update
This commit is contained in:
Morten Hjorth-Jensen
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"cells": [
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@@ -14,7 +14,7 @@
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@@ -27,7 +27,7 @@
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@@ -43,25 +43,26 @@
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"derivatives) of the chosen cost functions. Vectors are always written\n",
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"as boldfaced lower case letters and matrices as upper case boldfaced\n",
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"letters. You will find useful the notes from week 35 on derivatives of vectors and matrices.\n",
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"See also the textbook of Faisal at al, chapter 5 and in particular sections 5.3-5.5 at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/MathMLbook.pdf>\n",
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"\n",
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"Show that"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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},
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"source": [
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"$$\n",
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"\\frac{\\partial (\\boldsymbol{b}^T\\boldsymbol{a})}{\\partial \\boldsymbol{a}} = \\boldsymbol{b},\n",
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"\\frac{\\partial (\\boldsymbol{a}^T\\boldsymbol{x})}{\\partial \\boldsymbol{x}} = \\boldsymbol{a}^T,\n",
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"$$"
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]
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},
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{
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@@ -71,7 +72,7 @@
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@@ -83,7 +84,7 @@
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@@ -93,7 +94,7 @@
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@@ -105,7 +106,7 @@
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@@ -114,13 +115,14 @@
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"\n",
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"The second derivative of the mean squared error is then proportional to the so-called Hessian matrix $\\boldsymbol{H}=\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
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"\n",
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"**Hint**: In these exercises it is always useful to write out with summation indices the various quantities.\n",
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"**Hint**: In these exercises it is always useful to write out with summation indices the various quantities. Take also a look at the weekly slides from week 35 and the various examples included there.\n",
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"\n",
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"As an example, consider the function"
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]
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},
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{
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@@ -132,7 +134,7 @@
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@@ -142,7 +144,7 @@
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@@ -154,7 +156,7 @@
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@@ -164,7 +166,7 @@
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@@ -176,7 +178,7 @@
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@@ -186,7 +188,7 @@
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@@ -198,7 +200,7 @@
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@@ -216,7 +218,7 @@
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@@ -230,7 +232,7 @@
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@@ -244,7 +246,7 @@
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@@ -257,7 +259,7 @@
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@@ -268,7 +270,7 @@
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@@ -302,7 +304,7 @@
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{
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@@ -382,7 +384,7 @@
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"cell_type": "markdown",
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"metadata": {
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"editable": true
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},
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+422
-410
File diff suppressed because it is too large
Load Diff
@@ -13,11 +13,12 @@ derivatives as well as when we analyze various properties (like second
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derivatives) of the chosen cost functions. Vectors are always written
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as boldfaced lower case letters and matrices as upper case boldfaced
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letters. You will find useful the notes from week 35 on derivatives of vectors and matrices.
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See also the textbook of Faisal at al, chapter 5 and in particular sections 5.3-5.5 at URL:"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/MathMLbook.pdf"
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Show that
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!bt
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\[
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\frac{\partial (\bm{b}^T\bm{a})}{\partial \bm{a}} = \bm{b},
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\frac{\partial (\bm{a}^T\bm{x})}{\partial \bm{x}} = \bm{a}^T,
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\]
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!et
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and
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@@ -37,7 +38,8 @@ and finally find the second derivative of this function with respect to the vect
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The second derivative of the mean squared error is then proportional to the so-called Hessian matrix $\bm{H}=\bm{X}^T\bm{X}$.
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_Hint_: In these exercises it is always useful to write out with summation indices the various quantities.
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_Hint_: In these exercises it is always useful to write out with summation indices the various quantities. Take also a look at the weekly slides from week 35 and the various examples included there.
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As an example, consider the function
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!bt
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