From a99959483f2c51fd9a16c46af1ac28040dda07e4 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Fri, 22 Nov 2019 06:54:36 +0100 Subject: [PATCH] typo in svm --- doc/pub/svm/html/._svm-bs000.html | 2 +- doc/pub/svm/html/._svm-bs020.html | 7 ++++-- doc/pub/svm/html/._svm-bs022.html | 21 +++++++++++------ doc/pub/svm/html/svm-bs.html | 2 +- doc/pub/svm/html/svm-reveal.html | 30 ++++++++++++++++--------- doc/pub/svm/html/svm-solarized.html | 30 ++++++++++++++++--------- doc/pub/svm/html/svm.html | 30 ++++++++++++++++--------- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 182 -> 189 bytes doc/pub/svm/ipynb/svm.ipynb | 25 +++++++++++++-------- doc/pub/svm/pdf/svm-minted.pdf | Bin 348111 -> 266200 bytes doc/src/SupportVMachines/svm.do.txt | 25 ++++++++++++++------- 11 files changed, 114 insertions(+), 58 deletions(-) diff --git a/doc/pub/svm/html/._svm-bs000.html b/doc/pub/svm/html/._svm-bs000.html index 52d9034be..0ad4b19d0 100644 --- a/doc/pub/svm/html/._svm-bs000.html +++ b/doc/pub/svm/html/._svm-bs000.html @@ -174,7 +174,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 18, 2019

+

Nov 22, 2019


diff --git a/doc/pub/svm/html/._svm-bs020.html b/doc/pub/svm/html/._svm-bs020.html index 9e90df4cc..372609da7 100644 --- a/doc/pub/svm/html/._svm-bs020.html +++ b/doc/pub/svm/html/._svm-bs020.html @@ -166,7 +166,7 @@ $$

With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, $$ subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors @@ -189,7 +189,10 @@ $$ We note that this is nothing but the dot product of the two original vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). + +

+This leads to the so-called kernel trick and the result leads to the same as if we went through the trouble of performing the transformation \( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. diff --git a/doc/pub/svm/html/._svm-bs022.html b/doc/pub/svm/html/._svm-bs022.html index a529ff2e2..425544721 100644 --- a/doc/pub/svm/html/._svm-bs022.html +++ b/doc/pub/svm/html/._svm-bs022.html @@ -170,18 +170,25 @@ There are several popular kernels being used. These are and many other ones.

-An important theorem for us is Mercer's theorem. -The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then -there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space -(possibly with much higher dimensions) such that +An important theorem for us is Mercer's +theorem. The +theorem states that if a kernel function \( K \) is symmetric, continuous +and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there +exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into +another space (possibly with much higher dimensions) such that + $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ +

So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. +you don’t know what \( \phi \) is. + +

+Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice.

diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index 52d9034be..0ad4b19d0 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -174,7 +174,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 18, 2019

+

Nov 22, 2019


diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index 2b298cddc..eaa2e0cef 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -148,7 +148,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

 
-

Nov 18, 2019

+

Nov 22, 2019


@@ -988,7 +988,7 @@ $$ With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)

 
$$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, $$

 
@@ -1018,7 +1018,10 @@ $$ We note that this is nothing but the dot product of the two original vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). + +

+This leads to the so-called kernel trick and the result leads to the same as if we went through the trouble of performing the transformation \( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. @@ -1084,20 +1087,27 @@ There are several popular kernels being used. These are and many other ones.

-An important theorem for us is Mercer's theorem. -The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then -there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space -(possibly with much higher dimensions) such that +An important theorem for us is Mercer's +theorem. The +theorem states that if a kernel function \( K \) is symmetric, continuous +and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there +exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into +another space (possibly with much higher dimensions) such that +

 
$$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$

 
+

So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. +you don’t know what \( \phi \) is. + +

+Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice. diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index e8d78b39d..f9d5ef494 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -110,7 +110,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 18, 2019

+

Nov 22, 2019












@@ -831,7 +831,7 @@ $$

With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, $$ subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors @@ -854,7 +854,10 @@ $$ We note that this is nothing but the dot product of the two original vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). + +

+This leads to the so-called kernel trick and the result leads to the same as if we went through the trouble of performing the transformation \( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. @@ -913,18 +916,25 @@ There are several popular kernels being used. These are and many other ones.

