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dm.cs.tu-dortmund.de/mlbits/neural-nlp-positional-encoding/

Positional Encoding – Lecture Notes

compute position differences. \[ PE_{(pos,2i)} = sin\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \quad\quad PE_{(pos,2i+1)} = cos\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \] \(d_{\tex [...] \(\theta_i=10000^{-2(i-1)/d}\) similar to the sinusoidal embeddings. We now multiply these rotations onto the vectors, which preserves the dot products. Because the matrix is composed of dim/2 submatrixes of [...] of size 2×2, the rotation has only linear cost. Attention with Linear Biases (ALiBi) [ PrSmLe22 ] Position encodings make nearby tokens more similar than far away tokens. Idea: a similar effect can be achieved …
dm.cs.tu-dortmund.de/mlbits/cluster-density-dbscan-foundations/

Density-Based Clustering Fundamentals – Lecture Notes

iangleright }}{{\cdot }\mkern 2mu}}o_2 \mathrel {\centerOver{{\vartriangleright }}{{\cdot }\mkern 2mu}}\ldots \mathrel {\centerOver{{\vartriangleright }}{{\cdot }\mkern 2mu}}o_k=q\] Density-connectivity [...] \text{minPts}\Leftrightarrow \hat{\! f}(x) \geq \tfrac {\text{minPts}}{2n\varepsilon }\] Density-based Clustering: Foundations /2 Direct density reachability : An object \(q \in O\) is direct-density-reachable [...] (denote as \(q\mathrel {\centerOver{{\vartriangleleft }}{\mkern 2mu{\cdot }}}p\) or \(p\mathrel {\centerOver{{\vartriangleright }}{{\cdot }\mkern 2mu}}q\) ) if and only if \(p\) is a core point, and \(\text …
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www.akjstat.tu-dortmund.de/fileadmin/Forum_2_Personal_HzE_AKJStat_Fendrich_2022-09-28.pdf

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33 7 12 .9 79 3. 56 4 1. 28 0 12 .9 24 3. 54 4 1. 22 8 10 .4 68 3. 10 2 1. 22 6 10 .1 69 3. 53 4 1. 31 4 7. 94 2 2. 27 4 1. 22 4 0 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 Stationäre Hilfen [...] AKJStat Folie Nr. 7 17 .5 93 4. 99 7 2. 13 6 16 .9 35 5. 37 0 1. 97 9 16 .4 53 5. 39 9 1. 92 4 13 .1 26 4. 55 1 1. 93 6 12 .5 81 5. 32 8 2. 11 9 9. 85 8 3. 41 4 1. 89 9 0 2.000 4.000 6.000 8.000 10.000 12.000 [...] das Personal in den Hilfen zur Erziehung – Ergebnisse aus der Kinder- und Jugendhilfestatistik Forum 2: Personal in den Hilfen zur Erziehung – wie attraktiv ist der Arbeitsplatz? Input im Rahmen der digitalen …
dm.cs.tu-dortmund.de/en/mlbits/neural-nlp-positional-encoding/

Positional Encoding – Lecture Notes

compute position differences. \[ PE_{(pos,2i)} = sin\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \quad\quad PE_{(pos,2i+1)} = cos\left(\frac{pos}{10000^{2i/d_{\text{model}}}}\right) \] \(d_{\tex [...] \(\theta_i=10000^{-2(i-1)/d}\) similar to the sinusoidal embeddings. We now multiply these rotations onto the vectors, which preserves the dot products. Because the matrix is composed of dim/2 submatrixes of [...] of size 2×2, the rotation has only linear cost. Attention with Linear Biases (ALiBi) [ PrSmLe22 ] Position encodings make nearby tokens more similar than far away tokens. Idea: a similar effect can be achieved …
dm.cs.tu-dortmund.de/mlbits/foundations-distances/

