GaugeVLM: Structuring Spatial Supervision with Measured Geometric Interventions

• Answer-only supervision does not show how far an error is from truth or whether a relation should remain consistent across viewpoints, while ordinal preferences treat errors of different magnitudes alike.The paper motivates instance-level offsets that reflect geometric error while remaining unchanged for the same allocentric error across camera views.Source: §1 · p. 2
• GaugeDPO derives preference offsets from measured deviations in controlled 3D scenes, using a bounded metric over circular direction differences and distance ratios with explicit saturation.The metric is designed to make the offset depend on geometry rather than angular origin or distance units.Source: §1 · p. 2
• Controlled object and camera interventions link observations where true relations change or remain fixed; training directly supervises canonical rankings and relates answer-odds contrasts to measured changes using view-specific scales.GaugeVLM uses the resulting Gauge-50K data for training and a probe within Constancy-Bench.Source: §1 · p. 2
• Across three backbone families, GaugeVLM improves all ten established spatial metrics over SFT; its 7B model gains 15.0 points on MSMU distance and 18.9 on QSpatial+.The introduction also reports gains on driving scenarios, including an 11.4-point improvement on SURDS depth.Source: §1 · p. 2

2.1 GEOMETRY-GROUNDED OFFSETS

• GaugeVLM maps controlled 3D perturbations to bounded, unit-invariant offsets, supplying geometric information absent from deterministic ordinal labels.Prior work includes DPO’s ordinal preferences and methods that regulate gaps or offsets (Rafailov et al., 2023; Meng et al., 2024; Gheshlaghi Azar et al., 2024; Wu et al., 2025; Amini et al., 2024).Source: §2.1 · p. 3, §2.2 · p. 3

2.2 FORMAL SETUP AND DESIGN REQUIREMENTS

• A scene relation is quantized in a fixed anchor frame, then verbalized and parsed per camera; invalid answers map to ⊥.The camera supplies the image–prompt input, and the parser maps verbalized relations back to the quantized relation.Source: §2.2 · p. 3
• The prescribed offset γ shifts the gap required at a fixed loss level, with γ = 0 recovering DPO.Source: §2.2 · p. 3
• Margins should reflect geometric error, remain camera-invariant for the same displacement, and increase before saturation; ideal predictions recover the true relation in every identifiable view.The requirements distinguish margin magnitude from cross-view constancy; agreement alone permits consistent error.Source: §2.2 · p. 3

3.1 FROM GEOMETRY TO MATCHED PREFERENCES

• The shared relation metric combines shortest-arc clock difference and capped log-distance ratio, grounding preference offsets in measured geometric error across views.Clock direction is measured relative to the anchor’s canonical front, and distance is measured in meters.Source: §3.1 · pp. 3–4

3.2 LEARNING MAGNITUDE AND CORRECTNESS ACROSS VIEWS

• Measured pairs scale preference offsets with geometric error, while direct worst-view supervision guarantees correct canonical predictions when its loss is below δ0.The direct loss compares view–candidate scores against a margin containing both a classification term and geometric error.Source: §3.2 · pp. 4–5

3.3 LINKING INTERVENTION RESPONSES TO GEOMETRY

• Intervention profiles fit a view-specific scale to answer-odds contrasts after canceling separable score components; at zero losses, predictions are correct and contrasts scale with measured changes.The prescribed proportionality concerns the sum of two gaps and does not require them to be equal.Source: §3.3 · p. 6

3.4 THE GAUGE ENGINE

• Gauge Engine constructs measured pairs and matched intervention blocks from rendered 3D scenes, fixing anchor, context, object identities, and camera set across states.Blocks require co-visibility, an identifiable anchor frame, and sufficient metric-scale evidence; Gauge-50K and Constancy-Bench cover training pairs and held-out answer-flip evaluation, respectively.Source: Figure 2 · p. 5, §3.4 · p. 7

