Abstract In various pattern classification problems, semi-supervised learning methods have shown its effectiveness in utilizing unlabeled data to yield better performance than some supervised and unsupervised learning methods. Semi-supervised discriminant embedding (SDE) is a semi-supervised extension of local discriminant embedding (LDE). However, when dealing with high dimensional data, SDE often suffers from the small-sample-size (SSS) problem. In order to settle this problem, an exponential semi-supervised discriminant embedding (ESDE) method was proposed in [ F. Dornaika, Y. EI Traboulsi . Matrix exponential based semi-supervised discriminant embedding for image classification, Pattern Recognition, 61 (2017): 92–103], which makes use of the tool of matrix exponential. Despite its high discriminative ability, the computational overhead of ESDE is very large for high dimensional data. In order to cure this drawback, the first contribution of this paper is to propose a fast implementation on the ESDE method. The key is to equivalently transform the large matrix problem of size d into a much smaller one of size n, where d is the data dimension and n is the number of training samples, with d ≫ n in practice. On the other hand, in many real world applications, it is likely that whole labeled training set is unavailable beforehand, and the training data is obtained incrementally. Many incremental semi-supervised learning methods have been proposed to deal with this problem, to the best of our knowledge, however, there are no incremental algorithms for matrix exponential discriminant methods till now. To fill in this gap, the second contribution of this paper is to propose incremental ESDE algorithms for incremental learning problems. Numerical experiments on some real-world data sets show the numerical behavior of the proposed algorithms.

中文翻译：

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指数半监督判别嵌入的快速增量算法

摘要 在各种模式分类问题中，半监督学习方法在利用未标记数据方面表现出比一些监督和无监督学习方法更好的性能。半监督判别嵌入（SDE）是局部判别嵌入（LDE）的半监督扩展。然而，在处理高维数据时，SDE 经常会遇到小样本（SSS）问题。为了解决这个问题，[F. Dornaika, Y. EI Traboulsi 中提出了一种指数半监督判别嵌入 (ESDE) 方法。用于图像分类的基于矩阵指数的半监督判别嵌入，模式识别，61 (2017): 92–103]，它利用了矩阵指数工具。尽管具有很高的判别能力，对于高维数据，ESDE 的计算开销非常大。为了解决这个缺点，本文的第一个贡献是提出了一种对 ESDE 方法的快速实现。关键是将大小为 d 的大矩阵问题等效地转换为大小为 n 的小得多的问题，其中 d 是数据维度，n 是训练样本的数量，在实践中 d ≫ n。另一方面，在许多实际应用中，很可能整个标记训练集事先都不可用，而训练数据是增量获得的。据我们所知，已经提出了许多增量半监督学习方法来解决这个问题，但是，到目前为止，还没有用于矩阵指数判别方法的增量算法。为了填补这个空白，本文的第二个贡献是针对增量学习问题提出了增量ESDE算法。对一些真实世界数据集的数值实验显示了所提出算法的数值行为。