From 20deeb767e43470efcaec4c3cfa111639ca38095 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Tue, 26 Jan 2021 22:30:20 +0100 Subject: [PATCH] finished perzeptron und vc-dimension --- Acronyms.tex | 2 +- Packages.tex | 1 + chapters/Supervised Learning.tex | 41 ++++++++++++++++-- images/Perzeptron_Lernalgorithmus.png | Bin 0 -> 46581 bytes ...Perzeptron_Lernalgorithmus_symmetrisch.png | Bin 0 -> 40828 bytes images/XOR-Problem1.png | Bin 0 -> 51362 bytes images/XOR-Problem2.png | Bin 0 -> 61312 bytes images/vc-dimension_shattering.png | Bin 0 -> 35849 bytes 8 files changed, 40 insertions(+), 4 deletions(-) create mode 100644 images/Perzeptron_Lernalgorithmus.png create mode 100644 images/Perzeptron_Lernalgorithmus_symmetrisch.png create mode 100644 images/XOR-Problem1.png create mode 100644 images/XOR-Problem2.png create mode 100644 images/vc-dimension_shattering.png diff --git a/Acronyms.tex b/Acronyms.tex index e1941c4..cce7297 100644 --- a/Acronyms.tex +++ b/Acronyms.tex @@ -2,5 +2,5 @@ % list of acronyms %% \begin{acronym} - \acro{example}[EXMPL]{example of an acronym} + \acro{vc-dimension} [VC-Dimension] {Vapnik-Chervonenkis-Dimension} \end{acronym} \ No newline at end of file diff --git a/Packages.tex b/Packages.tex index 68bd819..7d47dc0 100644 --- a/Packages.tex +++ b/Packages.tex @@ -34,4 +34,5 @@ rightsub = \grq% \usepackage[square, numbers]{natbib} %math \usepackage{amsmath} +\usepackage{amssymb} \usepackage{bm} \ No newline at end of file diff --git a/chapters/Supervised Learning.tex b/chapters/Supervised Learning.tex index 7c4a5c3..8ff655c 100644 --- a/chapters/Supervised Learning.tex +++ b/chapters/Supervised Learning.tex @@ -52,6 +52,41 @@ $$g(\bm{m}) > 0 \forall \bm{m}\in C_2$$ $$g(\bm{m}) < 0 \forall \bm{m}\in C_1$$ - \subsection{Das technische Neuron} - Ein technisches Neuron besteht aus den Gewichten für die Eingangswerte und der Aktivierungsfunktion:\\ - \includegraphics[width=.8\textwidth]{technisches_neuron.png} \ No newline at end of file + \section{Das technische Neuron} + Ein technisches Neuron besteht aus den Gewichten für die Eingangswerte und der Aktivierungsfunktion:\\ + \includegraphics[width=.8\textwidth]{technisches_neuron.png} + + \section{Das Perzeptron} + Ein Perzeptron beschreibt eine lineare Maschine, die eine Datenmenge durch eine Hyper-Ebene (die Diskriminante) in zwei Cluster unterteilt. + Die Funktion für die Diskriminante ist hierbei $y(\bm{m})=\text{sng}(g(\bm{m})) = \text{sgn}(\bm{w}^T\bm{m}+w_0)$. + Da sich $\bm{w}$ durch $\bm{w} = \sum^n_{i=1}\alpha_i\cdot y_i \cdot \bm{m}_i$ (mit $n = $Anzahl der Datenpunkte und $\alpha_i = $ Anzahl, wie oft $\bm{m}_i$ ausgewählt wurde) definiert ist die Dimension von $\bm{m}$ unwichtig. + + \subsection{Beispiel: nicht-symmetrischer Lernalgorithmus} + \includegraphics[width=.8\textwidth]{Perzeptron_Lernalgorithmus.png} + + \subsection{Novikoff's Theorem} + Novikoff's Theorem besagt, dass der Lernalgorithmus des Perzeptrons bei einem linear trennbaren Datensatz unweigerlich eine Diskriminante findet. + Zudem besagt es, dass der Algorithmus bei einem nicht linear separierbaren Datensatz nicht konvergieren kann. + Um die Konvergenz zu beschleunigen ist es sinnvoll die sigmoid-Funktion anstatt der signum-Funktion zu verwenden. + $$ \text{sigm}(x)=\frac{2}{1+e^ {-x}}-1; -1\le\text{sigm}(x)\le 1$$ + + \subsection{Beispiel: symmetrischer Lernalgorithmus} + \includegraphics[width=.8\textwidth]{Perzeptron_Lernalgorithmus_symmetrisch.png} + + \section{\acs{vc-dimension}} + Die \acl{vc-dimension} gibt ein Maß für die \say{learning power} einer Klassifizierung. + + \subsection{Shattering} + \say{Here we will only consider functions that correspond to the two-class pattern recognition case, so that $g(\bm{m}, y) \in \{-1, 1\}\forall x, y$. + Now if a given set of n points can be labeled in all possible $2^n$ ways, + and for each labeling, a member of the set $\{g(y)\}$ can be found which correctly assigns those labels, we say that that set of points is shattered by that set of functions.}\\ + Hierbei zergliedert (shatters) eine Hyperebene in einem Feature Space $\mathbb{R}^d$ $h=d+1$ linear unabhängige Punkte. + + \subsubsection{Beispiel: Shattering im 2-dimensionalen Raum} + \includegraphics[width = .7\textwidth]{vc-dimension_shattering.png} + + \subsection{Das XOR-Problem} + Um das XOR-Problem zu zergliedern werden 2 Diskriminanten benötigt:\\ + \includegraphics[width=.8\textwidth]{XOR-Problem1.png}\\ + Um das XOR-Problem von einer linearen Maschine klassifizieren zu lassen muss diese aus mindestens 2 Schichten bestehen.\\ + \includegraphics[width=\textwidth]{XOR-Problem2.png} \ No newline at end of file diff --git a/images/Perzeptron_Lernalgorithmus.png b/images/Perzeptron_Lernalgorithmus.png new file mode 100644 index 0000000000000000000000000000000000000000..0b28b2e2dd80bdb230c84367c028be4ce3f697ac GIT binary patch literal 46581 zcmcG$_dl2a`#yfzdqt5^Dl>(UO+xl)AhM;X%*cwYjFK{vlu;-uE6Lt8L{=pknHd=o 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