Ideas inside computation
Field notes
Follow a mathematical idea into an experiment, then bring the evidence back to the idea.
001 — Computation is also movement
A matrix product contains opportunities for reuse. A reduction requires partial answers to meet. A neighborhood sum turns local relationships into a larger computation.
002 — Keeping values close to the arithmetic
A reduction ends in one value per row, so partial answers have to meet somewhere. A tile decides how often shared memory is read. Two Rust kernels were rearranged around both questions.
Read the second investigation.
003 — What the kernels were short of
A ratio between two register shapes separates shared-memory instructions from arithmetic. The count of resident threads decides how much memory traffic a grid can keep in flight. Both signals chose the next change, and several attempts measured worse than what they replaced.
004 — What the compiler chose
A tile compiler takes over the register tiles, the shared-memory layouts and the barriers. It wins one operation, matches another, and loses the two matmul-shaped ones — and most of the gap in the timing column turns out to be the launch path rather than the kernel. The library’s own search also beat the hand-picked tile ladder.
Read the fourth investigation.
005 — What the loop found, and what answered it
A benchmark that proposes its own variants rediscovered two changes that had been chosen by hand, and kept a third that turned out to be half of a combination faster than either half. Two of its own claims were withdrawn by better measurement, and the hand-written pipeline that landed afterwards went further still.
006 — The load in flight, and the shape Triton uses
A matrix multiply that waits for its own tile loads can be given a second buffer and an asynchronous copy, so the next tile arrives while the current one is multiplied. A layer norm can be rearranged into the shape the other kernel already uses, read from its generated code. The second rewrite narrows the gap and does not close it.
The code and detailed evidence are maintained in GitHub. The Superposition journal follows the motivation behind the work.