name: layout-general layout: true class: left, top <style> .remark-slide-number { position: inherit; } .remark-slide-number .progress-bar-container { position: absolute; bottom: 0; height: 4px; display: block; left: 0; right: 0; } .remark-slide-number .progress-bar { height: 100%; background-color: rgba(2, 70, 79, 0.874); } .remark-slide-number .numeric { position: absolute; bottom: 4%; height: 4px; display: block; right: 2.5%; font-weight: bold; } .remark-slide-number .numeric-out{ color: rgba(2, 70, 79, 0.874); } </style> --- count: false class: top, center, title-slide <img src="img/X.png" align="right" height="110" width="100"> <img src="img/cnrs.png" align="middle" height="95"> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="img/UniversiteParisCite_logo.jpg" align="left" height="90"> <hr class="L1"> ## `\(K\)`-means Clustering Algorithm<br> via Vector Quantization <hr class="L2"> #### Graduate School of Science-RUPP <img src="img/RUPP_logo.png" align="middle" height="100"> ### Sothea .textsc[Has] --- ### In word<h0br> .stress[Clustering] consists in _partitioning_ a set of points from some metric space in such a way that .stress[points within the same group are close enough] while .stress[points from different groups are distant]. It belongs to the .stress[unsupervised learning] branch of Machine Learning (ML).<hbr> -- .pull-left[ ```r faithful[1:7,] %>% knitr::kable(format = "markdown") ``` | eruptions| waiting| |---------:|-------:| | 3.600| 79| | 1.800| 54| | 3.333| 74| | 2.283| 62| | 4.533| 85| | 2.883| 55| | 4.700| 88| ] .pull-right[ <img src="img/faithful.png" align="left" height="370" width='340'> ] --- count:false ### In word<h0br> .stress[Clustering] consists in _partitioning_ a set of points from some metric space in such a way that .stress[points within the same group are close enough] while .stress[points from different groups are distant]. 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] --- count:false ### In word<h0br> .stress[Clustering] consists in _partitioning_ a set of points from some metric space in such a way that .stress[points within the same group are close enough] while .stress[points from different groups are distant]. It belongs to the .stress[unsupervised learning] branch of Machine Learning (ML).<hbr> .pull-left[ ```r df_iris[id,] %>% select(S.L, S.W, P.L, P.W, Species) %>% knitr::kable(format = "markdown") ``` | | S.L| S.W| P.L| P.W|Species | |:---|---:|---:|---:|---:|:----------| |1 | 5.1| 3.5| 1.4| 0.2|setosa | |2 | 4.9| 3.0| 1.4| 0.2|setosa | |51 | 7.0| 3.2| 4.7| 1.4|versicolor | |52 | 6.4| 3.2| 4.5| 1.5|versicolor | |101 | 6.3| 3.3| 6.0| 2.5|virginica | |102 | 5.8| 2.7| 5.1| 1.9|virginica | ] .pull-right[ <img src="img/iris.png" align="left" height="365" width='350'> ] --- count:false ### In word<h0br> .stress[Clustering] consists in _partitioning_ a set of points from some metric space in such a way that .stress[points within the same group are close enough] while .stress[points from different groups are distant]. 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] --- count:false ### In word<h0br> .stress[Clustering] consists in _partitioning_ a set of points from some metric space in such a way that .stress[points within the same group are close enough] while .stress[points from different groups are distant]. It belongs to the .stress[unsupervised learning] branch of Machine Learning (ML).<hbr> ### Clustering in ML applications<h0br> Clustering shows up in many Machine Learning applications, for example: <h0br> .left-column[ - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M157.52 272h36.96L176 218.78 157.52 272zM352 256c-13.23 0-24 10.77-24 24s10.77 24 24 24 24-10.77 24-24-10.77-24-24-24zM464 64H48C21.5 64 0 85.5 0 112v288c0 26.5 21.5 48 48 48h416c26.5 0 48-21.5 48-48V112c0-26.5-21.5-48-48-48zM250.58 352h-16.94c-6.81 0-12.88-4.32-15.12-10.75L211.15 320h-70.29l-7.38 21.25A16 16 0 0 1 118.36 352h-16.94c-11.01 0-18.73-10.85-15.12-21.25L140 176.12A23.995 23.995 0 0 1 162.67 