4.1 EXPERIMENTAL SETUP

• Experiments use Qwen2.5-VL-7B-Instruct, GLM-4.1V-9B, and Pixtral-12B, with Qwen as the main backbone, and evaluate spatial understanding and driving-oriented reasoning benchmarks.Training uses spatial SFT followed by preference optimization; the SFT stage uses 25,378 QA turns.Source: §4.1 · pp. 7–8

4.2 MAIN RESULTS

• GaugeVLM improves all 30/30 established spatial-task metrics over base and SFT checkpoints across backbones, with gains also extending to driving and embodied 3D benchmarks.At 7B, it leads prior open spatial VLMs on 9/10 established metrics under the evaluated input protocol.Source: §4.2 · p. 9
• With both direct and intervention terms enabled, GaugeVLM reaches 63.0% accurate and 11.5% wrong cross-view agreement, improving by 12.5 and 6.0 percentage points over the pair-only baseline.The tolerance-based audit keeps invalid groups in all denominators and uses separate tolerances for agreement and correctness.Source: §4.2 · p. 9, Table 8 · p. 9

4.3 ABLATIONS AND CONTROLS

• Measured margins outperform the zero-offset control by 12.5 points on MSMU distance and 14.9 on QSpatial, raising Rank ρ from .119 to .279.The comparison holds direct and interaction supervision fixed; constant, shuffled, and reward-model offsets also improve over zero offsets but trail measured margins.Source: Table 3 · p. 8, §4.3 · p. 9
• At each pair-draw budget, GaugeDPO outperforms DPO on all metrics, and its 25%-budget scores exceed the full-budget DPO baseline.The sensitivity analysis shows a broad intermediate plateau, with task accuracy more sensitive to the direct-supervision weight.Source: Table 4 · p. 8, Table 5 · p. 8, §4.3 · pp. 9–10
• Adding the interaction profile raises BlockAcc from 60.0% to 70.0% and lowers mean worst m from .186 to .134, with the BlockAcc gain remaining 11.0 points on unseen magnitudes and magnitude–camera combinations.The comparison also improves profile fit over the single-gap and symmetry controls.Source: Table 6 · p. 9, Table 7 · p. 9, §4.3 · p. 10

5 CONCLUSION

• GaugeVLM combines geometric preference offsets, direct cross-view supervision, and intervention profiles linking answer-odds changes to geometric magnitude.Across three backbone families, it improves over SFT on established spatial benchmarks, with gains extending to unseen magnitudes and magnitude–camera combinations.Source: §5 · p. 10

AI USE STATEMENT

• Claude-Fable-5 and GPT-6-Astra assisted with language editing, literature retrieval and discovery, and synthetic dataset generation; the authors state that they reviewed and verified the work and take full responsibility.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

• The supplementary material details theoretical assumptions and proofs, data construction and filtering, and training and evaluation protocols.It also specifies hyperparameters, scoring and aggregation procedures, and matched controls and repeated runs for assessing objective contributions and robustness.Source: REPRODUCIBILITY STATEMENT · p. 10, APPENDIX OVERVIEW · p. 14

A DERIVATIONS

• The derivations use σ(t) to denote the logistic function and take expectations over preference triples unless otherwise stated.Source: A DERIVATIONS · p. 15

A.1 RELATION METRICS AND CONSISTENCY CERTIFICATES

• The clock–distance margin m is an invariant metric when both coordinate weights are positive, combining cyclic direction error with capped absolute log-distance error.Margins computed from serialized labels approximate geometric errors up to quantization error; extending the bound to continuous azimuth requires a separate angular quantization term.Source: §A.1 · pp. 15–16
• For shared truths, cross-view spread lower-bounds the largest prediction error; known isometries extend this certificate across changed truths, and PairAcc requires both intervention answers to match their truths.The certificate requires a known scene-geometry transformation for object interventions.Source: §A.1 · pp. 16–17

