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ACM Program. Lang."],"published-print":{"date-parts":[[2025,10,9]]},"abstract":"<jats:p>\n                    This paper introduces the continuous tensor abstraction, allowing indices to take real-number values (e.g., A[3.14]). It also presents continuous tensor algebra expressions, such as\n                    <jats:italic toggle=\"yes\">\n                      C\n                      <jats:sub>x,y<\/jats:sub>\n                      = A\n                      <jats:sub>x,y<\/jats:sub>\n                      * B\n                      <jats:sub>x,y<\/jats:sub>\n                      ,\n                    <\/jats:italic>\n                    where indices are defined over a continuous domain. This work expands the traditional tensor model to include continuous tensors. Our implementation supports piecewise-constant tensors, on which infinite domains can be processed in finite time. We also introduce a new tensor format for efficient storage and a code generation technique for automatic kernel generation. For the first time, our abstraction expresses domains like computational geometry and computer graphics in the language of tensor programming. Our approach demonstrates competitive or better performance to hand-optimized kernels in leading libraries across diverse applications. Compared to hand-implemented libraries on a CPU, our compiler-based implementation achieves an average speedup of 9.20\u00d7 on 2D radius search with \u223c60\u00d7 fewer lines of code (LoC), 1.22\u00d7 on genomic interval overlapping queries (with \u223c18\u00d7 LoC saving), and 1.69\u00d7 on trilinear interpolation in Neural Radiance Field (with \u223c6\u00d7 LoC saving).\n                  <\/jats:p>","DOI":"10.1145\/3763146","type":"journal-article","created":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T08:51:31Z","timestamp":1759999891000},"page":"2681-2709","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["The Continuous Tensor Abstraction: Where Indices Are Real"],"prefix":"10.1145","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3082-4348","authenticated-orcid":false,"given":"Jaeyeon","family":"Won","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4963-0869","authenticated-orcid":false,"given":"Willow","family":"Ahrens","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology, Atlanta, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-7578-7683","authenticated-orcid":false,"given":"Teodoro Fields","family":"Collin","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3459-5466","authenticated-orcid":false,"given":"Joel S.","family":"Emer","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"},{"name":"NVIDIA, Westford, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7231-7643","authenticated-orcid":false,"given":"Saman","family":"Amarasinghe","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2025,10,9]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","unstructured":"Mart\u00edn Abadi Paul Barham Jianmin Chen Zhifeng Chen Andy Davis Jeffrey Dean Matthieu Devin Sanjay Ghemawat Geoffrey Irving Michael Isard et al. 2016. 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