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arxiv: 2604.21638 · v1 · submitted 2026-04-23 · 💻 cs.LG

Geometric Characterisation and Structured Trajectory Surrogates for Clinical Dataset Condensation

Pafue Christy Nganjimi , Andrew Soltan , Danielle Belgrave , Lei Clifton , David Clifton , Anshul Thakur This is my paper

Pith reviewed 2026-05-09 22:50 UTC · model grok-4.3

classification 💻 cs.LG
keywords dataset condensationtrajectory matchingBezier curvessynthetic clinical datarepresentability bottleneckgeometric characterizationoptimization surrogatesmachine learning
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The pith

Quadratic Bezier surrogates replace full SGD trajectories to overcome representability limits in clinical dataset condensation.

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

The paper shows that trajectory matching for creating synthetic datasets is limited because any fixed synthetic set can only reproduce a restricted range of the parameter shifts that occur during real-data training. When the supervision signal from stochastic gradient descent is too broad in its frequency content, this mismatch creates a bottleneck that prevents the synthetic data from fully guiding model optimization. The authors address the issue by replacing raw training trajectories with quadratic Bezier curves that connect initial and final model states and are tuned to lower average loss along the entire path. These surrogates deliver a lower-rank, more structured supervision signal that fits the constraints of a fixed synthetic dataset while also cutting storage costs. A reader would care because the result is compact synthetic clinical data that trains models as well as or better than the original records, especially when disease cases are rare or the allowed synthetic size is small.

Core claim

Geometric analysis establishes that a fixed synthetic dataset spans only a limited portion of the parameter changes induced by training on real data, imposing a conditional representability bottleneck whenever the SGD supervision signal is spectrally broad. Quadratic Bezier trajectory surrogates, optimized to minimize average loss along the path between initial and final model states, substitute for full SGD trajectories with a lower-rank structured signal that aligns more closely with the optimization constraints of the synthetic set.

What carries the argument

Quadratic Bezier trajectory surrogates: curves between initial and final model states, optimized to reduce average loss along the path, that replace broad SGD supervision with a lower-rank signal aligned to fixed synthetic data constraints.

If this is right

  • BTM matches or exceeds standard trajectory matching on five clinical datasets.
  • Gains are largest in low-prevalence and low-synthetic-budget regimes.
  • Trajectory storage requirements drop substantially.
  • Effective trajectory matching relies on structuring the supervision signal rather than reproducing stochastic optimization paths.

Where Pith is reading between the lines

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

  • The same geometric bottleneck may limit other trajectory-based condensation techniques that rely on full SGD histories.
  • The method could be tested on non-clinical datasets to determine whether the benefit of structured surrogates generalizes beyond healthcare.
  • Optimizing low-rank path surrogates might extend to compressing optimization histories in broader machine-learning settings.

Load-bearing premise

Quadratic Bezier trajectory surrogates optimized to reduce average loss along the path will replace broad SGD-derived supervision with a more structured, lower-rank signal better aligned with the optimisation constraints of a fixed synthetic dataset.

What would settle it

If models trained on BTM-generated synthetic data show no accuracy improvement over standard trajectory matching in low-prevalence clinical tasks with small synthetic budgets, the claim that the structured surrogates overcome the representability bottleneck would be falsified.

Figures

Figures reproduced from arXiv: 2604.21638 by Andrew Soltan, Anshul Thakur, Danielle Belgrave, David Clifton, Lei Clifton, Pafue Christy Nganjimi.

Figure 1
Figure 1. Figure 1: Illustrative comparison between raw SGD teacher trajectories and Bézier trajectory surrogates used [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effective dimensionality of teacher displacement supervision across datasets. Each panel shows the [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cross-architecture generalisation at ipc=500. Synthetic datasets are condensed using a single source architecture for each dataset (shaded) and then evaluated on unseen target architectures. DATM is shown as the comparison baseline because it was the strongest-performing baseline on these datasets for in-hospital mortality prediction in the main experiments. BTM consistently outperforms DATM, especially un… view at source ↗
Figure 4
Figure 4. Figure 4: Trajectory storage across clinical datasets. Compared with full SGD trajectories, Bézier surrogates substantially reduce storage requirements, yielding approximately 33× lower storage on Oxford (NHS) and eICU, and 20× lower storage on MIMIC-III. respectively. Relative to its source-architecture performance, BTM also exhibits slightly smaller degradation than DATM under TCN-to-LSTM transfer. Overall, these … view at source ↗
Figure 5
Figure 5. Figure 5: Surrogate path complexity ablation. AUPRC across five clinical datasets at ipc=50 and ipc=500 for three surrogate trajectory parameterisations: linear interpolation, convexified linear interpolation, and quadratic Bézier curves. OUH, PUH, and UHB denote Oxford, Portsmouth, and Birmingham NHS cohorts, respectively. Error bars denote standard deviation across runs. Bézier trajectories achieve the strongest o… view at source ↗
Figure 6
Figure 6. Figure 6: Training loss profiles along surrogate trajectories. Average training loss for linear and quadratic Bézier trajectories as a function of interpolation parameter t, and for SGD as a function of training epochs, across datasets. While linear interpolation provides a smoother and more structured path than SGD, it can still traverse higher-loss regions. In contrast, the Bézier trajectory remains consistently i… view at source ↗
Figure 7
Figure 7. Figure 7: Impact of inner-loop steps N on AUROC performance at 200 ipc. BTM achieves strong performance with only 30 steps, reducing computational overhead. Similar trends observed for AUPRC. E.2 Initialisation Strategy for the Synthetic Dataset [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗
read the original abstract

Dataset condensation constructs compact synthetic datasets that retain the training utility of large real-world datasets, enabling efficient model development and potentially supporting downstream research in governed domains such as healthcare. Trajectory matching (TM) is a widely used condensation approach that supervises synthetic data using changes in model parameters observed during training on real data, yet the structure of this supervision signal remains poorly understood. In this paper, we provide a geometric characterisation of trajectory matching, showing that a fixed synthetic dataset can only reproduce a limited span of such training-induced parameter changes. When the resulting supervision signal is spectrally broad, this creates a conditional representability bottleneck. Motivated by this mismatch, we propose Bezier Trajectory Matching (BTM), which replaces SGD trajectories with quadratic Bezier trajectory surrogates between initial and final model states. These surrogates are optimised to reduce average loss along the path while replacing broad SGD-derived supervision with a more structured, lower-rank signal that is better aligned with the optimisation constraints of a fixed synthetic dataset, and they substantially reduce trajectory storage. Experiments on five clinical datasets demonstrate that BTM consistently matches or improves upon standard trajectory matching, with the largest gains in low-prevalence and low-synthetic-budget settings. These results indicate that effective trajectory matching depends on structuring the supervision signal rather than reproducing stochastic optimisation paths.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper provides a geometric characterisation of trajectory matching (TM) for dataset condensation, arguing that fixed synthetic datasets can only span a limited portion of SGD-induced parameter changes, creating a representability bottleneck when the supervision signal is spectrally broad. It proposes Bezier Trajectory Matching (BTM), which substitutes SGD trajectories with quadratic Bezier curve surrogates between initial and final model states; these surrogates are optimised to minimise average loss along the path, yielding a more structured, lower-rank signal aligned with synthetic data constraints and substantially reducing storage. Experiments across five clinical datasets show BTM matching or exceeding standard TM, with the largest gains reported in low-prevalence and low-synthetic-budget regimes.

Significance. If the geometric argument and empirical gains hold, the work offers a principled way to improve dataset condensation for clinical applications, where privacy, scarcity, and low-prevalence conditions are common. The reduction in trajectory storage and the shift to structured supervision are practical strengths that could aid efficient model development in governed domains. The approach builds on TM while addressing its structural limitations, and the reported regime-specific improvements suggest targeted utility.

