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ADPTSK

ADPTSK is a high-dimensional Takagi-Sugeno-Kang fuzzy system built on two key ideas from Ma et al. (2025): - adaptive double-parameter softmin (ADP-softmin) for antecedent aggregation, - Gaussian PIMF membership functions with a positive infimum to avoid underflow.

Reference

Ma, M., Qian, L., Zhang, Y., Fang, Q., & Xue, G. (2025). An adaptive double-parameter softmin based Takagi-Sugeno-Kang fuzzy system for high-dimensional data. Fuzzy Sets and Systems, 521, 109582, doi: 10.1016/j.fss.2025.109582.

Mathematical Formulation

Antecedent

ADPTSK replaces the standard product or min T-norm with an adaptive softmin operator. For rule \(r\) and feature \(d\), membership values are computed using the Gaussian PIMF:

\[ \mu_{r,d}(x_d) = \exp\left(-K\left[1 - \exp\left(-\frac{(x_d - m_{r,d})^{2}}{2\sigma_{r,d}^{2}}\right)\right]\right) \]

The infimum of this membership is \(\exp(-K) > 0\), which prevents the antecedent from producing zero values in high-dimensional settings.

For each rule \(r\), ADP-softmin computes a firing coefficient using adaptive parameters \(\eta\) and \(q\):

\[ f_r = \left(\frac{1}{D} \sum_{d=1}^{D} (\eta\,\mu_{r,d})^{q}\right)^{1/q} \]

The parameters are chosen to avoid numeric underflow and fake-minimum behaviour:

  • \(\eta\) adapts to the current rule's minimum and maximum membership values,
  • \(q\) is selected as the tightest negative exponent that still remains numerically representable.

In practice, highFIS implements the ADP-softmin computation in a numerically stable log-space fashion, with recommended default values \(\kappa = 690.0\) and \(\xi = 730.0\).

Consequent and defuzzification

The consequent structure remains first-order TSK with a weighted linear output. The rule strengths are normalized and passed to the standard TSK consequent layer.

For classification, the model uses a classification consequent head and MSE loss (with one-hot encoded targets in training). For regression, it uses a regression consequent head and MSE loss.

Code ↔ Paper Correspondence

Paper concept highFIS class / method Description
Gaussian PIMF highfis.memberships.GaussianPiMF Gaussian membership with positive infimum exp(-K)
ADP-softmin antecedent highfis.layers.ADPSoftminRuleLayer Adaptive double-parameter softmin rule aggregation
ADPTSK classifier highfis.models.ADPTSKClassifierModel Classifier model using ADP-softmin antecedents
ADPTSK regressor highfis.models.ADPTSKRegressorModel Regressor model using ADP-softmin antecedents
Estimator wrapper highfis.estimators.ADPTSKClassifier sklearn-style wrapper using GaussianPiMF
Estimator wrapper highfis.estimators.ADPTSKRegressor sklearn-style wrapper using GaussianPiMF

Implementation notes

  • The model uses GaussianPiMF to ensure antecedent membership values have a nonzero positive lower bound.
  • ADPTSK still builds on the BaseTSK pipeline, but replaces the rule layer with ADPSoftminRuleLayer instead of a vanilla product T-norm.
  • Default hyperparameters mirror the paper:
  • three antecedent MFs per feature with centers [0.0, 0.5, 1.0] and sigma=1.0,
  • rule_base="coco" yielding a compact 3-rule structure,
  • epochs=200, learning_rate=0.001,
  • dynamic batch policy: full-batch when N < 500, else 0.2 * N,
  • Adam optimizer,
  • zero initialization of consequent weights and biases,
  • kappa=690.0, xi=730.0,
  • K=1.0 for the Gaussian PIMF lower bound.
  • The estimator wrapper converts initialized GaussianMF objects to GaussianPiMF before model construction.
  • Preprocessing remains external to the estimator by design: train/validation split and feature normalization should be handled by the user or by an sklearn pipeline.

Model classes

highfis.models.ADPTSKClassifierModel

  • Uses ADPSoftminRuleLayer for antecedent aggregation.
  • Uses ClassificationConsequentLayer and MSELoss.
  • Uses Adam by default when no external optimizer is provided.
  • Zero-initializes consequent parameters by default.

highfis.models.ADPTSKRegressorModel

  • Uses the same ADP-softmin antecedent.
  • Uses RegressionConsequentLayer and MSELoss.
  • Uses Adam by default when no external optimizer is provided.
  • Zero-initializes consequent parameters by default.

Estimator wrappers

highfis.estimators.ADPTSKClassifier

This estimator:

  • uses paper defaults (mf_init="grid", n_mfs=3) to initialize antecedents as Gaussian MFs with centers [0.0, 0.5, 1.0] and sigma=1.0,
  • wraps them as GaussianPiMF with the chosen K value,
  • defaults to rule_base="coco",
  • applies paper-style batch sizing when batch_size=None,
  • constructs ADPTSKClassifierModel with kappa, xi, eps, and zero consequent initialization. No automatic normalization is applied.

highfis.estimators.ADPTSKRegressor

This estimator is analogous to the classifier wrapper but builds ADPTSKRegressorModel for regression tasks.

Membership functions

highfis.memberships.GaussianPiMF

  • Implements the PIMF version of Gaussian membership.
  • Prevents crash caused by zero membership values in large dimensionality.
  • K controls the lower bound: membership infimum is exp(-K).

Training in the paper vs. highFIS

  • The paper optimizes ADPTSK end-to-end with gradient-based learning.
  • highFIS follows the same paradigm: the estimator builds the model and uses Adam optimization within BaseTSK.fit().
  • ADPTSKClassifier and ADPTSKRegressor expose the same training hyperparameters as other highFIS estimators.
  • As in the paper protocol, linear normalization to [0, 1] is expected; in highFIS this step is intentionally external to the model.

Alignment with the paper

  • ADPTSK in highFIS uses adaptive ADP-softmin aggregation, exactly as the paper describes.
  • The implementation also uses Gaussian PIMF membership functions to ensure a positive antecedent infimum, matching the paper's stability motivation.
  • Paper default initialization and training configuration are implemented as default estimator behavior.
  • Recommended default values kappa=690.0, xi=730.0, and K=1.0 are derived directly from the paper's suggested settings.