Robust Signature Moments

• Robust signature moments are defined as feature maps using tensor normalization to stabilize iterated integrals and mitigate divergence issues in high-order expansions.
• They are characterized by properties such as continuity, boundedness, and injectivity, ensuring reliable statistical inference and discrimination across complex sequential and density data.
• Applications span stochastic process law characterization, nucleon density analysis, and financial models, with computational strategies like dynamic-programming and kernelized evaluations ensuring robustness under truncation.

Robust signature moments are a class of feature maps and associated moment sequences designed to provide stable, discriminative, and theoretically well-founded summaries of objects with inherent sequential or functional structure, including paths of stochastic processes, probability laws on path spaces, and structural densities in physics. These moments generalize classical tensorial or scalar moments, harnessing the algebraic and analytic properties of the path (or density) “signature” to capture high-order structure while ensuring practical robustness under truncation, normalization, and model regularization.

1. Foundations and Definitions

The robust signature moment framework arises from the need to rigorously characterize complex objects—such as probability laws of stochastic processes or short-distance behavior of radial densities—using structured, well-behaved features.

For path-valued data, the classical signature of an absolutely continuous path $x \colon [0,T] \to V$ (with $V$ a Banach or Hilbert space) is the sequence of iterated integrals

$$S(x) = \left(1, \int_0^T dx, \int_{0 < t_1 < t_2 < T} dx_{t_1} \otimes dx_{t_2}, \ldots \right) \in \prod_{m \ge 0} V^{\otimes m}\,.$$

These signatures exhibit injectivity up to tree-like equivalence and therefore have the capacity to faithfully encode pathwise information (Chevyrev et al., 2018). However, raw signature features can be unbounded, especially for processes on non-compact domains, leading to instability and lack of universality.

To address this, a tensor normalization map $\Lambda$ is introduced, yielding the robust signature feature

$\Phi(x) = \Lambda(S(x)) \in T(V)\,,$

where $T(V)$ denotes the tensor algebra. The normalization ensures boundedness, continuity, and injectivity of the feature map.

For radial densities $\rho(r)$ in three dimensions, the negative-order radial moments

$$M_{-k} = \langle r^{-k} \rangle = 4\pi \int_0^\infty dr\, r^{2-k} \rho(r)$$

provide insight into short-distance structure. These moments are evaluated robustly via suitably regularized momentum-space integrals, linking them to the analytic structure of the density at $r \to 0$ (Atoui et al., 17 Oct 2024).

2. Theoretical Properties

Robust signature moments possess several theoretically significant properties:

• Characterization / Universality: The collection of all robust signature moments $\{M_\ell(\mu)\}_{\ell \in T(V)'}$ characterizes the law $\mu$ of a random path, i.e., the map $\mu \mapsto M(\mu)$ is injective on finite Borel measures (Chevyrev et al., 2018).
• Continuity and Boundedness: The normalized feature map $\Phi$ is continuous in the rough-path metric and is bounded; this assures well-posedness for metric and statistical procedures.
• Structural Stability: In physical contexts, regularization in momentum-space (for density moments) or norm constraints (for signatures) confers numerical and analytic robustness, even in the presence of singularities or truncation.

By careful construction, these moments overcome the pitfalls of divergence and sensitivity in naïve high-order expansions or moment sequences.

3. Methodologies for Extraction and Evaluation

The extraction of robust signature moments centers on three main technical methodologies, relevant both in stochastic process and density contexts:

1. Normalization of Feature Maps: The tensor normalization $\Lambda(t) = \delta_{\lambda(t)} t$ with a carefully chosen scaling function $\psi$ yields strong Lipschitz bounds, ensuring that discretization or sampling errors do not propagate catastrophically (Chevyrev et al., 2018).
2. Regularized Integral Representations: For radial densities, negative moments are computed not directly from possibly divergent coordinate-space integrals but via subtracted and cutoff-regularized dispersion-type integrals in momentum space,

$$\langle r^\lambda \rangle = \mathcal{N}_{\lambda,3} \;\mathrm{P.V.} \int_0^{\infty} \frac{F(q)}{q^{\lambda+1}}\,dq\,,$$

where the subtraction of the asymptotic series for $F(q)$ systematically removes divergences; the remainder is numerically stable (Atoui et al., 17 Oct 2024).

