Categorical Foundations for Cute Layouts
21 hours ago
- #Memory Layouts
- #Mathematical Theory
- #GPU Programming
- GPU performance depends on memory data storage and access, requiring multi-dimensional to one-dimensional mapping known as layouts.
- CUTLASS introduced CuTe layouts with shape and stride tuples, featuring a 'layout algebra' for operations like composition and complementation.
- A new mathematical theory connects CuTe layouts to categories and operads, introducing a graphical calculus for layout operations.
- Tractable layouts, including row-major, column-major, compact, projections, and dilations, are practical and can be encoded by diagrams.
- Layout functions are crucial invariants, derived from a realization functor in the category Tuple, linking layouts to finite sets.
- Key layout operations like coalesce, complement, and composition have analogues in Tuple, with theorems ensuring compatibility.
- Nested layouts extend the framework, with a category Nest for nested tuples and morphisms, supporting similar operations as Tuple.
- A composition algorithm for tractable layouts uses mutual refinements, pullbacks, and pushforwards to compute layout compositions.
- Connections to operad theory provide a mathematical foundation, with Tuple as a subcategory of operad categories and profiles forming an operad.
- Span categories incorporate refinements and tuple morphisms, offering a sophisticated framework for layout operations and their mathematical properties.