text:
\section{Introduction}

Due to the popularity and exponential growth of e-commerce  and online streaming websites, a compelling demand has been created for efficient recommender systems to guide users toward items of their interests (e.g. products, books, movies)~\cite{100}. In collaborative methods, either \textit{filtering} or \textit{ranking}, by relying on the low-rank assumption on the users' preferences, both users and items are mapped into a latent feature space based on partially observed ratings that are later used to make predictions. In collaborative filtering (CF) methods such as matrix factorization~\cite{koren2009matrix}, where the aim is to accurately predict the ratings, the latent features are extracted in a way to minimize the prediction error measured in terms of popular performance measures such as root mean square error (RMSE). In spark contrast to CF,   in collaborating ranking (CR) models~\cite{koren2009matrix,christakopoulou2015collaborative,volkovs2012collaborative,cremonesi2010performance}, where the goal is to rank the unrated items in the order of relevance to the user,  the popular ranking measures such as  as discounted cumulative gain (DCG), normalized
discounted cumulative gain (NDCG), and average precision (AP)~\cite{jarvelin2000ir}  are  often employed to collaboratively learn a ranking model for the latent features.

Recent studies have demonstrated  that CR models lead to significantly higher ranking
accuracy  over their traditional CF counterparts  that optimize rating prediction. This  is important considering the fact that what we really care in recommendation  is not the actual values of ratings, but the order of items to be recommended to a specific user.  Therefore, the error  measures such as RMSE are often hopelessly insufficient,  as their place equal emphasis on all the ratings.  Among ranking models,  the methods that mainly concentrate on the \textit{top of the list} have received a considerable amount of attention, due to the higher probability of examining the top portion of the list of recommendations by users.  Therefore, the introduction of ranking metrics such as push norm or infinite norm~\cite{rudin2009p,agarwal2011infinite,christakopoulou2015collaborative,lee2014local},  sparked a widespread interest in CR models and has been proven to be more effective in
practice~\cite{steck2015gaussian,christakopoulou2015collaborative}.

Although CR models for recommender systems has  been studied extensively and some progress has been made, however,  the state of affairs remains unsettled:  the issue of handling \textit{cold-start} items in ranking models  and coping with \textit{not missing at random} assumption of ratings  are  elusive open issues. First, in many real world applications, the rating data are very sparse (e.g., the density of the data is around 1\% for many publicly available datasets) or for a subset of users or items the rating data is entirely missing (knows as cold-start user and cold-start item problem, respectively)~\cite{schein2002methods}. Second, collaborative filtering and ranking models rely on the critical assumption that  the  missing ratings are sampled uniformly at random. However, in many real applications of recommender systems, this assumption is not believed to hold,  as invariably some users are more active than others and some items are rated by many people while others are rarely rated~\cite{steck2010training}.  These issues have been  investigated in factorization based methods,  nonetheless, it is not straightforward to adapt them to CR models and are  left open~\cite{christakopoulou2015collaborative}. Motivated by these challenges, we ask the following fundamental question in the context of collaborative ranking models:
\begin{quote}
\textit{Is it possible to effectively learn a collaborative ranking model in the presence of cold-start items/users that is  robust to  the sampling of observed ratings?}
\end{quote}

In this paper, we give an affirmative answer to the above question.  In particular, we introduce a   \textit{semi-supervised collaborative ranking} model, dubbed~\alg~,  by leveraging side information about \textit{both observed and missing ratings} in collaboratively learning the ranking model.  In the learned model, unrated items are conservatively pushed after the relevant  and before the irrelevant items in the ranked list of items for each individual user. This crucial difference greatly boosts the performance and limits the bias caused by learning only from sparse non-random observed ratings. We also introduce a graph regularization method to exploit the side information about users to overcome the cold-start users problem.  In summary, the key features of~\alg~are:

\begin{itemize}[noitemsep,topsep=10pt,parsep=5pt,partopsep=5pt]
\item Inspired by recent developments in ranking at top~\cite{rudin2009p,agarwal2011infinite,li2014top}, the proposed model is a collaborative ranking model   that   primarily focuses on  the top of the recommendation  list for each user. Moreover, in stark contrast to pairwise ranking models which have quadratic dependency on the number of items, the proposed ranking model has a linear dependency on the number of items, making it suitable for large-scale recommendation.
\item It leverages side information about items with both observed and missing ratings while collaboratively learning the ranking model, which enables it  to effectively incorporate the available side information in knowledge transduction.
\item By incorporating the unrated items in ranking, it limits the bias caused by learning solely based on the observed ratings and consequently deals with the  not missing at random issue of ratings.
\item It is also able to leverage the similarity information between users based on a graph regularization method to make high quality recommendations for users with few ratings or cold-start users without an rating information.
\end{itemize}