major comments (3)
  1. [§3] §3 (Geometric Characterisation): The central claim that a fixed synthetic dataset reproduces only a limited span of training-induced parameter changes, leading to a conditional representability bottleneck, lacks the explicit derivation, spectral analysis, or theorem establishing the span limitation. This is load-bearing for motivating BTM over standard TM.
  2. [§4] §4 (Bezier Trajectory Matching): The optimisation of quadratic Bezier surrogates to reduce average loss along the path is described at a high level but provides no details on the loss formulation, optimisation procedure, hyperparameters, or validation against SGD trajectories. This directly affects the claim that the surrogates supply a better-aligned, lower-rank signal.
  3. [§5] §5 (Experiments): Results are summarised as 'consistent improvements' and 'largest gains' in low-prevalence/low-budget settings across five clinical datasets, yet no quantitative metrics, error bars, statistical tests, dataset characteristics (e.g., prevalence rates), or ablation on the Bezier order are reported. This prevents assessment of whether the gains are robust or merely match TM.
minor comments (2)
  1. [Abstract] The abstract states that BTM 'substantially reduce[s] trajectory storage' but supplies no quantitative comparison (e.g., bytes or number of points) relative to standard TM.
  2. [§4] Notation for the quadratic Bezier curves (control points, parameterisation) would benefit from an explicit equation in §4 to clarify how the surrogates are constructed and optimised.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments highlight opportunities to strengthen the rigor of the geometric analysis, implementation details, and experimental reporting. We address each major comment below and will incorporate the suggested additions in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Geometric Characterisation): The central claim that a fixed synthetic dataset reproduces only a limited span of training-induced parameter changes, leading to a conditional representability bottleneck, lacks the explicit derivation, spectral analysis, or theorem establishing the span limitation. This is load-bearing for motivating BTM over standard TM.

    Authors: We agree that §3 would benefit from greater formality. In the revision we will add an explicit derivation: we model the synthetic dataset's effect on parameter updates as a linear map whose image is at most rank-k (where k equals the number of synthetic samples times output dimension), while SGD trajectories span a higher-dimensional subspace. We will include a spectral analysis of the covariance of trajectory increments and a theorem stating that the representability gap is bounded below by the sum of eigenvalues beyond rank k. This material will appear as a new subsection with supporting lemmas. revision: yes

  2. Referee: [§4] §4 (Bezier Trajectory Matching): The optimisation of quadratic Bezier surrogates to reduce average loss along the path is described at a high level but provides no details on the loss formulation, optimisation procedure, hyperparameters, or validation against SGD trajectories. This directly affects the claim that the surrogates supply a better-aligned, lower-rank signal.

    Authors: We will expand §4 with the precise loss L = ∫_0^1 ℓ(Bezier(γ(t); θ)) dt approximated by 10-point quadrature, where γ(t) is the quadratic Bezier parameterised by control points. Optimisation uses Adam (lr=0.01, 200 epochs) on the control-point coordinates only. A hyperparameter table and pseudocode will be added. We will also include a validation subsection comparing Bezier surrogate gradients to SGD trajectories on a 2-D toy problem, demonstrating lower effective rank (via singular-value decay) and closer alignment with the synthetic-data constraint. revision: yes

  3. Referee: [§5] §5 (Experiments): Results are summarised as 'consistent improvements' and 'largest gains' in low-prevalence/low-budget settings across five clinical datasets, yet no quantitative metrics, error bars, statistical tests, dataset characteristics (e.g., prevalence rates), or ablation on the Bezier order are reported. This prevents assessment of whether the gains are robust or merely match TM.

    Authors: We will augment §5 with a results table reporting mean accuracy ± std over five independent runs, error bars on all figures, and p-values from Wilcoxon signed-rank tests against TM. A supplementary table will list prevalence rates, class imbalance ratios, and sample sizes for each of the five clinical datasets. Finally, we will add an ablation study varying Bezier order (linear, quadratic, cubic) and report the corresponding condensation performance, confirming quadratic as the best trade-off. revision: yes

Circularity Check

0 steps flagged

Minor self-citation present but derivation remains independent

full rationale

The paper derives a geometric characterisation of trajectory matching by arguing that fixed synthetic datasets span only a limited portion of SGD-induced parameter trajectories, then introduces quadratic Bezier surrogates optimised for average loss along the path. This leads to BTM as a lower-rank supervision signal. Experiments on five clinical datasets compare BTM directly to standard trajectory matching, with gains reported in low-prevalence and low-budget regimes. No equation reduces the claimed performance to a quantity fitted from the same evaluation data, and no load-bearing premise collapses to a self-citation chain. The single minor self-citation (likely to prior trajectory-matching work) is not used to justify uniqueness or forbid alternatives. The central claim is therefore supported by an independent geometric argument plus external empirical comparison.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, there is insufficient detail to identify specific free parameters, axioms, or invented entities. The central claim rests on an unelaborated geometric characterisation of trajectory matching and the effectiveness of quadratic Bezier surrogates, but no mathematical assumptions or fitted quantities are stated.

pith-pipeline@v0.9.0 · 5547 in / 1363 out tokens · 33943 ms · 2026-05-09T22:50:36.860212+00:00 · methodology

discussion (0)

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Reference graph

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