160h26.66A23.99 23.99 0 0 1 212 176.13l53.69 154.62c3.61 10.4-4.11 21.25-15.11 21.25zM424 336c0 8.84-7.16 16-16 16h-16c-4.85 0-9.04-2.27-11.98-5.68-8.62 3.66-18.09 5.68-28.02 5.68-39.7 0-72-32.3-72-72s32.3-72 72-72c8.46 0 16.46 1.73 24 4.42V176c0-8.84 7.16-16 16-16h16c8.84 0 16 7.16 16 16v160z"/></svg> __Marketing__]: finding groups of customers with similar behavior given a large database of customer data containing their properties and past buying records - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M581.12 96.2c-27.67-.15-52.5 17.58-76.6 26.62C489.98 88.27 455.83 64 416 64c-11.28 0-21.95 2.3-32 5.88V56c0-13.26-10.75-24-24-24h-16c-13.25 0-24 10.74-24 24v48.98C286.01 79.58 241.24 64 192 64 85.96 64 0 135.64 0 224v240c0 8.84 7.16 16 16 16h64c8.84 0 16-7.16 16-16v-70.79C128.35 407.57 166.72 416 208 416s79.65-8.43 112-22.79V464c0 8.84 7.16 16 16 16h64c8.84 0 16-7.16 16-16V288h128v32c0 8.84 7.16 16 16 16h32c8.84 0 16-7.16 16-16v-32c17.67 0 32-14.33 32-32v-92.02c0-34.09-24.79-67.59-58.88-67.78zM448 176c-8.84 0-16-7.16-16-16s7.16-16 16-16 16 7.16 16 16-7.16 16-16 16z"/></svg> __Biology__]: classification of plants and animals given their features - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M143.25 220.81l-12.42 46.37c-3.01 11.25-3.63 22.89-2.41 34.39l-35.2 28.98c-6.57 5.41-16.31-.43-14.62-8.77l15.44-76.68c1.06-5.26-2.66-10.28-8-10.79l-77.86-7.55c-8.47-.82-11.23-11.83-4.14-16.54l65.15-43.3c4.46-2.97 5.38-9.15 1.98-13.29L21.46 93.22c-5.41-6.57.43-16.3 8.78-14.62l76.68 15.44c5.26 1.06 10.28-2.66 10.8-8l7.55-77.86c.82-8.48 11.83-11.23 16.55-4.14l43.3 65.14c2.97 4.46 9.15 5.38 13.29 1.98l60.4-49.71c6.57-5.41 16.3.43 14.62 8.77L262.1 86.38c-2.71 3.05-5.43 6.09-7.91 9.4l-32.15 42.97-10.71 14.32c-32.73 8.76-59.18 34.53-68.08 67.74zm494.57 132.51l-12.42 46.36c-3.13 11.68-9.38 21.61-17.55 29.36a66.876 66.876 0 0 1-8.76 7l-13.99 52.23c-1.14 4.27-3.1 8.1-5.65 11.38-7.67 9.84-20.74 14.68-33.54 11.25L515 502.62c-17.07-4.57-27.2-22.12-22.63-39.19l8.28-30.91-247.28-66.26-8.28 30.91c-4.57 17.07-22.12 27.2-39.19 22.63l-30.91-8.28c-12.8-3.43-21.7-14.16-23.42-26.51-.57-4.12-.35-8.42.79-12.68l13.99-52.23a66.62 66.62 0 0 1-4.09-10.45c-3.2-10.79-3.65-22.52-.52-34.2l12.42-46.37c5.31-19.8 19.36-34.83 36.89-42.21a64.336 64.336 0 0 1 18.49-4.72l18.13-24.23 32.15-42.97c3.45-4.61 7.19-8.9 11.2-12.84 8-7.89 17.03-14.44 26.74-19.51 4.86-2.54 9.89-4.71 15.05-6.49 10.33-3.58 21.19-5.63 32.24-6.04 11.05-.41 22.31.82 33.43 3.8l122.68 32.87c11.12 2.98 21.48 7.54 30.85 13.43a111.11 111.11 0 0 1 34.69 34.5c8.82 13.88 14.64 29.84 16.68 46.99l6.36 53.29 3.59 30.05a64.49 64.49 0 0 1 22.74 29.93c4.39 11.88 5.29 25.19 1.75 38.39zM255.58 234.34c-18.55-4.97-34.21 4.04-39.17 22.53-4.96 18.49 4.11 34.12 22.65 39.09 18.55 4.97 45.54 15.51 50.49-2.98 4.96-18.49-15.43-53.67-33.97-58.64zm290.61 28.17l-6.36-53.29c-.58-4.87-1.89-9.53-3.82-13.86-5.8-12.99-17.2-23.01-31.42-26.82l-122.68-32.87a48.008 48.008 0 0 0-50.86 17.61l-32.15 42.97 172 46.08 75.29 20.18zm18.49 54.65c-18.55-4.97-53.8 15.31-58.75 33.79-4.95 18.49 23.69 22.86 42.24 27.83 18.55 4.97 34.21-4.04 39.17-22.53 4.95-18.48-4.11-34.12-22.66-39.09z"/></svg> __Insurance__]: identifying groups of motor insurance policy holders with a high average claim cost; identifying frauds ] .right-column[ - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M448 360V24c0-13.3-10.7-24-24-24H96C43 0 0 43 0 96v320c0 53 43 96 96 96h328c13.3 0 24-10.7 24-24v-16c0-7.5-3.5-14.3-8.9-18.7-4.2-15.4-4.2-59.3 0-74.7 5.4-4.3 8.9-11.1 8.9-18.6zM128 134c0-3.3 2.7-6 6-6h212c3.3 0 6 2.7 6 6v20c0 3.3-2.7 6-6 6H134c-3.3 0-6-2.7-6-6v-20zm0 64c0-3.3 2.7-6 6-6h212c3.3 0 6 2.7 6 6v20c0 3.3-2.7 6-6 6H134c-3.3 0-6-2.7-6-6v-20zm253.4 250H96c-17.7 0-32-14.3-32-32 0-17.6 14.4-32 32-32h285.4c-1.9 17.1-1.9 46.9 0 64z"/></svg> __Bookshops__]: book ordering (recommendation) - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M616 192H480V24c0-13.26-10.74-24-24-24H312c-13.26 0-24 10.74-24 24v72h-64V16c0-8.84-7.16-16-16-16h-16c-8.84 0-16 7.16-16 16v80h-64V16c0-8.84-7.16-16-16-16H80c-8.84 0-16 7.16-16 16v80H24c-13.26 0-24 10.74-24 24v360c0 17.67 14.33 32 32 32h576c17.67 0 32-14.33 32-32V216c0-13.26-10.75-24-24-24zM128 404c0 6.63-5.37 12-12 12H76c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12H76c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12H76c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm128 192c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm160 96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12V76c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm160 288c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40zm0-96c0 6.63-5.37 12-12 12h-40c-6.63 0-12-5.37-12-12v-40c0-6.63 5.37-12 12-12h40c6.63 0 12 5.37 12 12v40z"/></svg> __City-planning__]: identifying groups of houses according to their type, value and geographical location - .stress[<svg aria-hidden="true" role="img" viewBox="0 0 640 512" style="height:1em;width:1.25em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M640 264v-16c0-8.84-7.16-16-16-16H344v-40h72c17.67 0 32-14.33 32-32V32c0-17.67-14.33-32-32-32H224c-17.67 0-32 14.33-32 32v128c0 17.67 14.33 32 32 32h72v40H16c-8.84 0-16 7.16-16 16v16c0 8.84 7.16 16 16 16h104v40H64c-17.67 0-32 14.33-32 32v128c0 17.67 14.33 32 32 32h160c17.67 0 32-14.33 32-32V352c0-17.67-14.33-32-32-32h-56v-40h304v40h-56c-17.67 0-32 14.33-32 32v128c0 17.67 14.33 32 32 32h160c17.67 0 32-14.33 32-32V352c0-17.67-14.33-32-32-32h-56v-40h104c8.84 0 16-7.16 16-16zM256 128V64h128v64H256zm-64 320H96v-64h96v64zm352 0h-96v-64h96v64z"/></svg> __Internet__]: document classification; clustering weblog data to discover groups of similar access patterns; topic modeling ... ] --- class: left, top ## What's "vector quantization"?<hbr> - A mathematical model of _data compression_ - Compression (encoding): original .stress[data points] `\(\to\)` less/simpler/compact .stress[objects] - Reconstruction (decoding): compressed object `\(\to\)` original data .center[<img src="./img/reconstruct.png" width="700px"/>] - Type: variable-rate & .stress[fixed-rate] quantization problem. --- template: inter-slide class: left, middle count: false ## <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:rgb(16, 111, 171);;overflow:visible;position:relative;"><path d="M0 117.66v346.32c0 11.32 11.43 19.06 21.94 14.86L160 416V32L20.12 87.95A32.006 32.006 0 0 0 0 117.66zM192 416l192 64V96L192 32v384zM554.06 33.16L416 96v384l139.88-55.95A31.996 31.996 0 0 0 576 394.34V48.02c0-11.32-11.43-19.06-21.94-14.86z"/></svg> .bold-blue[Outline] <br> .hhead[I. Theory of fixed-rate vector quantization] <br> .hhead[II. Empirical setting & K-means algorithm] <br> .hhead[III. Questions & conclusion] --- template: inter-slide class: left, middle count: false ## <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:rgb(16, 111, 171);;overflow:visible;position:relative;"><path d="M0 117.66v346.32c0 11.32 11.43 19.06 21.94 14.86L160 416V32L20.12 87.95A32.006 32.006 0 0 0 0 117.66zM192 416l192 64V96L192 32v384zM554.06 33.16L416 96v384l139.88-55.95A31.996 31.996 0 0 0 576 394.34V48.02c0-11.32-11.43-19.06-21.94-14.86z"/></svg> .bold-blue[Outline] <br> .section[I. Theory of fixed-rate vector quantization] <br> .hhead[II. Empirical setting & K-means algorithm] <br> .hhead[III. Questions & conclusion] --- ## Fixed-rate quantization problem<hbr> ### Notation<hbr> - Data: `\(X_1,X_2,..., X \sim \mu\)`, `\(iid\)` `\(\mathbb{R}^p\)`-valued random variable. -- - A .stress[fixed-rate] `\(K\)`.stress[-point quantizer] `\(q\)` associated with .stress[codebook] `\(\mathcal{C}=\{c_1, ..., c_K\}\subset\mathbb{R}^p\)`<br> ( `\(c_k\)` are called .stress[codevectors] ) is a Borel measurable mapping: `$$q:\mathbb{R}^p\to \mathcal{C}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\mapsto c_k=q(x), \text{ for some }k.$$` -- - Distortion measure: let `\(d\)` be a distance (Euclidean in this talk), `$$\mathcal{D}_K(\mu, q)=\mathbb{E}_{\mu}[d(X,q(X))]=\int_{\mathbb{R}^p}d(x,q(x))\mu(dx)=\int_{\mathbb{R}^p}\|x-q(x)\|^2\mu(dx).