-An important theorem for us is Mercer's theorem. -The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then -there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space -(possibly with much higher dimensions) such that +An important theorem for us is Mercer's +theorem. The +theorem states that if a kernel function \( K \) is symmetric, continuous +and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there +exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into +another space (possibly with much higher dimensions) such that + $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ +

So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. +you don’t know what \( \phi \) is. + +

+Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice.











diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index af6d4c338..804b3f318 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -115,7 +115,7 @@ MathJax.Hub.Config({

[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

-

Nov 18, 2019

+

Nov 22, 2019












@@ -836,7 +836,7 @@ $$

With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) $$ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, $$ subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \), and for the support vectors @@ -859,7 +859,10 @@ $$ We note that this is nothing but the dot product of the two original vectors \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). Instead of thus computing the product in the Lagrangian of \( \boldsymbol{z}_i^T\boldsymbol{z}_j \) we simply compute -the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). This leads to the so-called +the dot product \( (\boldsymbol{x}_i^T\boldsymbol{x}_j)^2 \). + +

+This leads to the so-called kernel trick and the result leads to the same as if we went through the trouble of performing the transformation \( (\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j) \) during the SVM calculations. @@ -918,18 +921,25 @@ There are several popular kernels being used. These are and many other ones.

-An important theorem for us is Mercer's theorem. -The theorem states that if a kernel function \( K \) is symmetric, continuous and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then -there exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into another space -(possibly with much higher dimensions) such that +An important theorem for us is Mercer's +theorem. The +theorem states that if a kernel function \( K \) is symmetric, continuous +and leads to a positive semi-definite matrix \( \boldsymbol{P} \) then there +exists a function \( \phi \) that maps \( \boldsymbol{x}_i \) and \( \boldsymbol{x}_j \) into +another space (possibly with much higher dimensions) such that + $$ K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). $$ +

So you can use \( K \) as a kernel since you know \( \phi \) exists, even if -you don’t know what \( \phi \) is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. +you don’t know what \( \phi \) is. + +

+Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice.











diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index 87cddf0b1e066d4bb9610f8524b431a30a8f2081..5ee1bf504f082ec7f3047a0f7141ca9e744333f0 100644 GIT binary patch literal 189 zcmV;u07CyCiwFSech_A21MSaC3c@fD2H>uHia9|@nkHQfcHu%0@d7DLZPX?;NyXmY zK0sFrE{X{GHa{~A!^}Qkt+zSycONYVW6aYO#>_aI5+`##AtV7p8D~7B86$uRrJO~8 zmpkdDb6(itl=>vKlltx4FjiF`_RO!qGylY)Rt`41-q%KhHeMFSoWb21q}pB=P#WpR rK3I9}l$9WK527rUS4N9V+*)(AqH!qx^(R3P1i?2R4kJu&00;m8xN%VV literal 182 zcmV;n07?HJiwFRt71Lb+1MSeU4uUWc24H5M;+}w9p>#Kq!GXjWU!a5{jnc!m7@M1cr%;*u#}-iVB)z_MI$!BW99z$tPb0dBX_ z+u*#k(<+u|^$Vu4rdjEK9yjNoIJert;W+x13>fmNBD#WmbcyaqQ$bD2&2_NqK4_~z knh|7OYp==3Tijc^SlPKG{`Dn65Cp+DUikG1000O807g+*e*gdg diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb index fe2c161a8..e5c37dc32 100644 --- a/doc/pub/svm/ipynb/svm.ipynb +++ b/doc/pub/svm/ipynb/svm.ipynb @@ -10,7 +10,7 @@ " \n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Nov 18, 2019**\n", + "Date: **Nov 22, 2019**\n", "\n", "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", "\n", @@ -1267,7 +1267,7 @@ "metadata": {}, "source": [ "$$\n", - "{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{z}_j,\n", + "{\\cal L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{z}_i^T\\boldsymbol{z}_j,\n", "$$" ] }, @@ -1327,7 +1327,10 @@ "We note that this is nothing but the dot product of the two original\n", "vectors $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$. Instead of thus computing the\n", "product in the Lagrangian of $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we simply compute\n", - "the dot product $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$. This leads to the so-called\n", + "the dot product $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$.\n", + "\n", + "\n", + "This leads to the so-called\n", "kernel trick and the result leads to the same as if we went through\n", "the trouble of performing the transformation\n", "$(\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n", @@ -1412,10 +1415,12 @@ "\n", "and many other ones.\n", "\n", - "An important theorem for us is [Mercer's theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). \n", - "The theorem states that if a kernel function $K$ is symmetric, continuous and leads to a positive semi-definite matrix $\\boldsymbol{P}$ then\n", - "there exists a function $\\phi$ that maps $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_j$ into another space\n", - "(possibly with much higher dimensions) such that" + "An important theorem for us is [Mercer's\n", + "theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). The\n", + "theorem states that if a kernel function $K$ is symmetric, continuous\n", + "and leads to a positive semi-definite matrix $\\boldsymbol{P}$ then there\n", + "exists a function $\\phi$ that maps $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_j$ into\n", + "another space (possibly with much higher dimensions) such that" ] }, { @@ -1433,8 +1438,10 @@ "source": [ "So you can use $K$ as a kernel since you know $\\phi$ exists, even if\n", "you don’t know what $\\phi$ is. \n", - "Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work\n", - "well in practice.\n", + "\n", + "Note that some frequently used kernels (such as the Sigmoid kernel)\n", + "don’t respect all of Mercer’s conditions, yet they generally work well\n", + "in practice.\n", "\n", "\n", "## The moons example" diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf index 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b/doc/src/SupportVMachines/svm.do.txt index 5bbb29bd3..f1417f311 100644 --- a/doc/src/SupportVMachines/svm.do.txt +++ b/doc/src/SupportVMachines/svm.do.txt @@ -708,7 +708,7 @@ z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) !bt \[ -{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\bm{x}_i^T\bm{z}_j, +{\cal L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\bm{z}_i^T\bm{z}_j, \] !et subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$, and for the support vectors @@ -734,7 +734,10 @@ K(\bm{x}_i,\bm{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_ We note that this is nothing but the dot product of the two original vectors $(\bm{x}_i^T\bm{x}_j)^2$. Instead of thus computing the product in the Lagrangian of $\bm{z}_i^T\bm{z}_j$ we simply compute -the dot product $(\bm{x}_i^T\bm{x}_j)^2$. This leads to the so-called +the dot product $(\bm{x}_i^T\bm{x}_j)^2$. + + +This leads to the so-called kernel trick and the result leads to the same as if we went through the trouble of performing the transformation $(\phi(\bm{x}_i)^T\phi(\bm{x}_j)$ during the SVM calculations. @@ -785,19 +788,25 @@ o Gaussian Radial Basis Function: $K(\bm{x},\bm{y})=\exp{\left(-\gamma\vert\vert o Tanh: $K(\bm{x},\bm{y})=\tanh{(\bm{x}^T\bm{y}+\gamma)}$, and many other ones. -An important theorem for us is "Mercer's theorem":"https://en.wikipedia.org/wiki/Mercer%27s_theorem". -The theorem states that if a kernel function $K$ is symmetric, continuous and leads to a positive semi-definite matrix $\bm{P}$ then -there exists a function $\phi$ that maps $\bm{x}_i$ and $\bm{x}_j$ into another space -(possibly with much higher dimensions) such that +An important theorem for us is "Mercer's +theorem":"https://en.wikipedia.org/wiki/Mercer%27s_theorem". The +theorem states that if a kernel function $K$ is symmetric, continuous +and leads to a positive semi-definite matrix $\bm{P}$ then there +exists a function $\phi$ that maps $\bm{x}_i$ and $\bm{x}_j$ into +another space (possibly with much higher dimensions) such that + !bt \[ K(\bm{x}_i,\bm{x}_j)=\phi(\bm{x}_i)^T\phi(\bm{x}_j). \] !et + So you can use $K$ as a kernel since you know $\phi$ exists, even if you don’t know what $\phi$ is. -Note that some frequently used kernels (such as the Sigmoid kernel) don’t respect all of Mercer’s conditions, yet they generally work -well in practice. + +Note that some frequently used kernels (such as the Sigmoid kernel) +don’t respect all of Mercer’s conditions, yet they generally work well +in practice. !split