Robust PCA is a popular anomaly detection technique and has been widely used in many applications. Although Robust PCA is promising, it is usually designed in a two-order matrix form, which is inferior to the tensor that can capture multilinearity features of data. Moreover, the detection accuracy under Robust PCA further suffers due to its sensitivity to the rank parameter which is hard to set in practice and the limitation of PCA method in capturing the non-linear feature in the data. To address the issues, we propose a Robust Tensor Convolutional Autoencoder (RTCAE) where the autoencoder instead of SVD is exploited to recover the normal data from the corrupted measurement tensor data. However, directly exploiting deep autoencoder may suffer from the problem of high memory consumption and computation overhead due to the large number of parameters used in autoencoder. To make our anomaly detection lightweight, we further design a Light Convolutional Autoencoder (LightCAE) which contains a compressed autoencoder by exploiting tensor factorization to largely compress the parameters while significantly reducing the computation complexity. We conduct extensive experiments on three real data traces to compare the performance of our proposed schemes (RTCAE and lightCAE) with that of seven baseline algorithms. The experiment results demonstrate that our proposed RTCAE achieves the highest anomaly detection accuracy. Moreover, our LightCAE requires over 60 times smaller memory storage than that required in RTCAE while achieving the similar anomaly detection accuracy.