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 approximates the minimum of the membership degrees with two adaptive parameters \(\hat\eta_r\) and \(\hat q_r\):

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

With \(\underline\mu_r\) and \(\bar\mu_r\) the smallest and the largest degree of the rule,

\[ \ln\hat\eta_r = -\frac{\xi\ln\underline\mu_r + (\kappa - \ln D)\ln\bar\mu_r}{\kappa - \ln D + \xi}, \qquad \hat q_r = \frac{-\xi}{\ln(\hat\eta_r\,\bar\mu_r)} \]

The scaling by \(\hat\eta_r\) centres the degrees around one, so that the powers neither overflow nor underflow, and lets \(\hat q_r\) be far more negative than the index of Ada-softmin, which brings the result closer to the minimum. The division by \(\hat\eta_r\) undoes the scaling.

highFIS evaluates the sum in the logarithmic domain, with the default values \(\kappa = 690.0\) and \(\xi = 730.0\) of the article, and treats \(\hat\eta_r\) and \(\hat q_r\) as constants in the backward pass.

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="auto" (0.001 above 1000 features, the value of the article; 0.01 below),
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 with the default batch_size="auto",
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.

Fidelity to the source article

The defaults of ADPTSKClassifier are the experimental settings of the article: the Gaussian membership function bounded below with \(K = 1\), the partition with three rules and spreads of 1, consequents at zero, and Adam on the whole training set for 200 iterations with a learning rate of 0.001. The published accuracies are reproduced; see the reproduction.

The firing strength is within 0.002 of the minimum of the membership degrees and gives the value of the numerical example of the article (\(1.1026 \times 10^{-26}\) for a minimum of \(1.1 \times 10^{-26}\)). The index is neither rounded nor bounded, as in the article.

Two points are not fixed by the article and were chosen here. The gradient treats \(\hat\eta_r\) and \(\hat q_r\) as constants, as the code of the same group does for Ada-softmin. The loss is the mean squared error instead of the sum over the classes halved, a constant factor that Adam compensates.