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ECA-BLS: An Efficient Complex-Augmented Broad Learning System

A. Rahaman, A. Quadir, M. Sajid, M. Akhtar, M. Tanveer

arXiv:2608.29763v1cs.LG

TL;DR

Existing real-valued BLS models do not exploit complete complex-domain second-order statistics, motivating a complex augmented extension. The paper develops CA-BLS with phase encoding and conjugate augmentation, then derives ECA-BLS as an equivalent real-domain reformulation. Across benchmark datasets, ECA-BLS is reported to outperform classical BLS and recent randomized neural networks while preserving CA-BLS’s decision function.

  • Problem

    Existing BLS variants are confined to real-valued representations and cannot exploit complete second-order statistics, including covariance and pseudo-covariance, in complex-domain data.

  • Method

    CA-BLS phase-encodes real-valued inputs and uses complex conjugate augmentation, while ECA-BLS reformulates the model with equivalent real-valued computations.

  • Results

    ECA-BLS consistently outperforms classical BLS and recent state-of-the-art randomized neural networks across diverse benchmark datasets.

  • Takeaways & Limitations

    Complex augmented second-order modeling adds a previously missing representational dimension to BLS while ECA-BLS preserves the exact CA-BLS decision function.

Abstract

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Broad Learning System (BLS) is an efficient alternative to deep architectures due to its fast training, analytical learning, and strong generalization under limited data. However, existing BLS variants are confined to real-valued representations, restricting their ability to capture nonlinear interactions and second-order statistical dependencies inherent in real-world data. Notably, no prior BLS model fully exploits the complete second-order statistics that naturally emerge when data are embedded in the complex domain. To address this limitation, this paper introduces the first complex augmented Broad Learning System (CA-BLS), which transforms real-valued inputs into phase-encoded complex representations and adopts widely linear modeling to jointly leverage covariance and pseudo-covariance information via complex conjugate augmentation. This enables effective modeling of latent nonlinearities, coherence structures, and second-order dependencies inaccessible to conventional BLS formulations. To mitigate the additional computational cost of complex augmentation, an Efficient Complex Augmented BLS (ECA-BLS) is further developed, reformulating CA-BLS entirely in the real domain while preserving its exact decision function, achieving up to 75\% fewer multiplications and over 60\% fewer additions. A rigorous theoretical analysis proves the mathematical equivalence between CA-BLS and ECA-BLS, ensuring zero theoretical loss. Extensive experiments on 26 benchmark datasets from the UCI and KEEL repositories demonstrate that ECA-BLS consistently outperforms classical BLS and recent state-of-the-art randomized neural networks in accuracy, average rank, and statistical significance, establishing augmented second-order modeling as a critical and previously missing dimension of BLS research.

I. INTRODUCTION AND MOTIVATION

BLS offers fast analytical learning but real-valued formulations struggle to represent complex nonlinear and second-order structure. This paper introduces CA-BLS and an equivalent efficient real-domain formulation, ECA-BLS.

  • Existing Broad Learning Systems: BLS uses a simple three-layer architecture and randomized learning to reduce training time and hardware requirements.Its layers are feature learning, enhancement, and output.
  • Existing Broad Learning Systems: Standard BLS hidden representations may inadequately capture complex nonlinear structures and higher-order dependencies because they rely on random projections.This limits representational capacity despite computational efficiency.
  • Motivation for Complex Augmentation: Complex-valued models can represent phase and amplitude interactions, while complete complex second-order statistics require both covariance and pseudo-covariance.Widely linear augmented models jointly process a complex signal and its conjugate.
  • Proposed Contributions: CA-BLS integrates complex augmented representations into feature and enhancement layers to capture covariance, pseudo-covariance, and nonlinear structures.The model uses complex weights and analytic activation functions.
  • Proposed Contributions: ECA-BLS replaces most complex-valued operations with equivalent real-valued computations while preserving CA-BLS modeling capability.The paper provides a mathematical equivalence proof and reports comprehensive evaluations against recent state-of-the-art models.

II. NOTATION

The paper represents training data as feature and label matrices, then phase-encodes normalized real-valued features into complex inputs for the BLS framework. CA-BLS additionally incorporates conjugate representations to model complete second-order statistics.

  • Notation: The training data contain D samples, B features, and C output classes, arranged as Z ∈ R^D×B and label matrix X ∈ R^D×C.The raw samples are indexed by j from 1 to D.
  • Transformation of Data from Real Field to Complex Field: Each normalized real-valued feature z_ij ∈ [0, 1] is independently transformed into a complex-valued representation using phase encoding.The transformation is applied across samples and features.
  • Transformation of Data from Real Field to Complex Field: Phase encoding maps features onto the complex unit circle, providing bounded magnitude and nonlinear angular separation without increasing input dimensionality.The resulting dataset cxZ ∈ C^D×B feeds the complex-valued BLS framework.
  • Complex Augmented BLS: Because phase-encoded representations are generally non-circular, CA-BLS incorporates conjugate representations to exploit covariance and pseudo-covariance jointly.This augmentation is intended to use complete second-order statistics.
  • Complex Augmented BLS: CA-BLS maps the phase-encoded complex input into feature and enhancement layers using randomly generated complex weights.The target labels remain represented by the real-valued matrix X.

