Geometry-Based Diagnostic Framework

• Geometry-based diagnostic frameworks are systematic methods that leverage intrinsic geometric and topological properties of data representations to diagnose system behaviors and failures.
• They integrate techniques from metric geometry, spectral graph theory, and information geometry to produce interpretable, robust metrics that outperform traditional diagnostics.
• Applications span diverse domains—from deep representation learning and astrophysics to brain connectomics and program testing—demonstrating versatile, practical insights.

A geometry-based diagnostic framework is a systematic approach that leverages the intrinsic geometric and topological structure of high-dimensional data representations, physical systems, or program executions to detect, quantify, and interpret system properties or failures. Such frameworks utilize mathematical and algorithmic constructions from metric geometry, spectral graph theory, information geometry, and related domains to furnish diagnostic metrics that are both interpretable and robust to confounding factors. Application domains span machine learning, astrophysical discs, molecular clouds, complex systems dynamics, brain connectomics, and program verification.

1. Geometric Diagnostics in Representation Learning and Out-of-Distribution Robustness

Geometry-based diagnostics in deep representation learning exploit the structure of learned feature spaces to assess and predict model robustness, particularly under distribution shift where in-distribution (ID) accuracy is insufficient for evaluating out-of-distribution (OOD) generalization. The "TorRicc" framework constructs class-conditional mutual k-nearest-neighbor graphs from ℓ₂-normalized embeddings $z_i$ (extracted from a mid/penultimate layer), with edges formed by mutual proximity and weighted using a self-tuning Gaussian kernel. Subsequent geometric invariants—specifically, a global spectral complexity proxy (reduced log-determinant of the normalized Laplacian) and mean Ollivier–Ricci (OR) curvature—quantify the global and local organization, respectively.

The spectral complexity for class %%%%1%%%% is computed as

$$\tau(\theta) = \frac{1}{|C|}\sum_{c} -\sum_{i=2}^n \log(\lambda_i + \epsilon),$$

where $\lambda_i$ are the nonzero eigenvalues of the symmetric normalized Laplacian $L = I - D^{-1/2} W D^{-1/2}$ and $\epsilon$ is a small ridge term. Lower $\tau$ indicates simpler, well-connected class manifolds, while higher $\tau$ denotes tangled, cycle-rich structure.

Mean curvature $\bar{\kappa}$ is defined through the OR curvature:

$\kappa(u, v) = 1 - W_1 (\mu_u, \mu_v),$

where $W_1$ is the 1-Wasserstein distance between neighbor-weight probability measures $\mu_u, \mu_v$. High $\bar{\kappa}$ signals local smoothness and contraction, while low or negative $\bar{\kappa}$ reflects rough, fragmented geometry.

Empirically, $\tau$ is strongly negatively correlated (Spearman $\rho\approx -0.88$) and $\bar{\kappa}$ positively correlated ($\rho\approx+0.68$) with OOD accuracy across training checkpoints. These geometric metrics outpredict classical diagnostics (CKA, feature norm, second-smallest Laplacian eigenvalue). Crucially, the signal vanishes under geometry-destroying controls (label shuffling, feature shuffling, edge rewiring), but not under isometries, demonstrating its invariance to spurious statistics. A normalized GeoScore $\mathrm{GeoScore}(t) = z_\tau(t) - z_\kappa(t)$ enables unsupervised checkpoint selection yielding near-oracle OOD accuracy (Zia et al., 3 Feb 2026).

2. Geometry-Based Diagnostics in Astrophysical Disc Structure

In protoplanetary disc astrophysics, geometry-based diagnostics use high-resolution molecular spectroscopy (notably CO ro-vibrational line profiles) to infer the presence and size of structural features such as inner gaps or holes. The line full width at half maximum (FWHM) as a function of rotational quantum number $J$ provides a direct kinematic probe: in a Keplerian disc, FWHM at $J$ traces orbital velocity at emission radius $r_J$,

$$\mathrm{FWHM}(J) \simeq 2 v_K(r_J) \sin i = 2 \sin i \sqrt{\frac{G M_\star}{r_J}},$$

where $i$ is inclination and $M_\star$ is stellar mass. In continuous discs, higher-$J$ lines originate closer in (smaller $r_J$), yielding an increasing FWHM with $J$. In discs with a central gap, all transitions are emitted from a fixed radius at the gap edge, producing a flat FWHM versus $J$ profile. This behavior is corroborated by both observed systems (e.g., HD 250550, Hen 2-80) and ProDiMo-based thermo-chemical modeling. The method is complementary to SED-based diagnostics and directly probes gas, bypassing dust/gas decoupling artifacts (Bertelsen et al., 2016).

