise discrimination (Betthauser et al., 2019).

Ongoing research addresses:

• Quantitative stability for functionals and geometric landscapes (Lenzen et al., 10 Dec 2025);
• Extension to higher codimension, more general parameter posets, or more intricate algebraic structures (e.g., multidimensional or labeled persistence modules) (Fu et al., 9 Dec 2025);
• Fast algorithms for multiparameter persistent landscapes (García-Redondo et al., 1 Apr 2025, Vipond, 2018);
• Advanced statistical techniques (inference, CLTs, confidence sets) for functional landscape summaries (Berry et al., 2018, García-Redondo et al., 1 Apr 2025).

These developments establish the generalized persistence landscape family as a flexible, stable, and expressive toolkit for rigorous topological representation and analysis across diverse data modalities and analytic objectives.