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This paper argues that the sequence of intermediate states a vision encoder computes for each image — the representation trajectory — carries reliability evidence that the final representation discards, and that an in-distribution-only 'tra

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Recording how an image's representation evolves block-by-block, relative to learned class routes, improves OOD detection in 131/152 comparisons and clean classification in 71/72 model–dataset cases.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection OOD trajectory surprise is a solid, well-controlled result; clean classification gains are real but likely generic ensembling more than trajectory-specific. the 2 major comments →

arxiv 2607.26565 v2 pith:VYDZ5CKG submitted 2026-07-29 cs.CV

Representation Trajectories Matters: Complementary Evidence for OOD Detection and Image Classification

Ignacio M. De la Jara , Cristian Rodriguez-Opazo , Hamed Damirchi , Stephen Gould , Damith Ranasinghe This is my paper
classification cs.CV
keywords representation trajectoriesout-of-distribution detectionintermediate representationsclass routestransition surpriseimage classificationdistribution shiftfrozen encoders
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the route a frozen vision encoder takes to produce a final representation is itself a reliability signal. For each image, the sequence of block states forms a trajectory; per-class routes are estimated from in-distribution data, and a sample's deviations from its assigned route are summarised by an ID-only transition-surprise score. Added as a small correction to a strong final-state density score, this score lowers FPR95 in 131 of 152 non-saturated benchmark comparisons, with a mean reduction of 4.33 points alongside Mahalanobis++ and positive results on every CIFAR comparison. The same trajectory evidence, read through linear probes on native block updates and fused with a final-state probe, improves clean classification in 71 of 72 model–dataset cases by 4.41 accuracy points on average. The paper concludes that final representations carry content, confidence carries uncertainty, and trajectories indicate whether the computation followed a familiar class route.

Core claim

The central claim is that computation paths contain evidence not present in the final representation. The paper separates each update into class-coherent transport and sample-specific innovation, models residual transitions with a shared depth-conditioned MLP, and defines a transition-surprise score D(x) as the summed, per-component-scaled prediction error over depth. Conditioning on the final state leaves a non-negative conditional-divergence term, so any route evidence must appear after conditioning on the endpoint; the experiments show it does, and that it is complementary rather than redundant with strong final-state detectors: path surprise still helps after Relative Mahalanobis is alre

What carries the argument

The central object is the representation trajectory τ(x) = (z1(x), …, zL(x)) together with the class route µ^c_l = E[z_l(X) | Y=c]. Subtracting the class route yields residual trajectories r_l(x) whose cross-depth continuity is validated by paired and identity-shuffled controls. The decisive mechanism is the transition-surprise score D(x): a shared MLP with a depth embedding predicts the next residual state from the current one; per-component error scales are estimated on in-distribution data; and D sums squared prediction errors across depth and dimensions. After ID-only standardisation and residualisation, a fixed small fraction of D is added to the final-state score, which lets the detect

Load-bearing premise

The trajectory-surprise score assumes a one-step Markov model: a finite-capacity MLP, conditioned on depth, can predict the next class-residual state from the current one, with per-component error scales estimated on in-distribution data; if residual transitions are higher-order or those Gaussian scales misestimate out-of-distribution transitions, the score adds noise rather than signal.

