<html><head><meta http-equiv="Content-Type" content="text/html; charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">I remember from my days doing systems and controls that the Kronecker product arises when you extend 1D methods for linear-quadradic problems to 2D. I actually used it in my thesis.<div class=""><br class=""></div><div class="">Ed<br class=""><div class="">
<span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div class="">_______________________</div><div class=""><br class="Apple-interchange-newline">Ed Angel<br class=""><br class=""></div><div class="">Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br class="">Professor Emeritus of Computer Science, University of New Mexico<br class=""><br class="">1017 Sierra Pinon</div><div class="">Santa Fe, NM 87501<br class="">505-984-0136 (home)<span class="Apple-tab-span" style="white-space: pre; "> </span> <span class="Apple-tab-span" style="white-space: pre; "> </span><a href="mailto:angel@cs.unm.edu" class="">angel@cs.unm.edu</a></div><div class="">505-453-4944 (cell) <span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><a href="http://www.cs.unm.edu/~angel" class="">http://www.cs.unm.edu/~angel</a><br class=""></div></span></span>
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<div><br class=""><blockquote type="cite" class=""><div class="">On Nov 4, 2021, at 1:36 PM, Jon Zingale <<a href="mailto:jonzingale@gmail.com" class="">jonzingale@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class=""><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">Thanks to everyone who helped pitch in on this. I was happy to finally track down a pdf of Van Loan & Pitsianis [1993]. As stated in this other article on "Automated Kronecker Product Approximation"[KoPA]:</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">"This was introduced in the matrix computation literature as the nearest Kronecker product problem in Van Loan and Pitsianis, who demonstrated its equivalence to the best rank-one approximation and therefore also to the SVD, after a proper rearrangement of the matrix entries."</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">As well as:</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">"Finding a low-rank approximation of a given matrix is closely related to the singular value decomposition, and the connection was revealed as early as Eckart and Yount (1936)]"</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">It is surprising that it took until the 90s for the problem to bite someone and for explicit algorithms to emerge. Also, I am amazed that the body of the literature seems to come from the data sciences, as I arrived at the problem from studying discrete dynamical systems. I am hoping to write up a paper on the connection between them and NKP soon. Would anyone know where I can find a python, C++, or Haskell implementation easily?</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">Thanks again for the help.</div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">[1993] <a href="https://www.cs.cornell.edu/cv/ResearchPDF/Approximation%20with%20Kronecker%20Products,%20from%20Linear%20Algebra%20for%20Large%20Scale%20and%20Real-%20Time%20Applications.pdf" class="">https://www.cs.cornell.edu/cv/ResearchPDF/Approximation%20with%20Kronecker%20Products,%20from%20Linear%20Algebra%20for%20Large%20Scale%20and%20Real-%20Time%20Applications.pdf</a></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333"><br class=""></div><div class="gmail_default" style="font-family:verdana,sans-serif;font-size:small;color:#333333">[KoPA] <a href="https://arxiv.org/pdf/1912.02392.pdf" class="">https://arxiv.org/pdf/1912.02392.pdf</a></div></div>
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