A Technical Note on a Construction Method for Neural Networks Without Activation Functions

This technical note presents a neural network (NN) construction method that achieves nonlinear approximation without using activation functions. The key idea is to fully categorize all input attributes and encode them as independent one-hot vectors, thereby transforming the input domain into a finite discrete set. Within such a domain, even a shallow linear network can realize arbitrary mappings. The proposed architecture assigns each category its own weight and its own bias, where the bias contributes even when the corresponding category is not selected. This enables flexible, pattern‑dependent responses beyond those of standard one-hot linear models. Although conceptually related to lookup‑table mappings and indicator‑based methods, this note provides a concise formulation within the NN framework. To illustrate the method, an implementation example is presented for anomaly detection on monochrome images of the traditional Japanese Umebachi crest. Four attributes describing conformity to a reference pattern were discretized into 50 categories each, producing a 200‑dimensional input vector composed of four 50‑dimensional one‑hot blocks. A network with only one hidden neuron and no activation functions achieved 100% accuracy on 10,000 evaluation images. This result demonstrates that complete categorization of input attributes enables shallow linear networks to approximate nonlinear functions effectively within finite discrete input spaces.