What would settle it

Run the same ID-only protocol on a frozen encoder whose layer order has been randomly permuted at test time, or pair each residual state with a different image's previous state to break the identity link; if the trajectory-surprise score still reduces FPR95 by a similar amount, the claimed sample-specific cross-depth continuity is not the operative mechanism, while a vanishing gain would support it.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If correct, any frozen vision encoder exposes a reliability signal that can be extracted without retraining the backbone, using only labelled in-distribution data and about 0.3 ms extra latency per image.
  • OOD gains are largest on visually disruptive and semantically far shifts, and remain positive on near-OOD for Mahalanobis++, Relative Mahalanobis, and kNN; the residual limitation concentrates in the hardest ImageNet near splits.
  • Linear probes on native block updates recover class evidence that the final state compresses, with the largest clean-classification gains on fine-grained datasets and for plain Vision Transformers.
  • There is no universal best layer or monotonic refinement curve: movement, relational deformation, class-evidence gain, and residual share peak at different depths per architecture, so trajectory analysis must be performed in each model's native block sequence.
  • Under label-preserving shift, trajectory fusion improves 169 of 180 CIFAR-100-C conditions and most PACS and Office-Home folds, though the benefit depends on architecture and corruption type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the trajectory signal is a genuine route-typicality measure, then a test-time estimator with access to the full path should be able to flag images whose intermediate states are unusual even when their final embeddings are typical; one could test this directly by matching shifted and in-distribution samples for final-state distance and comparing their transition surprise.
  • The paper's global image-level scope suggests a natural extension to patch-level or spatial trajectories, where per-location route surprise could localise anomalous regions rather than only flag the whole image.
  • Because the fusion rule residualises path evidence against the final-state score, the same transition-surprise score may complement other families of final-state detectors, not just the density-based scorers tested here.
  • If architecture-specific depth schedules are stable across scales and datasets, trajectory fingerprints could serve as a diagnostic for comparing encoders without probing downstream task performance.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies whether the sequence of intermediate representations (the 'trajectory') of an image through a frozen vision encoder contains evidence beyond the final representation. It decomposes layer-to-layer updates into class-coherent transport and sample-specific innovation, then uses the innovation residual for two tasks: an ID-only transition-surprise score fused with final-state OOD detectors, and a fusion of linear probes on native updates with a final-state probe for classification. Across OpenOOD v1.5 and 38 checkpoints the surprise score lowers FPR95 in 131/152 non-saturated backbone–benchmark pairs; on 12 clean datasets the update-probe fusion improves 71/72 model–dataset cells by +4.41 points on average; shifted-data gains are more mixed. Extensive controls (identity shuffling, random-score partners, sequential complementarity, fixed-weight audits, exact checkpoint identifiers) support the OOD side.

Significance. If the results hold, the paper provides a broadly useful and cheap reliability signal: it shows that the recorded computation path contains information that final representations compress away, and that this information can be exploited without OOD data. The empirical protocol is unusually strong for a vision paper: 42 exact public checkpoints, ID-only fitting, a fixed a priori fusion weight, random-noise and matched-state controls, full appendix tables, executable pseudocode, and frank limitation statements. The OOD-detection results are convincing. The clean-recognition contribution, however, is currently not isolated from generic ensembling, and the update-specific contribution is small for several architectures; this is the main gap.

major comments (2)
  1. [Section 5.3, Table 1] The headline clean-recognition result (+4.41 points, 71/72 cases) compares the path fusion against a single final-state linear probe, not against the matched final-state ensemble that Section 4.2 promises as the control for generic ensembling. That control is reported for PACS/Office-Home and CIFAR-100-C (Appendices K and O) but not for the clean 72-cell table. Without Δctl for Table 1, the recognition gains cannot be attributed to the trajectory/update representation: independently seeded final-state probes are well known to give ensemble gains, and the state-probe ablation does not separate ensembling from representation. Please add the matched final-state ensemble column to Table 1 (or an equivalent appendix table) and restate the recognition claim based on the control-adjusted gains.
  2. [Section 5.3, state-probe ablation] The ablation replacing update probes with state probes shows that Swin and ConvNeXt remain within 0.25 points, and only four of six backbones gain more than 0.63 points from representing updates explicitly. This substantially narrows what the recognition experiments establish about trajectories as transitions. The abstract's 'Frozen update probes improve 71/72 clean model–dataset cases' is technically true, but the mechanism claim should be softened to reflect that for hierarchical/CNN families the improvement is essentially obtainable from intermediate states, and the unique update/transition contribution is small. If the authors intend 'trajectory' to include states, this should be stated explicitly.
minor comments (4)
  1. [Equation (5)] Typo: 'It does not does not assert' should read 'It does not assert'.
  2. [Table 1 / Appendix I] The clean-recognition gains are reported as three-seed means without standard deviations or error bars. Given that some cells are small (+0.21 to +0.65), please provide seed-level spread or paired confidence intervals so the reader can judge which gains are beyond noise.
  3. [Figure P2 / Section 5.2] The term 'non-saturated' is used for the 152-count denominator, but its definition (MNIST FPR95 near zero) appears only in the appendix. Please define it in the main text and state explicitly how the 38 MNIST cases (28 exact ties, 10 tiny gains) are handled in the headline count.
  4. [Section 4.1, Eq. (7)] The residualization of D against S0 is linear. A nonlinear dependence of D on the final-state score could remain; the random-score control rules out the fusion rule itself, but not this specific nonlinearity. Consider adding a nonlinear residualization control or discussing this limitation.

