Essence of Linear Algebra (18): Frontiers and Summary

Wed, 30 Apr 2025 09:00:00 +0000

We have walked the long road of linear algebra together. We started with arrows in the plane and ended at the gates of quantum computers, the inner workings of large language models, and the topology of data clouds. The remarkable thing — the thing this series has tried to make visible — is that the same handful of ideas keeps coming back. A vector is a state. A matrix is a transformation. A decomposition is the structure hiding inside the transformation. A norm tells you when you can trust your computation. Once you internalise that loop, every “frontier” looks less like a foreign country and more like another dialect of a language you already speak.

PDE and ML (8): Reaction-Diffusion Systems and Graph Neural Networks

Wed, 14 Aug 2024 09:00:00 +0000

Anyone who has trained a deep GNN has seen it collapse — past a dozen or so layers, every node’s embedding becomes nearly identical and the model goes mush. There is a name for this — over-smoothing — and the underlying math is surprisingly clean: GNN message passing is essentially a diffusion equation on the graph, and diffusion’s long-time behavior is to flatten everything to a constant.

Graph Neural Networks on Chen Kai Blog

Essence of Linear Algebra (18): Frontiers and Summary

PDE and ML (8): Reaction-Diffusion Systems and Graph Neural Networks