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arxiv: 2605.22155 · v1 · pith:PP6N2Z2Tnew · submitted 2026-05-21 · 💻 cs.LG

Algebraic Machine Learning for Small-to-Medium Datasets Is Competitive against Strong Standard Baselines

David Mendez , Fernando Martin-Maroto , Gonzalo G. de Polavieja This is my paper

Pith reviewed 2026-05-22 08:02 UTC · model grok-4.3

classification 💻 cs.LG
keywords Algebraic Machine Learningsmall-to-medium datasetsimage classificationtabular classificationinductive biassupervised learningmachine learning baselines
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The pith

Algebraic Machine Learning matches or beats cross-validated CNNs and tree methods on small image and tabular datasets without any tuning or validation splits.

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

The paper evaluates Algebraic Machine Learning, which learns by decomposing algebraic structures rather than optimizing numbers. AML trained only on the training data, with no validation or cross-validation, outperforms cross-validated baselines including CNNs on image classification tasks using 50 to 2000 examples. On tabular data in the same size range AML performs comparably to LightGBM and random forests, though XGBoost is strongest overall. The result is interesting because AML applies one generic algebraic bias to both data types instead of relying on the modality-specific designs and hyperparameter searches built into the baselines.

Core claim

AML trained only on training data without using validation or cross-validation outperforms a family of cross-validated baseline methods including CNNs on small to medium image datasets with 50 to 2000 training examples; on tabular datasets in the same size range AML is comparable to LightGBM and random forests even though XGBoost performs best overall, all achieved with a generic algebraic inductive bias rather than modality-specific biases.

What carries the argument

Subdirect decomposition of algebraic structure, the mechanism AML uses to learn instead of numerical optimization.

If this is right

  • The same AML procedure succeeds on two very different data modalities without any modality-specific engineering.
  • Skipping cross-validation lets every training example contribute directly to the model rather than being held out for tuning.
  • A generic algebraic bias can match the results of methods that embed strong task-specific assumptions such as convolution or gradient boosting.

Where Pith is reading between the lines

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

  • If the algebraic approach continues to work at larger scales it could serve as a low-tuning complement to deep learning in data-scarce settings.
  • The result raises the question of whether other algebraic decompositions would show similar robustness across modalities.
  • Testing AML on regression or on data types such as sequences would clarify how far the generic bias extends.

Load-bearing premise

The algebraic inductive bias by itself produces competitive performance on both image and tabular data when baselines receive full cross-validation and task-specific tuning.

What would settle it

A new collection of small image datasets on which AML consistently underperforms a cross-validated CNN would disprove the outperformance claim.

Figures

Figures reproduced from arXiv: 2605.22155 by David Mendez, Fernando Martin-Maroto, Gonzalo G. de Polavieja.

Figure 1
Figure 1. Figure 1: For image datasets, per training-set size, count of how many times each method achieved the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: For image datasets, examples of distributions of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: For tabular datasets, per training-set size, count of how many times each method achieved [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: For tabular datasets, examples of distributions of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: For image datasets, and per each of the considered training set sizes, critical-difference dia [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For tabular datasets, and per each of the considered training set sizes, critical-difference dia [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
read the original abstract

Symbolic methods are generally not considered competitive with strong modern learners on realistic supervised tasks. We evaluate Algebraic Machine Learning (AML), a framework that learns through subdirect decomposition of algebraic structure rather than numerical optimization, against standard baselines on image and tabular classification across varying training-set sizes. We find that AML trained only on training data without using validation or cross-validation outperforms a family of cross-validated baseline methods including CNNs on small to medium image datasets (50--2000 training examples). On tabular datasets in the same size range, XGBoost is overall the best performing method, but AML is nonetheless comparable to methods incorporating task-specific biases such as LightGBM and random forests. AML achieves this competitive performance across two very different types of datasets using a generic algebraic inductive bias, rather than the modality-specific biases built into standard baselines like CNNs for images or XGBoost for tabular data, and requires no cross validation because it has no task-dependent hyperparameters to tune.

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

2 major / 2 minor

Summary. The manuscript evaluates Algebraic Machine Learning (AML), which learns via subdirect decomposition of algebraic structure, against standard baselines on image and tabular classification tasks with training sets of 50--2000 examples. It reports that AML trained only on the training data (no validation or cross-validation) outperforms cross-validated baselines including CNNs on the image datasets, while on tabular data AML is comparable to LightGBM and random forests even though XGBoost is overall strongest. The central claim is that a generic algebraic inductive bias suffices for competitive performance without modality-specific engineering or hyperparameter tuning.