1) Feature Learning Segment:

The model builds complex feature and enhancement representations through grouped random projections, concatenates them, and augments the resulting hidden matrix with conjugate information to capture complete second-order statistics.

  • Feature learning: Feature groups apply complex-valued activations to randomly weighted inputs and concatenate their outputs into an augmented feature matrix.Each group uses a randomly generated weight matrix, bias matrix, and complex-valued activation function.
  • Enhancement: Enhancement groups further project the augmented feature matrix through random complex transformations and nonlinear activation.The enhancement outputs are concatenated before forming the hidden-layer representation.
  • Hidden representation: The feature and enhancement outputs are concatenated to form the complex hidden-layer output matrix.
  • Augmented modeling: Conjugate augmentation incorporates both covariance and pseudo-covariance information into the hidden representation.The augmented matrix is used in the regularized least-squares output-learning problem.
  • Output learning: The output weights are obtained by regularized least squares using a closed-form solution involving Hermitian transposition and an identity matrix.The regularization parameter controls the closed-form output-weight solution.

C. Efficient Complex Augmented Broad Learning System (ECA-BLS)

ECA-BLS reformulates complex conjugate augmentation with real-valued computations to reduce the cost of CA-BLS while retaining its modeling structure.

  • Motivation: CA-BLS augmentation doubles hidden nodes and requires complex-valued matrix inversion, increasing computational cost.
  • Reformulation: ECA-BLS uses phase encoding and complex BLS representations, then replaces explicit complex augmentation with an equivalent real-valued augmentation strategy.The real-valued representation implicitly captures conjugate information.
  • Real-domain representation: The real-valued augmented representation is derived from the real and imaginary parts of the complex hidden matrix H.The reformulation preserves the conjugate information without explicitly constructing the complex augmented matrix.
  • Output learning: ECA-BLS output weights are learned by solving a real-valued regularized problem with a closed-form solution.The relevant algorithm and complexity analyses are provided in the supplementary material.

D. Theoretical Justification of Equivalence between Proposed CA-BLS and ECA-BLS

Theoretical analysis establishes that CA-BLS and ECA-BLS are mathematically equivalent while ECA-BLS reduces the cost of complex computation. Experiments on 26 datasets show superior accuracy, average rank, and statistically significant differences among models.

  • Equivalence proof: CA-BLS and ECA-BLS are algebraically isomorphic through a bijective transformation between complex augmented and real expanded feature spaces.Both formulations optimize the same objective in different coordinate systems.
  • Equivalence proof: ECA-BLS preserves the full phase-amplitude relationships of the widely linear model through cross-correlations between separated real and imaginary channels.The real-domain reformulation therefore does not discard the complex characteristics used by CA-BLS.
  • Equivalence proof: λa = 2λr guarantees identical CA-BLS and ECA-BLS outputs and the same decision boundary with zero theoretical loss in accuracy.This condition is required by the equivalence theorem.
  • Computational efficiency: ECA-BLS reduces computational cost by replacing complex arithmetic with optimized real-valued linear algebra.A complex multiplication requires four real multiplications and two additions, increasing the cost avoided by ECA-BLS.
  • Empirical evaluation: 84.5337% average classification accuracy was achieved by ECA-BLS across 26 UCI and KEEL benchmark datasets, exceeding KRPBLS at 80.9% and GEIB at 80.7861%.All remaining state-of-the-art methods achieved average accuracies below 80%.
  • Statistical evaluation: 2.9231 lowest average rank was achieved by ECA-BLS, while GEIB and F-BLS obtained average ranks of 4.0577 and 4.6923, respectively.The ranking analysis is presented as an aggregated comparison across all datasets.
  • Statistical evaluation: 2.2564 exceeded the 5% critical value of 2.1595 in the Friedman F-statistic, rejecting equal performance among the seven models evaluated on 26 datasets.The corresponding chi-squared statistic was 12.9142.

IV. CONCLUSION AND FUTURE WORK

The paper introduces complex augmented modeling to BLS and derives an equivalent real-domain implementation. Theoretical guarantees and benchmark analyses support ECA-BLS as an expressive and computationally practical BLS formulation.

  • Conclusion: CA-BLS introduces complex augmented modeling into BLS to jointly use covariance and pseudo-covariance information.The formulation targets nonlinear structures and coherence patterns inaccessible to conventional real-valued BLS.
  • Conclusion: ECA-BLS preserves CA-BLS's exact decision function through a rigorously proven real-domain reformulation.This formulation is intended to improve computational efficiency and scalability without changing the model's output.
  • Conclusion: Experiments with Friedman and win–tie–loss analyses consistently report superior generalization, robustness, and stability for ECA-BLS over classical BLS and recent variants.The evidence comes from diverse benchmark datasets.

