Center-Based Classifier

• Center-based classifiers are machine learning models that determine predictions by comparing sample features to learned class-specific centers.
• They employ techniques like multi-prototype modeling and specialized loss functions to enhance intra-class compactness and maximize inter-class separation.
• These methods are applied in diverse tasks such as image recognition, semantic segmentation, and feature selection, offering improved accuracy and interpretability.

A center-based classifier (CC) is a class of machine learning models that assign predictions by comparing sample features to one or more reference centers or prototypes representative of each class in a learned feature space. The fundamental principle is that classification is determined by the similarity (often distance-based or dot-product) between a sample and its class center(s), with auxiliary objectives to optimize both intra-class compactness and inter-class separation. Center-based classifiers underpin modern neural network training objectives, feature selection algorithms, multi-prototype encoding strategies, and specialized designs for structured outputs such as semantic segmentation.

1. Mathematical Foundations and Core Principles

Center-based classifiers formalize classification as a comparison between sample features and class-specific centers. For a $C$-class classification problem, the standard linear CC approach defines centers $\{w_1, \ldots, w_C\} \subset \mathbb{R}^d$ in feature space. For sample feature $x$, the class scores and probabilities are typically

$$f_c(x) = w_c^T x, \quad p(j|x) = \frac{\exp(f_j(x))}{\sum_{c=1}^C \exp(f_c(x))}$$

A CC may employ multiple centers per class, $\{w_{c}^{(k)}\}_{k=1}^{K}$, addressing class heterogeneity. The seminal centroid/center loss (Qi et al., 2017) augments the softmax objective with a penalty promoting intra-class compactness: $$L_{\text{center}} = \sum_{i=1}^n \|x_i - c_{y_i}\|_2^2$$ Contrastive-center loss (Qi et al., 2017) further divides by the sum of non-class center distances, explicitly maximizing inter-class separation: $$L_{ct-c} = \frac{1}{2}\sum_{i=1}^m \frac{ \|x_i - c_{y_i}\|_2^2 }{ \left( \sum_{j \neq y_i} \|x_i - c_j\|_2^2 \right) + \delta }$$ where $\delta$ stabilizes denominators.

CCs can replace trainable parameters (classification weights) with fixed, analytically determined “evenly-distributed” prototypes (Zhu et al., 2019), or sample multi-modal centers from conditional distributions (Zhang et al., 29 Jan 2024, Zhang et al., 2022). Extensions include probabilistic assignments over multiple centers (Wang et al., 2023), hard assignment with feature regularization (Zhang et al., 21 Mar 2025), and explicit feature selection constraints (Calafiore et al., 2019).

2. Extensions for Intra-Class Diversity and Multi-Center Modeling

Single-center models are insufficient when class data are distinctly multi-modal (due to pose, context, or fine-grained attributes). CCs have been augmented to represent each class as a union of multiple centers:

• Multi-center classifiers via conditional Gaussian sampling (Zhang et al., 29 Jan 2024): Instead of static prototypes, each class weight vector $w_c$ defines a Gaussian $\mathcal{N}(w_c, \sigma_c^2 I)$, and sub-centers $w_c^{(k)}$ are sampled during training. All are used for loss calculation; only the mean is retained at test time, incurring no additional parameter cost.
• Fixed Sub-Center (F-SC) method (Zhang et al., 2022): Sub-centers are sampled and fixed before training, decoupling parameter growth from the number of sub-centers and mitigating sub-center collapse. Samples are assigned via softmax over all sub-centers, with an added Euclidean compactness penalty per sub-cluster.
• Multifeature center classifiers (Wang et al., 2023): Learnable multi-prototypes restricted by assignment and regularization strategies, suitable for modeling data with high intra-class variance and small inter-class separation, as demonstrated in SAR ship recognition.

These multi-center strategies significantly improve modeling of intra-class variation, especially for large-scale image and fine-grained recognition tasks (Zhang et al., 29 Jan 2024, Zhang et al., 2022, Wang et al., 2023).