To build the intuition on how incorporating missing ratings in~\alg~is beneficial in handling cold-start problem and mitigating data sparsity issue,  we note that  in many real world applications the available feedback on items is extremely sparse, and therefore the ranking models fail to effectively leverage the available side information in transdcuting the knowledge from existing ratings to unobserved ones. This problem becomes especially eminent in cases where surrogate ranking models such as pairwise models are used due to their computational virtues, where the unobserved ratings do not play any role in learning the  model.  As a result,  by leveraging rich sources of information about  all items, one can potentially bridge the gap between existing items  and new  items  to overcome the cold-start problem.

Turning to the non-random sampling issue of observed ratings, we note that the non-randomness is observing the ratings creates a  bias in  learning the model that negatively impacts the future predictions and may degrade the resulting recommendation accuracy if ignored.  Therefore, the nature of missing ratings  has to be modeled precisely as to obtain correct results. To reduce the effect of bias, the proposed ranking model takes a conservative approach and pushes the items with unknown ratings to the middle of ranked list, i.e., after the relevant and before the irrelevant items. This is equivalent to assuming a prior about the unknown ratings which is believed  to perform well  as investigated in~\cite{devooght2015dynamic}. However, unlike~\cite{devooght2015dynamic}, the proposed ranking idea is free of deciding an explicit value for  missing ratings which makes it more valuable from a practical point of view.

We conduct thorough experiments on real datasets and compare our results with the state-of-the-art models for cold-start recommendation to demonstrate the effectiveness of our proposed algorithm in recommendation at the top of the list and mitigating the data sparsity issue. Our results indicate that our algorithm outperforms other algorithms and provides recommendation with higher quality compared to the state-of-the-art methods. \\

\noindent\textbf{Organization.} {This paper is organized as follows. We briefly
 review related work in Section~\ref{sec:related}. We establish the notation and formally define the problem in Section~\ref{sec:preliminaries}. In Section~\ref{sec:taco}, we propose the semi-supervised collaborative ranking model with a push at the top of the list. Section~\ref{sec:optimization} discusses  efficient convex and non-convex optimization algorithms for optimization. Section~\ref{sec:taco-users} generalizes the proposed algorithm to leverage similarity information about users. We empirically evaluate the proposed method in Section~\ref{sec:results}, and conclude in Section~\ref{sec:conclusions}.}

\section{Related Work}\label{sec:related}

There is now a vast body of literature on ranking models for recommendation, coping with non-random missing ratings and an even bigger body of literature on handling cold-start problem by leveraging side information; we restrict our literature review here to papers that are most directly related.

\noindent\paragraph{Collaborative ranking for recommendation.~}{The last few years have seen a resurgence in collaborative ranking   centered around the technique of exploiting low-rank structures, an approach we take as well. Several approaches to CR have recently been proposed that are mainly inspired by the analogy between query-document relations in IR and user-item relations in recommender systems. The PMF-based approach~\cite{balakrishnan2012collaborative} uses the latent representations produced by matrix factorization as user-item features and learns a ranking model on these features. CofiRank~\cite{weimer2007maximum} learns latent representations that minimize a ranking-based loss instead of the squared error. ListRankMF~\cite{shi2010list}  aims at minimizing the cross entropy between the predict item permutation probability and true item permutation probability. In~\cite{lee2014local} a method for Local Collaborative Ranking (LCR) where ideas of local low-rank matrix approximation were applied to
the pairwise ranking loss minimization framework is introduced. In~\cite{steck2015gaussian} a  framework that allows for pointwise as well as listwise training with respect to various ranking
metrics is proposed. Finally,~\cite{christakopoulou2015collaborative} proposed a CR model build on the recent developments in ranking methods~\cite{agarwal2011infinite,rudin2009p} that focus on accuracy at top and proposed CR methods with p-push and infinite push norms. Incorporating side information in their model which is left as open issue was the main motivation of the current work.
}

