From 3ce0a995913f068d5775489e33814a99165b1ce8 Mon Sep 17 00:00:00 2001 From: Stefan Dresselhaus Date: Sun, 1 Oct 2017 20:06:41 +0200 Subject: [PATCH] more graphs for evo1d --- arbeit/bibma.bib | 25 ++ arbeit/files/erklaerung.aux | 2 +- arbeit/ma.md | 197 +++++++++---- arbeit/ma.pdf | Bin 180653 -> 196317 bytes arbeit/ma.tex | 69 ++++- arbeit/settings/commands.tex | 14 +- ...n1D_5x5_100Times-added_one.gnuplot.fit.log | 6 +- ...on1D_5x5_100Times-added_one.gnuplot.script | 12 +- ...mes-added_one_improvement-vs-evo-error.png | Bin 5820 -> 5822 bytes ...00Times-added_one_improvement-vs-steps.png | Bin 5770 -> 5841 bytes ...100Times-added_one_regularity-vs-steps.png | Bin 6099 -> 6150 bytes .../20170830-evolution1D_5x5_100Times-all.csv | 201 ++++++++++++++ ...olution1D_5x5_100Times-all.gnuplot.fit.log | 138 +++++++++ ...0-evolution1D_5x5_100Times-all.gnuplot.log | 261 ++++++++++++++++++ ...volution1D_5x5_100Times-all.gnuplot.script | 20 ++ ..._100Times-all_improvement-vs-evo-error.png | Bin 0 -> 7284 bytes ..._5x5_100Times-all_improvement-vs-steps.png | Bin 0 -> 8280 bytes ...D_5x5_100Times-all_regularity-vs-steps.png | Bin 0 -> 8359 bytes ...0-evolution1D_5x5_100Times.gnuplot.fit.log | 6 +- ...30-evolution1D_5x5_100Times.gnuplot.script | 12 +- ..._5x5_100Times_improvement-vs-evo-error.png | Bin 5521 -> 5677 bytes ...on1D_5x5_100Times_improvement-vs-steps.png | Bin 5703 -> 5818 bytes ...ion1D_5x5_100Times_regularity-vs-steps.png | Bin 5579 -> 5614 bytes dokumentation/evolution1d/combine.sh | 33 +++ dokumentation/evolution1d/images.sh | 12 +- 25 files changed, 918 insertions(+), 90 deletions(-) create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all.csv create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all.gnuplot.fit.log create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all.gnuplot.log create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all.gnuplot.script create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-evo-error.png create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all_improvement-vs-steps.png create mode 100644 dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-all_regularity-vs-steps.png create mode 100755 dokumentation/evolution1d/combine.sh diff --git a/arbeit/bibma.bib b/arbeit/bibma.bib index 3fd1833..e54b214 100644 --- a/arbeit/bibma.bib +++ b/arbeit/bibma.bib @@ -50,3 +50,28 @@ publisher = {Springer-Verlag}, address = {Berlin, Heidelberg}, } +@book{golub2012matrix, + title={Matrix computations}, + author={Golub, Gene H and Van Loan, Charles F}, + volume={3}, + year={2012}, + publisher={JHU Press} +} +@article{weise2012evolutionary, + title={Evolutionary Optimization: Pitfalls and Booby Traps}, + author={Weise, Thomas and Chiong, Raymond and Tang, Ke}, + journal={J. Comput. Sci. \& Technol}, + volume={27}, + number={5}, + year={2012}, + url={http://jcst.ict.ac.cn:8080/jcst/EN/article/downloadArticleFile.do?attachType=PDF\&id=9543} +} +@inproceedings{thorhauer2014locality, + title={On the locality of standard search operators in grammatical evolution}, + author={Thorhauer, Ann and Rothlauf, Franz}, + booktitle={International Conference on Parallel Problem Solving from Nature}, + pages={465--475}, + year={2014}, + organization={Springer}, + url={https://www.lri.fr/~hansen/proceedings/2014/PPSN/papers/8672/86720465.pdf} +} diff --git a/arbeit/files/erklaerung.aux b/arbeit/files/erklaerung.aux index e175b91..599b851 100644 --- a/arbeit/files/erklaerung.aux +++ b/arbeit/files/erklaerung.aux @@ -22,7 +22,7 @@ \setcounter{ContinuedFloat}{0} \setcounter{float@type}{16} \setcounter{lstnumber}{1} -\setcounter{NAT@ctr}{5} +\setcounter{NAT@ctr}{8} \setcounter{AM@survey}{0} \setcounter{r@tfl@t}{0} \setcounter{subfigure}{0} diff --git a/arbeit/ma.md b/arbeit/ma.md index 65a5a3b..0c687c7 100644 --- a/arbeit/ma.md +++ b/arbeit/ma.md @@ -4,14 +4,16 @@ fontsize: 11pt \chapter*{How to read this Thesis} -As a guide through the nomenclature used in the formulas we prepend this chapter. +As a guide through the nomenclature used in the formulas we prepend this +chapter. Unless otherwise noted the following holds: - lowercase