$$` -- - Optimal performance: `$$\mathcal{D}_K^*(\mu)=\inf_{q\in\mathcal{Q}_K}\mathcal{D}_K(\mu,q), \text{ with }\mathcal{Q}_K=\{q:\mathcal{D}_K(\mu,q)<\infty\}.$$` --- ## Fixed-rate quantization problem<hbr> ### Objective<hbr> - Find `\(q^*\)` such that `$$\mathcal{D}_K(\mu,q^*)=\mathcal{D}_K^*(\mu) =\inf_{q\in\mathcal{Q}_K}\mathcal{D}_K(\mu,q)=\inf_{q\in\mathcal{Q}_K}\int_{\mathbb{R}^p}\|x-q(x)\|^2\mu(dx).$$` <h0br> -- ### Remark<h0br> .pull-left[ - `\(q\)` is completely defined by - its codebook `\(\mathcal{C}=\{c_k\}_{k=1}^K\)` - the associated .stress[cells] `\(S_k=\{x\in\mathbb{R}^p:q(x)=c_k\}\)`. - `\(\mathcal{S}=\{S_k\}_{k=1}^K\)` is a partition of `\(\mathbb{R}^p\)`. ] .pull-right[ <hbr> .right[<img src="./img/voro.png" width="300px" height ="210px"/>] ] --- ## <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M512 176.001C512 273.203 433.202 352 336 352c-11.22 0-22.19-1.062-32.827-3.069l-24.012 27.014A23.999 23.999 0 0 1 261.223 384H224v40c0 13.255-10.745 24-24 24h-40v40c0 13.255-10.745 24-24 24H24c-13.255 0-24-10.745-24-24v-78.059c0-6.365 2.529-12.47 7.029-16.971l161.802-161.802C163.108 213.814 160 195.271 160 176 160 78.798 238.797.001 335.999 0 433.488-.001 512 78.511 512 176.001zM336 128c0 26.51 21.49 48 48 48s48-21.49 48-48-21.49-48-48-48-48 21.49-48 48z"/></svg> Some key quantities<hbr> - Let `\(X\sim\mu\)` be an `\(\mathbb{R}^p\)`-valued random variable and `\(S\subset\mathbb{R}^p\)` be s.t `\(\mu(S)>0\)`, then - `\(\mathbb{E}(X)=\arg\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2]\)` - `\(\mathbb{V}(X)=\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2]\)` - `\(\mathbb{E}(X|X\in S)=\arg\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2|X\in S]\)` - `\(\mathbb{V}(X|X\in S)=\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2|X\in S]\)`.<h0br> .pull-right[ .right[<img src="./img/voro.png" width="300px" height ="210px"/>] ] --- count:false ## <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M512 176.001C512 273.203 433.202 352 336 352c-11.22 0-22.19-1.062-32.827-3.069l-24.012 27.014A23.999 23.999 0 0 1 261.223 384H224v40c0 13.255-10.745 24-24 24h-40v40c0 13.255-10.745 24-24 24H24c-13.255 0-24-10.745-24-24v-78.059c0-6.365 2.529-12.47 7.029-16.971l161.802-161.802C163.108 213.814 160 195.271 160 176 160 78.798 238.797.001 335.999 0 433.488-.001 512 78.511 512 176.001zM336 128c0 26.51 21.49 48 48 48s48-21.49 48-48-21.49-48-48-48-48 21.49-48 48z"/></svg> Some key quantities<hbr> - Let `\(X\sim\mu\)` be an `\(\mathbb{R}^p\)`-valued random variable and `\(S\subset\mathbb{R}^p\)` be s.t `\(\mu(S)>0\)`, then - `\(\mathbb{E}(X)=\arg\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2]\)` - `\(\mathbb{V}(X)=\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2]\)` - `\(\mathbb{E}(X|X\in S)=\arg\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2|X\in S]\)` - `\(\mathbb{V}(X|X\in S)=\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2|X\in S]\)`.<h0br> .pull-left[ - Conditional expectation and variance are defined by `$$\mathbb{E}(X|X\in S)=\frac{\int_{S}x\mu(dx)}{\mu(S)}\quad\quad\quad\quad\quad\quad\quad\ \ \ \\ \mathbb{V}(X|X\in S)=\frac{\int_{S}\|x-\mathbb{E}(X|X\in S)\|^2\mu(dx)}{\mu(S)}$$` ] .pull-right[ .right[<img src="./img/voro.png" width="300px" height ="210px"/>] ] --- ## Theoretical result<hbr> ### Lemma 1 (.textsc[.stress[Nearest Neighbor Condition]])<h0br> ---- - Let `\(q\)` and `\(\hat{q}\)` be two `\(K\)`-point quantizers with the same codebook `\(\mathcal{C}=\{c_k\}_{k=1}^K\)` s.t `$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \hat{q}(x)=\arg\min_{c_k\in\mathcal{C}}\|x-c_k\|^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$` where the ties are broken arbitrarily. Then one has `$$\mathcal{D}_K(\mu,\hat{q})\leq\mathcal{D}_K(\mu,q).$$` <h0br> ---- </h0br> <h3br> .center[  ] --- ## Theoretical result<hbr> ### Lemma 1 (.textsc[.stress[Centroid Condition]])<h0br> ---- - Let `\(q\)` be any `\(K\)`-point quantizer with partition cells `\(\mathcal{S}=\{S_k\}_{k=1}^K\)`. If `\(\hat{q}\)` is defined on the same cells with codevectors given by `$$\hat{c}_k=\arg\min_{c\in\mathbb{R}^p}\mathbb{E}[\|X-c\|^2|X\in S_k]=\mathbb{E}[X|X\in S_k], \text{ for }k=1,...,K,$$` then `$$\mathcal{D}_K(\mu,\hat{q})\leq \mathcal{D}_K(\mu,q).