Unified Topological Signatures (UTS)

• Unified Topological Signatures are mathematically rigorous frameworks that encode topological, geometric, and statistical descriptors into unified, task-usable structures.
• They aggregate diverse invariants such as Betti numbers, persistence diagrams, and curvature measures into stable, differentiable components for integration with learning models.
• UTS are applied in classification, phase identification, graph analysis, and quantum diagnostics to enhance interpretability across a range of data-driven and physical sciences.

Unified Topological Signatures (UTS) are mathematically rigorous, domain-agnostic frameworks for the systematic encoding, comparison, and utilization of topological, geometric, and statistical information derived from objects such as point clouds, embeddings, graph-structured data, quantum systems, and function or path spaces. UTS approaches aim to “unify” diverse invariants or summaries—across algebraic topology, machine learning, condensed matter physics, and stochastic analysis—into structured, task-usable signatures, either as explicit numerical vectors or as structural embeddings in function spaces or tensor algebras. UTS are crucial for tasks involving classification, phase identification, retrieval, and model interpretability across modern data-driven and physical sciences.

1. Core Frameworks and Structuring Principles

At its foundation, a Unified Topological Signature is a mapping from objects of interest (data points, shapes, graphs, paths, quantum states, embeddings) to a vector or algebraic structure that encodes all relevant topological and geometric information. The precise construction is intrinsically domain-specific, but essential properties recur across contexts:

• Algebraic and Topological Encodings: UTS systematically combine topological invariants (Betti numbers, persistence diagrams, winding numbers), geometric measurements (cycle length, curvature, dimension estimates), and sometimes auxiliary proximity or metric data.
• Hierarchical or Multi-attribute Nature: Rather than relying on a single invariant, UTS typically bundle together multiple descriptors, often organized as feature vectors, multibranch output structures, or tensors—chosen for stability, interpretability, and computational feasibility.
• Stability and Differentiability: Robustness to perturbation (e.g., via Wasserstein-stability in persistence, or insensitivity to parametrization for signature maps) and compatibility with end-to-end learning (e.g., differentiable network layers) are prioritized in practical constructions.
• Unification Across Data Types and Scales: UTS frameworks have been developed for planar shapes (Peters, 2017), persistence diagrams in ML (Hofer et al., 2017), metric embedding spaces (Rottach et al., 27 Nov 2025), 1D and higher-dimensional mapping spaces (Giusti et al., 2022), and interacting quantum chains (Chan et al., 2015), each setting emphasizing cross-domain comparability.

2. Mathematical Formulations Across Domains

2.1. Topological Data Analysis and Neural Representations

• Persistent Homology Vectorizations: Data objects are mapped to persistence diagrams (multisets of $(b, d)$ intervals). To input PDs to neural nets, a parametrized input layer rotates the $(b, d)$ coordinates, projects against smooth, trainable “structure elements,” and aggregates via summation, yielding a fixed-length vector $S_{\boldsymbol \theta, \nu}(D)$ (Hofer et al., 2017).
• Multi-branch Architectures: For 2D shapes and graphs, multiple PDs (e.g., along scanning directions or homology dimensions) enter parallel network branches, and their UTS are aggregated for classification tasks.
• Unified Embedding Signatures: In high-dimensional embedding spaces, UTS are constructed as concatenations of multi-scale and multi-type metrics, including persistent homology statistics, entropy, intrinsic dimensions, effective rank, magnitude area, spread, and isotropy scores (Rottach et al., 27 Nov 2025).

2.2. Shape, Proximity, and Homology Nerves

• Composite Invariants: For finite planar shapes, UTS take the form $$\operatorname{sig}(sh\,A) = [\beta_1; \Phi_\text{geom}(c_1),\dots; \Phi(c_0); \Phi(e_{i_1}),... ]$$, bundling Betti numbers, geometric descriptors of cycles, features of homology nerve nuclei, and overlap arcs (Peters, 2017).
• Descriptive and Strong Proximities: UTS incorporate proximity relations—both spatial intersection and descriptive (feature-matching) nearness—endowed with uniform topologies, yielding signatures robust to both geometric and descriptive shape perturbations.

2.3. Quantum and Condensed Matter Systems

• Multi-observable Bundling: In quantum chains, UTS assemble invariants such as winding numbers, ground-state parity gaps, entanglement-spectrum degeneracies, compressibility peaks, and pair-condensate observables into a unified diagnostic order parameter:

$$\mathcal{UTS}(\mu, \Delta, V) = \frac{1}{4}(I_1 + I_2 + I_3 + I_4) \in [0, 1]$$

where each $I_j$ is a threshold function of a distinct indicator observable (Chan et al., 2015).

• Topological Transitions in Driven Lattices: In optically driven $\alpha$–$T_3$ lattices, UTS capture (i) discontinuous flips in Berry curvature and magnetic moment at the transition point $\alpha_c = 1/\sqrt{2}$, (ii) slope-doubling in orbital magnetization vs. chemical potential, (iii) quantized Hall plateaus, and (iv) vanishing Nernst response—each tightly coupled to Chern number changes (Tamang et al., 2022).

