用凸凹算法训练形态感知机，解决非可微问题。
Morphological Perceptron with Competitive Layer: Training Using Convex-Concave Procedure
- 将形态感知机训练转为凸凹问题迭代求解
- 通过线性规划子问题实现多类分类优化
- 适合需避免梯度依赖的神经网络场景
形态感知机是一种基于数学形态学算子的多层前馈神经网络。针对多类分类任务，通过在标准形态架构中引入胜者为王输出层，构建了带竞争层的形态感知机（MPCL）。由于形态算子不可微，传统基于梯度的方法不适用，因此常采用不依赖梯度的替代策略。本文提出使用凸凹过程（CCP）训练MPCL网络，将训练问题表述为凸函数之差（DC），并通过迭代方式求解，每次生成线性规划子问题。计算实验表明，该方法在处理MPCL网络的分类任务中具有有效性。
原文摘要 · Abstract (English)
A morphological perceptron is a multilayer feedforward neural network in which neurons perform elementary operations from mathematical morphology. For multiclass classification tasks, a morphological perceptron with a competitive layer (MPCL) is obtained by integrating a winner-take-all output layer into the standard morphological architecture. The non-differentiability of morphological operators renders gradient-based optimization methods unsuitable for training such networks. Consequently, alternative strategies that do not depend on gradient information are commonly adopted. This paper proposes the use of the convex-concave procedure (CCP) for training MPCL networks. The training problem is formulated as a difference of convex (DC) functions and solved iteratively using CCP, resulting in a sequence of linear programming subproblems. Computational experiments demonstrate the effectiveness of the proposed training method in addressing classification tasks with MPCL networks.
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