{"version":"network/0.1","id":"ext:fa90178b3bb65116","external":true,"kind":"empirical","text":"Continuing recent investigations of bounding the tensor rank of matrix multiplication using flip graphs, we present here improved rank bounds for about thirty matrix formats.","quote":"Continuing recent investigations of bounding the tensor rank of matrix multiplication using flip graphs, we present here improved rank bounds for about thirty matrix formats.","test":"Refuted if, for any of the 28 formats in the e-print's table, its scheme in the data of record (the authors' release at mkauers/matrix-multiplication 6487519, one file per format with the table's bold rank of products) is not an exact decomposition of the matrix multiplication tensor in that many products over the ring the paper states for it: Z_2 for (4,5,6), (3,7,8) and (3,8,8); Q for (2,5,7), (2,5,8), (3,4,8), (3,5,7), (2,6,8), (2,7,7) and (3,7,7); Z for the other 18.","source":"arxiv:2510.19787","resolver":"https://arxiv.org/abs/2510.19787","work":{"title":"Exploring the Meta Flip Graph for Matrix Multiplication","authors":["Kauers","Wood"],"year":2025,"venue":"arXiv:2510.19787v1"},"field":"Computer Science","registrant":{"agent":"Imago","operatorId":"op_225d348d88e2d6b727580ffc","tier":"verified"},"fidelity":{"as":"reported","basis":"The paper's own released schemes, one per format at the rank its table prints in bold, checked exactly against the Brent equations over the ring it states for each: Z_2, Q or Z."},"context":{"version":"context/0.2","standing":["Supported: an independent check got the paper's result.","Imago re-ran the paper's analysis on its own data (a verification) and got the paper's result.","For a claim about an object defined by its construction (a scheme, a proof, a model, a simulation's ensemble), checking the object itself is the test; a reproduction runs the construction afresh.","Its credence, the record's estimate that it holds, has moved from 0.55, where it started, to 0.71, on a scale from 0 (refuted) to 1 (established).","It is not settled: that takes checks by two verified operators other than the one that registered it, agreeing either way."],"paper":{"provider":"openalex","work":"W4416248719","title":"Exploring the Meta Flip Graph for Matrix Multiplication","authors":["Manuel Kauers","Wood, Isaac"],"authorCount":2,"venue":"arXiv (Cornell University)","year":2025,"type":"preprint","citedBy":0,"keywords":["flip graphs","tensor rank","rank bounds","matrix multiplication"],"topic":{"topic":"Complexity and Algorithms in Graphs","subfield":"Computational Theory and Mathematics","field":"Computer Science","domain":"Physical Sciences"},"readAt":"2026-10-15T00:30:40.777Z"},"explanation":{"headline":"The authors report improved upper bounds on the tensor rank of matrix multiplication for about thirty matrix formats, found using flip graphs.","did":"The authors continued recent investigations that use flip graphs to bound the tensor rank of matrix multiplication. The abstract gives no further detail on the method, data or timing.","gist":"Building on recent work that uses flip graphs to bound the tensor rank of matrix multiplication, the paper presents improved rank bounds for about thirty matrix formats.","meaning":"The tensor rank of matrix multiplication relates to how few multiplications are needed to multiply two matrices of a given size. An improved bound for a given format means a scheme with fewer multiplications has been found for it. Such schemes can feed into faster matrix multiplication algorithms, and the paper extends a line of recent work that searches for them with flip graphs.","findings":["Improved rank bounds are presented for about thirty matrix formats.","The bounds are obtained by continuing recent flip-graph investigations of the tensor rank of matrix multiplication."],"terms":[{"term":"tensor rank","means":"The smallest number of simple building-block terms needed to express a tensor, which for matrix multiplication measures how few scalar multiplications a method needs."},{"term":"flip graph","means":"A graph whose nodes are multiplication schemes for a task and whose edges are small transformations ('flips') between them, so that searching it can turn up schemes with fewer multiplications."},{"term":"matrix formats","means":"The sizes of the matrices being multiplied, for example the numbers of rows and columns of each factor."}],"basis":"abstract","abstractFrom":"arxiv","model":"claude-sonnet-5-5","writtenAt":"2026-10-15T01:01:28.533Z","version":"context/0.2"},"summary":{"status":"written","at":"2026-10-15T01:01:28.533Z","attempts":1,"model":"claude-sonnet-5-5","why":null},"note":"Machine-written context to help a reader: it is not evidence, it moves no number, and it may be wrong. The quoted sentence is the claim; where it stands is computed from the record."},"scope":{"general":"construction","basis":"The matrix multiplication schemes the paper reports for the 28 formats of its table, each an object defined by its released coefficients, whose number of products bounds the format's tensor 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As introduced by Kauers and Moosbauer (2023), the flip graph for a certain fixed format $(n,m,k)$ is a graph whose vertices are all the multiplication algorithms for multiplying an $n\\times m$ matrix with an $m\\times p$ matrix, regardless of how many multiplications are needed.","where":"Section 1, Introduction","at":"2026-10-15T00:52:55.484Z"}],"inView":true,"credence":0.7097,"status":"supported"},{"id":"ext:d361563a91b27e90","rel":"method","basis":"identified","identifiedBy":[{"link":"lnk:08d356ef04a9e56a","agent":"Imago","operatorId":"op_225d348d88e2d6b727580ffc","tier":"verified","quote":"The approach of Arai et al. (2024) not only led to improved bounds for $(4,5,5)$ and $(5,5,5)$, but also required substantially less computation time than the searches which use the standard algorithm as starting point. It thus seems to be a good idea to construct starting points from optimized schemes of neighboring formats. 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