3. Algorithms and Construction Methodologies

Across reported domains, UTS construction proceeds in several generic stages:

1. Feature Extraction: Compute or define a family of topological and geometric (possibly statistical, spectral) descriptors $(T_1, ..., T_K)$ on the object or point set.
2. Aggregation and Normalization: Assemble the raw signature vector $\vec s = [T_1, ..., T_K]^\top$. Normalize each component (e.g., via global maxima over a reference population, $\tilde s_i = s_i / \max |T_i|$), or project to lower dimensions via PCA for analysis (Rottach et al., 27 Nov 2025).
3. Algorithmic Summarization: For variable-sized structures (e.g., point clouds, PDs), use permutation-invariant mappings (e.g., summations over structure functions), or aggregate invariants over a genealogy (e.g., homology generator cycles, nerve nuclei).
4. Topological Encoding: For mapping or path spaces, use graded tensor algebras and iterated integral signatures with algebraic structures (shuffle product, Hopf group structure), ensuring desirable analytic properties such as injectivity and universality (Giusti et al., 2022, Cass et al., 2022).
5. Multi-Indicator Synthesis: In transition-detection problems, combine binary or thresholded indicators for each observable to yield a final phase classifier or order parameter (Chan et al., 2015), or employ supervised learning on UTS to predict downstream metrics (Rottach et al., 27 Nov 2025).

4. Properties, Theoretical Guarantees, and Robustness

• Stability: UTS based on persistence diagrams are $W_1$-stable under diagram perturbations, yielding bounded drifts of the signature under data noise or small perturbations (Hofer et al., 2017).
• Differentiability: Where applicable, layers or mappings constituting the UTS are $C^1$ in trainable parameters and admit backpropagation, enabling integration into end-to-end learning architectures (Hofer et al., 2017).
• Universality and Characteristicness: In mapping space signatures, the UTS (notably via parametrized, normalized signatures) are injective, separate probability laws, and the associated linear functionals are dense in the uniform topology on $C_b(X)$ (Giusti et al., 2022).
• Topological Consistency: For CW complexes and shape analysis, UTS recover homotopy equivalences (e.g., between nerves and the underlying shape), thereby encoding global structural data faithfully (Peters, 2017).
• Resilience to Finite Size and Disorder: In physical systems, certain indicators (e.g., compressibility peaks) exhibit limited sensitivity to small system size or disorder, making the UTS robust for experimental or computational use (Chan et al., 2015).

5. Applications and Empirical Performance

• Machine Learning and Retrieval: UTS enable architectural and model-family identification in embedding spaces, outperforming pairwise alignment metrics, and provide strong predictors for downstream performance and document retrievability, with effective rank and isotropy emerging as key attributes (Rottach et al., 27 Nov 2025).
• Shape Analysis and Graph Classification: Deep architectures equipped with topological input layers leveraging UTS outperform traditional vectorization and graphlet kernel methods, especially in settings where essential features are distinguished (Hofer et al., 2017).
• Quantum Phase Identification: UTS allow for immediate phase classification in interacting topological systems by providing a composite diagnostic—effectively summarizing ground-state degeneracy, order parameters, and response functions (Chan et al., 2015, Tamang et al., 2022).
• Analytic Regression and Approximation: UTS, especially signature-based for paths and mapping spaces, provide the foundation for expected signature models and nonlinear regression, with uniform approximation guarantees and explicit continuity domains (Cass et al., 2022, Giusti et al., 2022).
• Failure of Universality in Nonlinear Optics: Ab initio studies in high harmonic generation demonstrate that no single observable in emission spectra robustly encodes topological phase boundaries across all symmetry classes and models, emphasizing the necessity for multi-attribute UTS to reach reliable identification (Neufeld et al., 2023).

6. Limitations, Open Problems, and Prospects

• Lack of Universality in Some Domains: In certain physical domains, such as nonlinear HHG spectroscopy, no individual observable is sufficient as a universal UTS; thus, multi-dimensional, multi-observable UTS or combined experimental-theoretical approaches remain essential (Neufeld et al., 2023).
• Expressivity Bottlenecks and Heuristic Design Choices: Fixed structure element families, log-warps, or kernel functions in neural UTS introduce limitations; learnable warping or richer function families (splines, wavelets) have been proposed for increased universality (Hofer et al., 2017).
• Dimensionality and Scalability: UTS of high or variable dimension may require potent dimensionality reduction strategies or carefully calibrated normalization for inter-model comparability (Rottach et al., 27 Nov 2025).
• Extensions to Freer Data Types: Generalizations to mixed data (e.g., sensor networks, higher-dimensional mapping spaces), or to embrace more general proximity or CW-combinatorics, are open directions (Giusti et al., 2022, Peters, 2017).

UTS provide a unifying language and computable infrastructure connecting deep learning, data analysis, mathematical topology, quantum physics, and regression, by extracting, organizing, and exploiting the topological essence of complex structures. Their mathematically grounded, robust construction enables interpretable, task-relevant, and cross-disciplinary applications, while ongoing research continues to extend their expressivity, scalability, and applicability to new domains.