3. Information-Geometric Diagnostic Frameworks for Interaction and Integration

The information geometry-based diagnostic paradigm systematically quantifies interactions in stochastic dynamical systems by measuring the Kullback–Leibler (KL) divergence between the system's probability distribution $p$ and constrained distributions $q^*$ reflecting severed interaction schemas. Important cases are:

• Mutual information: $I(X;Y)$ via projection onto independent submanifold.
• Transfer entropy: $$\mathrm{TE}(x_i \rightarrow y_j \mid X \setminus x_i)$$ via projection enforcing conditional independence.
• Geometric integrated information $\Phi_G$: minimum KL divergence when all inter-part causal links are cut, subject to partition ($\forall i, q(Y[P_i]|X) = q(Y[P_i]|X[P_i])$).

The diagnostic algorithm involves fitting $p(X,Y)$ (via empirical estimation or log-linear/Gaussian models), projecting onto relevant submanifolds using generalized iterative scaling or closed-form equations, and interpreting the divergence as the information-theoretic “cost” of the severed interactions. Classical and novel interaction measures emerge as geometric distances in the model manifold, supporting quantification and visualization of system-level integration (Oizumi et al., 2015).

4. Chemistry- and Morphology-Driven Geometry Diagnostics in Molecular Clouds

For prestellar core geometry, the diagnostic framework integrates hydrodynamic, chemical, and radiative-transfer simulations with targeted molecular line mapping. Three regimes of projected aspect ratio $R_\mathrm{proj}$ guide the diagnostic strategy:

• $R_\mathrm{proj} \leq 0.15$: Elongated objects distinguished by morphology in N$_2$H$^{+}$ maps—disk-like cores produce a characteristic "splitting" (two parallel filaments) absent in cylindrical filaments.
• $0.15 < R_\mathrm{proj} < 0.9$: Central-peakedness quantified by metric $\Delta_X$ for CN distinguishes disk-like ($\Delta_\mathrm{CN} \approx 1$, flat) from cylindrical ($\Delta_\mathrm{CN}\gg1$, peaked) and spherical structures.
• $R_\mathrm{proj} \geq 0.9$: Statistical profiles $(\Delta_X, \Delta_{\sigma^2,X})$ for OH, CO, H$_2$CO robustly classify disk-like (flat), spherical (depressed, low variance), and cylindrical (peaked, high variance) cores.

The framework is robust against variations in temperature and cosmic-ray ionization rate. Diagnostic steps proceed from dust continuum mapping to tracer selection, profile measurement, and geometry inference (Tritsis et al., 2016).

5. Geometry-Guided Diagnostics in Functional Brain Connectomics

Geometry-based diagnostic identifiability in functional connectomics employs a two-stage protocol: first, denoise subject-specific functional connectivity (FC) matrices via principal component analysis (PCA) maximizing test–retest differential identifiability; second, select and reconstruct in a diagnostic subspace defined by components that best separate diagnostic groups (e.g., Alzheimer’s disease vs. non-dementia). Each subject's FC is vectorized in $\mathbb{R}^M$ and projected onto PCA-derived connectivity modes. The diagnostic identifiability index

$$D_g = \mathrm{corr}(g, g) - \mathrm{corr}(g, \mathrm{AD})$$

and analogous distance-based metrics quantify the geometric separation between group subspaces. Reconstruction in this subspace enhances group clustering and links geometric weights to longitudinal neurocognitive trajectories via regression. The methodology generalizes to other disorders and networked modalities (Svaldi et al., 2018).

6. Geometric Frameworks for Program Testing and Fault Localization

The geometric theory for program testing encodes program executions as labeled two-dimensional diagrams (stick diagrams) where vertical axes track objects (variables, channels, threads) and horizontal axes encode atomic actions. Edges represent causality (vertical sequencing, horizontal synchronization). The framework identifies violations of geometric constraints—e.g., causal edges crossing cut boundaries, forbidden operations—in the diagram, and traces causal chains backward to minimal "blame candidate" events. This enables automated localization and visualization of synchronization errors or contract violations in concurrent code. Integration with toolchains enables interactive GUI-based navigation of execution traces and code-linked fix suggestions. The approach scales efficiently with program size and is grounded in algebraic-combinatorial models of concurrent processes (Moller et al., 2022).

7. Synthesis and Domain-General Principles

Across applications, geometry-based diagnostic frameworks share several defining features:

• Representation mapping: Construction of geometric objects (graphs, manifolds, diagrams) encoding system structure.
• Invariant extraction: Computation of metrics reflecting complexity (spectral, curvature, divergence, statistical profile) invariant to isometric or spurious transformations.
• Interpretability and robustness: Diagnostic metrics have interpretable physical or algorithmic meaning, with empirical or theoretical guarantees of their robustness to artifacts or nuisance variations.
• Protocolization: Stepwise methodologies specify input data requirements, metric computation, statistical evaluation, and interpretive criteria.
• Complementarity: Geometric diagnostics frequently outperform or complement classical, heuristic, or low-order statistical methods, particularly in unsupervised or label-free contexts.

A plausible implication is that further unification of geometric diagnostics across domains may yield theoretically principled, domain-agnostic techniques for system monitoring, anomaly detection, and active model selection. The empirically demonstrated outperformance of geometry-derived invariants over traditional metrics in several domains underlines their value as next-generation probes of high-dimensional systems.