{"paper":{"title":"Almost Optimal Tensor Sketch","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.PR","stat.ML"],"primary_cat":"cs.DS","authors_text":"Jakob B. T. Knudsen, Thomas D. Ahle","submitted_at":"2019-09-03T08:56:59Z","abstract_excerpt":"We construct a matrix $M\\in R^{m\\otimes d^c}$ with just $m=O(c\\,\\lambda\\,\\varepsilon^{-2}\\text{poly}\\log1/\\varepsilon\\delta)$ rows, which preserves the norm $\\|Mx\\|_2=(1\\pm\\varepsilon)\\|x\\|_2$ of all $x$ in any given $\\lambda$ dimensional subspace of $ R^d$ with probability at least $1-\\delta$. This matrix can be applied to tensors $x^{(1)}\\otimes\\dots\\otimes x^{(c)}\\in R^{d^c}$ in $O(c\\, m \\min\\{d,m\\})$ time -- hence the name \"Tensor Sketch\". (Here $x\\otimes y = \\text{asvec}(xy^T) = [x_1y_1, x_1y_2,\\dots,x_1y_m,x_2y_1,\\dots,x_ny_m]\\in R^{nm}$.)\n  This improves upon earlier Tensor Sketch const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.01821","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.01821/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}