Wasserstein-Barycentric Interaction Fields for Spatial Factor Models: Evidence from Language-Model Representations

Marcus Gawronsky; Chun-Sung Huang

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Wasserstein-Barycentric Interaction Fields for Spatial Factor Models: Evidence from Language-Model Representations

Marcus Gawronsky, Chun-Sung Huang

arXiv:2608.29669v1Aug 30, 2026q-fin.ST

TL;DR

Spatial return models typically take the interaction matrix as given, leaving its economic construction and the meaning of feedback unresolved. This paper builds a bandwidth-free target-anchored Wasserstein field from firms’ language-model article distributions and embeds it in a quadratic exposure-adjustment model. The frozen field improves conditional fit and yields a 3.46 adjustment index, while joint estimation assigns indices of 2.33 and 0.86 to barycentric and news co-mention fields.

Problem
Spatial return models condition on supplied interaction matrices, leaving where the interaction field comes from and what ρ measures beyond conditional return dependence unresolved.
Method
The paper represents firms by information distributions, constructs a target-anchored Wasserstein barycentric field, and maps peer feedback into exposure adjustment through a quadratic closure.
Results
The frozen barycentric field yields a 3.46 adjustment index and higher conditional fit than equal-weighted peer support or RBF weighting, while joint indices are 2.33 for barycentric and 0.86 for news co-mention fields.
Takeaways & Limitations
The barycentric field provides a distinct, incrementally informative peer structure, and joint estimation reallocates model-implied adjustment across fields rather than materially increasing total feedback.
Takeaways & Limitations
The estimates are conditional working-model quantities: freezing fields prevents same-sample feedback but does not establish text exogeneity, causal peer effects, or structural adjustment costs.

Abstract

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Spatial return models take the interaction matrix as given and leave feedback uninterpreted. We construct a bandwidth-free field from firms' language-model article embedding distributions using target-anchored Wasserstein barycentric reconstruction. A quadratic exposure-adjustment problem maps feedback into a peer-misalignment penalty ratio. For 52 firms, the field, frozen from 2018-2022 news, yields a 2023-2026 penalty ratio of 3.46 (95% interval [2.89, 4.17]) and higher conditional quasi-likelihood than equal-weighted peer support or RBF weighting of the same distances. Joint penalty ratios for the barycentric and news co-mention fields are 2.33 and 0.86 with boundary calibrated tests which reject both exclusions.

Introduction

The paper replaces researcher-supplied interaction matrices with a target-anchored Wasserstein field built from firms’ text distributions, then interprets spatial feedback through exposure adjustment. In a 52-firm application, the frozen field produces a 2023–2026 adjustment index of 3.46 and improves conditional fit when combined with a news field.

Spatial models usually condition on a researcher-supplied interaction matrix, leaving the field’s origin and the meaning of ρ outside the model.
Firms are represented by probability measures over embedding positions, preserving spread, multimodality, and internal composition beyond centroid-based representations.
Target-anchored Wasserstein reconstruction combines aligned peer distributions with nonnegative unit-sum weights to approximate each fixed target without a conventional bandwidth.
The quadratic adjustment problem maps peer misalignment into λ, the penalty on peer misalignment relative to departing from stand-alone exposure, with λ = ρ/(1 −ρ).
3.46 is the working-model adjustment index for 52 firms using a field frozen from 2018–2022 text and evaluated on 885 aligned trading days from 2023 through incomplete 2026.
Joint estimation gives λB = 2.33 and λN = 0.86, with each field improving conditional fit once the other is included.

1 Related Literature

Related work commonly represents firms as points, links, or first-order text features; this paper instead uses distribution-valued information and target-specific reconstruction weights. The resulting multi-field framework treats candidate interaction fields as distinct adjustment channels and freezes text-derived fields before return evaluation.

Spatial asset-pricing models use supplied matrices to organize local dependence, often with point-valued locations and inverse geographic distance weights.
Network games derive propagation conditional on a given graph, whereas this paper applies adjustment one layer below returns to peer-adjusted factor exposures.
Text-finance research often maps information into attention measures, textual factors, embeddings, or learned network exposures rather than distribution-valued firm objects.
The paper represents each firm by a probability distribution that preserves dispersion and multimodality, with point locations recovered as a special case.
Pairwise quadratic Wasserstein transport measures least aggregate squared displacement in embedding space, not literal monetary transport.
Target-anchored reconstruction estimates target-specific spanning weights rather than converting pairwise distances directly through a kernel or bandwidth.
Freezing text-derived fields before the return window removes same-sample feedback but does not identify causal peer effects under omitted factors affecting text and returns.
The multi-field model assigns separate coefficients to barycentric and news-link fields, asking how adjustment divides across channels rather than which single W wins.

