95–99% of the gradient norm is lost at the LM head during backprop.
Authors show the softmax bottleneck (D≪V) hurts optimization, not just expressivity: raising head rank 32→4096 made 2B-model pretraining converge 16× faster.
https://t.co/oGwL0GSIWM
Linear algebra wasn't built in a day. It took nearly 200 years of ideas, from Leibniz's determinants in 1693 to Peano's formal definition of vector spaces in 1888.
Along the way, Cramer found a way to solve linear systems, Gauss developed elimination methods, and Sylvester gave matrices their name.
One equation captures much of this journey: Ax = b. Behind those three symbols lies a mathematical language that now powers computer graphics, engineering, AI and modern physics.