\paragraph{Cold-start recommendation with side information.~}{Due in part to its importance,  there has been an active line of work to address difficulties associated with cold-start users and items, where a  common  theme  among  them  is  to  exploit  auxiliary  information  about  users  or  items  besides  the  rating data  that  are  usually  available~\cite{shi2014collaborative}. A feature based regression ranking model for predicting the values (rates) of user-item matrix in cold-start scenarios by leveraging all information available for users and items is proposed in~\cite{park2009pairwise}. The kernelized matrix
factorization approach studied in~\cite{zhou2012kernelized}, which incorporates the auxiliary information into the MF. In~\cite{saveski2014item} joint factorization of the user-item and item-feature matrices by using the same item latent feature matrix in both decompositions is utilized. The FBSM model is introduced in~\cite{sharmafeature2015b}, which learns factorized
bilinear similarity model for new items, given the rating information as well as the features of these items. Recently,~\cite{barjasteh2015cold} proposed a decoupling approach to transduct knowledge from side information to rating prediction which is able to handle both cold-start items and users problems in factorization based models.

}

\noindent\paragraph{Recommendation with not missing at random ratings.~}{
Substantial evidence for violations of the missing at random condition in recommender systems is reported in~\cite{missing-at-random2007} and it has been  showed that incorporating an explicit model of the missing data mechanism can lead to significant improvements in prediction performance.The first study of the effect of non-random missing data on collaborative ranking is presented in~\cite{marlin2009collaborative}. In~\cite{sindhwani2010one}  an EM algorithm to optimize in turn the factorization and the estimation of missing values. Recently, in~\cite{devooght2015dynamic} a novel dynamic matrix factorization framework that allows to set an explicit prior on unknown values is introduced. However their algorithm requires a careful setting of the prior rating to be practical.}

\section{Preliminaries}\label{sec:preliminaries}

In this section we establish the notation used throughout the paper and formally describe our problem setting.

Scalars are denoted by  lower case letters and vectors by  bold face lower case letters such as $\bu$. We use  bold face upper case letters such as $\bM$ to denote matrices.   The Frobenius norm of a matrix $\bM \in \mathbb{R}^{n \times m}$ is denoted by $\|\bM\|_\text{F}$,  i.e,     $\|\mathbf{M}\|_{\rm{F}}=\sqrt{\sum_{i=1}^n\sum_{j=1}^m |M_{ij}|^2}$ and its $(i,j)$th entry is denoted by $A_{i,j}$. The trace norm of a matrix is denoted by $\|\bM\|_{*}$ which is defined as the sum of its singular values. The transpose of  a vector and a matrix denoted by $\bu^{\top}$ and $\bU^{\top}$, respectively. We use $[n]$ to denote the set  on integers $\{1,2, \cdots, n\}$. The set of non-negative real numbers is denoted by $\mathbb{R}_{+}$. The indicator function is denoted by $\mathbb{I}[\cdot]$. For a vector $\bu \in \mathbb{R}^p$ we use $\|\bu\|_1 = \sum_{i=1}^{p}{|\bu_i|}$, $\|\bu\|_2 = \left(\sum_{i=1}^{p}{|\bu_i|^2}\right)^{1/2}$, and $\|\bu\|_{\infty} = \max_{1 \leq i \leq p } \bu_i$ to denote its $\ell_1$, $\ell_2$, and $\ell_{\infty}$ norms, respectively.  The dot product between two vectors $\bu$ and $\bu'$ is denoted by either $\dd{\bu}{\bu'}$ or $\bu^{\top}\bu'$.

In collaborative filtering we assume that there is  a set of $n$ users $\mathcal{U} = \{u_1, \cdots, u_n\}$ and a set of $m$ items $\mathcal{I} = \{i_1, \cdots, i_m\}$ where  each user $u_i$ expresses opinions about a set of items. The rating information is summarized in an $n \times m$ matrix $\bR \in \{-1, +1, ?\}^{n \times m}, 1 \leq i \leq n, 1 \leq j \leq m$ where the rows correspond to the users and the columns correspond to the items and $(p,q)$th entry is the  rate given by user $u_p$ to the item $i_q$. We note that the rating matrix is partially observed and it is sparse in most cases.  We are mainly interested in recommending a set of items for an active user such that  the user has not rated these items before.

\section{Transductive Collaborating Ranking}\label{sec:taco}

We now turn our attention to the main thrust of the paper where  we  present  our  transductive collaborative ranking  algorithm  with accuracy at top by exploiting the features of unrated data.  We begin with the basic formulation and then extend it to incorporate the unrated items. The pseudo-code of the resulting learning algorithm is provided in Algorithm~\ref{alg:taco}.

\subsection{A basic formulation
$\quad$  We consider a ranking problem, where, given a set of users $\mathcal{U}$ and known user feedback on a set of  items $\mathcal{I}$, the goal is to generate rankings of  unobserved items, adapted to each of the users' preferences. Here we consider the bipartite setting in which items are either relevant (positive) or irrelevant (negative). Many ranking methods have been developed for bipartite ranking, and most of them are essentially based on pairwise ranking. These algorithms reduce the ranking problem into a binary classification problem by treating each relevant/irrelevant instance pair as a single object to be classified~\cite{liu2009learning}.

As mentioned above, most research has concentrated on the rating prediction problem in CF where the aim is to accurately predict the ratings for the unrated items for each user. However, most applications that use CF typically aim to recommend only a small ranked set of items to each user. Thus rather than concentrating on rating prediction we instead approach this problem from the ranking viewpoint where the goal is to rank the unrated items in the order of relevance to the user. Moreover,  it is desirable  to concentrate aggressively on top portion of the ranked list to include mostly relevant items and  push irrelevant items down from the top. Specifically,   we propose an algorithm that maximizes  the number of relevant items which are pushed to  the absolute top of the list by utilizing the P-Norm Push ranking measure which is specially designed for this purpose~\cite{rudin2009p} .

\begin{algorithm}[t]
\center \caption{\texttt{\alg}}
\begin{algorithmic}[1] \label{alg:taco}
     \STATE \textbf{input:}
 $\lambda \in \mathbb{R}_{+}$: the regularization parameter, and $\{\eta_t\}_{t \geq 1}$: the sequence of scalar step sizes
	\STATE Initialize $\bW_0 \in \mathbb{R}^{n \times d}$
	\STATE Choose an appropriate step size
	\FOR{$t = 1, \ldots, T$}
		\STATE Compute the sub-gradient of $\mtx{G}_t \in \partial \mathcal{L}(\bW_{t})$ using Eq.~(\ref{eqn-grad-u})
		\STATE $[\mtx{U}_t, \mtx{\Sigma}_t, \mtx{V}_t] \leftarrow \texttt{SVD}(\bW_{t-1} - \frac{1}{\eta_{t-1}}\mtx{G}_t))$
		\STATE $\bW_t \leftarrow \mtx{U}_t \left[ \mtx{\Sigma} - \frac{\lambda}{\eta_{t-1}} \mtx{I}\right]_{+} \mtx{V}_t^{\top}$
	 \ENDFOR
    \STATE  \textbf{output:}
\end{algorithmic}
\end{algorithm}

For simplicity of exposition, let us first consider the ranking model for a single user $u$. Let $\mathcal{X}^{+} =\{\bx_1^{+}, \cdots, \bx_{n_{+}}^{+}\}$ and $\mathcal{X}^{-} =\{\bx_1^{-}, \cdots, \bx_{n_{-}}^{-}\}$ be the set of feature vectors of $n_{+}$ relevant and $n_{-}$ irrelevant items to user $u$, respectively.
We consider linear ranking functions where each  item features  vector $\bx \in \mathbb{R}^d$ is mapped to a score $\bw^{\top}\bx$ . The goal is to find parameters $\bw$ for each user such that the ranking function best captures past feedback from the user. The goal of ranking is to maximize the number of relevant items ranked above the highest-ranking irrelevant item.  We cast this idea for each user $u$ individually into the following optimization problem:
\begin{equation}\label{}
\begin{aligned}
\min_{\bw \in \mathbb{R}^d}  \frac{1}{n^{+}} \sum_{i = 1}^{n^{+}}{\mathbb{I} \left[\dd{\bw}{\bx_i^{+}} \leq \max \limits_{1 \leq j \leq n^{-}} \dd{\bw}{\bx_j^{-}}\right]}
\end{aligned}
\end{equation}
where $\mathbb{I}[\cdot]$ is the indicator function which returns 1 when the input is true and 0 otherwise, $n^{+}$ and $n^{-}$ are the the number of relevant and irrelevant items to user $u$, respectively.

Let us now derive the general form of our objective. We hypothesize that most users base their decisions about items based on a  number of latent features about the items.  In order to uncover these latent feature dimensions, we impose a low-rank constraint on the set of parameters for all users. To this end,  let $\bW = [\bw_1, \bw_2, \cdots, \bw_n]^{\top} \in \mathbb{R}^{n \times d}$ denote the matrix of all parameter vectors for $n$ users. Let $\mathcal{I}^{+}_{i} \subseteq \{1,2,\ldots, m\}$ and $\mathcal{I}^{-}_{i} \subseteq \{1,2,\ldots, m\}$ be the set of  relevant and irrelevant items of $i$th user, respectively.     The overall objective for all users is formulated as follows:
\begin{equation}\label{eqn-non-convex-1}
\begin{aligned}
\mathcal{F}(\bW) &=  \lambda \|\bW\|_{*} \\
&+    \sum_{i = 1}^{n}{\left( \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{j \in \mathcal{I}^{+}_{i}}^{}{{\mathbb{I} \left[\dd{\bw_i}{\bx_j} \leq \max \limits_{k \in \mathcal{I}^{-}_{i}} \dd{\bw_i}{\bx_k} \right]}} \right)},
\end{aligned}
\end{equation}
where $\|\cdot\|_*$ is the trace norm (also known as nuclear norm) which is the sum of the singular values of the input matrix.

The objective in~Eq.~(\ref{eqn-non-convex-1}) is composed of two terms. The first term is the regularization term and is introduced  to capture the factor model intuition discussed above. The premise behind a factor model is that there is only a small number of factors influencing the preferences, and that a user's preference vector is determined by how each factor applies to that user.  Therefore, the parameter vectors of all users must lie in a low-dimensional subspace. Trace-norm  regularization  is  a  widely-used  and  successful  approach  for  collaborative filtering and matrix completion. The trace-norm regularization is well-known to be a convex surrogate to the matrix rank, and has repeatedly shown good performance in practice~\cite{srebro2004maximum,candes2010power}.  The second term is introduced to push the relevant items of each user to the top of the list when ranked based on the parameter vector of the user and features of items.

The above optimization problem is intractable due to the non-convex indicator function. To design practical learning algorithms, we replace the indicator function in~(\ref{eqn-non-convex-1})  with its convex surrogate. To this end, define the convex loss function $\ell: \mathbb{R} \mapsto \mathbb{R}_{+}$ as $\ell(x) = [1-x]_{+}$. This is the widely used hinge loss in SVM classification (see e.g.,~\cite{burges1998tutorial})~\footnote{We note that other convex loss functions such as exponential loss $\ell(x) = \exp(-x)$, and logistic loss $\ell(x) = \log(1+\exp(-x))$ also can be used as the surrogates of indicator function, but for the simplicity of derivation we only consider the hinge loss here.}. This loss function reflects the amount by which the constraints are not satisfied. By replacing the non-convex indicator function with this convex surrogate leads to the following tractable convex optimization problem:
\begin{equation}\label{}
\begin{aligned}
\mathcal{F}(\bW) &=  \lambda \|\bW\|_{*}  \\
& +    \sum_{i = 1}^{n}{\left( \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{j \in \mathcal{I}^{+}_{i}}^{}{\ell\left(\dd{\bw_i}{\bx_{j}} - \|\bX_i^{-}\bw_i\|_{\infty}\right)} \right)}
\end{aligned}
\end{equation}
where $\bX_i^{-} =[\bx_{1}, \ldots, \bx_{n^{-}_{i}}]^{\top}$ is the matrix of features of $n^{-}_{i}$ irrelevant items in  $\mathcal{I}^{-}_{i}$ and $\|\cdot\|_{\infty}$ is the max norm of a vector.