2 Economic Environment and Stand-Alone Exposures

The economic environment distinguishes information distributions, stand-alone exposures, and peer-adjusted exposures linked to returns through a maintained factor bridge. Peer adjustment is modeled as a reduced-form transformation of firm-specific exposure rather than as a direct property of observed text.

Each firm has a characteristic law Ci over information positions, while its stand-alone exposure ξi is the factor-loading vector implied by its own information.
The quadratic criterion represents operational, financing, or portfolio reconfiguration costs as an as-if adjustment problem, without requiring firms literally to choose factor loadings each period.
A measurable transmission map T converts information draws and idiosyncratic transmission shocks into stand-alone exposures.
The return bridge uses peer-adjusted exposure Bi, while the return panel does not identify T, its latent inputs, or cross-firm coupling of stand-alone exposures.
Peer adjustment uses an admissible matrix W with nonnegative off-diagonal weights whose rows sum to one.
The peer-weighted exposure remains in the same factor-exposure units as Bi, while W determines how firms weight other firms.

3 Exposure Adjustment and Spatial Closure

A quadratic exposure-adjustment objective yields a spatial closure in which peer-adjusted exposure balances stand-alone exposure against admissible peer averages. The framework extends naturally to two fields, allowing total adjustment to be allocated between barycentric and news-based channels.

3 Exposure Adjustment and Spatial Closure: The quadratic criterion balances fidelity to stand-alone exposure against alignment with peer-weighted exposure, yielding rather than assuming a spatial autoregression in exposures.
3 Exposure Adjustment and Spatial Closure: B = ρ WB + (1 −ρ) ξ, with ρ = λ/(1 + λ), makes peer-adjusted exposure a convex balance between peer average and stand-alone exposure.
3 Exposure Adjustment and Spatial Closure: The adjustment index λ is the penalty on peer misalignment relative to the penalty on departing from stand-alone exposure, and 0 ≤ρ < 1 corresponds to nonnegative adjustment intensity.
3 Exposure Adjustment and Spatial Closure: Under ∥ρW∥<1, the equilibrium is unique and iterated peer feedback remains anchored by stand-alone exposures.
3 Exposure Adjustment and Spatial Closure: Two admissible fields combine through λB and λN, with ρB and ρN allocating total model-implied adjustment across channels.
3 Exposure Adjustment and Spatial Closure: The mixture of two admissible row-stochastic fields remains admissible, so one-field closure results apply to the joint-field problem.
3 Exposure Adjustment and Spatial Closure: The barycentric field is constructed by aligning candidate firms to a fixed target and using distributional spanning weights, without a conventional kernel bandwidth.

4 Constructing the Barycentric Interaction Field

The paper constructs a target-anchored barycentric interaction field by aligning each target with candidate peers and jointly fitting simplex reconstruction weights. The resulting directed field supplies admissible peer averages without conventional bandwidth weighting.

Target-anchored reconstruction: A target firm’s article distribution is represented by jointly aligned peer clouds rather than by pairwise distances alone.Transport establishes article-level correspondence, after which one common set of peer weights represents the target footprint.
Target-anchored reconstruction: 0.60 and 0.40 are barycentric coordinates for peers B and C in the example row W ♭A· = (0, 0.60, 0.40).The weights describe coordinates on aligned peer positions, not probabilities of drawing articles from either firm.
Two-stage construction: Target-anchored reconstruction fixes pairwise transport assignments, then minimizes joint squared reconstruction loss over nonnegative unit-sum peer weights.Self-links are excluded and every eligible other firm enters the candidate set.
Interpretation: The coefficients are approximate distributional spanning weights within the investable universe, not causal influence, tradable replicating portfolios, or literal arbitrage relations.A small residual supports approximate semantic substitutability only within the maintained representation, metric, and candidate universe.
Two-stage construction: The fitted rows are directed because target-specific alignments and reconstruction objectives make W ♭ generally asymmetric.A large W ♭ij need not imply a large W ♭ji because direction records reconstruction relevance for the target.
Comparators: The field avoids a conventional kernel-bandwidth choice but still depends on researcher-selected representation, candidate set, and reconstruction rules.The RBF comparator makes the alternative bandwidth choice explicit through h set to the median off-diagonal squared Wasserstein distance.

