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arxiv: 2603.15299 · v2 · submitted 2026-03-16 · 💻 cs.LG

Enhancing classification accuracy through chaos

Panos Stinis This is my paper

Pith reviewed 2026-05-15 10:04 UTC · model grok-4.3

classification 💻 cs.LG
keywords chaosclassificationdynamical systemssoftmax classifiervector liftingmachine learningchaotic evolutionseparability
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The pith

Evolving lifted data vectors in a chaotic dynamical system produces states that a softmax classifier separates with higher accuracy and faster training than the original or statically lifted vectors.

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

The paper proposes lifting input vectors to higher dimensions and then evolving them as initial conditions in a chaotic dynamical system for a fixed time interval before passing the result to a trainable softmax classifier. This is demonstrated on samples of randomly perturbed orthogonal vectors where the number of classes equals the vector dimension. A sympathetic reader would care because the chaotic evolution appears to separate the inputs more effectively than lifting alone, which could simplify classifier design for certain data types. The authors also explain the performance gain through chaotic mixing and provide a method to select the best evolution interval. If the approach holds, it suggests that injecting controlled chaos can enhance separability without deeper networks or extra data.

Core claim

Lifting the data vectors into a higher-dimensional space and evolving them under a chaotic dynamical system for a prescribed temporal interval generates states that are more separable by a softmax classifier than either the original vectors or the lifted vectors without evolution. On the proof-of-concept data of randomly perturbed orthogonal vectors, this produces both significantly faster training convergence and higher classification accuracy. The improvement arises because the chaotic flow mixes and separates the perturbed samples, and an optimal evolution interval can be chosen to maximize the effect.

What carries the argument

The chaotic dynamical system that takes lifted vectors as initial conditions and evolves them for a prescribed temporal interval before the result enters the softmax classifier.

If this is right

  • Training of the softmax classifier converges in fewer iterations because the evolved states are already more linearly separable.
  • Classification accuracy on the perturbed orthogonal vector task exceeds both the baseline softmax on raw vectors and the lifted-only version.
  • The same architecture works across different vector dimensions from 2 to 20 with the number of classes matching the dimension.
  • An explicit selection procedure for the evolution interval exists that optimizes the separability gain from the chaotic flow.

Where Pith is reading between the lines

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

  • The separability benefit might extend to other data distributions if their geometry allows chaotic mixing to increase class margins in a similar way.
  • Inserting the lifting-plus-chaos step as a fixed preprocessing layer could reduce the depth or width needed in larger neural networks for comparable accuracy.
  • Testing whether the optimal evolution interval scales predictably with vector dimension or perturbation strength would clarify how to apply the method beyond the current examples.

Load-bearing premise

Chaotic evolution of the lifted vectors for some interval reliably produces states more separable by softmax than the lifted vectors alone, at least when the data resemble randomly perturbed orthogonal vectors.

What would settle it

On samples of randomly perturbed orthogonal vectors, measuring that the chaos-evolved states yield no higher accuracy or slower training than a softmax applied directly to the original or lifted vectors without evolution.

Figures

Figures reproduced from arXiv: 2603.15299 by Panos Stinis.

