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浅谈范数正则化
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<div id="cnblogs_post_body" class="blogpost-body blogpost-body-html"><a rel="nofollow" name="_labelTop"></a><div id="navCategory"><p style="font-size:18px"><b>阅读目录(Content)</b></p><ul class="first_class_ul"><li><a rel="nofollow" href="#_label0">浅谈范数正则化</a></li><ul class="second_class_ul"><li><a rel="nofollow" href="#_lab2_0_0">1. 向量范数与矩阵范数</a></li><li><a rel="nofollow" href="#_lab2_0_1">2. 为什么要添加正则项？</a></li><li><a rel="nofollow" href="#_lab2_0_2">3. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>0</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.699em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="msubsup" id="MathJax-Span-3"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-5" style="font-size: 70.7%; font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.29em; border-left: 0px solid; width: 0px; height: 1.227em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-1">L_0</script>范数</a></li><li><a rel="nofollow" href="#_lab2_0_3">4. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-6" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-7"><span class="msubsup" id="MathJax-Span-8"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-9" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-10" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-2">L_1</script>范数</a></li><li><a rel="nofollow" href="#_lab2_0_4">5. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-11" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-12"><span class="msubsup" id="MathJax-Span-13"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-14" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-15" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-3">L_2</script>范数</a></li><li><a rel="nofollow" href="#_lab2_0_5">6. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-16" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-17"><span class="msubsup" id="MathJax-Span-18"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-19" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-20" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-4">L_1</script>范数与<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-21" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-22"><span class="msubsup" id="MathJax-Span-23"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-24" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-25" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-5">L_2</script>范数作为正则项的区别</a></li><li><a rel="nofollow" href="#_lab2_0_6">7. 用概率解释传统线性回归模型</a></li><li><a rel="nofollow" href="#_lab2_0_7">8. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-26" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-27"><span class="msubsup" id="MathJax-Span-28"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-29" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-30" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-6">L_2</script>范等价于Gauss先验</a></li><li><a rel="nofollow" href="#_lab2_0_8">9. <span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-31" style="width: 1.47em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.133em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.13em, 2.683em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-32"><span class="msubsup" id="MathJax-Span-33"><span style="display: inline-block; position: relative; width: 1.11em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-34" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="mn" id="MathJax-Span-35" style="font-size: 70.7%; font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.271em; border-left: 0px solid; width: 0px; height: 1.208em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-7">L_1</script>范数等价于Laplace先验</a></li><li><a rel="nofollow" href="#_lab2_0_9">10. 矩阵的<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-36" style="width: 2.137em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.667em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.67em, 2.821em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-37"><span class="msubsup" id="MathJax-Span-38"><span style="display: inline-block; position: relative; width: 1.66em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="texatom" id="MathJax-Span-40"><span class="mrow" id="MathJax-Span-41"><span class="mn" id="MathJax-Span-42" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="mo" id="MathJax-Span-43" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mn" id="MathJax-Span-44" style="font-size: 70.7%; font-family: MathJax_Main;">1</span></span></span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.442em; border-left: 0px solid; width: 0px; height: 1.379em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-8">L_{2, 1}</script>范数及<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-45" style="width: 2.137em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.667em; height: 0px; font-size: 125%;"><span style="position: absolute; clip: rect(1.45em, 1001.67em, 2.821em, -1000em); top: -2.333em; left: 0em;"><span class="mrow" id="MathJax-Span-46"><span class="msubsup" id="MathJax-Span-47"><span style="display: inline-block; position: relative; width: 1.634em; height: 0px;"><span style="position: absolute; clip: rect(3.117em, 1000.65em, 4.2em, -1000em); top: -4em; left: 0em;"><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math; font-style: italic;">L</span><span style="display: inline-block; width: 0px; height: 4em;"></span></span><span style="position: absolute; top: -3.85em; left: 0.681em;"><span class="texatom" id="MathJax-Span-49"><span class="mrow" id="MathJax-Span-50"><span class="mi" id="MathJax-Span-51" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">p</span><span class="mo" id="MathJax-Span-52" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-53" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">q<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.01em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.333em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.442em; border-left: 0px solid; width: 0px; height: 1.379em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow class="MJX-TeXAtom-ORD"><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-9">L_{p, q}</script>范数</a></li><li><a rel="nofollow" href="#_lab2_0_10">11. 矩阵的核范数及Schatten范数</a></li><li><a rel="nofollow" href="#_lab2_0_11">12. MATLAB程序：Laplace分布与Gauss分布的概率密度函数图</a></li><li><a rel="nofollow" href="#_lab2_0_12">13. 参考文献</a></li></ul></ul></div>
浅谈范数正则化
作者：凯鲁嘎吉 - 博客园 http://www.cnblogs.com/kailugaji/
这篇博客介绍不同范数作为正则化项时的作用。首先介绍了常见的向量范数与矩阵范数，然后说明添加正则化项的原因，之后介绍向量的，，范数及其作为正则化项的作用，对三者进行比较分析，并用贝叶斯观点解释传统线性模型与正则化项。随后，介绍矩阵的范数及其推广形式范数，以及矩阵的核范数及其推广形式Schatten范数。最后，用MATLAB程序编写了Laplace分布与Gauss分布的概率密度函数图。有关矩阵范数优化求解问题可参考：一类涉及矩阵范数的优化问题 - 凯鲁嘎吉 - 博客园
1. 向量范数与矩阵范数
2. 为什么要添加正则项？
3. 范数
4. 范数
5. 范数
6. 范数与范数作为正则项的区别
7. 用概率解释传统线性回归模型
8. 范等价于Gauss先验
9. 范数等价于Laplace先验
10. 矩阵的范数及范数
11. 矩阵的核范数及Schatten范数
12. MATLAB程序：Laplace分布与Gauss分布的概率密度函数图
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | %% Demo of Laplace Density Function% x : variable% lambda : size para%miu: location paraclearclcx = -10:0.1:10;y_1=Laplace_distribution(x, 0, 1);y_2=Laplace_distribution(x, 0, 2);y_3=Laplace_distribution(x, 0, 4);y_4=Laplace_distribution(x, -5, 4);y_5=Laplace_distribution(x, 5, 4);y_6=normpdf(x,0,1);plot(x, y_1, 'r-', x, y_2, 'g-', x, y_3, 'c-', x, y_4, 'm-', x, y_5, 'y-', x, y_6, 'b-', 'LineWidth',1.2);legend('\mu =0, \lambda=1','\mu=0, \lambda=2','\mu=0, \lambda=4','\mu=-5, \lambda=4','\mu=5, \lambda=4', '\mu=0, \sigma=1'); %图例的设置xlabel('x');ylabel('f(x)');title('Laplace vs Gauss pdf');set(gca, 'FontName', 'Times New Roman', 'FontSize',11);saveas(gcf,sprintf('demo_Laplace_Gauss.jpg'),'bmp'); %保存图片%% Laplace Density Functionfunction y=Laplace_distribution(x, miu, lambda) y = 1 / (2*lambda) * exp( -abs(x-miu)/lambda);end |
13. 参考文献
[1] 证明核范数是矩阵秩的凸包络
EJ Candès, Recht B . Exact Matrix Completion via Convex Optimization[J]. Foundations of Computational Mathematics, 2009, 9(6):717.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.312.1183&rep=rep1&type=pdf
[2] 关于说明范数是范数的凸包络的文献及教案
Donoho D L , Huo X . Uncertainty Principles and Ideal Atomic Decomposition[J]. IEEE Transactions on Information Theory, 2001, 47(7):2845-2862.
Learning with Combinatorial Structure Note for Lecture 12
http://people.csail.mit.edu/stefje/fall15/notes_lecture12.pdf
L1-norm Methods for Convex-Cardinality Problems
https://web.stanford.edu/class/ee364b/lectures/l1_slides.pdf
[3] 有关过拟合的教案及图片来源
2017 Lecture 2: Overfitting. Regularization
https://www.cs.mcgill.ca/~dprecup/courses/ML/Lectures/ml-lecture02.pdf
[4] 一些可供参考的资料
The difference between L1 and L2 regularization
https://explained.ai/regularization/L1vsL2.html
Why L1 norm for sparse models
https://stats.stackexchange.com/questions/45643/why-l1-norm-for-sparse-models
Why L1 regularization can “zero out the weights” and therefore leads to sparse models? [duplicate]
What are L1, L2 and Elastic Net Regularization in neural networks?
Introduction. Sharpness Enhancement and Denoising of Image Using L1-Norm Minimization Technique in Adaptive Bilateral Filter.
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