5 Cross-Sectional Wasserstein Dispersion and Spatial Attenuation

The paper uses an unrestricted Wasserstein barycentre to measure cross-sectional distributional dispersion, then derives how peer adjustment attenuates exposure dispersion under maintained transfer and stationarity conditions. The resulting bounds preserve a coefficient-dependent share of stand-alone dispersion.

Cross-sectional dispersion: The unrestricted Wasserstein barycentre summarizes cross-sectional heterogeneity through a free centre distribution Q, unlike the fixed-target field construction.Its dispersion functional reduces to the familiar pairwise Wasserstein distance for two firms.
Transfer restriction: Characteristic dispersion supplies a floor for stand-alone exposure dispersion, while peer adjustment determines how much of that floor survives.This is a supporting closure result for the field, not a second field construction or an empirical calibration of the carrier restrictions.
Transfer restriction: The transfer restriction uses a common carrier and firm-specific slack to relate embedding-space separation to exposure-space dispersion.The carrier, scale factor, and slack terms are maintained inputs rather than estimated objects.
Transfer restriction: At zero slack, the exposure-dispersion bound becomes Dq(P1, . . . , PN) ≥ L−2Dq(C1, . . . , CN).Observable separation is informative when carrier-adjusted magnitude exceeds root-mean-square transmission slack.
Spatial attenuation: Peer adjustment cannot increase cross-sectional dispersion, and at least {(1 −ρ)/(1 + ρ)}2 of stand-alone dispersion remains.The result assumes a nonnegative row-stochastic field, a strictly positive stationary distribution, and 0 ≤ρ < 1.
Spatial attenuation: The retained-share expression is a structural bound rather than the exact attenuation or a calibration of the carrier, scale, or slack parameters.At ρ = 0, peer adjustment leaves dispersion unchanged; the bound need not be sharp for the fitted operator.

6 Return Bridge, Empirical Design, and Data

The empirical design tests whether a predetermined barycentric field organizes conditional return dependence and adds information beyond alternative distance-based or equal-weighted constructions. It builds the field from frozen language-model article distributions, bridges latent exposure adjustment to returns through scalar projection and a maintained return restriction, and evaluates the resulting working-model estimates on a survivor-conditioned 52-firm panel.

Empirical questions: The analysis asks whether the barycentric field organizes conditional return dependence and adds information beyond RBF proximity, equal weighting, and a persistent news co-mention field.These questions distinguish field existence, reconstruction beyond pairwise distance decay, and information beside another network.
Return bridge: Scalar projection preserves the spatial operator and structural coefficient for systematic returns, but the observable equation is estimated under a spherical working quasi-likelihood.The return bridge connects latent peer-adjusted exposures to centered excess returns, while QMLE targets a working-model coefficient.
Estimand and identification: The working-model index is λ̂ = ρ̂/(1−ρ̂), whereas QMLE does not separately identify structural adjustment costs, latent exposures, or a causal peer effect.The adjustment interpretation requires the maintained return bridge and quadratic closure; without that restriction, the estimate describes conditional spatial dependence for the specified geometry and likelihood.
Data and timing: The evaluation uses a frozen 52-firm intersection over 885 common dates, selected from a Nasdaq-100-based frame with continuous 2018–2022 article coverage.The panel is survivor-conditioned and not representative of listed firms; balanced article clouds also remove information about true news volume.
Representation construction: Each firm is represented by a balanced empirical distribution of normalized language-model article embeddings rather than a single averaged point.The construction uses the fixed, full-width Qwen3-Embedding-8B representation and treats each firm’s article cloud as an equal-mass probability distribution.
Design sensitivity: Representation-width and encoder-vintage contrasts are paired, multiplicity-corrected sensitivity diagnostics rather than representation-selection tests.The post-period encoder vintage is explicitly a negative control, and the vintage comparisons remain descriptive because encoder capacity differs materially.
Field interpretation: The field’s illustrative diagnostics describe sourcing breadth and concentration, not economic substitutability or evidence that the field organizes returns.The displayed entries are only the five largest coefficients in each row, while diagnostics use the complete operator.

7 Results

The frozen barycentric field shows substantial conditional return dependence, with distinct peer selection and weighting from the persistent news-link field. Joint estimation finds both channels informative, while annual refits support persistence but not equality of magnitudes.

