Euler Characteristic Profiles in Topological Data Analysis

• Euler characteristic profiles are integer-valued functions that capture the evolution of topological invariants across multidimensional filtrations, bridging classical TDA and scalable data analysis.
• They are computed using efficient algorithms that update cell counts with local rules, making them practical for high-dimensional imaging, shape analysis, and dynamical systems.
• While compressing detailed homological data, ECPs provide robust stability guarantees and computational tractability, enabling applications in biomedical imaging and machine learning.

Euler characteristic profiles (ECPs) are integer-valued functions or multidimensional arrays that summarize the evolution of the Euler characteristic—a classical topological invariant—across a filtration of subcomplexes defined by one or several parameters. ECPs and their higher parameter extensions, such as Euler characteristic surfaces (ECS), provide computationally tractable, stable, and interpretable summaries of topological information in data, bridging classical topological data analysis (TDA) and scalable big-data pipelines. Their utility spans from shape analysis and machine learning to multiparameter TDA, biomedical imaging, and dynamical system regime detection (Beltramo et al., 2021, Dłotko et al., 2022, Munch, 2023, Hacquard et al., 2023, Majhi et al., 16 Apr 2026, Kirveslahti et al., 2024, George et al., 24 Jun 2025).

1. Mathematical Formulation and Generalization

Given a finite cell complex $K$ (simplicial, cubical, or CW), the Euler characteristic is defined as

$$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$

where $n_i$ counts the $i$-cells and $\beta_i$ are the Betti numbers. A filtration on $K$ is a nested family of subcomplexes parameterized by a function or tuple of functions. For a single parameter $h:K\to\mathbb{R}$, the sublevel sets $K_t = \{\sigma\in K | h(\sigma)\le t\}$ yield the one-parameter ECP: $E_h(t) = \chi(K_t).$ For a bifiltration $h_1,h_2:K\to\mathbb{R}$, the joint sublevel sets

$$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$0

define the two-parameter ECP, or Euler characteristic surface (ECS): $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$1 For $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$2-parameter filtrations, the ECP generalizes to $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$3 over a parameter grid (Beltramo et al., 2021, Dłotko et al., 2022).

2. Computation and Algorithmic Aspects

ECPs benefit from exceptional computational efficiency:

• The Euler characteristic depends only on cell counts, allowing its computation by sequentially streaming through sorted simplices or voxels and updating $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$4 by $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$5 at each cell's entrance.
• For cubical complexes (e.g., grayscale images), efficient algorithms utilize local update rules and 1D/2D cumulative summation; in dimension $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$6, updating the ECS matrix scales as $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$7 for an $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$8-voxel image with two parameter grids of size $$\chi(K) = \sum_{i=0}^d (-1)^i n_i = \sum_{i=0}^d (-1)^i \beta_i(K),$$9 (Beltramo et al., 2021).
• For simplicial complexes (e.g., Vietoris–Rips, α, or Delaunay), distributed algorithms operate in vertex-local fashion, emitting update streams for each vertex, which are then merged via prefix-sum or multi-dimensional bucketing (Dłotko et al., 2022).
• Grid resolution is a trade-off: finer grids yield higher fidelity but with storage and compute scaling as $n_i$0 for $n_i$1 parameters (Beltramo et al., 2021).

The digital Euler characteristic transform framework implements an exact, non-discretized calculation by partitioning the sphere of directions according to vertex orderings and representing the ECP as a collection of step-function gains over these regions (Kirveslahti et al., 2024).

3. Stability, Injectivity, and Theoretical Guarantees

ECPs inherit both stability and, under certain conditions, injectivity:

• For one-parameter filtrations, the $n_i$2-distance between ECCs is bounded above by twice the sum of the 1-Wasserstein distances of the persistent diagrams across all homological dimensions (Dłotko et al., 2022, Hacquard et al., 2023).
• For multiparameter ECPs, if the filtration vectors of all cells are perturbed by at most $n_i$3, the $n_i$4 difference of ECPs is bounded above by a constant times $n_i$5, where $n_i$6 is the number of filtration parameters (Dłotko et al., 2022).
• In the case of the Euler characteristic transform (ECT)—the continuous analog—injectivity is established on large classes of compact, definable sets: if two shapes share all ECPs for all directions, then the shapes are identical (Munch, 2023, George et al., 24 Jun 2025).
• The ECP is piecewise constant and right-continuous; small perturbations of the filtration function, as long as they do not reorder cell entry events, leave the ECP unchanged (Beltramo et al., 2021, Marsh et al., 2023, George et al., 24 Jun 2025).
• Robust sup-norm stability theorems are available, with explicit combinatorial constants relating pointwise perturbations of geometric embeddings to the maximal change in ECPs (George et al., 24 Jun 2025).

