A fuzzy restricted Boltzmann machine (FRBM) is extended from a restricted Boltzmann machine (RBM) by replacing all the real-valued parameters with fuzzy numbers. A new FRBM that employs the crisp possibilistic mean value of a fuzzy number to defuzzify the fuzzy free energy function is presented. This approach is much clearer and easier to obtain the expression of the defuzzified free energy function and its approximation than the centroid method. Several theorems that discuss the error bounds of the approximation to ensure the rationality and validity are also investigated. Learning algorithms are given for the designed FRBM with symmetric triangular fuzzy numbers (STFNs), asymmetric triangular fuzzy numbers, and Gaussian fuzzy numbers. By appropriately choosing the parameters, a theorem is concluded that all FRBMs with symmetric fuzzy numbers will have identical learning algorithm to that of FRBMs with STFNs. This is illustrated by a case of FRBM with Gaussian fuzzy numbers. Two experiments including the MNIST handwriting recognition and the Bars-and-Stripes benchmark are carried out. The results show that the proposed FRBMs significantly outperform RBMs in learning accuracy and generalization ability, especially when encountering unlearned samples and recovering incomplete images.

中文翻译：

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模糊限制玻尔兹曼机：基于模糊数的清晰可能均值的新颖学习算法


模糊受限玻尔兹曼机（FRBM）是由受限玻尔兹曼机（RBM）扩展而来，用模糊数替换所有实值参数。提出了一种新的FRBM，它利用模糊数的清晰可能平均值来对模糊自由能函数进行去模糊化。该方法比质心法更清晰、更容易获得去模糊化自由能函数的表达式及其近似值。还研究了讨论近似误差范围以确保合理性和有效性的几个定理。给出了所设计的带有对称三角模糊数（STFN）、非对称三角模糊数和高斯模糊数的 FRBM 的学习算法。通过适当选择参数，得出一个定理：所有具有对称模糊数的 FRBM 都将具有与具有 STFN 的 FRBM 相同的学习算法。这可以通过具有高斯模糊数的 FRBM 案例来说明。进行了 MNIST 手写识别和 Bars-and-Stripes 基准测试两个实验。结果表明，所提出的 FRBM 在学习精度和泛化能力方面显着优于 RBM，特别是在遇到未学习的样本和恢复不完整图像时。