1. Recursive and Kernelized Computation: For path signatures, dynamic-programming algorithms enable fast computation of truncated robust signature moments, and the associated robust signature kernel

$k(x,y) = \langle \Phi(x), \Phi(y) \rangle_{T(V)}$

permits efficient evaluation of statistical metrics such as Maximum Mean Discrepancy (MMD) (Chevyrev et al., 2018).

4. Applications Across Domains

Stochastic Processes and Law Characterization

Robust signature moments underpin a universal and characteristic feature map for probability laws on geometric rough path spaces $P^p(V)$, empowering statistical inference on path-space and non-parametric testing:

• The MMD built on robust signature features defines a metric on the space of laws, consistent for two-sample testing and with theoretical guarantees on metrization of weak convergence in finite-dimensional settings (Chevyrev et al., 2018).
• Empirical evaluation of these kernels is feasible, with explicit discretization error bounds and computational complexity scaling as $O((nn')M)$ for two sequences of lengths $n, n'$ and order $M$.
• The moments are universal in the strict topology and their coordinate functionals separate points (up to tree-like equivalence).

Quantum and Nuclear Densities

In nucleon structure, negative-order radial moments serve as “signature moments” of the underlying three-dimensional charge or magnetization density:

• By fitting the singular behavior $\rho(r) = r^{-\beta}(g_0 + g_1 r + \cdots)$ near $r=0$, one reconstructs the leading Maclaurin coefficients using residues of $\langle r^\lambda \rangle$ at its characteristic pole $\lambda_c$ (Atoui et al., 17 Oct 2024).
• Robustness is checked by varying the cutoff and the number of subtracted asymptotic terms, with reported numerical stability at the percent level when the momentum cutoff $K \gg 1$.
• The method is largely model-independent and sensitive to the large-$Q^2$ (momentum transfer) regime, with limitations tied to the availability and precision of high-$Q^2$ experimental form factors.

Financial Models with Signature Volatility

In the signature-volatility framework, robust signature moments characterize model structure and moment (non-)explosions:

• True martingality of the price process requires that the signature truncation order $N$ is odd and that the sign constraint $\rho\theta_{2^N} \le 0$ is enforced (Jaber et al., 21 Mar 2025).
• Finiteness of higher moments (absence of moment explosions) depends entirely on the condition $|\rho|>\sqrt{1-1/p}$ for order-$p$ moments.
• Violating these constraints leads to instantaneous blow-up of moments or failure of the martingale property, providing sharp necessary and sufficient tests for robust model design.

5. Numerical and Practical Considerations

The practical utility of robust signature moments is anchored in their computability and stability in empirical contexts:

• Discretization and Truncation: Explicit error estimates are available both for time-discretization (scaling as mesh$^{1/2}$) and for signature or moment truncation (factorial tail decay) (Chevyrev et al., 2018).
• Regularization and Error Quantification: Regularized integral representations (for radial moments) and cutoff variation protocols provide systematic methods to control and estimate uncertainties. Combined errors due to experimental form factor uncertainties, asymptotic subtractions, and residual cutoff dependence typically yield relative uncertainties in the range $\lesssim 10\text{–}20\%$ (Atoui et al., 17 Oct 2024).
• Model Calibration: In signature volatility and rough path modeling, the algebraic constraints on truncation order and correlation parameters translate into explicit guidelines for calibration and model validation (Jaber et al., 21 Mar 2025).

6. Limitations, Open Problems, and Outlook

Despite their strengths, robust signature moments are subject to several notable limitations:

• For density moments, failure to saturate correct asymptotic behavior in form factor fits or insufficiency of high-$Q^2$ data results in bias and amplified uncertainties (Atoui et al., 17 Oct 2024).
• In path-space applications, while the robust signature feature kernel metrizes weak convergence on weakly compact sets, the induced topology is in general strictly weaker than the topology of weak convergence.
• In signature volatility models, the non-existence of intermediate finite moment-explosion times results in strict dichotomies for moment finiteness. Calibration constraints are sharp but may restrict model flexibility for certain implied volatility surface specifications (Jaber et al., 21 Mar 2025).

Open directions include the development and deployment of higher-precision large-momentum data (in hadronic physics), extension to spin-dependent densities or two-photon exchange corrections, and the integration of robust signature moments into broader machine learning pipelines for functional, financial, and physical data.

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Relevant Papers by arXiv ID:

• "Signature moments to characterize laws of stochastic processes" (Chevyrev et al., 2018)
• "Negative moments as the signature of the radial density at small distances" (Atoui et al., 17 Oct 2024)
• "Martingale property and moment explosions in signature volatility models" (Jaber et al., 21 Mar 2025)