Multiplying monochrome images as matrices: A*B and softmax

We can interpret monochrome images as matrices KxM and MxN and combine them via matrix multiplication. It turns out that results are often visually interesting, especially if we normalize rows of the left-hand-side matrix and columns of the right-hand-side matrix with softmax before taking the product.

It would be great to understand the properties of matrix multiplication better. This seems to be particularly worthwhile because matrix multiplication plays a prominent role in Transformers, a class of machine learning models invented in 2017 and responsible for many recent breakthroughs including GPT-3. Sometimes, the rows of the left-hand-side matrix are softmax-normalized in order to use them as probabilities.

One can interpret a monochrome image as a matrix (the size of the matrix depends on the resolution of the image in question, so one should rescale the image in question as desired). I decided to explore whether matrix products of monochrome images are visually interesting. I used JuliaImages packages, Julia LinearAlgebra facilities and Julia Jupyter notebooks.

I looked at standard Julia test images, such as "mandrill" and "jetplane", and discovered that there is plenty of visually interesting information in their matrix products. I used the scaling of pixel values which is also used by ImageView.imshow() methods.

It turned out that the matrix products were particularly informative and had a lot of visible fine structure, if one softmax-normalized rows of the left-hand-side matrix and columns of the right-hand-side matrix before taking the product. The images looked slightly toned-down and striped after normalization, but not too different visually. However the products were drastically different. Note that in Transformer models one usually applies softmax only on one side, but this turns out to be insufficient for our visual exploration of matrix products.

I like the resulting images as visual art, and I think this might point to some interesting novel ways to obtain visual art by mathematical transformations.

I also hope this might eventually be of help as people try to achieve better understanding and more fine-grained control of our machine learning models.

The markdown file commenting the Julia notebook and elaborating on machine learning connections is posted at

After this proposal was submitted, I have explored composing matrix multiplications with other image transformations. The resulting compact neural machines produce visually interesting results.

I have conducted first experiments in solving machine learning problems formulated in terms of those compact machines taking advantage of flexibility of differentiable programming in Julia Flux.

I have created a repository containing materials relevant to this poster: