optimality.tex

Paper VIII: Packet Operator Algorithms, Exact Compilation, and GPU Realization

optimality.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \node[font=\small,align=center,font=\small] at (1.73,4.5) {$LR^T$};
4 \path[draw=black!30,fill=black!8] (0.65,3.08) rectangle (1.37,3.8);
5 \node[font=\small,align=center,font=\small] at (1.01,3.44) {$3$};
6 \path[draw=black!30,fill=white] (1.37,3.08) rectangle (2.09,3.8);
7 \node[font=\small,align=center,font=\small] at (1.73,3.44) {$0$};
8 \path[draw=black!30,fill=white] (2.09,3.08) rectangle (2.81,3.8);
9 \node[font=\small,align=center,font=\small] at (2.45,3.44) {$0$};
10 \path[draw=black!30,fill=black!8] (0.65,2.36) rectangle (1.37,3.08);
11 \node[font=\small,align=center,font=\small] at (1.01,2.72) {$4$};
12 \path[draw=black!30,fill=white] (1.37,2.36) rectangle (2.09,3.08);
13 \node[font=\small,align=center,font=\small] at (1.73,2.72) {$0$};
14 \path[draw=black!30,fill=white] (2.09,2.36) rectangle (2.81,3.08);
15 \node[font=\small,align=center,font=\small] at (2.45,2.72) {$0$};
16 \path[draw=black!30,fill=white] (0.65,1.64) rectangle (1.37,2.36);
17 \node[font=\small,align=center,font=\small] at (1.01,2) {$0$};
18 \path[draw=black!30,fill=black!8] (1.37,1.64) rectangle (2.09,2.36);
19 \node[font=\small,align=center,font=\small] at (1.73,2) {$2$};
20 \path[draw=black!30,fill=white] (2.09,1.64) rectangle (2.81,2.36);
21 \node[font=\small,align=center,font=\small] at (2.45,2) {$0$};
22 \path[draw=black!30,fill=white] (0.65,0.92) rectangle (1.37,1.64);
23 \node[font=\small,align=center,font=\small] at (1.01,1.28) {$0$};
24 \path[draw=black!30,fill=white] (1.37,0.92) rectangle (2.09,1.64);
25 \node[font=\small,align=center,font=\small] at (1.73,1.28) {$0$};
26 \path[draw=black!30,fill=white] (2.09,0.92) rectangle (2.81,1.64);
27 \node[font=\small,align=center,font=\small] at (2.45,1.28) {$0$};
28 \node[font=\small,align=center,font=\small] at (6.13,4.5) {rank-one output};
29 \path[draw=black!30,fill=black!8] (5.05,3.08) rectangle (5.77,3.8);
30 \node[font=\small,align=center,font=\small] at (5.41,3.44) {$3$};
31 \path[draw=black!30,fill=white] (5.77,3.08) rectangle (6.49,3.8);
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33 \path[draw=black!30,fill=white] (6.49,3.08) rectangle (7.21,3.8);
34 \node[font=\small,align=center,font=\small] at (6.85,3.44) {$0$};
35 \path[draw=black!30,fill=black!8] (5.05,2.36) rectangle (5.77,3.08);
36 \node[font=\small,align=center,font=\small] at (5.41,2.72) {$4$};
37 \path[draw=black!30,fill=white] (5.77,2.36) rectangle (6.49,3.08);
38 \node[font=\small,align=center,font=\small] at (6.13,2.72) {$0$};
39 \path[draw=black!30,fill=white] (6.49,2.36) rectangle (7.21,3.08);
40 \node[font=\small,align=center,font=\small] at (6.85,2.72) {$0$};
41 \path[draw=black!30,fill=white] (5.05,1.64) rectangle (5.77,2.36);
42 \node[font=\small,align=center,font=\small] at (5.41,2) {$0$};
43 \path[draw=black!30,fill=white] (5.77,1.64) rectangle (6.49,2.36);
44 \node[font=\small,align=center,font=\small] at (6.13,2) {$0$};
45 \path[draw=black!30,fill=white] (6.49,1.64) rectangle (7.21,2.36);
46 \node[font=\small,align=center,font=\small] at (6.85,2) {$0$};
47 \path[draw=black!30,fill=white] (5.05,0.92) rectangle (5.77,1.64);
48 \node[font=\small,align=center,font=\small] at (5.41,1.28) {$0$};
49 \path[draw=black!30,fill=white] (5.77,0.92) rectangle (6.49,1.64);
50 \node[font=\small,align=center,font=\small] at (6.13,1.28) {$0$};
51 \path[draw=black!30,fill=white] (6.49,0.92) rectangle (7.21,1.64);
52 \node[font=\small,align=center,font=\small] at (6.85,1.28) {$0$};
53 \draw[line width=.6pt] (9.7,0.9) -- (13.5,0.9);
54 \node[font=\small,align=center] at (11.6,4.25) {spectral error};
55 \path[draw=black,fill=black!12,line width=.6pt] (10.004,0.9) rectangle (10.662667,3.5);
56 \node[font=\small,align=center,font=\scriptsize] at (10.333333,0.6) {rank 0};
57 \node[font=\small,align=center,font=\scriptsize] at (10.333333,3.72) {$5$};
58 \path[draw=black,fill=black!12,line width=.6pt] (11.270667,0.9) rectangle (11.929333,1.94);
59 \node[font=\small,align=center,font=\scriptsize] at (11.6,0.6) {rank 1};
60 \node[font=\small,align=center,font=\scriptsize] at (11.6,2.16) {$2$};
61 \path[draw=black,fill=black!12,line width=.6pt] (12.537333,0.9) rectangle (13.196,0.9);
62 \node[font=\small,align=center,font=\scriptsize] at (12.866667,0.6) {rank 2};
63 \node[font=\small,align=center,font=\scriptsize] at (12.866667,1.12) {$0$};
64 \node[font=\small,align=center,font=\scriptsize] at (7,0.1) {Frobenius errors: $\sqrt{29},\ 2,\ 0$. The rank-one choice loses only the second mode.};
65 \end{tikzpicture}

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