A.2 MEASURED PREFERENCE OBJECTIVE AND MIXED SUPERVISION

• Measured preferences use a geometric-error offset in a graded comparison loss; larger margins raise scalar gradient pressure, while valid ordinal-only pairs use zero offset and invalid pairs are excluded.Image-side comparisons evaluate the same response against each image’s ground truth and are interpreted as conditional discrimination.Source: §A.2 · pp. 17–18

A.3 IDENTIFIABILITY AND INTERPRETATION OF MEASURED MARGINS

• A measured margin shifts the fixed-loss separation threshold and increases scalar gradient magnitude; under equal weights and a fixed total separation budget, the shifted optimum offsets each m_i by a common constant.The linear choice preserves increments in the declared geometric-error scale without adding a nonlinear shape parameter.Source: §A.3 · pp. 19–20
• Stochastic Bradley–Terry labels identify separations h_i = m_i, but deterministic preference labels contain no margin information and allocate an equal fixed budget uniformly.The equal-allocation result assumes equal pair weights, free separations, and a linear budget.Source: §A.3 · pp. 19–20

A.4 CANONICAL PREDICTION AND CROSS-VIEW SUPERVISION

• The direct group objective promotes correct same-relation predictions across cameras, with a canonical-predictor bound linking prediction failure and geometric error to maximal cross-view disagreement.The bound assumes a shared truth and a finite candidate space containing it; canonical inference fixes the prompt, candidate domain, quantizer, verbalizer, and tie rule.Source: §A.4 · p. 20
• Zero direct loss guarantees a unique correct prediction in every view, although a correct argmax may still violate a margin.The bound remains informative at nonzero loss.Source: §A.4 · p. 20
• The margin-rescaling bound also applies to empirical training error; approximate-search guarantees need a missed-violation upper bound, and greedy or random decoding need not preserve canonical-scoring guarantees.Negative sampling or beam search alone does not provide the required search-error bound.Source: §A.4 · p. 21

A.5 INTERVENTION PROFILES AND JOINT FEASIBILITY

• The intervention loss links answer-support changes to measured geometric changes; its fitted camera-specific scale accommodates confidence differences while the direct objective ensures correct predictions.The two individual gaps may differ.Source: §A.5 · pp. 21–22
• For finite groups and blocks with consistent truths on identical canonical inputs, a finite-score conditional distribution can make all direct and interaction losses vanish simultaneously.For any finite t ≥ ξ, the construction uses ηv = 2t, including for overlapping blocks and shared inputs.Source: §A.5 · p. 22
• The single-gap control fits two independent nonnegative scales per camera and block, while the EqSim-style control enforces gap symmetry without constraining dependence on geometric magnitude.The two controls retain the same pair and direct terms.Source: §A.5 · p. 22

B.1 PAIR SELECTION AND PERTURBATION

• Object pairs are filtered by visibility, camera distance, and projected area before controlled perturbations create measured alternatives.The final graded pool is reciprocal/sign balanced, with 2,701 minor, 4,021 moderate, and 2,671 fatal perturbations.Source: §B.1 · pp. 22–23

B.2 TASK SPACES, RESPONSE GENERATION AND VERIFICATION

• The task metric covers clock direction and positive inter-object distance; the parser reads terminal relations independently of intermediate response steps.The measured training branch retains only exactly correct chosen labels and valid rejected labels with positive geometric margins.Source: §B.2 · p. 23

B.3 CORPUS COMPOSITION AND MARGIN STATISTICS

• 50,436-pair release and 30,000-pair preference pool are allocated by task family; serialized graded labels occupy distinct ranges and have positive margins.Ordering-only preferences use zero offset.Source: §B.3 · p. 23, Table 9 · p. 24, Table 10 · p. 24