Circularity Check

0 steps flagged

No load-bearing circularity: the trajectory score and probe fusion are fitted on ID data only, validated on held-out OpenOOD/test splits with explicit controls, and the paper's identities are labeled as non-assertive.

full rationale

The paper's derivation chain contains no step where a prediction reduces to its inputs by construction. Section 3.2 explicitly disclaims the KL decomposition as mere localization: "This is an identity: the residual term is non-negative by construction... It does not assert that a computation path carries additional evidence; it specifies where such evidence would have to reside." Equation 4 is likewise a definitional decomposition of the update into class transport and innovation; it is not used to force any empirical gain. The OOD score (Eq. 6) and fusion rule (Eq. 7) are fitted exclusively on ID data: "The method is fitted using labelled ID data only with three random seeds and no OOD exposure" (Sec. 5.1), and "The coefficient 0.3 is fixed a priori and never adjusted per checkpoint, benchmark, or split" (Sec. 4.1). Residualizing D against S0 on ID data leaves an orthogonal component whose OOD value is genuinely uncertain, and the paper's controls show the combination rule does not mechanically help: random noise improves no cell (Appendix H, "Random noise improves no cell and worsens both benchmarks, ruling out the combination rule itself as the source of improvement"); sequential addition of Relative Mahalanobis and path surprise in both orders reaches the same total reduction, demonstrating complementarity rather than forced gain. The 131/152 FPR95 result is an out-of-sample measurement on OpenOOD splits never used in fitting. The recognition branch (Eqs. 8-9) trains probes on source labels, selects fusion weights and gate on a disjoint validation split, and reports test-set accuracy; the +4.41 gain is thus a genuine holdout prediction. The skeptic's point that clean classification gains lack a matched final-state-ensemble control is a comparison/rigor limitation (addressed for shift experiments via the matched control in Appendix I.2), not circularity: no reported accuracy is equivalent to a fitted quantity by construction. Self-citations (Damirchi et al. 2026; M. De la Jara et al. 2025; M. De la Jara et al. 2023; Rodriguez-Opazo et al. 2025) appear only in Related Work and positioning, are not used to justify any load-bearing premise, uniqueness claim, or ansatz, and do not inject the target result into the derivation. The central claims are externally benchmark-validated and supported by the paper's own identity-shuffled, random-noise, and matched-ensemble controls, so no circular step can be exhibited from the paper's equations.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The method is empirical; no new physical or theoretical entity is introduced. Free parameters are either ID-calibrated statistics or a fixed a priori fusion weight, keeping the circularity burden low: no parameter is fit to OOD data or to the target outcome.

free parameters (5)
  • Fusion weight w = 0.3
    Fixed a priori in Eq. 7, not tuned per benchmark; robustness sweep in Fig L3 covers 0.1–1.0 and shows positive mean benefit throughout.
  • Transition error scales sigma^2_{l,d} = mean squared ID fitting residuals + 1e-6 per component and transition
    Normalizes the surprise score in Eq. 6; estimated from ID fitting data only, so it is a legitimate calibration statistic rather than an OOD-fitted parameter.
  • Residualization coefficients (a,b) in Eq. 7 = least squares on held-out ID data
    Remove overlap between the path surprise and the final-state score; estimated without OOD exposure.
  • Transition MLP hyperparameters = hidden 1024, lr 3e-4, batch 512, 80 epochs
    Chosen in Appendix E and held fixed across benchmarks; the central method depends on this capacity, but the paper's continuity controls show the predictor captures real signal.
  • Seed-0 Gaussian common-width stage projection = seed-0 random matrix
    Aligns variable-width stages for Swin/CNN models; Appendix L.3 shows interpolation is comparable, so this choice is not a critically fitted parameter.
axioms (5)
  • standard math KL chain rule, data-processing inequality, and conditional-expectation projection identities used in Eq. 5 and Eqs. 10–13.
    Unproved background identities; the paper explicitly states they do not guarantee gains and that empirical controls are required.
  • domain assumption Vision encoders are frozen and deterministic; states at each native block are recorded in block order.
    Required for the trajectory object; determinism implies the path cannot create distributional information absent from earlier states (Sec. 6, F.1).
  • ad hoc to paper The class-conditional mean trajectory is a sufficient anchor; subtracting it isolates sample-specific innovation.
    Modeling choice in Eqs. 2–4; supported empirically by Table L1 but not proven for all distributions.
  • ad hoc to paper One-step Markov assumption: next residual is predictable from current residual plus depth embedding, with per-component Gaussian error scale.
    Definition of D in Eq. 6; Appendix L.2 shows full-prefix history adds only +0.34 FPR95 points, bounding but not eliminating this assumption.
  • domain assumption OpenOOD v1.5 splits and the balanced 38-checkpoint grid are representative of OOD detection practice.
    Benchmark protocol choice; exhaustive tables show broad coverage, but the headline counts are conditional on this grid and the non-saturated exclusion.

