Existing learning methods often struggle to balance interpretability and predictive performance. While models like nearest neighbors and non-negative matrix factorization (NMF) offer high interpretability, their predictive performance on supervised learning tasks is often limited. In contrast, neural networks based on the multi-layer perceptron (MLP) support the modular construction of expressive architectures and tend to have better recognition accuracy but are often regarded as black boxes in terms of interpretability. Our approach aims to strike a better balance between these two aspects through the use of a building block based on NMF that incorporates supervised neural network training methods to achieve high predictive performance while retaining the desirable interpretability properties of NMF. We evaluate our Predictive Factorized Coupling (PFC) block on small datasets and show that it achieves competitive predictive performance with MLPs while also offering improved interpretability. We demonstrate the benefits of this approach in various scenarios, such as continual learning, training on non-i.i.d. data, and knowledge removal after training. Additionally, we show examples of using the PFC block to build more expressive architectures, including a fully-connected residual network as well as a factorized recurrent neural network (RNN) that performs competitively with vanilla RNNs while providing improved interpretability. The PFC block uses an iterative inference algorithm that converges to a fixed point, making it possible to trade off accuracy vs computation after training but also currently preventing its use as a general MLP replacement in some scenarios such as training on very large datasets. We provide source code at https://github.com/bkvogel/pfc