B.4 DATA QUALITY AND SPLIT CHECKS

• Quality checks separately assess 1,000 candidate preference pairs and 200 camera groups, and inspect train/test duplicates among the inspected units.The candidate-pair audit checks template parseability and agreement with scene ground truth; camera groups are checked for observability properties.Source: §B.4 · p. 24

C TRAINING AND EVALUATION PROTOCOLS

• Evaluation protocols specify training and sampling budgets, scoring procedures, camera audits, intervention evaluation, and applicable input and invalid-prediction rules.View groups and intervention blocks are counted separately from preference pairs.Source: C TRAINING AND EVALUATION PROTOCOLS · p. 24

C.1 TRAINING CONFIGURATION AND IMPLEMENTATION

• SFT uses LoRA with a frozen vision encoder, followed by full-parameter preference fine-tuning of the model and vision encoder against a frozen merged-SFT reference.The reported end-to-end training times on four A100 80GB GPUs are 7.7, 8.4, and 12.2 hours for Qwen, GLM, and Pixtral, respectively.Source: §C.1 · p. 24
• 7,500 pair draws define the 25% preference-budget setting, sampled from the same fixed 30,000-pair pool rather than a quarter of its distinct examples.View-group and intervention-block exposure are tracked separately from pair draws.Source: §C.1 · p. 24
• Data-quality intervals apply to checks on 1,000 candidate pairs and 200 camera groups, while duplicate rates cover inspected train/test units.The table distinguishes the two checked sets rather than pooling their pass rates.Source: Table 11 · p. 25
• Hyperparameters are selected on scenes disjoint from training and testing, and matched comparisons share SFT initialization, reference model, data, update budget, and seed.Across seeds, objectives use aligned pair/block schedules and stochastic-operation streams where applicable; removing a loss does not alter exposure of remaining branches.Source: §C.1 · p. 25
• The canonical candidate space contains 1,200 clock–distance candidates, with out-of-domain relations counted separately and native benchmarks using their own scoring protocols.Sequence scores include answer tokens through termination without length normalization.Source: §C.1 · p. 25

• Ratio accuracy uses r× < 1.25 for MSMU and r× < 2 for QSpatial, while Fig. 3's ±25% band uses relative error ≤0.25.The corresponding ratio intervals are (0.8, 1.25) and (0.5, 2), respectively; the ±25% band is [0.75, 1.25].Source: §C.2 · p. 25
• Spearman measures rank agreement but is undefined for constant predictions, while VisC. measures agreement under brightness and contrast changes with geometry and pose fixed.Cross-camera correctness is evaluated separately.Source: §C.2 · p. 25
• GaugeVLM-7B answers both states correctly in 136 of 205 relocation pairs, yielding 66.3% PairAcc.Each pair shares the scene, camera, and anchor while the target move reverses the left/right relation.Source: §C.2 · p. 25
• Each relocation pair contributes one left and one right label; missing or invalid answers count as incorrect, and independent uniform binary predictions yield 25% expected PairAcc.A random first prediction followed by a forced reversal yields 50%.Source: §C.2 · p. 26

C.3 FIXED-RELATION CAMERA EVALUATION

• Fixed-relation camera groups share an anchor frame and allocentric ground truth, with questions asking for clock direction and positive distance in that common frame.The evaluation compares answers expressed in the common frame.Source: §C.3 · p. 26
• Agreement requires pairwise direction and distance spreads within their thresholds, while invalid or incomplete groups score zero and remain in the denominator.Correctness tolerances are independent of agreement tolerances; exact canonical correctness implies agreement.Source: §C.3 · p. 26
• Fallback conventions define targets but do not establish that they are identifiable from inputs; six incomplete groups remain in the denominator and score invalid for every model.Fixed-truth camera groups complement fixed-camera object-relocation pairs, which test whether answers follow a changed relation.Source: §C.3 · p. 27