$$` <h0br> ---- .center[  ] --- ## Lemma 1 suggests that...<hbr> - .textsc[.stress[Nearest Neighbor Condition]]: given .stress[codebook] `\(\mathcal{C}=\{c_k\}_{k=1}^K\)`, we know an optimal way to assign `\(x \mapsto c_j\Leftrightarrow\)` .stress[partition] `\(S=\{S_k\}_{k=1}^K\)`. - .textsc[.stress[Centroid Condition]]: given a .stress[partition] `\(S=\{S_k\}_{k=1}^K\)`, we know a way to define an optimal .stress[codebook] `\(\mathcal{C}=\{c_k\}_{k=1}^K\)`. <h0br> .center[ <video width="600" height="375" controls> <source src="./img/quantizer.mp4" type="video/mp4"> Your browser does not support the video tag. </video> ] --- count:false ## Lemma 1 suggests that...<hbr> - .textsc[.stress[Nearest Neighbor Condition]]: given .stress[codebook] `\(\mathcal{C}=\{c_k\}_{k=1}^K\)`, we know an optimal way to assign `\(x \mapsto c_j\Leftrightarrow\)` .stress[partition] `\(S=\{S_k\}_{k=1}^K\)`. - .textsc[.stress[Centroid Condition]]: given a .stress[partition] `\(S=\{S_k\}_{k=1}^K\)`, we know a way to define an optimal .stress[codebook] `\(\mathcal{C}=\{c_k\}_{k=1}^K\)`. <hbr> .pull-left[ ### Remark<hbr> - Such a `\(\hat{q}\)` (.stress[Lemma 1]) with codebook `\(\hat{\mathcal{C}}=\{\hat{c}_k\}_{k=1}^K\)` is called .stress[nearest neighbor] quantizer (.stress[NNQ]). - The associated partition `\(\hat{S}=\{\hat{S}_k\}\)` is called .stress[Voronoï] or .stress[nearest neighbor] of `\(\mathbb{R}^p\)` associated with `\(\hat{\mathcal{C}}\)`. ] .pull-right[ <img src="./img/voronoi.png" width="380px" height ="260px"/> ] --- ## Existence of optimal NNQ<hbr> - From .stress[Lemma 1], it suffices to consider .stress[NNQ] i.e., `$$\mathcal{D}_K^*(\mu)=\inf_{q\in\mathcal{Q}_K}\int_{\mathbb{R}^p}\|x-q(x)\|^2\mu(dx) =\inf_{\hat{\mathcal{C}}:|\hat{\mathcal{C}}|=K}\int_{\mathbb{R}^p}\min_{\hat{c}_k\in\hat{\mathcal{C}}}\|x-\hat{c}_k\|^2\mu(dx).$$` <h0br> -- ### Theorem 1 (.textsc[Existence of optimal] NNQ)<h0br> ---- There exists a nearest neighbor quantizer `\(q^*\in\mathcal{Q}_K\)` such that `$$\mathcal{D}_K(\mu, q^*)=\mathcal{D}_K^*(\mu)=\inf_{(\hat{c}_1,...,\hat{c}_K)\in(\mathbb{R}^p)^K}\int_{\mathbb{R}^p}\min_{1\leq k\leq K}\|x-\hat{c}_k\|^2\mu(dx).$$` <h0br> ---- -- - .stress[Sketch of proof]: let `\(g_K(y_1,...,y_K)=\int_{\mathbb{R}^p}\min_{1\leq k\leq K}\|x-y_k\|^2\mu(dx)\)`, thus: - `\(g_K\)` is continuous on `\((\mathbb{R}^p)^K\)`. - Let `\(\overline{B}_{r}=\{x\in\mathbb{R}^p:\|x\|\leq r\}\)` be the closed ball of `\(\mathbb{R}^p\)` with radius `\(r>0\)`, centered at the origin, then `$$\exists R>0: \mathcal{D}_K^*(\mu)=\inf_{(y_1,...,y_K)\in(\overline{B}_{R})^K}g_K(y_1,...,y_K).$$` --- template: inter-slide class: left, middle count: false ## <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:rgb(16, 111, 171);;overflow:visible;position:relative;"><path d="M0 117.66v346.32c0 11.32 11.43 19.06 21.94 14.86L160 416V32L20.12 87.95A32.006 32.006 0 0 0 0 117.66zM192 416l192 64V96L192 32v384zM554.06 33.16L416 96v384l139.88-55.95A31.996 31.996 0 0 0 576 394.34V48.02c0-11.32-11.43-19.06-21.94-14.86z"/></svg> .bold-blue[Outline] <br> .hhead[I. Theory of fixed-rate vector quantization] <br> .section[II. Empirical setting & K-means algorithm] <br> .hhead[III. Questions & conclusion] --- ## Empirical setting<hbr> - In practice, everything is intractable unless `\(\mu\)` is known! Unfortunately, it's not! -- - Observed training data: `\(\mathcal{T}_n=\{X_1,...,X_n\}\)` with `\(X_i\overset{iid}{\sim}\mu\)` of size `\(n\)`. -- - Empirical distribution: if `\(A\)` is a Borel measurable subset of `\(\mathbb{R}^p\)`, let `$$\mu_n(A)=\frac{1}{n}\sum_{i=1}^n\mathbb{1}_{\{X_i\in A\}}$$` - Empirical distortion measure: `$$\mathcal{D}_K(\mu_n, q)=\frac{1}{n}\sum_{i=1}^n\|X_i-q(X_i)\|^2$$` -- - .stress[Empirically optimal NNQ]: `\(q_n^*(.)=f(.