Chaos-Based Feature Extraction

• Chaos-based feature extraction is a method that uses deterministic chaotic maps to derive nonlinear, statistically rich features from raw data across various applications.
• It employs maps like logistic, tent, and circle to generate chaotic orbits that improve high-dimensional feature selection and time-frequency analysis.
• Advanced frameworks such as Neurochaos Learning and AutochaosNet utilize these techniques to achieve robust, hyperparameter-free performance in image and signal processing.

Chaos-based feature extraction encompasses a family of techniques that leverage deterministic chaotic dynamical systems to derive informative representations from raw data. Unlike random or conventional linear methods, these approaches exploit sensitive dependence on initial conditions, ergodicity, and the rich temporal or spatial diversity of chaotic maps to uncover nonlinear, statistically complex structures in data. Chaos-based feature extraction is deployed across diverse domains, including high-dimensional feature selection, time-frequency analysis, neuro-inspired machine learning, and image texture characterization.

1. Chaotic Maps as Feature Extractors

Chaotic maps are the core engines of chaos-based feature extraction. A one-dimensional chaotic map $\mathcal{M}: [0,1] \rightarrow [0,1]$ is defined recursively, e.g., $x_{k+1} = \mathcal{M}(x_k)$, with the canonical logistic map $x_{k+1} = \lambda x_k (1-x_k)$ at $\lambda=4$ yielding full chaos. Other maps include the tent, circle, Singer, sine, and skew-tent forms, each with tunable parameters controlling ergodic and Lyapunov characteristics. In image or signal processing, these maps are applied pointwise to either data features or embedded representations (e.g., spatial–intensity vectors for images), generating chaotic orbits whose statistical properties underpin subsequent feature calculations (Vivek et al., 2024, Florindo, 2020, NB et al., 2020).

Key properties:

• Ergodicity ensures coverage of the invariant set.
• Topological transitivity and dense orbits guarantee that, over time, all neighborhoods of the attractor are visited.
• Positive Lyapunov exponents indicate exponential divergence of nearby trajectories, central to producing decorrelated and rich features.

2. Chaos-Informed Feature Selection in Optimization

Chaos-based feature selection harnesses deterministic chaotic variables to replace standard (pseudo-)random elements in metaheuristic algorithms. In quantum-inspired Differential Evolution (QDE), each individual in the population encodes its state as quantum amplitudes, and random decisions in mutation, crossover, and measurements are supplied by iterates of a chaotic map, typically the logistic map in the fully developed regime ($\lambda=4$) (Vivek et al., 2024).

Summary of the procedure:

• Lyapunov-guided chaos: Initial transients (e.g., first 5000 iterates) are discarded to ensure only the stationary chaotic regime is used, leveraging Lyapunov time ($T_L \approx 1/L$) estimates.
• Chaotic replacement: All uniform random draws for mutation factors, crossover rates, and bit measurements are replaced by chaotic orbit values $\{x_t\}$.
• Population spread: The deterministically ergodic nature of the chaotic sequence encourages exploration and helps avoid premature convergence to local optima, yielding consistently higher AUC scores and more compact feature sets in high-dimensional biomedical benchmarks.
• Lasso-assisted pruning: Post hoc, LASSO regression is applied to selected subsets to sparsify further and collapse weak features (Vivek et al., 2024).

3. ChaosFEX and Neurochaos Learning Architectures

The Neurochaos Learning (NL) framework explicitly models neurons as one-dimensional chaotic maps, typically the skew-tent map $C_b(x)$. Each neuron receives a normalized stimulus $x$, initializes at a generic point $q$, and iterates until its chaotic trajectory approaches $x$ within some $\epsilon$ (NB et al., 2020). Four ChaosFEX features are extracted from this trajectory:

• Firing time $N$: number of steps to reach the target,
• Firing rate $F$: proportion of time above threshold $b$,
• Energy $E$: sum of squared trajectory values,
• Entropy $H$: Shannon entropy of the symbolic firing sequence.

These features encode nonlinear responses to input and are demonstrably sufficient: a single chaotic layer with $L$ neurons can approximate any discrete function with finite support (Universal Approximation Theorem for chaotic neurons) (NB et al., 2020). This property hinges on the dense orbits and topological transitivity of the map.