S.I. RELATED WORKS

The related-work material describes BLS as a flat, analytically trained alternative to deep networks and reviews its feature, enhancement, and output components. It also presents the equivalence theorem underlying the proposed real-domain formulation.

  • BLS background: BLS is a flat network that expands width rather than depth, using random connections and analytically computed output weights.Its four components are the input, feature learning, enhancement, and output layers.
  • Feature learning: The feature learning layer applies nonlinear activation to randomly projected inputs, and concatenates the outputs of multiple feature groups into an augmented feature matrix.Each group contains b nodes, with randomly generated weights and bias matrices.
  • Enhancement layer: The enhancement layer further projects the augmented feature matrix through random mappings and nonlinear activation before concatenating enhancement-group outputs.There are c enhancement groups, each containing d nodes.
  • Output layer: The output layer combines feature and enhancement representations, with output weights obtained analytically using the Moore–Penrose pseudoinverse.The pseudoinverse of E is denoted E†.
  • Equivalence theorem: The equivalence theorem assumes distinct training samples and shared hidden representations, and requires λa = 2λr for identical CA-BLS and ECA-BLS outputs.The proof uses augmented representations, a linear transformation, and corresponding output-weight solutions.

S.III ALGORITHM OF PROPOSED ECA-BLS

The proposed-model section identifies the algorithm of ECA-BLS, with its detailed procedure provided as the model's algorithm.

  • Proposed algorithm: The paper presents an algorithm for the proposed ECA-BLS model.The supplied passage marks the beginning of the proposed-model algorithm.

S. IV COMPLEXITY ANALYSIS OF PROPOSED MODELS

ECA-BLS reformulates the complex augmented computation using real-valued operations, yielding lower dominant training complexity while preserving the same decision function as CA-BLS.

  • Real-domain reformulation: CA-BLS instead uses a complex augmented hidden matrix with Hermitian products and complex-valued matrix inversion.These operations motivate the real-domain reformulation analyzed for ECA-BLS.
  • Real-domain reformulation: ECA-BLS constructs a real-augmented hidden matrix from the real and imaginary parts of the complex hidden matrix.The algorithm describes phase encoding, complex feature and enhancement outputs, and real augmentation before output-weight computation.
  • Complexity reduction: 75% fewer dominant cubic multiplications result from reducing CA-BLS complexity from 32n^3 to 8n^3.The reduction is exact for the dominant multiplication term.
  • Complexity reduction: 66.7% fewer dominant cubic additions result from reducing the corresponding complexity from 24n^3 to 8n^3.The addition reduction is approximate.
  • Complexity reduction: A fourfold reduction in computational complexity is achieved because the cubic term dominates overall training cost.The identical decision function is preserved by the theoretical equivalence analysis.

S.VII. WIN–TIE–LOSS (W–T–L) SIGN TEST

The Win–Tie–Loss sign test evaluates whether ECA-BLS wins significantly across the benchmark datasets. ECA-BLS reaches the significance threshold against several strong baselines, while overall pairwise outcomes favor it.

  • Sign-test criterion: 18 wins are required for statistical significance across 26 datasets under the stated sign-test criterion.Ties are divided evenly before applying the threshold.
  • Significant comparisons: ECA-BLS achieves at least 18 wins against F-BLS, GEIB, and H-ELM, demonstrating statistically significant superiority under this criterion.The threshold is not reached against a few other baselines.
  • Overall comparison: The pairwise W–T–L outcomes consistently favor ECA-BLS across the compared baseline models.Each table entry reports wins, ties, and losses for the row model against the corresponding column model.

S.VIII. SENSITIVITY ANALYSIS

The sensitivity analysis examines regularization and structural parameters across representative datasets. It identifies stable or favorable parameter ranges while showing that d and parameter interactions remain dataset dependent.

  • Regularization sensitivity: λr stabilizes accuracy at values ≥0.01, whereas values above 1000 degrade accuracy across the examined datasets.Subsequent experiments select λr from [0.01, 1000].
  • Structural-parameter sensitivity: a improves accuracy up to an optimum at a ≥15, after which performance declines beyond a =45.The selected range for a is [15, 45].
  • Structural-parameter sensitivity: b achieves optimal performance consistently within the range [7, 11].This range is adopted for the experiments.
  • Structural-parameter sensitivity: d has dataset-dependent optimal values, unlike the preceding parameters.Its sensitivity is evaluated across four representative datasets.
  • Parameter interactions: The optimal regions for (a,d) and (b,d) interactions differ across datasets, making systematic hyperparameter tuning necessary.The joint influence of a and b is also examined, with medium-to-high values producing favorable performance.
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