letters $x,y,z$ refer to real variables and represent a point in 3D-Space. - lowercase letters $u,v,w$ - refer to real variables between $0$ and $1$ used as coefficients in a 3D B-Spline grid. + refer to real variables between $0$ and $1$ used as coefficients in a 3D + B-Spline grid. - other lowercase letters refer to other scalar (real) variables. - lowercase **bold** letters (e.g. $\vec{x},\vec{y}$) @@ -21,12 +23,13 @@ Unless otherwise noted the following holds: # Introduction -In this Master Thesis we try to extend a previously proposed concept of predicting -the evolvability of \acf{FFD} given a Deformation-Matrix\cite{anrichterEvol}. -In the original publication the author used random sampled points weighted with -\acf{RBF} to deform the mesh and defined three different criteria that can be -calculated prior to using an evolutional optimisation algorithm to asses the -quality and potential of such optimisation. +In this Master Thesis we try to extend a previously proposed concept of +predicting the evolvability of \acf{FFD} given a +Deformation-Matrix\cite{anrichterEvol}. In the original publication the author +used random sampled points weighted with \acf{RBF} to deform the mesh and +defined three different criteria that can be calculated prior to using an +evolutional optimization algorithm to asses the quality and potential of such +optimization. We will replicate the same setup on the same meshes but use \acf{FFD} instead of \acf{RBF} to create a deformation and evaluate if the evolution-criteria still @@ -35,12 +38,14 @@ work as a predictor given the different deformation scheme. ## What is \acf{FFD}? First of all we have to establish how a \ac{FFD} works and why this is a good -tool for deforming meshes in the first place. For simplicity we only summarize the -1D-case from \cite{spitzmuller1996bezier} here and go into the extension to the 3D case in chapter \ref{3dffd}. +tool for deforming meshes in the first place. For simplicity we only summarize +the 1D-case from \cite{spitzmuller1996bezier} here and go into the extension to +the 3D case in chapter \ref{3dffd}. Given an arbitrary number of points $p_i$ alongside a line, we map a scalar value $\tau_i \in [0,1[$ to each point with $\tau_i < \tau_{i+1} \forall i$. -Given a degree of the target polynomial $d$ we define the curve $N_{i,d,\tau_i}(u)$ as follows: +Given a degree of the target polynomial $d$ we define the curve +$N_{i,d,\tau_i}(u)$ as follows: \begin{equation} \label{eqn:ffd1d1} N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases} @@ -52,12 +57,17 @@ and N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u) \end{equation} -If we now multiply every $p_i$ with the corresponding $N_{i,d,\tau_i}(u)$ we get the contribution of each -point $p_i$ to the final curve-point parameterized only by $u \in [0,1[$. -As can be seen from \eqref{eqn:ffd1d2} we only access points $[i..i+d]$ for any given $i$^[one more for each recursive step.], which -gives us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local interference of $d+1$ points. +If we now multiply every $p_i$ with the corresponding $N_{i,d,\tau_i}(u)$ we get +the contribution of each point $p_i$ to the final curve-point parameterized only +by $u \in [0,1[$. As can be seen from \eqref{eqn:ffd1d2} we only access points +$[i..i+d]$ for any given $i$^[one more for each recursive step.], which gives +us, in combination with choosing $p_i$ and $\tau_i$ in order, only a local +interference of $d+1$ points. -We can even derive this equation straightforward for an arbitrary $N$^[*Warning:* in the case of $d=1$ the recursion-formula yields a $0$ denominator, but $N$ is also $0$. The right solution for this case is a derivative of $0$]: +We can even derive this equation straightforward for an arbitrary +$N$^[*Warning:* in the case of $d=1$ the recursion-formula yields a $0$ +denominator, but $N$ is also $0$. The right solution for this case is a +derivative of $0$]: $$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$ @@ -65,30 +75,42 @@ For a B-Spline $$s(u) = \sum_{i} N_{i,d,\tau_i}(u) p_i$$ these derivations yield $\frac{\partial^d}{\partial u} s(u) = 0$. -Another interesting property of these recursive polynomials is that they are continuous (given $d \ge 1$) as every $p_i$ gets -blended in linearly between $\tau_i$ and $\tau_{i+d}$ and out linearly between $\tau_{i+1}$ and $\tau_{i+d+1}$ -as can bee seen from the two coefficients in every step of the recursion. +Another interesting property of these recursive polynomials is that they are +continuous (given $d \ge 1$) as every $p_i$ gets blended in linearly between +$\tau_i$ and $\tau_{i+d}$ and out linearly between $\tau_{i+1}$ and +$\tau_{i+d+1}$ as can bee seen from the two coefficients in every step of the +recursion. ### Why is \ac{FFD} a good deformation function? -The usage of \ac{FFD} as a tool for manipulating follows directly from the properties of the polynomials and the correspondence to -the control points. -Having only a few control points gives the user a nicer high-level-interface, as she only needs to move these points and the -model follows in an intuitive manner. The deformation is smooth as the underlying polygon is smooth as well and affects as many -vertices of the model as needed. Moreover the changes are always local so one risks not any change that a user cannot immediately see. +The usage of \ac{FFD} as a tool for manipulating follows directly from the +properties of the polynomials and the correspondence to the control points. +Having only a few control points gives the user a nicer high-level-interface, as +she only needs to move these points and the model follows in an intuitive +manner. The deformation is smooth as the underlying polygon is smooth as well +and affects as many vertices of the model as needed. Moreover the changes are +always local so one risks not any change that a user cannot immediately see. -But there are also disadvantages of this approach. The user loses the ability to directly influence vertices and even seemingly simple tasks as -creating a plateau can be difficult to achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}. +But there are also disadvantages of this approach. The user loses the ability to +directly influence vertices and even seemingly simple tasks as creating a +plateau can be difficult to +achieve\cite[chapter~3.2]{hsu1991dmffd}\cite{hsu1992direct}. -This disadvantages led to the formulation of \acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly interacts with the surface-mesh. -All interactions will be applied proportionally to the control-points that make up the parametrization of the interaction-point -itself yielding a smooth deformation of the surface *at* the surface without seemingly arbitrary scattered control-points. -Moreover this increases the efficiency of an evolutionary optimization\cite{Menzel2006}, which we will use later on. +This disadvantages led to the formulation of +\acf{DM-FFD}\cite[chapter~3.3]{hsu1991dmffd} in which the user directly +interacts with the surface-mesh. All interactions will be applied +proportionally to the control-points that make up the parametrization of the +interaction-point itself yielding a smooth deformation of the surface *at* the +surface without seemingly arbitrary scattered control-points. Moreover this +increases the efficiency of an evolutionary optimization\cite{Menzel2006}, which +we will use later on. -But this approach also has downsides as can be seen in \cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the tessellation of -the invisible grid has a major impact on the deformation itself. +But this approach also has downsides as can be seen in +\cite[figure~7]{hsu1991dmffd}\todo{figure hier einfügen?