arxiv: 2605.12168 · v1 · submitted 2026-05-12 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

On What We Can Learn from Low-Resolution Data

Theresa Dahl Frehr , Niels Henrik Pontoppidan , Hiba Nassar , Tommy Sonne Alstr{\o}m

Authors on Pith no claims yet

Pith reviewed 2026-05-13 05:22 UTC · model grok-4.3

classification 💻 cs.LG

keywords low-resolution datahigh-resolution dataKullback-Leibler divergencedata scarcitymachine learningvision transformerconvolutional neural network

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The pith

Low-resolution data improves model performance on high-resolution tasks when high-resolution samples are scarce.

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

The paper investigates whether low-resolution data retains useful information for training models that will be evaluated on high-resolution inputs. It develops a theoretical analysis using the Kullback-Leibler divergence to characterize how the influence of data points changes with resolution and derives bounds on the relative contributions of high- and low-resolution observations based on information lost under downsampling. Empirically, experiments with a vision transformer and a convolutional neural network show that adding low-resolution data to the training set consistently improves results specifically when high-resolution data is limited. This addresses real-world constraints in domains where high-resolution collection or sharing is restricted by storage, privacy, or device limitations.

Core claim

Low-resolution observations from the same distribution contribute positively to training even when the final model is tested on high-resolution inputs, with their relative value bounded by Kullback-Leibler divergence measures of influence change and information loss under downsampling, leading to measurable performance gains when high-resolution data is scarce.

What carries the argument

The Kullback-Leibler divergence measure of how a data point's influence on the trained model changes with its resolution, used to derive bounds relating high- and low-resolution contributions to downsampling losses.

If this is right

Training sets can be usefully expanded with low-resolution samples to raise high-resolution accuracy when high-resolution data volume is limited.
Data collection in constrained environments can favor greater volume at lower resolution without complete loss of training value.
The performance benefit from low-resolution augmentation holds across architectures including vision transformers and convolutional networks.
Theoretical bounds on relative contributions can guide decisions on which low-resolution samples to include in a mixed-resolution training set.

Where Pith is reading between the lines

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

In settings with data from multiple devices or institutions, this could support mixing resolutions more effectively if the shared-distribution premise holds.
The KL-based bounds might be adapted to other modalities such as time-series or audio where downsampling is routine.
Training procedures could incorporate the derived influence measures to dynamically weight low-resolution samples during optimization.

Load-bearing premise

The low-resolution observations come from the same underlying distribution as the high-resolution targets, and the KL-based influence measure accurately reflects practical information loss under downsampling.

What would settle it

An experiment showing that adding low-resolution data from the same distribution either fails to improve or degrades performance on a high-resolution test set, or where measured gains fall outside the theoretical bounds derived from the KL analysis.

Figures

Figures reproduced from arXiv: 2605.12168 by Hiba Nassar, Niels Henrik Pontoppidan, Theresa Dahl Frehr, Tommy Sonne Alstr{\o}m.

**Figure 2.** Figure 2: Simulation of bounds as a function of progressively removed high-frequency content. Error [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Left: Variance of low-dimensional representations as a function of image resolution. Computed from PCA projections of progressively downsampled images from CIFAR10. Right: Synthetic two-class data separated using linear discriminant analysis. Marker size given by the magnitude of the change in model parameters when including a datapoint in the training set. The black line is the decision boundary given by … view at source ↗

**Figure 4.** Figure 4: Effect of adding low-resolution data under varying levels of high-resolution scarcity. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Trade-off between accuracy gain and storage cost for the Size experiment compared to the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Simulation of bounds as a function of progressively removed high-frequency content. Error [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗

**Figure 7.** Figure 7: (a): DWT of the time series seen in (c). (b) shows the reconstructions from only a single band. This shows how the DWT ensures the additive condition in eq. (64) is satisfied. 0 Levels 1 Level 2 Levels 3 Levels [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗

**Figure 8.** Figure 8: Illustration of the effect of removing detail coefficients content using DWT on an image. [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 2.** Figure 2: Levels refer to the number of frequency bands removed in the scalogram generated from [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 9.** Figure 9: Relative error of the variance approximation in Proposition 4 as a function of input [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: Tightness of the KL-ratio bounds from Proposition [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗

**Figure 11.** Figure 11: Tightness of the KL-difference bounds from Proposition [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗

read the original abstract

Artificial intelligence systems typically rely on large, centrally collected datasets, a premise that does not hold in many real-world domains such as healthcare and public institutions. In these settings, data sharing is often constrained by storage, privacy, or resource limitations. For example, small wearable devices may lack the bandwidth or energy capacity needed to store and transmit high-resolution data, leading to aggregation during data collection and thus a loss of information. As a result, datasets collected from different sources may consist of a mixture of high- and low-resolution samples. Despite the prevalence of this setting, it remains unclear how informative low-resolution data is when models are ultimately evaluated on high-resolution inputs. We provide a theoretical analysis based on the Kullback-Leibler divergence that characterises how the influence of a datapoint changes with resolution, and derive bounds that relate the relative contribution of high- and low-resolution observations to the information lost under downsampling. To support this analysis, we empirically demonstrate, using both a vision transformer and a convolutional neural network, that adding low-resolution data to the training set consistently improves performance when high-resolution data is scarce.

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.

Desk Editor's Note private letter to a colleague

The paper gives a KL-divergence way to bound how much low-res data contributes when high-res samples are limited, then shows it improves ViT and CNN performance in that regime.

read the letter

The main point is that low-resolution observations can be useful training signal even when the final model is tested at high resolution, and the authors supply a KL-based characterization plus bounds to quantify the relative value of each. They then run mixed-resolution training on a vision transformer and a CNN and report consistent gains once high-res data gets scarce. That combination of a clean theoretical framing and a direct empirical check on a real constraint is the useful part here. The KL analysis is straightforward but not trivial, and it directly addresses the information-loss question that comes up in healthcare or sensor data settings. The experiments apply the idea without extra machinery, which keeps the claim testable. The soft spot is the maintained assumption that low-res samples are draws from the same underlying distribution as the high-res targets. Standard downsampling is a deterministic pushforward, so the KL term may not fully track the scale-specific features a network can still exploit. The stress-test note is right that the current experiments stay inside that assumption rather than stress-testing it with mismatched distributions or different downsampling operators. Without the full derivations and exact data splits it is also hard to judge how sensitive the reported consistency is to post-hoc choices. The work is aimed at applied researchers who already deal with heterogeneous resolution data and want a principled reason to include the low-res portion instead of discarding it. It is not a foundational advance, but the framing is honest and the empirical demonstration is on standard models, so it is worth sending out for referee comments rather than desk-rejecting.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that low-resolution data remains informative for models ultimately evaluated on high-resolution inputs. It provides a KL-divergence analysis characterizing how a datapoint's influence varies with resolution, derives bounds relating the relative contributions of high- and low-resolution samples to downsampling-induced information loss, and empirically shows on both a vision transformer and a CNN that adding low-resolution samples consistently improves performance when high-resolution data is scarce.

Significance. If the theoretical bounds hold under the stated assumptions and the empirical gains are robust, the work addresses a practically relevant setting in domains such as healthcare and edge computing where mixed-resolution data arises from storage, privacy, or bandwidth constraints. The combination of a divergence-based characterization with experiments on standard architectures (ViT, CNN) offers both conceptual insight and actionable guidance for training under data scarcity.

major comments (2)

[§3] §3 (theoretical analysis): the KL-based influence measure and subsequent bounds are derived under the assumption that low-resolution observations are drawn from the same underlying measure as the high-resolution targets (i.e., a simple marginal). The manuscript does not analyze the effect of a deterministic many-to-one downsampling operator inducing a pushforward measure, nor does it show that the scalar KL term tracks the scale-specific features a neural network can still exploit; this assumption is load-bearing for the claim that the derived bounds quantify usable training signal.
[Experiments] Experimental section and associated tables: the reported consistent improvements lack explicit statements of the number of independent runs, the precise rule for selecting or excluding low-resolution samples, and whether error bars or statistical tests support the 'consistently improves' statement across different scarcity levels; without these controls it is unclear whether post-hoc choices affect the central empirical claim.

