Multi-way data analysis has become an essential tool for capturing underlying structures in higher-order data sets where standard two-way analysis techniques often fail to discover the hidden correlations between variables in multi-way data. We propose a multi-objective variational autoencoder (MVA) method for smart infrastructure damage detection and diagnosis in multi-way sensing data based on the reconstruction probability of autoencoder deep neural network (ADNN). Our method fuses data from multiple sensors in one ADNN at which informative features are being extracted and utilized for damage identification. It generates probabilistic anomaly scores to detect damage, asses its severity and further localize it via a new localization layer introduced in the ADNN. We evaluated our method on multi-way datasets in the area of structural health monitoring for damage diagnosis purposes. The data was collected from our deployed data acquisition system on a cable-stayed bridge in Western Sydney and from a laboratory based building structure obtained from Los Alamos National Laboratory (LANL). Experimental results show that the proposed method can accurately detect structural damage. It was also able to estimate the different levels of damage severity, and capture damage locations in an unsupervised aspect. Compared to the state-of-the-art approaches, our proposed method shows better performance in terms of damage detection and localization.