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L1 norm - Revision history
2026-07-14T22:28:22Z
Revision history for this page on the wiki
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https://calculus.subwiki.org/w/index.php?title=L1_norm&diff=2718&oldid=prev
Vipul at 18:43, 11 May 2014
2014-05-11T18:43:03Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:43, 11 May 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition in terms of the signum vector function===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition in terms of the signum vector function===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\<del style="font-weight: bold; text-decoration: none;">vec</del>{\operatorname{sgn}}(\vec{x}) \cdot \vec{x}</math> where <math>\<del style="font-weight: bold; text-decoration: none;">vec</del>{sgn}</math> denotes the [[signum vector function]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\<ins style="font-weight: bold; text-decoration: none;">overline</ins>{\operatorname{sgn}}(\vec{x}) \cdot \vec{x}</math> where <math>\<ins style="font-weight: bold; text-decoration: none;">overline{\operatorname</ins>{sgn<ins style="font-weight: bold; text-decoration: none;">}</ins>}</math> denotes the [[signum vector function]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition as a piecewise linear function===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition as a piecewise linear function===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of pieces of the domain involved is <math>2^n</math> interior regions, and an additional <math>3^n - 2^n</math> boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of pieces of the domain involved is <math>2^n</math> interior regions, and an additional <math>3^n - 2^n</math> boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions).</div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=L1_norm&diff=2717&oldid=prev
Vipul at 18:42, 11 May 2014
2014-05-11T18:42:29Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:42, 11 May 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition as a piecewise linear function===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition as a piecewise linear function===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of <del style="font-weight: bold; text-decoration: none;">linear </del>pieces involved is <math>2^n</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of pieces <ins style="font-weight: bold; text-decoration: none;">of the domain </ins>involved is <math>2^n</math> <ins style="font-weight: bold; text-decoration: none;">interior regions, and an additional <math>3^n - 2^n</math> boundary regions and boundary intersections (these can be included in the piece definitions for any of the bordering interior regions)</ins>.</div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=L1_norm&diff=2716&oldid=prev
Vipul at 18:41, 11 May 2014
2014-05-11T18:41:40Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:41, 11 May 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition in terms of the signum vector function===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Definition in terms of the signum vector function===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\vec{sgn}(\vec{x}) \cdot \vec{x}</math> where <math>\vec{sgn}</math> denotes the [[signum vector function]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\vec<ins style="font-weight: bold; text-decoration: none;">{\operatorname</ins>{sgn<ins style="font-weight: bold; text-decoration: none;">}</ins>}(\vec{x}) \cdot \vec{x}</math> where <math>\vec{sgn}</math> denotes the [[signum vector function]]<ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Definition as a piecewise linear function===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The <math>L^1</math>-norm can be defined as a piecewise linear function. The number of linear pieces involved is <math>2^n</math></ins>.</div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=L1_norm&diff=2715&oldid=prev
Vipul at 18:40, 11 May 2014
2014-05-11T18:40:07Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:40, 11 May 2014</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\| x \|_1 = \sum_{i=1}^n |x_i|</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\| x \|_1 = \sum_{i=1}^n |x_i|</math></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Definition in terms of the signum vector function===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The <math>L^1</math>-norm <math>\| x \|_1</math> can be defined as the [[dot product]] <math>\vec{sgn}(\vec{x}) \cdot \vec{x}</math> where <math>\vec{sgn}</math> denotes the [[signum vector function]].</ins></div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=L1_norm&diff=2714&oldid=prev
Vipul: Created page with "==Definition== Suppose <math>n</math> is a positive integer. The <math>L^1</math>-norm, denoted <math>\| \cdot \|_1</math>, is a function from <math>\R^n</math> to <math>\R</..."
2014-05-11T18:38:41Z
<p>Created page with "==Definition== Suppose <math>n</math> is a positive integer. The <math>L^1</math>-norm, denoted <math>\| \cdot \|_1</math>, is a function from <math>\R^n</math> to <math>\R</..."</p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
Suppose <math>n</math> is a positive integer. The <math>L^1</math>-norm, denoted <math>\| \cdot \|_1</math>, is a function from <math>\R^n</math> to <math>\R</math> defined as:<br />
<br />
<math>\| x \|_1 = \sum_{i=1}^n |x_i|</math></div>
Vipul