\subsection{Semi-supervised collaborative ranking}$\quad$

In this part, we extend the proposed ranking idea  to  learn both from rated as well as
unrated items. The motivation of incorporating unrated items comes from the following key observations.  First,  we note that commonly there is a small set of rated (either relevant or irrelevant) items for each user and a large number of unrated items.   As it can be seen from Eq.~(\ref{eqn-non-convex-1}), the unrated items do not play any role in learning the model for each user as the learning is only based on the pair of rated items. When the feature information for items is available, it would be very helpful if one can leverage such unrated items in the learning-to-rank process to effectively leverage the available side information. By leveraging both types of rated and unrated items, we can compensate  for the lack of rating data.  Second, the non-randomness in observing the observed ratings creates a  bias in  learning the model that  may degrade the resulting recommendation accuracy.  Therefore, finding a  precise model  to reduce the effect of bias  introduced by non-random missing ratings seems essential.

To address  these two issues, we extend the basic formulation in Eq.~(\ref{eqn-non-convex-1}) to incorporate items with missing ratings in ranking of items for individual users.  A conservative solution is to  push the items with unknown ratings to the middle of ranked list, i.e., after the relevant and before the irrelevant items. To do so, let $\mathcal{I}^{\circ}_i = \mathcal{I} \setminus \left( \mathcal{I}^{+}_i \cup \mathcal{I}^{-}_i\right)$ denote the set of items unrated for user $i \in \mathcal{U}$. We introduce two extra terms in the objective in Eq.~(\ref{eqn-non-convex-1})  to push the unrated items $\mathcal{I}_{\circ}^{i}$ below the relevant items and above the irrelevant items, which yilelds the following objective:
\begin{equation}\label{eqn-pnn-indicator}
\begin{aligned}
\mathcal{L}(\bw)  &=& \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{i \in \mathcal{I}^{+}_{i}}^{}{\ell \left(\dd{\bw}{\bx_i} \leq \max \limits_{j \in \mathcal{I}^{-}_i} \dd{\bw}{\bx_j}\right)} \\
&+ &  \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{i \in \mathcal{I}^{+}_{i}}^{}{\ell \left(\dd{\bw}{\bx_i} \leq \max \limits_{j \in \mathcal{I}^{\circ}_i} \dd{\bw}{\bx_j}\right)}\\
&+ &  \frac{1}{|\mathcal{I}^{\circ}_{i}|} \sum_{i \in \mathcal{I}^{\circ}_{i}}^{}{\ell \left(\dd{\bw}{\bx_i} \leq \max \limits_{j \in \mathcal{I}^{-}_i} \dd{\bw}{\bx_j}\right)}
\end{aligned}
\end{equation}
Equipped with the objective  of individual users, we now turn to  the final collaborating ranking  objective as:
\begin{equation}\label{eqn-primal-social-final}
\begin{aligned}
&\mathcal{F}(\bW) = \lambda \|\bW\|_{*} \\
& +  \sum_{i = 1}^{n}{\left( \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{j \in \mathcal{I}^{+}_{i}}^{}{\ell\left(\dd{\bw_i}{\bx_{j}} - \|\bX_i^{-}\bw_i\|_{\infty}\right)} \right)}\\
& +   \sum_{i = 1}^{n}{\left(  \frac{1}{|\mathcal{I}^{+}_{i}|} \sum_{j \in \mathcal{I}^{+}_{i}}^{}{ \ell\left(\dd{\bw_i}{\bx_{j}} - \|\bX_i^{\circ} \bw_i\|_{\infty}\right)} \right)}\\
&+  \sum_{i=1}^{n}{ \left(\frac{1}{|\mathcal{I}^{\circ}_{i}|} \sum_{j \in \mathcal{I}^{\circ}_{i}}^{}{\ell\left(\dd{\bw_i}{\bx_{j}} - \|\bX^{-}_i\bw_i\|_{\infty}\right)}\right)},
\end{aligned}
\end{equation}
where  $\bX_i^{\circ} =[\bx_{1}, \ldots, \bx_{n^{\circ}_{i}}]^{\top}$ is the matrix of $n^{\circ}_{i}$ unrated items in  $\mathcal{I}^{\circ}_{i}$.

\begin{remark} We emphasize that beyond the accuracy considerations of push norm at top of the list,  the  push norm has a clear advantage  to the pairwise ranking models from a computational point of view. In particular,  the push norm has  a linear $O(m)$ dependency on the number of items which is quadratic $O(m^2)$ for pairwise ranking models.
\end{remark}

\section{The Optimization}\label{sec:optimization}
We now turn to solving the optimization problem in~(\ref{eqn-primal-social-final}). We start by discussing a gradient descent method with shrinkage operator followed by its accelerated version,  and then propose a non-convex  formulation with alternative minimization for more effective optimization of objective in~\alg.

