• First, we 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. Numerical results demonstrate that the proposed algorithms are at least two orders of magnitude faster than their original counterparts, with no recognition accuracy lost. • Second, to the best of our knowledge, 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. • Third, the proposed fast implementation strategy and the incremental techniques also apply to other exponential discriminant analysis methods. 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. [ABSTRACT FROM AUTHOR]