Deep Neural Networks (DNNs) have achieved remarkable success in addressing many previously unsolvable tasks. However, the storage and computational requirements associated with DNNs pose a challenge for deploying these trained models on resource-limited devices. Therefore, a plethora of compression and pruning techniques have been proposed in recent years. Low-rank decomposition techniques are among the approaches most utilized to address this problem. Compared to post-training compression, compression-promoted training is still under-explored. In this paper, we present a theoretically-justified novel approach, termed Low-Rank Induced Training (LoRITa), that promotes low-rankness through the composition of linear layers and compresses by using singular value truncation. This is achieved without the need to change the structure at inference time or require constrained and/or additional optimization, other than the standard weight decay regularization. Moreover, LoRITa eliminates the need to (i) initialize with pre-trained models and (ii) specify rank selection prior to training. Our experimental results (i) demonstrate the effectiveness of our approach using MNIST on Fully Connected Networks, CIFAR10 on Vision Transformers, and CIFAR10/100 on Convolutional Neural Networks, and (ii) illustrate that we achieve either competitive or SOTA results when compared to leading structured pruning methods in terms of FLOPs and parameters drop.