From 7ba520e0231b24e263737f65aef5a6e304cbed92 Mon Sep 17 00:00:00 2001 From: feger <marc.feger@hhu.de> Date: Thu, 5 Dec 2019 14:49:47 +0100 Subject: [PATCH] Add more text for recommender systems --- recommender.tex | 12 ++++++++++++ submission.pdf | Bin 204677 -> 209278 bytes 2 files changed, 12 insertions(+) diff --git a/recommender.tex b/recommender.tex index d07e556..f70837f 100644 --- a/recommender.tex +++ b/recommender.tex @@ -28,3 +28,15 @@ The first and easiest way to solve \textit{matrix-factorization} is to connect t \subsubsection{Regulated Matrix-Factorization} This problem extends the \textit{basic matrix factorization} by a \textit{regulation factor} $\lambda$ in the corresponding \textit{minimization problem}. Since $\mathcal{R}$ is thinly occupied, the effect of \textit{overfitting} may occur due to learning from the few known values. The problem with \textit{overfitting} is that the generated \textit{ratings} are too tight. To counteract this, the magnitudes of the previous vectors is taken into account. High magnitudes are punished by a factor $\lambda(\lVert q_i \rVert^2 + \lVert p_u \lVert^2)$ in the \textit{minimization problem}. Overall, the \textit{minimization problem} $\min_{p_u, q_i}{\sum_{(u,i) \in \mathcal{B}} (r_{ui} - \hat{r}_{ui})^{2}} + \lambda(\lVert q_i \rVert^2 + \lVert p_u \lVert^2)$ is to be solved. + +\subsubsection{Weighted Regulated Matrix-Factorization} +A \textit{regulation factor} $\lambda$ is introduced in analogy to \textit{regulated matrix-factorization}. Additional \textit{weights} $\alpha$ and $\beta$ are introduced to take into account the individual magnitude of a vector. The \textit{minimization problem} then corresponds to $\min_{p_u, q_i}{\sum_{(u,i) \in \mathcal{B}} (r_{ui} - \hat{r}_{ui})^{2}} + \lambda(\alpha\lVert q_i \rVert^2 + \beta\lVert p_u \lVert^2)$. + +\subsubsection{Biased Matrix-Factorization} +A major advantage of \textit{matrix-factorization} is the ability to model simple relationships according to the application. Thus, an excellent data source cannot always be assumed. Due to the \textit{natural interaction} of the \textit{users} with the \textit{items}, \textit{preferences} arise. Such \textit{preferences} lead to \textit{behaviour patterns} which manifest themselves in the form of a \textit{bias} in the data. In principle, a \textit{bias} is not bad, but it must be taken into account when modeling the \textit{recommender system}. +The most popular model that takes \textit{bias} into account is called \textit{biased matrix-factorization}. +In addition, the \textit{missing rating} is no longer determined only by the \textit{inner product} of the two vectors $q_i$ and $p_u$. Rather, the \textit{bias} is also considered. Accordingly, a \textit{missing rating} is calculated by $\hat{r}_{ui} = b_{ui} + q_i^Tp_u$, where $b_{ui}$ is the \textit{bias} of a \textit{user} $u$ and an \textit{item} $i$. The \textit{bias} is determined by $b_{ui}=\mu + b_u + b_i$. The parameter $\mu$ is the \textit{global average} of all \textit{ratings} $r_{ui} \in \mathcal{R}$. +Furthermore, $b_u = \mu_u - \mu$ and $b_i = \mu_i - \mu$. +Here $\mu_u$ denotes the \textit{average} of all \textit{assigned ratings} of the \textit{user} $u$. Similarly, $\mu_i$ denotes the \textit{average} of all \textit{received ratings} of an \textit{item} $i$. +Thus $b_u$ indicates the \textit{deviation} of the \textit{average assigned rating} of a \textit{user} from the \textit{global average}. Similarly, $b_i$ indicates the \textit{deviation} of the \textit{average rating} of an item from the \textit{global average}. +In addition, the \textit{minimization problem} can be extended by the \textit{bias}. Accordingly, the \textit{minimization problem} is then $\min_{p_u, q_i}{\sum_{(u,i) \in \mathcal{B}} (r_{ui} - \hat{r}_{ui})^{2}} + \lambda(\lVert q_i \rVert^2 + \lVert p_u \lVert^2 + b_u^2 + b_i^2)$. diff --git a/submission.pdf b/submission.pdf index ac8586d4e91fc3f5ba09587a889413239cff6601..40918daa327a14f5ad122b9dacdcf980edfa02d6 100644 GIT binary patch delta 49463 zcmZqO&-3pQkDzLRi<_>Yo++2=g)2vIaG6+c6f|REGBKarI8}af29p7ksp;fdOg5^f zW(o!hLHe#0CGNo`i6yBDrsiOQfTH|@)S{Bi)M5owi_O28jLewKOeX7F<}sO>Zk}%` zXuxD)ws}(~D<jjtNt<_Og^01t`<=09`bTF*VQ~YH5&213TsAiPKB;->B^e5)hQ=n7 zH$GOK9N&19>Hnn3YE8TAmlVm^nD91i>fz<hxSuZ*m~lex?R6J{i<Nu=qAzc@d4%3f zeRKQV!-oG8M0EFExBK=zVXCN_&xV-R#a*|g#EsbI&od65Ryy+xQ^$hop+ar*Uwe7h z`6TL_BrjM|vn>Dj`#vuVr`~TY4@Dx594a)?iM_DxsML-vGktDVJLk<$(5h!ke9WR^ zc(=iGR+av@olPHG(?0Bs{IpNnJWFR`hy1+zcQ^C7Wr!}bwclag{77N4hp}JLYi3bv z(>E*XCNY%BPU=1q^nU50>XolUOpIs$aoX<HT(CgLkL&qa!SlXFPV?`oi^gdj3wbuH zu2eGQ$V%A=ChowBTuZSNmqb+!y%tE<znM`az3{<O&SU?sP5dmlKk0R;`}!4DvL{S- 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