|X_1,...X_n)\in\mathcal{Q}_K\)` satisfying `$$\mathcal{D}_K(\mu_n,q_n^*)=\inf_{q\in\mathcal{Q}_K}\mathcal{D}_k(\mu_n,q)=\inf_{q\in\mathcal{Q}_K}\frac{1}{n}\sum_{i=1}^n\|X_i-q(X_i)\|^2$$` --- ## Consistency of empirical design<hbr> Roughly speaking, we want to show that `\(q_n^*\leadsto q^*\)` as `\(n\to\infty\)` in some sense. -- ### Theorem 2 (.textsc[Consistency of empirical design])<h0br> ---- For any `\(K\geq 1\)`, the sequence of the .stress[empirically optimal NNQ] `\((q_n^*)_{n\in\mathbb{N}}\)` satisfies `$$\lim_{n\to\infty}\mathcal{D}_K(\mu,q_n^*)=\mathcal{D}_K^*(\mu)\quad \mu\text{-}a.s.$$` <h0br> ---- -- .stress[Sketch of proof]: let `\(\mathcal{L}_2(\mathbb{R}^p)=\{\mu:\mathbb{E}_{\mu}(\|X\|^2)=\int_{\mathbb{R}^p}\|x\|^2\mu(dx)<\infty\}\)`.<hbr> - `\(L_2\)` Wasserstein distance: `\(\rho(\mu,\nu)=\displaystyle\inf_{X\sim\mu,Y\sim\nu}[\mathbb{E}(\|X-Y\|^2)]^{1/2}, \mu,\nu\in\mathcal{L}_2(\mathbb{R}^p)\)`. - If `\(q\)` is a .stress[NNQ]: `\(|(\mathcal{D}_K(\mu,q))^{1/2}-(\mathcal{D}_K(\nu,q))^{1/2}|<\rho(\mu, \nu),\forall \mu, \nu\in\mathcal{L}_2(\mathbb{R}^p).\)`<br><br> And consequently, `\(|(\mathcal{D}_K(\mu, q_n^*))^{1/2}-(\mathcal{D}_K^*(\mu))^{1/2}|<2\rho(\mu, \mu_n),\forall \mu\in\mathcal{L}_2(\mathbb{R}^p).\)` - One has: `\(\lim_{n\to\infty}\rho(\nu,\nu_n)=0\Leftrightarrow \nu_n\underset{n\to\infty}{\rightharpoonup}\nu\)` and `\(\mathbb{E}_{\nu_n}(\|X\|^2)\underset{n\to\infty}{\rightarrow}\mathbb{E}_{\nu}(\|X\|^2)\)`. --- ## Finite sample upper bounds The previous result does not provide any information about the convergence rate of `\(\mathcal{D}_K(\mu,q_n^*)\to\mathcal{D}_K^*(\mu)\)` or any bound on a finite resource. -- ### Theorem 3 (.textsc[Rate of convergence])<h0br> ---- Let `\(\mathcal{P}(R)=\{\mu:\mathbb{P}_{\mu}(X\in \overline{B}_ R)=\mu(\overline{B}_ R)=1\}\)`, thus for every `\(\mu\in\mathcal{P}(R)\)`, for `\(n\)` large enough, `$$\mathcal{D}_K(\mu,q_n^*)-\mathcal{D}_K^*(\mu)=O\Big(\sqrt{\frac{\log(n)}{n}}\Big)\quad \mu\text{-}a.s.$$` <h0br> ---- -- .stress[Sketch of proof]: --- count: false ## Finite sample upper bounds The previous result does not provide any information about the convergence rate of `\(\mathcal{D}_K(\mu,q_n^*)\to\mathcal{D}_K^*(\mu)\)` or any bound on a finite resource. ### Theorem 3 (.textsc[Rate of convergence])<h0br> ---- Let `\(\mathcal{P}(R)=\{\mu:\mathbb{P}_{\mu}(X\in \overline{B}_ R)=\mu(\overline{B}_ R)=1\}\)`, thus for every `\(\mu\in\mathcal{P}(R)\)`, for `\(n\)` large enough, `$$\mathcal{D}_K(\mu,q_n^*)-\mathcal{D}_K^*(\mu)=O\Big(\sqrt{\frac{\log(n)}{n}}\Big)\quad \mu\text{-}a.s.$$` <h0br> ---- .stress[Sketch of proof]: .stress[let not talk about it!!!] .stress[For a complete proof, see [Tamás Linder (2001)](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://mast.queensu.ca/~linder/pdf/cism.pdf)]. --- ## _K_-means algorithm<hbr> Given `\(\mathcal{T}_n=\{X_1,...,X_n\}\)`, we want to group this data into `\(K\)` groups such that:<h0br> - data points in the same group are similar enough - data points of different groups tend to be distant. <h0br> -- ### Algorithm<h0br> ---- 1. .stress[Initialization]: codebook `\(\mathcal{C}=\{c_1,...,c_K\}\subset\mathcal{T}_n\)` 2. .stress[Assigning step]: `for i = 1,2,...,n:`<br> `assign` `\(X_i\)` `to group` `\(k\)` `if` `\(\|X_i-c_k\|^2=\min_{1\leq j\leq K}\|X_i-c_j\|^2\)` 3. .stress[Centroid recomputation]: let `\(S_k=\{X_i\in\mathcal{T}_n:\|X_i-c_k\|\leq\|X_i-c_j\|,\forall j\neq k\}\)`<br> for `\(k=1,2,...,K\)`, then the codevectors are updated by `$$c_k\leftarrow \frac{1}{|S_k|}\sum_{X_i\in S_k}X_i.$$` 4. Alternatively repeat step 2. and 3. until any stopping criterion is met. <hbr> ---- --- ## _K_-means algorithm<h0br> ---- 1. .stress[Initialization]: codebook `\(\mathcal{C}=\{c_1,...,c_K\}\subset\mathcal{T}_n\)` 2. .stress[Assigning step]: `\(\mathcal{C}=\{c_1,...,c_K\}\to \hat{\mathcal{S}}=\{\hat{S}_1,...,\hat{S}_K\}\)`. 3. .stress[Centroid recomputation]: `\(\mathcal{S}=\{S_1,...,S_K\}\to\hat{\mathcal{C}}=\{\hat{c}_1,...,\hat{c}_K\}\)`. 4. 