Dense Matrices Are Alike; Sparse Matrices Are Sparse in Their Own Way: A Structure-Adaptive Tile Cholesky Factorization

Profiling shows attention computation is a bottleneck in your self-hosted service.

What you would change

Check the supported GPU, tensor shapes, and precision, then replace only the relevant kernel.

Check the supported GPU, tensor shapes, and precision, then replace only the relevant kernel.

Compare against

The existing kernel inside the same complete serving pipeline.

1. Locate the paper's implementation and check its license, model support, data needs, and hardware requirements. Stop here if you cannot obtain the required artifacts.
2. Save a small set of representative product tasks and expected outcomes. Keep evaluation tasks out of any training or tuning, and choose quality and cost limits before running.
3. Run the baseline and the proposed change on identical inputs. Keep other settings fixed and repeat paired runs to expose variation.
4. Save per-task outputs, failures, timings, and all usage costs. Review regressions before expanding the test.

What to measure

Numerical differences, end-to-end latency, throughput, and memory at realistic shapes.

When to keep it

Keep the change only if it meets your preselected quality and latency limits and improves the metric you care about after setup and operating costs. Otherwise retain the baseline. This is an exploration plan, not a validated recommendation.

recommendation

watch

research area

attention and kernels

technical change

Structure-adaptive tile Cholesky factorization

published

2026-09-24

This is a numerical linear algebra solver for repeated sparse SPD factorization, not an ML inference or training optimization, so it is only relevant to the reader if their production workload actually includes such factorizations, for example Bayesian spatial modeling with INLA.

1. claim 01Paperraft inference
   Read the brief interpretation

   The method replaces the fixed data structure used by conventional sparse direct Cholesky solvers with a per-matrix, per-tile selector that routes computation to dense, semisparse (active-column), or sparse tile representations on one static shared-memory schedule. It costs a one-time symbolic analysis of the factor's sparsity pattern, which only amortizes when the same pattern is factorized repeatedly (it pulls ahead by the third factorization), and the GPU path is limited to the dense regime with the factor resident on the device. It can fail to pay off when sparsity patterns change between factorizations, when matrices are small enough that analysis overhead dominates, or when the workload is inference-centric deep learning rather than SPD systems such as INLA or spatial statistics.

   This is a Paperraft inference. It is not pinned to a verified PDF passage.

   Inspect the source
2. claim 02Paperraft inference

   Summed over 60 SPD matrices, the adaptive selector beats the best fixed single-structure mode by 1.6 to 2.6x and all compared solvers (MUMPS, PaStiX, CHOLMOD, symPACK, PARDISO) by 1.8 to 12.5x on Intel and 2.6 to 10.1x on AMD; a single-A100 dense route is 1.2 to 6.3x faster than the faster CPU node.

   This is a Paperraft inference. It is not pinned to a verified PDF passage.

   Inspect the source