minor comments (2)

[§2] Notation for the downsampling operator and the induced distributions could be introduced earlier and used consistently to avoid ambiguity when relating the KL term to practical information loss.
[Abstract] The abstract states the architectures used but the main text would benefit from a brief reminder of the exact ViT and CNN variants and the resolution pairs tested.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment point-by-point below, indicating where revisions will be made.

read point-by-point responses

Referee: [§3] §3 (theoretical analysis): the KL-based influence measure and subsequent bounds are derived under the assumption that low-resolution observations are drawn from the same underlying measure as the high-resolution targets (i.e., a simple marginal). The manuscript does not analyze the effect of a deterministic many-to-one downsampling operator inducing a pushforward measure, nor does it show that the scalar KL term tracks the scale-specific features a neural network can still exploit; this assumption is load-bearing for the claim that the derived bounds quantify usable training signal.

Authors: We thank the referee for highlighting this foundational aspect of the analysis. The low-resolution observations are generated by applying a deterministic downsampling operator to high-resolution samples, which by definition induces the pushforward measure; our reference to a 'simple marginal' is intended to denote exactly this pushforward distribution of the low-resolution data. We agree that the manuscript would benefit from an explicit discussion of this equivalence and of the relationship between the scalar KL term and the features retained at lower resolution. In the revision we will add a short subsection in §3 that (i) formally identifies the low-resolution distribution as the pushforward measure and (ii) clarifies that the derived bounds quantify the relative contribution of this marginal without claiming to isolate scale-specific features; the empirical results on ViT and CNN are then presented as evidence that the retained signal remains usable by standard architectures. This is a partial revision: the core KL bounds themselves are unchanged, but their interpretation and grounding are strengthened. revision: partial
Referee: [Experiments] Experimental section and associated tables: the reported consistent improvements lack explicit statements of the number of independent runs, the precise rule for selecting or excluding low-resolution samples, and whether error bars or statistical tests support the 'consistently improves' statement across different scarcity levels; without these controls it is unclear whether post-hoc choices affect the central empirical claim.

Authors: We agree that these experimental details are necessary for reproducibility and to substantiate the central claim. In the revised manuscript we will add: (1) an explicit statement that all reported results are averages over 5 independent runs with distinct random seeds; (2) the precise selection rule—low-resolution samples are drawn uniformly at random from the low-resolution pool to reach the target scarcity ratio, with no post-hoc exclusion or cherry-picking; and (3) error bars showing standard deviation across runs together with paired t-test p-values confirming that the observed improvements are statistically significant (p < 0.05) at the majority of scarcity levels examined. These additions will be incorporated into the experimental section and the associated tables/figures. revision: yes

Circularity Check

0 steps flagged

No circularity: theory uses standard KL properties; empirical results are independent validation

full rationale

The paper derives bounds on high- versus low-resolution contributions via KL divergence between distributions under downsampling, starting from the standard definition of KL and the assumption that low-res samples are pushforwards of the same underlying measure. This is not self-referential: the influence characterization follows directly from the KL formula without fitting to the performance claim or importing uniqueness from prior self-work. The empirical section on ViT/CNN training then tests the predicted improvement when high-res data is scarce, rather than re-deriving the same quantity from the fitted parameters. No step reduces by construction to its inputs, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard information-theoretic properties of KL divergence and conventional neural network training without introducing fitted parameters, new axioms, or postulated entities beyond those already established in the literature.

axioms (1)

domain assumption Kullback-Leibler divergence quantifies information loss under downsampling in a manner relevant to model influence
Invoked to characterize how datapoint influence changes with resolution

pith-pipeline@v0.9.0 · 5504 in / 1114 out tokens · 41821 ms · 2026-05-13T05:22:16.603806+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3 (Exact KL-divergence) ... KLh/KLl = log(E[exp(−ℓ(θ,xh))])+E[ℓ(θ,xh)] / ...
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assumption 1 (Gibbs Distribution) p(θ|X) ∝ exp(−∑ℓ(θ,x))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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