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] .pull-right[ <br> <br> .center[ .stress[K-means algorithm]<br><br> .stress[with] `\(K=3\)`<br><br> .stress[(Lloyd's method).] ] ] --- template: inter-slide class: left, middle count: false ## <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:rgb(16, 111, 171);;overflow:visible;position:relative;"><path d="M0 117.66v346.32c0 11.32 11.43 19.06 21.94 14.86L160 416V32L20.12 87.95A32.006 32.006 0 0 0 0 117.66zM192 416l192 64V96L192 32v384zM554.06 33.16L416 96v384l139.88-55.95A31.996 31.996 0 0 0 576 394.34V48.02c0-11.32-11.43-19.06-21.94-14.86z"/></svg> .bold-blue[Outline] <br> .hhead[I. Theory of fixed-rate vector quantization] <br> .hhead[II. Empirical setting & K-means algorithm] <br> .section[III. Questions & conclusion] --- ## <svg aria-hidden="true" role="img" viewBox="0 0 384 512" style="height:1em;width:0.75em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M202.021 0C122.202 0 70.503 32.703 29.914 91.026c-7.363 10.58-5.093 25.086 5.178 32.874l43.138 32.709c10.373 7.865 25.132 6.026 33.253-4.148 25.049-31.381 43.63-49.449 82.757-49.449 30.764 0 68.816 19.799 68.816 49.631 0 22.552-18.617 34.134-48.993 51.164-35.423 19.86-82.299 44.576-82.299 106.405V320c0 13.255 10.745 24 24 24h72.471c13.255 0 24-10.745 24-24v-5.773c0-42.86 125.268-44.645 125.268-160.627C377.504 66.256 286.902 0 202.021 0zM192 373.459c-38.196 0-69.271 31.075-69.271 69.271 0 38.195 31.075 69.27 69.271 69.27s69.271-31.075 69.271-69.271-31.075-69.27-69.271-69.27z"/></svg> Question around _K_-means algorithm<hbr> - Choosing `\(K\)`? - Assessing clustering quality? - Scaling or not scaling? - Choosing a distance? (See also, [Banerjee et al. (2005)](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://jmlr.org/papers/volume6/banerjee05b/banerjee05b.pdf) and [Fisher (2010)](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://www.lpsm.paris/_media/users/fischer/quantization_and_clustering_with_bregman_divergences.pdf)) - Initialization methods? - Step Assigning/Centoid recomputation fast update? - Stopping rules? ... -- .stress[<svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M569.517 440.013C587.975 472.007 564.806 512 527.94 512H48.054c-36.937 0-59.999-40.055-41.577-71.987L246.423 23.985c18.467-32.009 64.72-31.951 83.154 0l239.94 416.028zM288 354c-25.405 0-46 20.595-46 46s20.595 46 46 46 46-20.595 46-46-20.595-46-46-46zm-43.673-165.346l7.418 136c.347 6.364 5.609 11.346 11.982 11.346h48.546c6.373 0 11.635-4.982 11.982-11.346l7.418-136c.375-6.874-5.098-12.654-11.982-12.654h-63.383c-6.884 0-12.356 5.78-11.981 12.654z"/></svg>] If there are no restrictions on the dimension of the input space, on `\(K\)`, or on sample size, computing an optimal codebook is a [NP-hard problem](https://en.wikipedia.org/wiki/NP-hardness). -- .stress[<svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M569.517 440.013C587.975 472.007 564.806 512 527.94 512H48.054c-36.937 0-59.999-40.055-41.577-71.987L246.423 23.985c18.467-32.009 64.72-31.951 83.154 0l239.94 416.028zM288 354c-25.405 0-46 20.595-46 46s20.595 46 46 46 46-20.595 46-46-20.595-46-46-46zm-43.673-165.346l7.418 136c.347 6.364 5.609 11.346 11.982 11.346h48.546c6.373 0 11.635-4.982 11.982-11.346l7.418-136c.375-6.874-5.098-12.654-11.982-12.654h-63.383c-6.884 0-12.356 5.78-11.981 12.654z"/></svg>] It suffers in high dimensional input spaces ( `\(p\)` large) due to [curse of dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality). -- .stress[<svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M569.517 440.013C587.975 472.007 564.806 512 527.94 512H48.054c-36.937 0-59.999-40.055-41.577-71.987L246.423 23.985c18.467-32.009 64.72-31.951 83.154 0l239.94 416.028zM288 354c-25.405 0-46 20.595-46 46s20.595 46 46 46 46-20.595 46-46-20.595-46-46-46zm-43.673-165.346l7.418 136c.347 6.364 5.609 11.346 11.982 11.346h48.546c6.373 0 11.635-4.982 11.982-11.346l7.418-136c.375-6.874-5.098-12.654-11.982-12.654h-63.383c-6.884 0-12.356 5.78-11.981 12.654z"/></svg>] The empirical distortion measure `\(\mathcal{D}_K(\mu_n,\hat{q})=\frac{1}{n}\sum_{i=1}\min_{1\leq k\leq K}\|X_i-\hat{c}_k\|^2\)` is not convex; therefore, the algorithm might stuck in a local optimal solution. --- ## <svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M576 240c0-23.63-12.95-44.04-32-55.12V32.01C544 23.26 537.02 0 512 0c-7.12 0-14.19 2.38-19.98 7.02l-85.03 68.03C364.28 109.19 310.66 128 256 128H64c-35.35 0-64 28.65-64 64v96c0 35.35 28.65 64 64 64h33.7c-1.39 10.48-2.18 21.14-2.18 32 0 39.77 9.26 77.35 25.56 110.94 5.19 10.69 16.52 17.06 28.4 17.06h74.28c26.05 0 41.69-29.84 25.9-50.56-16.4-21.52-26.15-48.36-26.15-77.44 0-11.11 1.62-21.79 4.41-32H256c54.66 0 108.28 18.81 150.98 52.95l85.03 68.03a32.023 32.023 0 0 0 19.98 7.02c24.92 0 32-22.78 32-32V295.13C563.05 284.04 576 263.63 576 240zm-96 141.42l-33.05-26.44C392.95 311.78 325.12 288 256 288v-96c69.12 0 136.95-23.78 190.95-66.98L480 98.58v282.84z"/></svg> Conclusion<hbr> - We went briefly from theories to algorithm of the `\(K\)`-means clustering method. - The `\(K\)`-means algorithm adapts the theoretical property of .stress[NNQ] into an iterative method that constructs a sequence of .stress[Voronoï] partitions. - The number of clusters `\(K\)` is supposed to be a known input parameter. In practice, .stress[cross-validation] method can be used to approximate the optimal `\(K\)`. - Convergence to a local minimum may produce counterintuitive ("wrong") results. In practice, several initializations can be done, and the best solution is retained. - Dimensional reduction such as [PCA](https://en.wikipedia.org/wiki/Principal_component_analysis) or random projection such as [Johnson-Lindenstrauss Lemma](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma) can be used alongside with or in addition to `\(K\)`-means. - `\(K\)`-means does not work well in clustering data points of mixing classes or of spiral shapes... In this case, one might be interested in [Spectral clustering](https://en.wikipedia.org/wiki/Spectral_clustering). .center[<img src="./img/spiral.png" width="140px" height ="130px"/> <img src="./img/spiral2.png" width="150px" height ="132px"/>] ??? Squared Euclidean distance is very sensitive to outliers An inappropriate choice of k may yield poor results. That is why, when performing k-means, it is important to run diagnostic checks for determining the number of clusters in the data set. --- count: false template: inter-slide class: left, middle count: false .center[# References]<h0br> &#128218; [Linder, T. (2002). Learning-Theoretic Methods in Vector Quantization. In: Györfi, L. (eds) Principles of Nonparametric Learning. International Centre for Mechanical Sciences, vol 434. Springer, Vienna.](https://link.springer.com/chapter/10.1007/978-3-7091-2568-7_4) &#128218; [Banerjee, S. Merugu, I.S. Dhillon, and J. Ghosh. Clustering with Bregman divergences. Journal of Machine Learning Research, 6:1705–1749, 2005](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://jmlr.org/papers/volume6/banerjee05b/banerjee05b.pdf) &#128218; [A. Fischer. Quantization and clustering with Bregman divergences. Journal of Multivariate Analysis, 101(10):2207–2221, 2010.](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://www.lpsm.paris/_media/users/fischer/quantization_and_clustering_with_bregman_divergences.pdf) &#128218; [S. Has, A. Fischer, and M. Mougeot. Kfc: A clusterwise supervised learning procedure based on the aggregation of distances. Journal of Statistical Computation and Simulation, 0(0):1–21, 2021. doi: 10.1080/00949655.2021. 1891539.](https://www.tandfonline.com/doi/abs/10.1080/00949655.2021.1891539) &#128218; [Aarti Singh. Spectral Clustering. Machine Learning 10-701/15-781](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf) <svg aria-hidden="true" role="img" viewBox="0 0 496 512" style="height:1em;width:0.97em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg> [https://github.com/hassothea/KFC-Procedure](https://github.com/hassothea/KFC-Procedure) <h0br> .pull-right[<h0br> # Thank you 🤓 ]