7.1 Conditional return dependence and the reconstruction mechanism: 3.46 is the pooled barycentric adjustment index, with 95 per cent interval [2.89, 4.17].The equivalent feedback coefficient is ˆρ = 0.776, and the benchmark λ = 0 removes conditional peer alignment.
7.1 Conditional return dependence and the reconstruction mechanism: The barycentric field retains higher conditional fit than equal weighting of its selected peers or RBF proximity using the same Wasserstein distances.The equal-active-support comparator keeps the selected peers but replaces distributional spanning weights with equal weights.
7.2 Are the barycentric and news-link fields distinct?: 38.8 active barycentric peers versus 27.1 co-mention peers, with overlap 0.579 against matched-random 0.531, establishes above-chance overlap rather than equivalent peer sets.The observed excess is about five percentage points and has p = 0.001.
7.2 Are the barycentric and news-link fields distinct?: 0.669 off-diagonal weight correlation falls to 0.467 for rank correlation on shared edges averaging 22.3 per row.The lower shared-edge rank correlation indicates only partial agreement in cardinal weighting.
7.2 Are the barycentric and news-link fields distinct?: 0.898 correlation between induced peer-return series exceeds the matched-random floor of 0.700, showing greater post-application similarity than similarity in peer maps.The observed excess has p = 0.001, while per-firm correlations range from 0.553 to 0.973.
7.3 Does the barycentric field remain informative beside a persistent co-mention field?: The joint feedback total is 0.761 versus 0.776 for the barycentric-only boundary, with joint 95 per cent interval [0.729, 0.793].The estimates reallocate working-model feedback across channels rather than materially increasing its total.
7.3 Does the barycentric field remain informative beside a persistent co-mention field?: 2.33 and 0.86 are the joint adjustment indices for the barycentric and news-link channels, with intervals [2.00, 2.70] and [0.61, 1.18].Both intervals exclude zero, and boundary-calibrated QLR tests reject ρN = 0 and ρB = 0 with p = 0.0005.
7.4 Descriptive persistence under a frozen field: Annual adjustment indices span 2.89–3.94, supporting descriptive persistence but not equality of magnitudes.The 2026 estimate uses only 133 dates, so its precision is not directly comparable with a complete year.

8 Discussion and Limitations

The paper reports that its barycentric field produces bounded, conditional evidence of peer-adjustment fit, while emphasizing identification, measurement, multi-field, and external-validity limits.

Discussion: Joint estimation reallocates model-implied adjustment between the barycentric and news co-mention fields while leaving total adjustment close to its single-field level.The two fields select largely different peers but generate correlated peer-return series.
Identification: The estimates measure conditional field fit rather than causal peer effects because text-based information positions and returns may share determinants.Evaluation returns are excluded from field construction, but text is not assumed exogenous.
Measurement and design: The field depends on a fixed representation, ground metric, balanced article clouds, target-specific alignment, and simplex restrictions.Balancing improves comparability but removes news volume as an information source.
Measurement and design: Cross-geometry likelihood comparisons are evidence about competing measurement designs rather than geometry-free rankings.Weighted or unbalanced clouds, alternative ground metrics, and held-out windows are proposed as robustness checks.
Measurement and multi-field conclusions: The dispersion and multi-field conclusions remain conditional because key transmission quantities are latent and the two fields are highly correlated.The peer-return correlation is 0.898, while the bootstrap correlation between the two estimated coefficients is −0.683.
External validity: External validity is limited by a survivor-conditioned sample of 52 large-cap firms and a time design using 2018–2022 text with evaluation through 15 July 2026.A broader or changing firm universe could alter feasible peer sets and stationary weighting.

9 Conclusion

The conclusion presents a distribution-based interaction field that gives spatial feedback an exposure-adjustment interpretation and improves conditional return dependence modeling within the working setup.

Conclusion: Firms are represented by distributions of information positions, and target-anchored Wasserstein reconstruction converts aligned peer positions into a directed field W ♭.The field has nonnegative unit-sum rows and a zero diagonal.
Conclusion: ρ = λ/(1 + λ) interprets spatial feedback as the relative intensity of peer alignment within the quadratic exposure-adjustment closure.Peer-adjusted exposure balances stand-alone exposure against the peer average.
Conclusion: The barycentric field delivers stronger conditional fit than pairwise RBF proximity using the same Wasserstein distances and remains incrementally informative alongside persistent news co-mention links.The field is constructed without using evaluation returns and selects a distinct peer structure.
Conclusion: The method bridges distribution-valued representation and spatial econometrics by converting target-specific joint representability in Wasserstein space into an admissible field.The estimates remain conditional working-model quantities rather than causal peer effects or geometry-invariant structural parameters.

