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Change-point detection (CPD), which detects abrupt changes in the data distribution, is recognized as one of the most significant tasks in time series analysis. Despite the extensive literature on offline CPD, unsupervised online CPD still suffers from major challenges, including scalability, hyperparameter tuning, and learning constraints. To mitigate some of these challenges, in this paper, we propose a novel deep learning approach for unsupervised online CPD from multi-dimensional time series, named Adaptive LSTM-Autoencoder Change-Point Detection (ALACPD). ALACPD exploits an LSTM-autoencoder-based neural network to perform unsupervised online CPD. It continuously adapts to the incoming samples without keeping the previously received input, thus being memory-free. We perform an extensive evaluation on several real-world time series CPD benchmarks. We show that ALACPD, on average, ranks first among state-of-the-art CPD algorithms in terms of quality of the time series segmentation, and it is on par with the best performer in terms of the accuracy of the estimated change-points. The implementation of ALACPD is available online on Github\footnote{\url{https://github.com/zahraatashgahi/ALACPD}}.