}, as the tessellation +of the invisible grid has a major impact on the deformation itself. -All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon mesh albeit the downsides. +All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon +mesh albeit the downsides. ## What is evolutional optimization? @@ -96,26 +118,83 @@ All in all \ac{FFD} and \ac{DM-FFD} are still good ways to deform a high-polygon ## Wieso ist evo-Opt so cool? -The main advantage of evolutional algorithms is the ability to find optima of general functions just with the help of a given -error-function (or fitness-function in this domain). This avoids the general pitfalls of gradient-based procedures, which often -target the same error-function as an evolutional algorithm, but can get stuck in local optima. +The main advantage of evolutional algorithms is the ability to find optima of +general functions just with the help of a given error-function (or +fitness-function in this domain). This avoids the general pitfalls of +gradient-based procedures, which often target the same error-function as an +evolutional algorithm, but can get stuck in local optima. -This is mostly due to the fact that a gradient-based procedure has only one point of observation from where it evaluates the next -steps, whereas an evolutional strategy starts with a population of guessed solutions. Because an evolutional strategy modifies -the solution randomly, keeps the best solutions and purges the worst, it can also target multiple different hypothesis at the same time -where the local optima die out in the face of other, better candidates. +This is mostly due to the fact that a gradient-based procedure has only one +point of observation from where it evaluates the next steps, whereas an +evolutional strategy starts with a population of guessed solutions. Because an +evolutional strategy modifies the solution randomly, keeps the best solutions +and purges the worst, it can also target multiple different hypothesis at the +same time where the local optima die out in the face of other, better +candidates. -If an analytic best solution exists (i.e. because the error-function is convex) an evolutional algorithm is not the right choice. Although -both converge to the same solution, the analytic one is usually faster. But in reality many problems have no analytic solution, because -the problem is not convex. Here evolutional optimization has one more advantage as you get bad solutions fast, which refine over time. +If an analytic best solution exists (i.e. because the error-function is convex) +an evolutional algorithm is not the right choice. Although both converge to the +same solution, the analytic one is usually faster. But in reality many problems +have no analytic solution, because the problem is not convex. Here evolutional +optimization has one more advantage as you get bad solutions fast, which refine +over time. +## Criteria for the evolvability of linear deformations +### Variability -## Evolvierbarkeitskriterien +In \cite{anrichterEvol} variability is defined as +$$V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},$$ +whereby $\vec{U}$ is the $m \times n$ deformation-Matrix used to map the $m$ +control points onto the $n$ vertices. -- Konditionszahl etc. +Given $n = m$, an identical number of control-points and vertices, this +quotient will be $=1$ if all control points are independent of each other and +the solution is to trivially move every control-point onto a target-point. -# Hauptteil +In praxis the value of $V(\vec{U})$ is typically $\ll 1$, because as +there are only few control-points for many vertices, so $m \ll n$. + +Additionally in our setup we connect neighbouring control-points in a grid so +each control point is not independent, but typically depends on $4^d$ +control-points for an $d$-dimensional control mesh. + +### Regularity + +Regularity is defined\cite{anrichterEvol} as +$$R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}$$ +where $\sigma_{min}$ and $\sigma_{max}$ are the smallest and greatest right singular +value of the deformation-matrix $\vec{U}$. + +As we deform the given Object only based on the parameters as $\vec{p} \mapsto +f(\vec{x} + \vec{U}\vec{p})$ this makes sure that $\|\vec{Up}\| \propto +\|\vec{p}\|$ when $\kappa(\vec{U}) \approx 1$. The inversion of $\kappa(\vec{U})$ +is only performed to map the criterion-range to $[0..1]$, whereas $1$ is the +optimal value and $0$ is the worst value. + +This criterion should be characteristic for numeric stability on the on +hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of evolutional +algorithms on the other hand\cite{anrichterEvol} as it is tied to the notion of +locality\cite{weise2012evolutionary,thorhauer2014locality}. + +### Improvement Potential + +In contrast to the general nature of variability and regularity, which are +agnostic of the fitness-function at hand the third criterion should reflect a +notion of potential. + +As during optimization some kind of gradient $g$ is available to suggest a +direction worth pursuing we use this to guess how much change can be achieved in +the given direction. + +The definition for an improvement potential $P$ is\cite{anrichterEvol}: +$$ +P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F +$$ +given some approximate $n \times d$ fitness-gradient $\vec{G}$, normalized to +$\|\vec{G}\|_F = 1$, whereby $\|\cdot\|_F$ denotes the Frobenius-Norm. + +# Implementation of \acf{FFD} ## Was ist FFD? \label{3dffd} @@ -124,41 +203,43 @@ the problem is not convex. Here evolutional optimization has one more advantage - Wieso Newton-Optimierung? - Was folgt daraus? -## Szenarien vorstellen +## Test Scenario: 1D Function Approximation -### 1D - -#### Optimierungszenario +### Optimierungszenario - Ebene -> Template-Fit -#### Matching in 1D +### Matching in 1D - Trivial -#### Besonderheiten der Auswertung +### Besonderheiten der Auswertung - Analytische Lösung einzig beste - Ergebnis auch bei Rauschen konstant? - normierter 1-Vektor auf den Gradienten addieren - Kegel entsteht -### 3D +## Test Scenario: 3D Function Approximation -#### Optimierungsszenario +### Optimierungsszenario - Ball zu Mario -#### Matching in 3D +### Matching in 3D - alternierende Optimierung -#### Besonderheiten der Optimierung +### Besonderheiten der Optimierung - Analytische Lösung nur bis zur Optimierung der ersten Punkte gültig - Kriterien trotzdem gut -# Evaluation +# Evaluation of Scenarios + +## Results of 1D Function Approximation + + ## Spearman/Pearson-Metriken @@ -166,7 +247,7 @@ the problem is not convex. Here evolutional optimization has one more advantage - Wieso sollte uns das interessieren? - Wieso reicht Monotonie? - Haben wir das gezeigt? -- Stastik, Bilder, blah! +- Statistik, Bilder, blah! # Schluss diff --git a/arbeit/ma.pdf b/arbeit/ma.pdf index 8280bea1beee17255244e2b298950d66f721967e..bda829dfc2b5d75a5f725fe11dcbc69d6abd1bd9 100644 GIT binary patch delta 122317 zcmZs>Q*__YAMG7BR+Gl~290gow%x|I^NDTSZk#l>?Z&o~#?I;Qzs_3cxp;0y7c*;a z-t(HhcU2(5#ydjXcW~CkS!Swu1r&6|CS&* zG`?x3@0xlc(aypS`71pG0E1*1Y60h;cMp1BOlH+m+KQpYhXdZ@XnL8!&U4}EgDw^m zy-S$FQ>F8(Xd$<6+)L>n-d6hSVQJ)nqPTltPv^Q^TaF}iOp;TFCkmZc8L{IHmgChl zXVe0E7JVQ!%VWxGAgv{aic!_m?@9aa!Q*SP&vm>C<~02z#-|$B9PYbIDoD(APg@{6 z1{u|(Gh^yyo&xGH04BDI0J~pwSqscoP9-`rcvKd8D5ie3i>5-z=ORTo_B7V3g?ZR> zz+>dqQdn!T`0N_5riXQD5hHmmONVslv)9161%DUb#P+9CVkpEmI9s9+A37u>BXeRt z2PJTkF6UIpj_r5#@eO|vTs;?7p1=&f`NY|*8Bf7$`9aWna1kRAfyCxZP#%aOlzu>89__6(ZFd z@6<#t;Pd?8>~J$DS`6$3>*thWWU;izEe(Q(Q}_|npdZ<+RxVneK`Pc3BdLICncqf= 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zD{7Oavvjw&&k6KP@JnXY^T&51r5m~h{O_p2Z7^_|A;8^i60L^ZkLk!uwdi||(T@Gb)svv~nh}F1hPw@E!c4e$ge>v0 z%(zgoWyGah4!28Me5})~ht&lz3tL8ke?JMEe-VI1jTsOR$8QPMpT%64e)*qc&fFL& zZv|xS&IQYhy8?PRPrL(fRkR8=+P)WCxk5!qp@=R!Nj55XI%+NcqmbQ1) Vhm9=`NC+$d0^=|<%WEj${2w4#Zyx{v diff --git a/arbeit/ma.tex b/arbeit/ma.tex index c84e9bf..7b20021 100644 --- a/arbeit/ma.tex +++ b/arbeit/ma.tex @@ -151,8 +151,8 @@ predicting the evolvability of \acf{FFD} given a Deformation-Matrix\cite{anrichterEvol}. In the original publication the author used random sampled points weighted with \acf{RBF} to deform the mesh and defined three different criteria that can be calculated prior -to using an evolutional optimisation algorithm to asses the quality and -potential of such optimisation. +to using an evolutional optimization algorithm to asses the quality and +potential of such optimization. We will replicate the same setup on the same meshes but use \acf{FFD} instead of \acf{RBF} to create a deformation and evaluate if the @@ -265,13 +265,64 @@ in reality many problems have no analytic solution, because the problem is not convex. Here evolutional optimization has one more advantage