3. Loss Functions for Center-Based Classifiers

Different loss functions have been developed to optimize center-based objectives:

• Contrastive-center loss (Qi et al., 2017): Simultaneously enforces intra-class compactness and inter-class center separation, outperforming pure center loss and softmax on classification and face recognition benchmarks.
• PEDCC-Loss (Zhu et al., 2019): PEDCC (Predefined Evenly-Distributed Class Centroids) assigns fixed, maximally spaced class weights; the loss combines AM-Softmax classification with direct regression to centroids, maximizing inter-class distance and minimizing intra-class variance by design.
• Centralized Coordinate Learning (CCL) (Qi et al., 2018): Jointly normalizes features and classifier weights (placing the latter on a hypersphere), incorporates adaptive angular margin for further separation, and disperses embeddings across the coordinate space for superior class discrimination in face recognition.
• Sparse Center Classifiers (Calafiore et al., 2019): Incorporate hard sparsity constraints selecting a fixed number of discriminative features, solving the selection/classification jointly and efficiently.

4. Specialized Center-Based Classifiers: Structured Prediction and Segmentation

CCs have been adapted for tasks beyond multiclass classification:

• Semantic segmentation via pixel-to-center similarity (Wu et al., 2023): The Class Center Similarity (CCS) layer computes scene-adaptive class centers, evaluating pixel-to-center similarity per image, with specialized losses for intra-class compactness and inter-class separation.
• Remote sensing segmentation with CenterSeg (Zhang et al., 21 Mar 2025): CenterSeg constructs multiple prototypes per class, with patch-level aggregation and regularization on the Grassmann manifold to ensure prototype diversity and inter-class separation, enabling interpretable, robust semantic segmentation under large intraclass variance.

Both methods report superior mIoU and F1 compared to vanilla softmax heads, especially under challenging intraclass variance scenarios.

5. Feature Selection and Linear Center-Based Discriminants

Center-based classifiers have demonstrated competitive performance as interpretable, scalable feature selectors:

• Sparse $\ell_2$ and $\ell_1$ center discriminants (Calafiore et al., 2019): Class center differences are constrained to $k$-sparse vectors; feature selection is performed optimally by identifying features with maximal inter-class centroid separation or reduction in within-class dispersion.
• Linear Centralization Classifier (LCC) (Bonyadi et al., 2017): Finds a linear transformation that centralizes data toward class centroids and maximizes center separation, formulated as a linear program, extending smoothly to non-linear boundaries via kernels.
• Binary tree SVM architectures (Govada et al., 2015): CBTS-SVM employs centroid-based K-means clustering to build hierarchical binary SVMs, drastically reducing the number of required binary classifiers and yielding superior scalability and efficiency over OVO/OVA approaches.

6. Empirical Evidence and Impact

Center-based classifiers have demonstrated systematic advantages:

• Classification tasks: Contrastive-center loss achieves 99.17% on MNIST and 98.68% on LFW face recognition (Qi et al., 2017), while PEDCC-Loss attains superior accuracies on EMNIST, CIFAR100, and LFW (Zhu et al., 2019).
• Segmentation tasks: CCSNet yields higher mIoU on ADE20K and PASCAL Context (Wu et al., 2023). CenterSeg achieves elevated mIoU and F1 across multiple remote sensing benchmarks (Zhang et al., 21 Mar 2025).
• Efficiency: Sparse center classifiers yield exact feature selection with quasi-linear computational complexity, outpacing traditional methods in both speed and accuracy (Calafiore et al., 2019). CBTS-SVM matches OVO in accuracy but with far fewer SVMs and evaluations (Govada et al., 2015).
• Multicenter approaches consistently enhance accuracy, especially for large-scale and fine-grained data, without increasing inference overhead (Zhang et al., 29 Jan 2024, Zhang et al., 2022).

7. Limitations, Open Issues, and Future Directions

Key challenges for center-based classifiers include:

• Determining center count and modality: Properly selecting the number of centers per class is context-dependent; overparameterization may induce redundancy, while underparameterization limits modeling capacity.
• Efficient learning of multiple centers: Fixed sub-centers (Zhang et al., 2022) and sampled Gaussian centers (Zhang et al., 29 Jan 2024) offer practical solutions, but learnable center collapse and memory constraints remain areas for future work.
• Structured predictions for complex tasks: Incorporating adaptive centers and regularization for segmentation requires careful balancing of compactness, diversity, and interpretability (Zhang et al., 21 Mar 2025, Wu et al., 2023).
• Extension to multi-class and multi-label settings: Linear centralization classifiers (Bonyadi et al., 2017) and tree-based approaches (Govada et al., 2015) suggest promising architectures, but require additional theory for natively handling complex output spaces.

A plausible implication is that, as high-dimensional datasets grow in scale and complexity, the utility of center-based classifiers—especially those modeling multi-modal structure and enforcing explicit inter- and intra-class objectives—will continue to expand, especially in domains requiring interpretability, efficiency, and robustness to variance.