reviewed 2026-08-01 · how reviews work

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Cite this review

Pith. "Pith review of Representation Trajectories Matters: Complementary Evidence for OOD Detection and Image Classification." pith.science (2026). https://pith.science/paper/VYDZ5CKG

@misc{pith2026260726565,
  author       = {Pith},
  title        = {Pith review of: Representation Trajectories Matters: Complementary Evidence for OOD Detection and Image Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYDZ5CKG}},
  note         = {Machine review of arXiv:2607.26565}
}
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read the original abstract

Vision models do not form a representation at once; each block revises it. We ask whether the resulting computation path contains evidence that the final representation discards, and whether that evidence improves OOD detection and image classification on clean and shifted data. Unlike approaches that treat intermediate layers as separate snapshots, we retain sample identity across depth and study the transformations connecting successive states. We separate class-coherent transport from input-specific innovation, and coordinate movement from relational reorganization. Across supervised, self-supervised, vision--language, hierarchical, and convolutional encoders, these paths show strong sample-specific continuity and architecture-specific depth profiles that recur across datasets. They are also practically useful. An ID-only transition-surprise score complements strong final-state detectors, reducing FPR95 in 131/152 non-saturated comparisons on a balanced OpenOOD grid; gains are largest for visually disruptive and semantically far shifts, and remain positive on near-OOD for most detectors. Frozen update probes improve 71/72 clean model--dataset cases, while shifted-data gains vary with architecture and corruption type. Computation paths therefore provide a broadly useful reliability signal whose value is determined jointly by model organization and the shift encountered.

Figures

Figures reproduced from arXiv: 2607.26565 by Cristian Rodriguez-Opazo, Damith Ranasinghe, Hamed Damirchi, Ignacio M. De la Jara, Stephen Gould.

Figure 1
Figure 1. Figure 1: The trajectories matters.Images with similar final representations can follow different computational routes. Typical ID samples remain close to learned class trajectories, whereas OOD samples deviate despite similar endpoints. Trajectory evidence complements the final representation for both OOD detection and image classification; bars report average gains across architecture families. models (Kornblith e… view at source ↗
Figure 2
Figure 2. Figure 2: Architectures allocate work differently. Each strip follows one checkpoint–dataset pair across its native blocks. Movement, relational deformation √ 1 − CKA, class-evidence gain, and residual share peak at different depths; colour is normalised within each row except residual share. only the endpoint of this process. We represent the forward pass as a representation trajectory, τ (x) = (z1(x), . . . , zL(x… view at source ↗
Figure 3
Figure 3. Figure 3: Matched trajectories preserve sample-specific continuity. (a) Matched transitions yield higher held-out prediction R2 than identity-shuffled pairs in all 16 cases. (b) History adds predictive information. (c) Predictability decays with layer gap. Full values are in Appendix Table J1. and all standardisation statistics are estimated from ID data only. This tests whether the trajectory provides evidence be￾y… view at source ↗
Figure 4
Figure 4. Figure 4: Trajectory gains depend on score and architecture. (a) Final-layer-only (open) and trajectory-augmented (filled) FPR95; (b) 152 paired reductions with improved-case counts, interquartile ranges, and mean ∆; (c) mean reduction and improved/total cases by architecture and score. rather than a temporal arrow. Consistent performance under fixed layer permutations further argues against privileged ordering. 5 E… view at source ↗
Figure 5
Figure 5. Figure 5: Path evidence is complementary. It gives the largest gain over Maha++ (a), and remains useful with RelMaha in either order (b). Counts report improvements out of 24 cases. 5.3 Recognition: Trajectory Information for Classification Image classification provides a complementary test of whether trajectory information extends beyond the final rep￾resentation [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory information under label-preserving shift remains architecture dependent. For the same six checkpoints, (a–b) report gain beyond a matched final-state ensemble and (c) reports disagreement by severity. Full nat￾ural and synthetic shift tables are in Appendices K and O. proves 71/72 model–dataset cases by +4.41 accuracy points on average. The largest dataset means occur on Cars (+9.81), Flowers (+… view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 1, 2026.