as you get bad solutions fast, which refine over time. -\section{Evolvierbarkeitskriterien}\label{evolvierbarkeitskriterien} +\section{Criteria for the evolvability of linear +deformations}\label{criteria-for-the-evolvability-of-linear-deformations} -\begin{itemize} -\tightlist -\item - Konditionszahl etc. -\end{itemize} +\subsection{Variability}\label{variability} + +In \cite{anrichterEvol} variability is defined as +\[V(\vec{U}) := \frac{\textrm{rank}(\vec{U})}{n},\] whereby \(\vec{U}\) +is the \(m \times n\) deformation-Matrix used to map the \(m\) control +points onto the \(n\) vertices. + +Given \(n = m\), an identical number of control-points and vertices, +this quotient will be \(=1\) if all control points are independent of +each other and the solution is to trivially move every control-point +onto a target-point. + +In praxis the value of \(V(\vec{U})\) is typically \(\ll 1\), because as +there are only few control-points for many vertices, so \(m \ll n\). + +Additionally in our setup we connect neighbouring control-points in a +grid so each control point is not independent, but typically depends on +\(4^d\) control-points for an \(d\)-dimensional control mesh. + +\subsection{Regularity}\label{regularity} + +Regularity is defined\cite{anrichterEvol} as +\[R(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}}\] +where \(\sigma_{min}\) and \(\sigma_{max}\) are the smallest and +greatest right singular value of the deformation-matrix \(\vec{U}\). + +As we deform the given Object only based on the parameters as +\(\vec{p} \mapsto f(\vec{x} + \vec{U}\vec{p})\) this makes sure that +\(\|\vec{Up}\| \propto \|\vec{p}\|\) when \(\kappa(\vec{U}) \approx 1\). +The inversion of \(\kappa(\vec{U})\) is only performed to map the +criterion-range to \([0..1]\), whereas \(1\) is the optimal value and +\(0\) is the worst value. + +This criterion should be characteristic for numeric stability on the on +hand\cite[chapter 2.7]{golub2012matrix} and for convergence speed of +evolutional algorithms on the other hand\cite{anrichterEvol} as it is +tied to the notion of +locality\cite{weise2012evolutionary,thorhauer2014locality}. + +\subsection{Improvement Potential}\label{improvement-potential} + +In contrast to the general nature of variability and regularity, which +are agnostic of the fitness-function at hand the third criterion should +reflect a notion of potential. + +As during optimization some kind of gradient \(g\) is available to +suggest a direction worth pursuing we use this to guess how much change +can be achieved in the given direction. + +The definition for an improvement potential \(P\) +is\cite{anrichterEvol}: \[ +P(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec(G)\|^2_F +\] given some approximate \(n \times d\) fitness-gradient \(\vec{G}\), +normalized to \(\|\vec{G}\|_F = 1\), whereby \(\|\cdot\|_F\) denotes the +Frobenius-Norm. \chapter{Hauptteil}\label{hauptteil} @@ -372,7 +423,7 @@ Optimierung}\label{besonderheiten-der-optimierung} \item Haben wir das gezeigt? \item - Stastik, Bilder, blah! + Statistik, Bilder, blah! \end{itemize} \chapter{Schluss}\label{schluss} diff --git a/arbeit/settings/commands.tex b/arbeit/settings/commands.tex index ba7d454..c2666d1 100644 --- a/arbeit/settings/commands.tex +++ b/arbeit/settings/commands.tex @@ -87,13 +87,13 @@ \renewcommand\lq{\text{\textgravedbl}\!\!}\renewcommand\rq{\!\!\text{\textacutedbl}} % ##### tables ##### -\newcommand\rl[1]{\multicolumn{1}{r|}{#1}} % item| -\renewcommand\ll[1]{\multicolumn{1}{|r}{#1}} % |item -% \newcommand\cc[1]{\multicolumn{1}{|c|}{#1}} % |item| -\newcommand\lc[2]{\multicolumn{#1}{|c|}{#2}} % -\newcommand\cc[1]{\multicolumn{1}{c}{#1}} -% \renewcommand{\arraystretch}{1.2} % Tabellenzeilen ein bischen h?her machen. -\newcommand\m[2]{\multirow{#1}{*}{$#2$}} +% \newcommand\rl[1]{\multicolumn{1}{r|}{#1}} % item| +% \renewcommand\ll[1]{\multicolumn{1}{|r}{#1}} % |item +% % \newcommand\cc[1]{\multicolumn{1}{|c|}{#1}} % |item| +% \newcommand\lc[2]{\multicolumn{#1}{|c|}{#2}} % +% \newcommand\cc[1]{\multicolumn{1}{c}{#1}} +% % \renewcommand{\arraystretch}{1.2} % Tabellenzeilen ein bischen h?her machen. +% \newcommand\m[2]{\multirow{#1}{*}{$#2$}} % ##### Text symbole ##### % \newcommand\subdot[1]{\lower0.5em\hbox{$\stackrel{\displaystyle #1}{.