Data availability statement

The paper identifies its news, market-data, and embedding-model sources while noting that provider terms constrain redistribution.

Data availability statement: Firm news records come from Nasdaq’s public ticker-indexed news archive, and adjusted-price histories come from Yahoo Finance through the pinned yfinance client.Article vectors use qwen/qwen3-embedding-8b through OpenRouter, documented by Zhang et al. (2025).

Funding

The supplied material states that the research received no financial support and describes maintained matrix conditions supporting the appendix’s peer-averaging and boundary results.

Funding: The authors received no financial support for the research, authorship, or publication of the article.
Appendix: The appendix examines stable peer averaging and proximity to the strong-interaction boundary using residual, Dobrushin, finite-ρ, and rank-one diagnostics.Its matrix implications apply to row-stochastic interaction matrices satisfying the stated conditions.
Maintained conditions: The main argument requires a nonnegative row-stochastic interaction matrix, a strictly positive stationary probability vector, and a stable coefficient ρ.These conditions define directed peer averages, stationary weighting, and a well-defined spatial multiplier.
Application: In the application, the fitted matrix is the barycentric interaction field W ♭, on which the spectral results operate.
Perron limit: A geometric Perron certificate implies W^k → P, so repeated peer averaging converges to the π-weighted cross-sectional mean.The stationary weights are invariant under one application of the operator, with π⊤W = π⊤.

A.3 Fitted operator and finite-ρ diagnostics

The fitted operator satisfies the required stochastic and stability conditions, supporting the peer-average and equilibrium mappings. Its finite-ρ behavior differs materially from the rank-one boundary, which is therefore a theoretical endpoint rather than an empirical approximation.

Fitted operator: The fitted 52 × 52 operator is row stochastic, nonnegative, and has maximum absolute row-sum norm one.Its row-sum residual is 0.000 000 00.
Fitted operator: The pooled estimate ˆρ = 0.776 satisfies the stability condition, making the spatial multiplier well defined.The stationary distribution has minimum mass 0.0105 and stationarity residual 3.38×10−13.
Finite-ρ diagnostics: The fitted operator has a geometric Perron certificate: its second power is strictly positive and its Dobrushin coefficient is 0.862.The minimum entry of the second power is 0.001 630, so directed peer profiles converge.
Finite-ρ diagnostics: At the pooled fitted coefficient, the exact normalized-resolvent error is 0.675, versus bounds of 1.353 and 3.786.The nonzero exact error shows that the rank-one limit is not a close approximation to the fitted normalized multiplier.
Rank-one boundary: As ρ ↑1, the normalized spatial multiplier converges to the stationary projection, and the rescaled spatial covariance becomes rank one.The boundary removes all cross-sectional directions except the common one.

B Supporting Interaction-Field Results and Implementation

Supporting analyses assess whether the interaction-field results depend on representation, weighting, annual sample, and implementation choices. They find descriptive persistence and formal algebraic compatibility, while retaining explicit limits on equivalence, constancy, and economic interpretation.

Supporting questions: The appendix tests whether joint reconstruction remains distinct from pairwise proximity and whether results survive alternative text representations.It also examines annual recurrence and verifies estimation and operator conditions.
Representation sensitivity: The representation sensitivities show zero-containing intervals for moderate Qwen3-8B truncation, while further compression lowers fitted barycentric feedback and its gap relative to RBF.The full-width 4B comparison does not isolate capacity, and BGE-large is an external sensitivity check.
Representation sensitivity: All three EttaX pairs satisfy the joint equivalence rule, but the post-window V3 encoder remains a negative control rather than a point-in-time specification.Log-likelihood is descriptive throughout.
Annual persistence: Both channel coefficients remain positive annually, with the barycentric estimate larger throughout; the incomplete 2026 period is not directly comparable with a complete year.The annual exercise documents descriptive persistence, not structural constancy or a formal equality test.
Field admissibility: The target-anchored reconstruction produces nonnegative, row-stochastic, zero-diagonal operators compatible with the SAR resolvent when |ρ| < 1.These properties follow from the leave-one-out simplex construction and imply ∥W ♭∥≤1.
Formal verification: Formal verification machine-checks the finite operator contract and algebraic results, but not empirical identification, representation validity, or causal interpretation.The finite-ρ bound is an algebraic bound, not a statistical confidence interval.