Significance. If the baseline comparisons are shown to be fair, the result would be significant for low-data supervised learning: it would indicate that an algebraic approach can match or exceed methods that incorporate strong task-specific biases (CNNs for images, tree ensembles for tabular) while requiring no cross-validation. The dual-modality evaluation and the explicit contrast between AML's lack of tunable hyperparameters and the fully cross-validated baselines are strengths that would make the finding relevant beyond a single domain.

major comments (2)
  1. [Experimental setup] Experimental setup (likely §4 or §5): the CNN baseline implementations must be described in sufficient detail to confirm they constitute strong small-data methods. In particular, the architectures, regularization (e.g., dropout rates, weight decay), data-augmentation policies, and hyperparameter search ranges should be stated explicitly; if the search spaces exclude small-data adaptations such as shallower networks or aggressive augmentation, the reported outperformance on image tasks (50--2000 examples) may reflect under-tuned baselines rather than superiority of the algebraic bias.
  2. [Results] Results sections (image and tabular tables): the manuscript should report the precise train/validation/test splits used for each dataset, the number of independent runs, and any statistical tests (e.g., paired t-tests or Wilcoxon) for the performance differences. Without these, it is difficult to assess whether the claimed superiority of AML over cross-validated CNNs is robust to split variability or post-selection effects.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'strong standard baselines' would be clearer if the specific methods (CNN variants, LightGBM, XGBoost, random forests) were named in the abstract itself.
  2. [Preliminaries] Notation: ensure that 'AML' and the algebraic decomposition operators are defined at first use and used consistently; a short table summarizing the algebraic primitives would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. The comments highlight important aspects of experimental transparency that will strengthen the paper. We address each major comment below and indicate the revisions we plan to incorporate.

read point-by-point responses

  1. Referee: [Experimental setup] Experimental setup (likely §4 or §5): the CNN baseline implementations must be described in sufficient detail to confirm they constitute strong small-data methods. In particular, the architectures, regularization (e.g., dropout rates, weight decay), data-augmentation policies, and hyperparameter search ranges should be stated explicitly; if the search spaces exclude small-data adaptations such as shallower networks or aggressive augmentation, the reported outperformance on image tasks (50--2000 examples) may reflect under-tuned baselines rather than superiority of the algebraic bias.

    Authors: We agree that explicit details on the CNN baselines are required to demonstrate they are appropriately strong for the small-data setting. In the revised manuscript we will add a new subsection in the experimental setup that fully specifies the CNN architectures (including layer counts, kernel sizes, and activation functions), regularization parameters (exact dropout rates and weight decay values), data-augmentation policies (including the specific transformations and their probabilities), and the complete hyperparameter search grids used for cross-validation. Our original tuning already considered small-data adaptations such as reduced network depth and moderate augmentation; these choices will now be stated explicitly so readers can judge their suitability. revision: yes

  2. Referee: [Results] Results sections (image and tabular tables): the manuscript should report the precise train/validation/test splits used for each dataset, the number of independent runs, and any statistical tests (e.g., paired t-tests or Wilcoxon) for the performance differences. Without these, it is difficult to assess whether the claimed superiority of AML over cross-validated CNNs is robust to split variability or post-selection effects.

    Authors: We acknowledge that greater precision in reporting splits, run counts, and statistical tests will improve assessment of robustness. The revised results sections will explicitly list the train/validation/test split sizes or indices for every dataset, state that all methods were evaluated over 5 independent runs with distinct random seeds, and include paired t-tests (with p-values) on the per-run accuracies to quantify the significance of differences between AML and the baselines. These additions will directly address concerns about split variability and post-selection effects. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical claims rest on direct experimental comparisons

full rationale

The paper's central claims consist of empirical performance measurements of AML versus cross-validated baselines on image and tabular datasets of varying sizes. No derivation chain, equations, or fitted parameters are presented that reduce by construction to inputs defined inside the paper itself. The abstract and description emphasize direct out-of-sample evaluation without validation or hyperparameter tuning for AML, contrasted with full cross-validation for baselines; this structure is self-contained against external benchmarks and contains no self-definitional, fitted-input, or self-citation load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption that subdirect algebraic decomposition supplies a sufficiently strong generic inductive bias; no free parameters or invented entities are mentioned because the method is described as having no task-dependent hyperparameters.

axioms (1)
  • domain assumption AML learns through subdirect decomposition of algebraic structure rather than numerical optimization and requires no task-dependent hyperparameters.
    This is the core premise stated in the abstract that enables the no-validation, no-tuning claim.

pith-pipeline@v0.9.0 · 5699 in / 1428 out tokens · 41992 ms · 2026-05-22T08:02:01.354163+00:00 · methodology

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