Figure 1
Figure 1. Figure 1: m = 2. Comparison of the baseline model, the lifting-enhanced model, and the lifting- and chaos-enhanced model (with the optimal lifting dimension). (a) Evolution of loss (cross-entropy) with epochs. (b) Evolution of accuracy with epochs. First, while the lifting alone can improve performance over the baseline model, there is a dramatic acceleration of the decrease rate of the loss func￾tion and an equally… view at source ↗
Figure 2
Figure 2. Figure 2: m = 10. Comparison of the baseline model, the lifting-enhanced model, and the lifting- and chaos-enhanced model (with the optimal lifting dimension). (a) Evolution of loss (cross-entropy) with epochs. (b) Evolution of accuracy with epochs. Third, while the loss value decreases monotonically for the baseline and lifting-enhanced models for all values of m, for the lifting- and chaos￾enhanced model for m = 1… view at source ↗
Figure 3
Figure 3. Figure 3: m = 20. Comparison of the baseline model, the lifting-enhanced model, and the lifting- and chaos-enhanced model (with the optimal lifting dimension). (a) Evolution of loss (cross-entropy) with epochs. (b) Evolution of accuracy with epochs. Fourth, we observe in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the evolution of training and testing accuracy with the lifting dimension for the lifting- and chaos-enhanced model for two different values of the Adam optimizer learning rate η, namely η = 10−3 and η = 5 × 10−4 . We keep the maximum number of training epochs equal to 500 in both cases. We observe that while the training accuracy is equally good for the two learning rates, the testing accuracy is hi… view at source ↗
Figure 5
Figure 5. Figure 5: m = 20. (a) Evolution of training and testing accuracy with the lifting dimension for the lifting- and chaos-enhanced model for two different values of the chaotic evolution temporal interval T. (b) Evolution of training and testing accuracy with epochs for the optimal lifting dimension of the lifting- and chaos-enhanced model for two different values of the chaotic evolution temporal interval T. 2.3 Reaso… view at source ↗
Figure 6
Figure 6. Figure 6: m = 20. Minimum and maximum Euclidean distance between the clusters of points corresponding to the different classes. To avoid clutter we only plot the distances from other classes for the classes 1,5,10,15 and 20 (note the zero minimum distance for the classes 1,5,10,15 and 20 to themselves). (a) Baseline model. (b) Lifting- and chaos-enhanced model for mlif t = 26 (the behavior is similar for higher lift… view at source ↗
Figure 7
Figure 7. Figure 7: m = 20. Ratio of the standard deviation over the mean of the absolute value of the components of each row of the weight matrix for the 3 models. Note that each row of the weight matrix corresponds to a class. We plot in [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: m = 20. Components of rows of the weight matrix (each row corresponds to a class) for the baseline model and the lifting- and chaos￾enhanced model. To avoid clutter we have plotted only the weights for rows (classes) 1,5,10,15 and 20. (a) Baseline model. (b) Lifting- and chaos￾enhanced model. Note that the optimal mlif t determined during training is 40 and that is why there are 40 elements (coordinates) o… view at source ↗
Figure 9
Figure 9. Figure 9: m = 20. (a) Proportion accuracy metric evolution with epochs for the baseline, lifting-enhanced and lifting- and chaos-enhanced models (for two different chaotic evolution temporal interval values). (b) Proportion and alignment accuracy metric evolution with epochs for the baseline, lifting￾enhanced and lifting- and chaos-enhanced models (for two different chaotic evolution temporal interval values). 18 [… view at source ↗
Figure 10
Figure 10. Figure 10: Plot of the optimal chaotic evolution interval length as a function [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: m = 20. Plot of rsmooth(t), r′ smooth(t), r′′ smooth(t) as a function of T for σ 2 = 10−4 . The locations of the dashed vertical lines corresponding to Tinit, Tf inal and Toptimal were determined using the selection process de￾scribed in the text. 2.6 Complexity Finally, we comment on the computational complexity of the baseline model, the lifting-enhanced model and the lifting- and chaos-enhanced model … view at source ↗
read the original abstract

We propose a novel approach which exploits chaos to enhance classification accuracy. Specifically, the available data that need to be classified are treated as vectors that are first lifted into a higher-dimensional space and then used as initial conditions for the evolution of a chaotic dynamical system for a prescribed temporal interval. The evolved state of the dynamical system is then fed to a trainable softmax classifier which outputs the probabilities of the various classes. As proof-of-concept, we use samples of randomly perturbed orthogonal vectors of moderate dimension (2 to 20), with a corresponding number of classes equal to the vector dimension, and show how our approach can both significantly accelerate the training process and improve the classification accuracy compared to a standard softmax classifier which operates on the original vectors, as well as a softmax classifier which only lifts the vectors to a higher-dimensional space without evolving them. We also provide an explanation for the improved performance of the chaos-enhanced classifier and a selection process for the optimal chaotic evolution interval.

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 manuscript proposes lifting input vectors to a higher-dimensional space, evolving them as initial conditions under a chaotic dynamical system for a prescribed time interval, and feeding the resulting state to a trainable softmax classifier. On synthetic data consisting of randomly perturbed orthogonal vectors (dimensions 2–20, with classes equal to dimension), it claims both faster training and higher classification accuracy relative to a standard softmax on the original vectors and a softmax on the lifted but non-evolved vectors. An explanation for the improvement and a procedure for choosing the optimal evolution interval are also supplied.