4. Connections to Persistent Homology and Other Invariants

ECPs offer a coarser summary than persistent homology but with key computational and stability advantages:

• ECPs collapse all Betti numbers to a single integer at each threshold, discarding the multi-dimensional “birth–death” information encoded by persistence diagrams.
• While persistent homology faces algebraic and computational complexity barriers in the multiparameter setting (e.g., lack of complete discrete invariants), ECPs are readily computed and interpreted in any parameter dimension (Dłotko et al., 2022, Beltramo et al., 2021).
• The Euler Characteristic Transform (ECT) unifies ECPs by viewing the full collection of ECCs in all directions; a fine enough sampling of directions yields a (near-)injective shape signature. Finite, regularly spaced direction grids yield discrete ECP matrices (Munch, 2023, Kirveslahti et al., 2024).
• ECPs may be regarded as pushforwards or projections of higher complexity topological invariants, providing practical “compressions” of the multi-scale topological structure (Hacquard et al., 2023).

5. Applications Across Data Types

ECPs and ECSs are used as feature representations in supervised and unsupervised machine learning, biomarker discovery, graph classification, dynamical systems, and more:

• In image analysis, one-parameter ECCs and two-parameter ECSs allow both classification (e.g., diabetic retinopathy angiogram images, achieving AUC up to 0.91 and classification rates of 80–90%) and interpretability, via visualization of regions where disease and control populations differ (Beltramo et al., 2021).
• ECSs reveal hidden correlations or patterns invisible to one-parameter summaries (e.g., synthetic images with correlated noise, Poisson vs. Hawkes process point clouds) (Beltramo et al., 2021).
• In topological regime detection for dynamical systems, the Mixup ECP statistic quantifies the topological overlap between pre- and post-transition time windows, enabling detection of bifurcations and transitions (e.g., monsoon onset timing, Lorenz and logistic systems) with built-in stability and permutation-based hypothesis testing (Majhi et al., 16 Apr 2026).
• ECPs underpin shape analysis pipelines for biological morphology (plant, bone, protein), graph classification, and digital pathology, often exceeding or matching the performance of more complex TDA-based features at several orders of magnitude lower computational cost (Hacquard et al., 2023, Dłotko et al., 2022, Munch, 2023).
• ECPs are also used in rotation-invariant analysis by integrating over the sphere of directions, and in deep learning architectures as input to graph neural networks or loss functions (Munch, 2023).

6. Limitations, Extensions, and Practical Considerations

While ECPs are extremely efficient and robust, certain limitations and extension strategies are notable:

• By summing over Betti numbers, ECPs cannot distinguish specific homological dimensions—loops and cavities contribute indistinguishably. Full persistence modules or diagrams are needed where this granularity is essential (Beltramo et al., 2021, Dłotko et al., 2022).
• As parameter dimension increases, storage and computational costs grow exponentially. Remedies include low-rank tensor approximation, parameter-space random projections, or streaming algorithms (Beltramo et al., 2021).
• Adaptive grid schemes for filtration parameters improve sensitivity where $n_i$7 changes rapidly, balancing noise resistance and resolution (Beltramo et al., 2021).
• Inverse problems, such as reconstructing a shape from finitely many ECPs, face fundamental ambiguities (e.g., loss of information due to degree-2 vertices in planar graphs (Fasy et al., 2018)); stability bounds provide partial reassurance for approximate inversion (George et al., 24 Jun 2025).
• Software implementations include the “euchar” Python package and Ectoplasm for digital ECT computation (Beltramo et al., 2021, Kirveslahti et al., 2024).

7. Statistical and Empirical Guarantees

ECPs possess strong theoretical guarantees regarding convergence and consistency:

• Law of large numbers and central limit theorems for ECPs and their integral (hybrid) transforms hold for random point processes, giving asymptotic control on their behavior in large-sample or thermodynamic limits (Hacquard et al., 2023).
• Statistical consistency results are established for ECP estimators from noisy data via Gaussian processes, provided appropriate curvature or smoothness bounds are satisfied (Marsh et al., 2023).
• Empirical benchmarks show state-of-the-art or competitive performance in curvature regression, orbit classification, and graph data sets across a variety of filtration and feature vectorization regimes (Hacquard et al., 2023).

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References:

(Beltramo et al., 2021) Euler Characteristic Surfaces (2021) (Dłotko et al., 2022) Euler Characteristic Curves and Profiles (2022) (Munch, 2023) An Invitation to the Euler Characteristic Transform (2023) (Hacquard et al., 2023) Euler Characteristic Tools For Topological Data Analysis (2023) (Majhi et al., 16 Apr 2026) Detecting Regime Transitions in Dynamical Systems via the Mixup Euler Characteristic Profile (2026) (Kirveslahti et al., 2024) Digital Euler Characteristic Transform (2024) (George et al., 24 Jun 2025) On the Stability of the Euler Characteristic Transform for a Perturbed Embedding (2025) (Fasy et al., 2018) Challenges in Reconstructing Shapes from Euler Characteristic Curves (2018)