  Gated Linear Units (GLUs) have become a common building block in modern
foundation models. Bilinear layers drop the non-linearity in the "gate" but
still have comparable performance to other GLUs. An attractive quality of
bilinear layers is that they can be fully expressed in terms of a third-order
tensor and linear operations. Leveraging this, we develop a method to decompose
the bilinear tensor into a set of sparsely interacting eigenvectors that show
promising interpretability properties in preliminary experiments for shallow
image classifiers (MNIST) and small language models (Tiny Stories). Since the
decomposition is fully equivalent to the model's original computations,
bilinear layers may be an interpretability-friendly architecture that helps
connect features to the model weights. Application of our method may not be
limited to pretrained bilinear models since we find that language models such
as TinyLlama-1.1B can be finetuned into bilinear variants.