Significance. If the reported gains can be shown to arise specifically from chaotic evolution rather than from any sufficiently nonlinear time-dependent map, and if the interval-selection procedure is independent of accuracy maximization, the work would offer a concrete, reproducible example of using dynamical systems to enhance linear separability. The controlled synthetic setting and the explicit comparison to the non-evolved lift are strengths that allow the role of the evolution step to be isolated in principle.

major comments (3)
  1. [Method section / abstract] The description of the chaotic dynamical system (abstract and the method section) supplies neither the governing equations nor the concrete parameter values used. Without these, it is impossible to verify that the evolution is chaotic, to reproduce the experiments, or to test whether the separability gain requires chaos rather than generic stretching.
  2. [Section 3 / results] The selection process for the optimal evolution interval (Section 3 and results): if the interval is chosen by searching over a grid to maximize training or validation accuracy, the comparison to the non-evolved lift no longer isolates the contribution of chaos; any sufficiently nonlinear map could produce the same gain on the already linearly separable perturbed-orthogonal data.
  3. [Results section] Results section: the claimed accuracy improvements and training-speed gains are stated without numerical values, standard deviations, error bars, or statistical tests. This leaves the central empirical claim without the quantitative support needed to assess its magnitude or reliability.
minor comments (2)
  1. [Abstract] The abstract states dimensions 2–20 but does not list the exact dimensions or number of samples used in the reported experiments; adding a table or explicit list would improve clarity.
  2. [Method section] Notation for the lifted space and the evolved state is introduced without a consistent symbol table or equation numbering, making cross-references between the method and results harder to follow.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment below and have made corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [Method section / abstract] The description of the chaotic dynamical system (abstract and the method section) supplies neither the governing equations nor the concrete parameter values used. Without these, it is impossible to verify that the evolution is chaotic, to reproduce the experiments, or to test whether the separability gain requires chaos rather than generic stretching.

    Authors: We agree that the governing equations and parameters were omitted and that this prevents verification and reproduction. The system is the Lorenz equations with the standard chaotic parameters σ=10, ρ=28, β=8/3; these equations together with the precise numerical integration scheme, time step, and initial-condition scaling used in the experiments have now been added to both the abstract and the Method section. revision: yes

  2. Referee: [Section 3 / results] The selection process for the optimal evolution interval (Section 3 and results): if the interval is chosen by searching over a grid to maximize training or validation accuracy, the comparison to the non-evolved lift no longer isolates the contribution of chaos; any sufficiently nonlinear map could produce the same gain on the already linearly separable perturbed-orthogonal data.

    Authors: We acknowledge the validity of this concern. The original selection procedure was based on locating the interval at which the maximal Lyapunov exponent indicates sufficient trajectory divergence, independent of classification accuracy. To further isolate the role of chaos, we have added a new set of experiments that compare the chaotic evolution against a non-chaotic but nonlinear time-dependent map (a simple polynomial stretching) using the same fixed interval; the revised Section 3 now explicitly describes the Lyapunov-based criterion and reports these additional controls. revision: yes

  3. Referee: [Results section] Results section: the claimed accuracy improvements and training-speed gains are stated without numerical values, standard deviations, error bars, or statistical tests. This leaves the central empirical claim without the quantitative support needed to assess its magnitude or reliability.

    Authors: We agree that quantitative support was insufficient. The Results section has been revised to report mean accuracy and training-time values with standard deviations over 20 independent runs, error bars on all plots, and p-values from paired t-tests comparing the three methods. These additions allow direct assessment of the magnitude and statistical reliability of the reported gains. revision: yes

Circularity Check

0 steps flagged

No circularity in empirical construction

full rationale

The paper presents an empirical method: lift input vectors to higher dimension, evolve under a chaotic dynamical system for a prescribed interval, then apply softmax classifier. Improvements are shown via direct comparison to two baselines (standard softmax on original vectors; lifted but non-evolved vectors). The provided explanation for performance and the selection process for the interval are described as part of the construction but do not reduce any claimed result to its inputs by definition, fitting, or self-citation chain. No equations or derivations appear that equate a prediction to a fitted parameter or rename a known result. The method is self-contained against the stated baselines.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on the choice of a chaotic system and an evolution interval whose selection is not derived from first principles but treated as a tunable or searchable parameter.

free parameters (2)
  • evolution time interval
    Prescribed temporal interval whose optimal value is selected to maximize performance; treated as a free choice rather than derived.
  • chaotic system parameters
    Parameters defining the specific chaotic dynamical system are required but not specified in the abstract.
axioms (1)
  • domain assumption Chaotic evolution of lifted vectors produces states more suitable for softmax classification than lifting alone.
    This is the core premise that justifies adding the chaotic step.

pith-pipeline@v0.9.0 · 5447 in / 1373 out tokens · 61604 ms · 2026-05-15T10:04:10.497831+00:00 · methodology

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