The Hidden Geometry of Astrophysical Spectra: Path-Signatures of Line Profiles

Technical Summary: The Hidden Geometry of Astrophysical Spectra

Problem Statement
Traditional summaries of astrophysical spectral-line profiles—such as integrated flux, centroid velocity, full width at half maximum (FWHM), non-parametric velocity widths (W80), and low-order moments (e.g., Gauss-Hermite coefficients)—often fail to capture the full informational content of a spectrum. These scalar metrics reduce a profile to global statistics, potentially rendering distinct physical configurations indistinguishable. For instance, a double-peaked profile arising from dual active nuclei, disk rotation, or biconical flows may share similar FWHM and asymmetry measures with a single broad component. Similarly, P-Cygni profiles (blueshifted absorption, redshifted emission) and inverse P-Cygni profiles (redshifted absorption, blueshifted emission) can appear similar under unsigned summaries or if the ordering of features is lost. The authors argue that the ordered sequence of flux variations across the velocity axis encodes critical physical information regarding outflows, inflows, and kinematic substructure that scalar summaries discard.

Methodology
The paper introduces a geometric representation of line profiles based on path signatures, a concept derived from rough path theory and iterated integrals (Chen 1958; Lyons 1998).

1. Path Formulation: A continuum-subtracted spectral line profile is treated not as a discrete vector of flux values, but as a directed trajectory $X(v) = (v, F(v))$ in a two-dimensional velocity–flux space. The path is traversed from blue to red velocity.
2. Signature Computation: The path signature maps this ordered curve into a hierarchy of tensor coordinates. The authors utilize the log-signature, which provides a non-redundant description of the path's geometry.
3. Low-Order Descriptors: To ensure interpretability and computational feasibility, the authors define a compact set of low-order scalar descriptors derived from the log-signature:
   • $p_2$: The signed velocity–flux area (antisymmetric second-order term), sensitive to net line area and blue–red imbalance.
   • $p_{3v}$ and $p_{3F}$: Third-order terms describing the localization of asymmetry along the velocity axis and its association with flux intensity (core vs. wings).
   • $p_{4F}$ and $p_{4T}$: Fourth-order terms capturing higher-order flux modulations (shoulders, broad bases) and twist-like structures (double peaks, multi-components).
   • $p_{\pm}$: A coefficient derived from the cumulative positive and negative parts of the profile, specifically measuring the ordering of emission versus absorption (distinguishing P-Cygni from inverse P-Cygni).
4. Implementation: The authors developed a minimalist, MIT-licensed Python package, spectropath, tailored to astrophysical conventions (handling blue-to-red mapping, normalization, and specific descriptor selection).

Key Contributions and Results

• Synthetic Profile Analysis: The authors tested the descriptors on synthetic line profiles with controlled morphological variations.
  • Separability: In experiments where profile families shared similar FWHM, W80, and global moments (Hermite coefficients) but differed in the ordering of features (e.g., blue vs. red wings, single vs. double peaks, emission-absorption sequence), the path-signature descriptors achieved significantly higher separability (measured by Area Under the ROC Curve, AUC).
  • Stress Tests: The method proved robust even when using complex Moffat-like components. Crucially, in a "signed emission-absorption" stress test where the absolute flux morphology was identical but the signed ordering was reversed (P-Cygni vs. inverse P-Cygni), conventional summaries computed on $|F|$ failed to separate the families, whereas the signed path coordinates successfully distinguished them.
• Application to MaNGA Data: The framework was applied to integral-field spectroscopy data from the MaNGA survey (five diverse galaxies).
  • Clustering: H$\alpha$ profiles were computed for individual spaxels and mapped into a low-dimensional feature space using the descriptors $(p_{3v}, p_{3F}, p_{4T})$.
  • Kinematic Recovery: Spaxels were clustered based solely on these morphological descriptors, without inputting external velocity maps. When spectra within these clustered regions were stacked, the resulting velocity maps recovered coherent large-scale kinematic structures. These structures were broadly consistent with the fiducial MaNGA Data Analysis Pipeline (DAP) velocity fields, demonstrating that ordered line morphology contains implicit kinematic information.

Significance and Claims
The paper positions this work as a change of perspective rather than a replacement for existing physical classification schemes. The authors claim that:

• Representation Matters: Compression should be viewed as a choice of representation that preserves information relevant to specific scientific questions (in this case, ordered morphology) while discarding incidental degrees of freedom.
• Interpretability: Unlike black-box machine learning or variance-optimal methods like PCA (where components mix flux, centroid, and width depending on the dataset), path-signature descriptors have fixed geometric interpretations (e.g., "signed area," "twist," "ordering").
• Intermediate Layer: The method serves as an effective intermediate representation for grouping spaxel-level spectra before kinematic or physical analysis, allowing spectra to be clustered by their ordered structure.
• Limitations: The authors explicitly note that the coefficients depend on preprocessing (continuum subtraction, normalization, sampling) and are not noise-invariant observables. They acknowledge that formal error propagation for these non-linear descriptors remains a challenge and that the current application is a proof of concept, deferring detailed astrophysical interpretation of the recovered regions to future work.

The paper concludes that treating astronomical data as paths offers a way to move from describing isolated measurements to describing the "geometry of change," leveraging mathematical structures (rough paths) to make the intrinsic ordering of spectral data explicit.