Distance Functions – Lecture Notes

Distance Functions /2 Many more variants: Squared Euclidean distance ( sum of squares ): \[\begin{align*} d_{\operatorname {Euclidean}}^2(x,y) =& \kern 5pt \sum \nolimits _d (x_d - y_d)^2 \quad = \left\lVert [...] y\right\rVert } = \tfrac { \sum _i x_i y_i }{ \sqrt{\sum _i x_i^2} \cdot \sqrt{\sum _i y_i^2} }\] only (!) on data normalized to \(L_2\) length \(\left\lVert x\right\rVert =\left\lVert y\right\rVert =1\) [...] where \(o_1\) has 1 and \(o_2\) has 0 \(f_{00}\) = the number of attributes where \(o_1\) has 0 and \(o_2\) has 0 \(f_{11}\) = the number of attributes where \(o_1\) has 1 and \(o_2\) has 1 Simple matching …
dm.cs.tu-dortmund.de/en/mlbits/foundations-distances/

Distance Functions – Lecture Notes

Distance Functions /2 Many more variants: Squared Euclidean distance ( sum of squares ): \[\begin{align*} d_{\operatorname {Euclidean}}^2(x,y) =& \kern 5pt \sum \nolimits _d (x_d - y_d)^2 \quad = \left\lVert [...] y\right\rVert } = \tfrac { \sum _i x_i y_i }{ \sqrt{\sum _i x_i^2} \cdot \sqrt{\sum _i y_i^2} }\] only (!) on data normalized to \(L_2\) length \(\left\lVert x\right\rVert =\left\lVert y\right\rVert =1\) [...] where \(o_1\) has 1 and \(o_2\) has 0 \(f_{00}\) = the number of attributes where \(o_1\) has 0 and \(o_2\) has 0 \(f_{11}\) = the number of attributes where \(o_1\) has 1 and \(o_2\) has 1 Simple matching …
dm.cs.tu-dortmund.de/en/mlbits/cluster-hac-acceleration/

Accelerating Hierarchical Clustering – Lecture Notes

O}(n^2)\) time and memory. finding the minimum distance / maximum similarity: \(\mathcal{O}(n^2)\cdot i\) updating the matrix: \(\mathcal{O}(n)\cdot i\) (with Lance-Williams) or \(\mathcal{O}(n^2)\cdot [...] \) time and \(\mathcal{O}(n^2)\) memory! Better algorithms can run in guaranteed \(O(n^2)\) time [ Defa77 ; Sibs73 ] for special cases, with priority queues we get \(O(n^2 \log n)\) [ DaEd84 ; Epps98 ; [...] ; Müll11 ; Murt83 ] , \(O(n^2)\) for reducible linkages [ Murt83 ] , or usually \(n^2\) time with caching [ Ande73 ] . Instead of agglomerative (bottom-up), we can also begin with one cluster, and divide …
dm.cs.tu-dortmund.de/mlbits/cluster-hac-acceleration/

Accelerating Hierarchical Clustering – Lecture Notes

O}(n^2)\) time and memory. finding the minimum distance / maximum similarity: \(\mathcal{O}(n^2)\cdot i\) updating the matrix: \(\mathcal{O}(n)\cdot i\) (with Lance-Williams) or \(\mathcal{O}(n^2)\cdot [...] \) time and \(\mathcal{O}(n^2)\) memory! Better algorithms can run in guaranteed \(O(n^2)\) time [ Defa77 ; Sibs73 ] for special cases, with priority queues we get \(O(n^2 \log n)\) [ DaEd84 ; Epps98 ; [...] ; Müll11 ; Murt83 ] , \(O(n^2)\) for reducible linkages [ Murt83 ] , or usually \(n^2\) time with caching [ Ande73 ] . Instead of agglomerative (bottom-up), we can also begin with one cluster, and divide …
www.mathematik.tu-dortmund.de/fileadmin/user_upload/0000_FKM/S_L-Studiengaenge/Hinweisblatt-Pruefungen-M-TM-WM-Bachelor-BPO2019.pdf

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