Limitations of k-Means Clustering – Lecture Notes

starting conditions we keep {c,d}. The optimum solution then has cost \(x^2/2\) , the bad solution has \(z^2/2\) . So the ratio \(z^2/x^2\) can be made arbitrarily bad. ➜ in the worst case, a \(k\) -means solution [...] \smash {\mu _T-x}\mathstrut \right\rVert ^2 - \tfrac {|S|}{|S|-1}\left\lVert \smash {\mu _S-x}\mathstrut \right\rVert ^2 = \tfrac {1}{2} 4^2 - \tfrac {4}{3} 3^2 = 8 - 12 = -4 \end{align*}\] ➜ The standard [...] If we have chosen \(a\) and \(d\) , it chooses the better solution with probability \(\frac{z^2}{x^2+z^2}\) . Nearest Center Assignment is Not Optimal [ HaWo79 ; TeVa10 ] The standard algorithm converges …

Limitations of k-Means Clustering – Lecture Notes

starting conditions we keep {c,d}. The optimum solution then has cost \(x^2/2\) , the bad solution has \(z^2/2\) . So the ratio \(z^2/x^2\) can be made arbitrarily bad. ➜ in the worst case, a \(k\) -means solution [...] \smash {\mu _T-x}\mathstrut \right\rVert ^2 - \tfrac {|S|}{|S|-1}\left\lVert \smash {\mu _S-x}\mathstrut \right\rVert ^2 = \tfrac {1}{2} 4^2 - \tfrac {4}{3} 3^2 = 8 - 12 = -4 \end{align*}\] ➜ The standard [...] If we have chosen \(a\) and \(d\) , it chooses the better solution with probability \(\frac{z^2}{x^2+z^2}\) . Nearest Center Assignment is Not Optimal [ HaWo79 ; TeVa10 ] The standard algorithm converges …

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L(x ; y) = f (x) + yT (Ax − b) Lρ(x ; y) = f (x) + yT (Ax − b) + ρ 2 kAx − bk2 2 ρ > 0 : penalty parameter min x f (x) + ρ 2 kAx − bk2 s.t. Ax = b MM: Method of Multipliers Dual ascent with augmented L [...] xk+1 i = argmin xi ⇣ fi (xi ) + (yk i )T (xi − x̄k) + ρ 2 kxi − x̄kk22 ⌘ yk+1 i = yk i + ρ(xk+1 i − x̄k+1) x1 x2 xN . . . . . . y f1 f2 . . . fN x1 x2 xN x̄ x̄ x̄ Message-Passing Algorithm •  Communication [...] R p×n , B ∈ R p×m Augmented Lagrangian: Lρ(x , z ; y) = f (x) + g(z) + yT (Ax + Bz − c) + ρ 2 kAx + Bz − ck2 2 “Alternating direction” method of multipliers xk+1 ∈ argmin x Lρ(x , z k ; yk) zk+1 ∈ argmin …

k-NN 1 2 3 4 5 0.6 0.8 1.0 1.2 earn 95.9 96.1 96.1 97.3 98.2 98.4 98.5 98.4 98.3 98.5 98.5 98.4 98.3 acq 91.5 92.1 85.3 92.0 92.6 94.6 95.2 95.2 95.3 95.0 95.3 95.3 95.4 money-fx 62.9 67.6 69.4 78.2 66.9 [...] 67.9 73.1 76.2 74.4 75.0 76.2 76.1 ship 78.7 83.1 80.9 79.2 82.0 85.4 86.0 86.5 86.0 85.4 86.5 87.6 87.1 wheat 60.6 79.4 85.5 76.6 83.1 84.5 85.2 85.9 83.8 85.2 85.9 85.9 85.9 corn 47.3 62.2 87.7 77.9 86 [...] k-NN 1 2 3 4 0.6 0.8 1.0 Pathology 52.7 50.8 47.6 53.4 50.3 54.9 57.2 58.2 56.7 57.4 58.1 Cardiovascular 72.4 70.1 70.5 72.6 71.1 76.2 77.6 77.3 77.2 77.5 77.6 Immunologic 61.7 58.0 58.8 66.8 69.7 73.2 73 …

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k-NN 1 2 3 4 5 0.6 0.8 1.0 1.2 earn 95.9 96.1 96.1 97.3 98.2 98.4 98.5 98.4 98.3 98.5 98.5 98.4 98.3 acq 91.5 92.1 85.3 92.0 92.6 94.6 95.2 95.2 95.3 95.0 95.3 95.3 95.4 money-fx 62.9 67.6 69.4 78.2 66.9 [...] 67.9 73.1 76.2 74.4 75.0 76.2 76.1 ship 78.7 83.1 80.9 79.2 82.0 85.4 86.0 86.5 86.0 85.4 86.5 87.6 87.1 wheat 60.6 79.4 85.5 76.6 83.1 84.5 85.2 85.9 83.8 85.2 85.9 85.9 85.9 corn 47.3 62.2 87.7 77.9 86 [...] k-NN 1 2 3 4 0.6 0.8 1.0 Pathology 52.7 50.8 47.6 53.4 50.3 54.9 57.2 58.2 56.7 57.4 58.1 Cardiovascular 72.4 70.1 70.5 72.6 71.1 76.2 77.6 77.3 77.2 77.5 77.6 Immunologic 61.7 58.0 58.8 66.8 69.7 73.2 73 …

operationalisiert. Hierfür wird zunächst in Abschn. 2.2.1 theore- tisch fundiert, was Verständnis und Prozesse des Verständnisaufbaus ausmacht. In Abschn. 2.2.2 werden daraus Anforderungen an qualitativ gute [...] 15 26 10 8 20 Fokusvideos Variable 13 11 9 10 6 11 6 0 1 Term 18 16 11 2 1 11 7 2 7 Gleichung 8 6 5 6 4 1 0 1 2 Kalkülvideos – 5 2 1 0 0 1 0 3 4 Der untere Teil von Tab. 5 zeigt, dass auch in den Kalkülvideos [...] Kalkül sind, haben Glade und Prediger (2017) am Beispiel des Brüche-Multiplizierens herausgearbeitet. 2.2.2 Ableitung von Qualitätsanforderungen an Verstehensangebote (in Erklärungen und Erklärvideos) Entsprechend …