\subsection{Gradient descent with shrinkage operator}

Due to the presence of trace norm of the parameters matrix,  this objective function falls into the general category of composite  optimization, which can be solved by  stochastic gradient or gradient descent methods. In this part we propose a projected gradient decent method to solve the optimization problem.
First we write the objective as:

\begin{equation}\label{eqn-primal-social-final-1}
\begin{aligned}
\min_{\bW \in \mathbb{R}^{n \times d}}  \mathcal{F}(\bW) =   \lambda \|\bW\|_{*} + \mathcal{L}(\bW),
\end{aligned}
\end{equation}
where $\mathcal{L}(\bW) = \sum_{i=1}^{n}{\mathcal{L}(\bw_i)}$.

A simple way to solving the above optimization problem is  gradient descent algorithm~\cite{nesterov2013gradient}, which needs to evaluate the gradient of objective at each iteration. To deal with the non-smooth  trace norm $\|\bW\|_{*}$ in the objective, we  first note that the optimization problem in Eq.~(\ref{eqn-primal-social-final-1}) can be reformulated under the framework of proximal regularization or composite gradient mapping~\cite{nesterov2013gradient}. By taking advantage of the composite structure it is possible to retain the same convergence rates of the gradient method for the smooth optimization problems. In particular,  the  optimization problem in~(\ref{eqn-primal-social-final-1}) can be solved iteratively  by:

\begin{equation}\label{eqn-proximal-term}
\begin{aligned}
\bW_t = \arg \min_{\bW} \;&\mathcal{L}(\bW_{t-1}) + \text{\texttt{tr}} \left((\mtx{W} - \mtx{W}_{t-1})^{\top}\nabla \mathcal{L}(\mtx{W}_{t-1})\right)  \\
&+\frac{\eta_t}{2}\|\bW - \bW_{t-1}\|_{\rm{F}}^2 +  \lambda \|\bW\|_{*},
\end{aligned}
\end{equation}
where $\{\eta_t\}_{t \geq 1}$ is a sequence of scalar step sizes and $\text{\texttt{tr}}(\cdot)$ is the trace of input matrix. \\

By ignoring the constant terms, Eq.~(\ref{eqn-proximal-term}) can also be rewritten as:
\begin{equation}\label{eqn-proximal-term-8}
\begin{aligned}
\frac{\eta_t}{2} \left\| \bW - \left( \bW_{t-1} - \frac{1}{\eta_{t}} \nabla \mathcal{L}(\bW_{t-1})\right) \right\|_{\rm{F}}^2 + \lambda \|\bW\|_{*}.
\end{aligned}
\end{equation}

We use the singular value shrinkage operator introduced in~\cite{cai2010singular} to find the optimal solution to Eq.~(\ref{eqn-proximal-term-8}). To this end, consider the singular value decomposition (SVD) of a matrix $\mtx{M} \in \mathbb{R}^{n \times d}$ of rank $r$ as $\mtx{M} = {\bU}  \mtx{\Sigma}   \mtx{V}^*,\quad   \mtx{\Sigma}  = {\diag}(\{\sigma_i\}_{1 \le i \le r}),$
where $\mtx{U}$ and $\mtx{V}$ are respectively $n \times r$ and $d
\times r$ matrices with orthonormal columns, and the singular values
$\sigma_i$ are positive. For a scalar $\tau \in \mathbb{R}_{+}$, define the singular value shrinkage operator  $\mathscr{P}_\tau$ as:
\begin{equation} \label{eqn:DlamM2}
\mathscr{P}_{\tau}(\mtx{M}):=  \mtx{U}
\mathscr{P}_{\tau}(\mtx{\Sigma}) \mtx{V}^*, \quad
\mathscr{P}_{\tau}(\mtx{\Sigma}) = {\diag}([\sigma_i - \tau]_+\}),
\end{equation}
where $[x]_+$ is the positive part of $x$, namely, $[x]_+ = \max(0, x)$. The shrinkage operator basically  applies a soft-thresholding rule to the singular values of $\mtx{M}$, effectively shrinking these towards zero.

