Restructure the maths before optimizing the code
The biggest wins are algorithmic. A well-tiled implementation of the wrong algorithm loses to a restructured one. Work through this before touching tile sizes.
Questions to ask
- Can I avoid materializing the large intermediate? (attention's N×N score matrix)
- Can a sequential dependence become an associative reduction? (scan, recurrence)
- Can I trade compute for memory, or recompute instead of store? (checkpointing, backward recompute)
- Can I change what stays resident? (block the loop so the working set fits smem/registers)
- Does the output have structure the algorithm ignores? (sparsity, low rank, symmetry, banding)
Catalog
| transform | what it buys | seen in |
|---|---|---|
| online softmax (running max + rescaled sum) | O(N²)→O(N) memory, single pass | FlashAttention |
| chunked/blocked recurrence | turns a sequential scan into a sequence of GEMMs | Mamba-2 SSD, Gated DeltaNet |
| WY representation | expresses a product of Householder-like updates as one matmul | DeltaNet family |
| associative / Blelloch scan | parallelizes a prefix dependence | cumsum, linear attention |
| radix partitioning | avoids a full sort for top-k / grouping | radix top-k, MoE token permute |
| counting sort + exclusive scan | stable grouping in two passes, no comparison sort | MoE dispatch |
| low-rank factorization | shrinks a GEMM's K | LoRA, MLA's compressed KV, PowerSGD |
| absorb / fold weights | removes an entire GEMM by folding it into a neighbour | MLA V-absorb |
| split-K / stream-K | fills the machine when M·N is too small for the SM count | skinny GEMMs |
| recompute in backward | trades FLOPs for HBM traffic | activation checkpointing |
| two-pass → one-pass | fuse statistics with the transform | fused LayerNorm/RMSNorm |
| persistent kernel | removes launch and pipeline-drain cost between stages | megakernels |
| symmetry exploitation | halves the work | X @ X.T is symmetric (Muon/Newton-Schulz) |
Sanity check before implementing
Compute the arithmetic intensity of the restructured algorithm and re-run the roofline. If the restructure does not move you toward the compute-bound side (or cut absolute traffic), it is not actually a win — write it down in the ledger and move on.