}$}} diff --git a/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.fit.log b/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.fit.log index c803c2c..75670fd 100644 --- a/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.fit.log +++ b/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.fit.log @@ -1,7 +1,7 @@ ******************************************************************************* -Wed Aug 30 22:31:49 2017 +Sun Oct 1 19:59:33 2017 FIT: data read from "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 2:5 @@ -47,7 +47,7 @@ b -0.996 1.000 ******************************************************************************* -Wed Aug 30 22:31:49 2017 +Sun Oct 1 19:59:33 2017 FIT: data read from "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:5 @@ -93,7 +93,7 @@ bb -1.000 1.000 ******************************************************************************* -Wed Aug 30 22:31:49 2017 +Sun Oct 1 19:59:33 2017 FIT: data read from "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:6 diff --git a/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.script b/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.script index 18cfebc..f607525 100644 --- a/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.script +++ b/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one.gnuplot.script @@ -2,13 +2,19 @@ set datafile separator "," f(x)=a*x+b fit f(x) "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 2:5 via a,b set terminal png +set xlabel 'regularity' +set ylabel 'steps' set output "20170830-evolution1D_5x5_100Times-added_one_regularity-vs-steps.png" -plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 2:5 title "regularity vs. steps", f(x) lc rgb "black" +plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 2:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:5 via aa,bb +set xlabel 'improvement potential' +set ylabel 'steps' set output "20170830-evolution1D_5x5_100Times-added_one_improvement-vs-steps.png" -plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:5 title "improvement potential vs. steps", g(x) lc rgb "black" +plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:6 via aaa,bbb +set xlabel 'improvement potential' +set ylabel 'evolution error' set output "20170830-evolution1D_5x5_100Times-added_one_improvement-vs-evo-error.png" -plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:6 title "improvement potential vs. evolution error", h(x) lc rgb "black" +plot "20170830-evolution1D_5x5_100Times-added_one.csv" every ::1 using 4:6 title "data", h(x) title "lin. fit" lc rgb "black" diff --git a/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one_improvement-vs-evo-error.png b/dokumentation/evolution1d/20170830-evolution1D_5x5_100Times-added_one_improvement-vs-evo-error.png index 0ac3c3ec9c892747609841d9c1ba74bac39d6bc2..dc1c599c8ea0f96017a50f8e6b4161ed39472337 100644 GIT binary patch delta 5493 zcmb7Ic{o(>+n)_H)?w`H3^JB5mXOfQgzRK(L1PVxY+t5iCM4Oj%UU#&l*m%Hv6Nl1 zBq3v8l3kST9l!VYUcc*lulJw#T<1E^`F!r>xz2q*_c`}@>g5~d%Y%U=WkX$@bx6*y 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every ::1 using 2:5 title "data", f(x) title "lin. fit" lc rgb "black" g(x)=aa*x+bb fit g(x) "$data" every ::1 using 4:5 via aa,bb +set xlabel 'improvement potential' +set ylabel 'steps' set output "${png}_improvement-vs-steps.png" -plot "$data" every ::1 using 4:5 title "improvement potential vs. steps", g(x) lc rgb "black" +plot "$data" every ::1 using 4:5 title "data", g(x) title "lin. fit" lc rgb "black" h(x)=aaa*x+bbb fit h(x) "$data" every ::1 using 4:6 via aaa,bbb +set xlabel 'improvement potential' +set ylabel 'evolution error' set output "${png}_improvement-vs-evo-error.png" -plot "$data" every ::1 using 4:6 title "improvement potential vs. evolution error", h(x) lc rgb "black" +plot "$data" every ::1 using 4:6 title "data", h(x) title "lin. fit" lc rgb "black" EOD ) > "${png}.gnuplot.script" gnuplot "${png}.gnuplot.script" 2> "${png}.gnuplot.log"