4. Hyperparameter-Free and Universal-Orbit Feature Extraction

AutochaosNet demonstrates that high-performance, chaos-based feature extraction can be realized entirely without hyperparameter tuning or stochastic training (Henry et al., 2 Aug 2025). The framework uses the decimal-shift map $f(x) = \text{frac}(10x)$ and the Champernowne constant as a normal base-10 number, guaranteeing that any finite sequence of digits (input features) will arise in its orbit. Each normalized feature is mapped to a finite block of three digits; the time until its first occurrence in the orbit determines the firing-bound $T$. Over this time, simple statistics (trace mean, firing rate) are computed for each input.

Performance characteristics:

• Stateless and training-free: No parameter search or learned weights.
• Universal orbit: The intrinsic normality of the Champernowne constant ensures arbitrariness of the feature space is reflected in the orbit.
• Empirical efficiency: TM-FR AutochaosNet matches or surpasses earlier neurochaos methods (e.g., ChaosNet) in macro F1 while reducing typical run times from thousands to single-digit seconds on standard benchmarks (Henry et al., 2 Aug 2025).

5. Chaos-Based Time-Frequency Decomposition

Chaotic behavior is also addressed explicitly in time-frequency signal analysis via the linear and hyperbolic chirp transforms (LCT and HCT) (Ricaud et al., 2010). The LCT projects a signal onto orthonormal bases of linear chirps of controllable slope: $$\psi_x^\theta(t) = \frac{1}{\sqrt{2\pi |\sin\theta|}} \exp\left( -i \frac{t^2}{2\tan\theta} + i\frac{x t}{\sin\theta} \right )$$ with $\theta$ controlling the time-frequency tradeoff. The HCT extends this idea for processes with scale invariance (e.g., self-similar chaotic signals), using Mellin-like bases.

Principal features:

• Adaptive resolution to chirp-like content in data (e.g., turbulent plasma, nonstationary biosignals).
• Mode separation: Peak detection in the transform domain isolates modes corresponding to individual chaotic/oscillatory components.
• Invertibility and computational efficiency: Both LCT and HCT admit $O(N \log N)$ implementations via FFT or log-FFT.
• Comparison to classical techniques: LCT/HCT supersede STFT and wavelets in cases where dominant features are chirps or scale-local, not stationary sinusoids (Ricaud et al., 2010).

6. Applications in Image Texture and Local Descriptor Construction

Chaos-based feature extraction robustly supports image texture analysis, particularly in regimes where limited training data and model interpretability are prioritized over deep learning. A key method constructs a joint spatial–intensity $\mathbb{R}^3$ embedding for each image, applies iterated 1D chaotic maps (e.g., circle, logistic, Singer) to each coordinate, and reconstructs intermediate images via a one-to-one mapping (Florindo, 2020). On each blended intermediate state, local binary pattern (LBP) histograms are extracted and concatenated.

Key operational aspects:

• Map choice and hyperparameter control: Polynomial maps yield rapid mixing but may over-scramble textures if too aggressive; circle maps suit homogeneous patterns.
• Blending: Gradual interpolation between orbits ($\delta$ parameter) modulates the degree of nonlinear spatial mixing.
• Empirical findings: The chaos+LBP model achieves near or better than state-of-the-art accuracy on diverse grayscale texture benchmarks, especially in low-supervision or interpretable settings (Florindo, 2020).

7. Comparative Evaluation and Practical Implications

Multiple research efforts demonstrate that chaos-based feature extraction is competitive with deep and classical machine learning baselines in various contexts:

• High-dimensional feature selection: Chaos-guided metaheuristics deliver higher AUC and sparser feature subsets, outperforming both standard binary DE and earlier chaotic quantum-inspired variants (Vivek et al., 2024).
• Neurochaos and universality: ChaosFEX and AutochaosNet yield high macro F1 in low-training regimes and are robust to noise, supported theoretically by universal approximation claims (NB et al., 2020, Henry et al., 2 Aug 2025).
• Signal and image domains: LCT/HCT exhibit sharper mode separation than STFT/wavelets for chaotic/nonstationary signals, while chaos+LBP descriptors rival CNNs in texture recognition given limited annotation (Ricaud et al., 2010, Florindo, 2020).

Limitations and future research directions include handling color and multi-channel data, robust hyperparameter selection for chaotic maps, integration of chaos-based extraction into deep-learning pipelines, and extension to online and streaming feature extraction (Henry et al., 2 Aug 2025, Florindo, 2020). A plausible implication is that the theoretical underpinnings of chaos—ergodicity, dense orbits, Lyapunov exponents—systematically enable nonlinear structure discovery, especially in high-dimensional, poorly structured, or low-signal regimes.