{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,12]],"date-time":"2024-01-12T21:05:41Z","timestamp":1705093541220},"reference-count":0,"publisher":"Association for the Advancement of Artificial Intelligence (AAAI)","issue":"01","license":[{"start":{"date-parts":[[2019,7,17]],"date-time":"2019-07-17T00:00:00Z","timestamp":1563321600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.aaai.org"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AAAI"],"abstract":"<jats:p>We show how to solve a number of problems in numerical linear algebra, such as least squares regression, lp-regression for any p \u2265 1, low rank approximation, and kernel regression, in time T(A)poly(log(nd)), where for a given input matrix A \u2208 Rn\u00d7d, T(A) is the time needed to compute A \u00b7 y for an arbitrary vector y \u2208 Rd. Since T(A) \u2264 O(nnz(A)), where nnz(A) denotes the number of non-zero entries of A, the time is no worse, up to polylogarithmic factors, as all of the recent advances for such problems that run in input-sparsity time. However, for many applications, T(A) can be much smaller than nnz(A), yielding significantly sublinear time algorithms. For example, in the overconstrained (1+\u03b5)-approximate polynomial interpolation problem, A is a Vandermonde matrix and T(A) = O(n log n); in this case our running time is n \u00b7 poly (log n) + poly (d\/\u03b5) and we recover the results of Avron, Sindhwani, and Woodruff (2013) as a special case. For overconstrained autoregression, which is a common problem arising in dynamical systems, T(A) = O(n log n), and we immediately obtain n\u00b7 poly (log n) + poly(d\/\u03b5) time. For kernel autoregression, we significantly improve the running time of prior algorithms for general kernels. For the important case of autoregression with the polynomial kernel and arbitrary target vector b \u2208 Rn, we obtain even faster algorithms. Our algorithms show that, perhaps surprisingly, most of these optimization problems do not require much more time than that of a polylogarithmic number of matrix-vector multiplications.<\/jats:p>","DOI":"10.1609\/aaai.v33i01.33014918","type":"journal-article","created":{"date-parts":[[2019,9,1]],"date-time":"2019-09-01T07:39:14Z","timestamp":1567323554000},"page":"4918-4925","source":"Crossref","is-referenced-by-count":2,"title":["Sublinear Time Numerical Linear Algebra for Structured Matrices"],"prefix":"10.1609","volume":"33","author":[{"given":"Xiaofei","family":"Shi","sequence":"first","affiliation":[]},{"given":"David P.","family":"Woodruff","sequence":"additional","affiliation":[]}],"member":"9382","published-online":{"date-parts":[[2019,7,17]]},"container-title":["Proceedings of the AAAI Conference on Artificial Intelligence"],"original-title":[],"link":[{"URL":"https:\/\/ojs.aaai.org\/index.php\/AAAI\/article\/download\/4421\/4299","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/ojs.aaai.org\/index.php\/AAAI\/article\/download\/4421\/4299","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,7]],"date-time":"2022-11-07T06:38:08Z","timestamp":1667803088000},"score":1,"resource":{"primary":{"URL":"https:\/\/ojs.aaai.org\/index.php\/AAAI\/article\/view\/4421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,17]]},"references-count":0,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2019,7,23]]}},"URL":"https:\/\/doi.org\/10.1609\/aaai.v33i01.33014918","relation":{},"ISSN":["2374-3468","2159-5399"],"issn-type":[{"value":"2374-3468","type":"electronic"},{"value":"2159-5399","type":"print"}],"subject":[],"published":{"date-parts":[[2019,7,17]]}}}