C.4 BLOCK GENERALIZATION AND MATCHED CONTROLS

• The block evaluation uses held-out scenes and three equally sized settings of 200 blocks: familiar factors, unseen magnitudes, and unseen magnitude–camera combinations.Unseen magnitudes are held out within the training range, while combination splits hold out joint pairs but retain each factor in training.Source: §C.4 · p. 27
• Matched supervision variants share scenes, inputs, updates, selection data, and evaluation blocks; direct computation uses scene-state labels without estimator fitting.Supplementary experiments examine transfer, loss-weight sensitivity, input and seed robustness, and baseline-input and reward-estimation effects.Source: §C.4 · pp. 27–28, D ADDITIONAL EXPERIMENTAL RESULTS · p. 28

D.1 ADDITIONAL EVALUATION

• 18.9 points on QSpatial+ and 8.7 points on SPAR-Bench overall are among the gains over SFT across additional spatial benchmarks.GaugeVLM also gains 5.4 points on SPBench-SI size estimation and reaches 60.5% on Omni3D-Bench MC, a 5.8-point gain; benchmark metrics are not aggregated.Source: §D.1 · p. 28, Table 15 · p. 29
• Relative to SFT, GaugeVLM gains 14.1 points on SPAR-Bench dist oc and 16.5 points on dist oo; RoboSpatial configuration accuracy rises from 74.8% to 78.1%.RefSpatial-Bench overall rises from 16.0 to 17.0, with a smaller improvement on spatial referring.Source: §D.1 · p. 29

D.2 CAMERA OUTCOMES AND SHORTCUT CONTROLS

• GaugeVLM reaches 69.5% camera group accuracy and 63.0% accurate agreement, versus 50.0% and 40.0% for SFT; incorrect agreement falls from 20.0% to 11.5%.Improved agreement is accompanied by improved correctness; the six incomplete groups contribute a 3.0% invalid rate for every model.Source: §D.2 · p. 29
• Masking images or using text alone lowers PairAcc. from 66.3% to 25.9% and 26.8%, while option permutation or state reversal changes it by less than one point.Predictions are mapped back to semantic labels before scoring.Source: Table 17 · p. 30, §D.2 · p. 30
• Combining reciprocal/sign balancing with coherent reasoning steps lowers text-only preference accuracy from 88.0% to 54.5% while raising accurate camera agreement from 54.0% to 63.0%.The combined construction reduces detectable textual shortcuts while improving spatial performance.Source: §D.2 · p. 30

D.3 TRAINING-SEED ROBUSTNESS AND PAIRED UNCERTAINTY

• Matched preference-stage runs used five seeds from the same SFT checkpoint, fixed data splits and evaluation manifests, and aligned schedules across seven objectives.Prompts, preprocessing, decoding, candidate search, and invalid-output rules were also held constant across runs.Source: §D.3 · p. 30
• All reported full-GaugeDPO gains over controls were positive across five seeds, and joint seed-and-scene intervals remained above zero.The paired bootstrap resampled both seed indices and test-scene clusters.Source: §D.3 · p. 31, Table 20 · p. 32
• Across three backbones, five independent SFT–preference replications showed consistent gains when variation in both training stages was included.This involved 15 SFT and 30 preference runs across the three backbones.Source: §D.3 · p. 31, Table 21 · p. 32

D.4 BASELINE INPUTS AND REWARD-ESTIMATOR COMPARISONS

• 62.5% MSMU distance and 64.4% QSpatial+ let GaugeVLM exceed both native-input baselines; native cues raise SpatialReasoner MSMU distance from 25.0% to 30.0% and SpatialRGPT to 32.5%.Source: §D.4 · p. 32, Table 22 · p. 33
• The image-conditioned regressor reaches 0.087 margin MAE and 0.72 rank correlation against serialized targets; direct scoring costs 0.02 GPU-hours and algebraically matches the target formula.That algebraic agreement does not measure error against continuous physical geometry.Source: §D.4 · p. 32, Table 23 · p. 33