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zm7`4kSJBQC?o{pJo80c*HNcfsC7W80qOAyX;g5Bn1%t<8Lb0G1$oYO~nggbJK&e@^MUL4-ms$`$|X)e7r{+Tedjrgsf=&NS`f1wM2qZ zHCrdoTrg5w(Qfz(g8WGpIj?wnuO|JlD5P;)mPJ#^rG(Fom&8sJS35=0zdOGPO`&v6 z;8bkp(Fo)8?~wG|{mU9KtX@9nwC~{1hY4y|RsVoIvu0t^Qr^yEjf0hEKF8Z8uod~M zBc(HCligVJ7a0+Ym&`*mk_X>@MpP|ZcD!8%R%jaVPaBnJ*8J0P#M~fb=d-K1B7bKs zg~omjW?qx)cOQS77e7}<|z2fF8+*g4N zwhJ_D(V5rMg{{$Y_n*wqlY><)#~o~=1)ifS>*5aUMr!*^J){rv0Rvxv-3H>rm8h|z zegHB|Z202=A_#hWV4L0$3+x_MBJLaj$PgE!FSH!<5LLk*{PwdC(ElEoTP7J!A+&_n zf_4w^juox|4km$K{iX12>NJ3`G8yIt4iAfH0S*5Uyr+EI^6BaFHZBMYOk#STP;$Uk zJXl!l=h6>wgdh~AG|>b{ct{EU5W)76LbsM~3|Q?*&UTz9L{U~JDsUixhBdB&N*U \ldots > \lambda_d > 0$,\\ +eigenvectors $v_i$ forming an orthonormal Basis of $\R^d$} + +\theorem \textbf{PCA Solution for $k=1$}\\ +\smalltext{$v$ is the Eigenvector associated with the largest $\lambda$ for $\Sigma$.} +$$ + w^* = v_1 \quad \text{solves} \quad w^* = \underset{\Vert w \Vert_2=1}{\text{arg max}}\ w^\top \Sigma w +$$ + +\subsubsection{Generalized PCA} + +Luckily, the case $k=1$ generalizes: + +\theorem \textbf{PCA General Solution}\\ +\smalltext{Here, $\mathbf{W}\in\R^{d\times k}$ and $z_i^* \in \R^k$ for parameter $k$} +$$ + \mathbf{W}^*,z_1^*,\ldots,z_n^* = \underset{\mathbf{W}\in\R^{d\times k}: \mathbf{W}^\top\mathbf{W}=\mathbf{I}}{\text{arg min}} \sum_{i=1}^{n} \Bigl\Vert x_i - \mathbf{W}z_i \Bigr\Vert^2 +$$ +is solved using +$$ + \mathbf{W}^* = \Bigl( v_1 \| \cdots \| v_k \Bigr), \qquad z_i^* = \mathbf{W}^{*\top}x_i +$$ + +{\footnotesize + \textbf{Intuition}: In words, the basis for the $k$-dim. subspace minimizing reconstruction error is given using the first $k$ eigenvectors of $\Sigma$. +} + +\lemma \textbf{PCA is an orthohgonal projection} +$$ + P: \R^d\to\R^d,\qquad x \mapsto \mathbf{W}^*\mathbf{W}^*x +$$ + +\lemma \textbf{Connection to SVD}\\ +\smalltext{The First $k$ EV of $\Sigma$ are the first $k$ right singular vectors of $\mathbf{X}$.} +$$ + \mathbf{X} = \mathbf{USV}^\top \qquad n\cdot\Sigma = \mathbf{VS}^\top \mathbf{SV}^\top +$$ + +{\footnotesize + \remark Non-linear methods and kernels can be applied to PCA the same way they are applied to regression. +} + +\newpage +\subsubsection{Connection to k-Means} + +k-Means (Clustering) can be reformulated: $z_1,\ldots,z_n \in E_k$ where $E_k = \{e_1,\ldots,e_n\}$ holds unit vectors $e_i$. +$$ + \mathbf{W}^*,z_1^*,\ldots,z_n^* = \underset{\mathbf{W}\in\R^{d\times k}: \mathbf{W}^\top\mathbf{W}=\mathbf{I}}{\text{arg min}} \sum_{i=1}^{n} \Bigl\Vert x_i - \mathbf{W}z_i \Bigr\Vert^2 +$$ +This is the same as PCA, only the space for $z^*$ changes. Both PCA and k-Means are special forms of a more general technique: + +\definition \textbf{Matrix Factorization Methods}\\ +\smalltext{General class of techniques for decomp. of $\mathbf{X}$} +$$ + \mathbf{X} \approx \mathbf{WZ} +$$ + +{\footnotesize + \remark \textbf{PCA}. For PCA, $\mathbf{W}$ is an orthonormal Basis of the optimal $k$-dimenional subspace, $\mathbf{Z}$ are the coefficients for the projection. + + \remark \textbf{k-Means}. 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