\begin{theorem}[Theorem~2.1,~\cite{cai2010singular}]\label{thm:prox}
For each $\tau \ge 0$ and $\mtx{W} \in \mathbb{R}^{n \times d}$, the  singular value shrinkage operator $\eqref{eqn:DlamM2}$ obeys
\begin{equation}\label{eqn:DlamM}
\mathscr{P}_{\tau}(\mtx{W}) = \arg\min_{\mtx{X}} \left\{\frac12\|\mtx{X}-\mtx{W}\|_{\rm{F}}^2 + \tau\|\mathbf{X}\|_{*}\right\}.
\end{equation}
\end{theorem}

The above theorem shows that the singular value shrinkage operator of a matrix is the solution to Eq.~(\ref{eqn:DlamM}).  Equipped with this result, the  optimization problem in Eq.~(\ref{eqn-proximal-term-8}) can be solved by first computing the SVD of updated $\mtx{W}_{t-1}$ and then applying soft thresholding on the singular values as:
$$\bW_{t} = \mathscr{P}_{\frac{\lambda}{\eta_{t-1}}} \left( \bW_{t-1} - \frac{1}{\eta_{t-1}} \nabla \mathcal{L}(\bW_{t-1})\right)$$
Now we need to evaluate the gradient of $\mathcal{L}(\bW)$ at $\bW_{t-1}$. The convex function $\mathcal{L}(\bW)$ is not differentiable, so we use its subgradient in updating the solutions which can be computed for $i$th parameter vector $\bw_i$,  as follows:
\begin{equation}
\begin{aligned}\label{eqn-grad-u}
 \bg_i &=  {\partial \mathcal{L}}/{\partial \bw_i} \\
& = \sum_{j \in \mathcal{I}^{+}_{i}}^{}{\mathbb{I}\left[\dd{\bw_i}{\bx_{j}} - \|\bX_i^{-}\bw_i\|_{\infty} \leq 1\right] \left( \partial \|\bX_i^{-} \bw_i\|_{\infty} - \bx_{j}\right)}\\
&+  \sum_{j \in \mathcal{I}^{+}_{i}}^{}{\mathbb{I}\left[\dd{\bw_i}{\bx_{j}} - \|\bX_i^{\circ}\bw_i\|_{\infty} \leq 1\right] \left( \partial \|\bX_i^{\circ} \bw_i\|_{\infty} - \bx_{j}\right)}\\
&+  \sum_{j \in \mathcal{I}^{\circ}_{i}}^{}{\mathbb{I}\left[\dd{\bw_i}{\bx_{j}} - \|\bX_i^{-}\bw_i\|_{\infty} \leq 1\right] \left( \partial \|\bX_i^{-} \bw_i\|_{\infty} - \bx_{j}\right)},
\end{aligned}
\end{equation}
where  $\partial \|\bX_i^{-} \bw_i\|_{\infty}$ is the subdifferential of the function $\|\bX_i^{-} \bw_i\|_{\infty}$ at point $\bw_i$.    Since the subdifferential of the maximum of functions is the convex hull of the union of subdifferentials of the active functions at point $\bw$~\cite{nesterov2004introductory}, we have:
\begin{align*}
\partial \|\bX_i^{-} \bw_i\|_{\infty} &= \partial \max_{1 \leq j \in \mathcal{I}^{-}_{i}} \dd{\bw_i}{\bx_j} \\
& = \text{conv} \left\{\bx_j| \dd{\bw_i}{\bx_j} = \|\bX_i^{-} \bw_i\|_{\infty}, j \in \mathcal{I}^{-}_{j} \right\}.
\end{align*}
Then, $\mtx{G} = [\bg_1, \bg_2, \ldots, \bg_n]^{\top} \in \mathbb{R}^{n \times d}$ is a subgradient at  $\bW$, i.e. $\mtx{G} \in \partial \mathcal{L}(\bW)$.

\begin{remark}
We  note that here, for the ease of exposition, we only consider the non-differentiable convex hinge loss as the surrogate of non-convex due to its computational virtues. However, by using other smooth convex surrogate losses such as smoothed hinge loss or exponential loss, one can apply the accelerated gradient descent methods~\cite{nesterov2004introductory} to solve the optimization problem which results in significantly faster convergence rate compared to the naive gradient descent (i.e, $O(1/\sqrt{\epsilon})$ convergence rate for accelerated method compared to the $O(1/\epsilon^2)$ rate for gradient descent for non-smooth optimization, where $\epsilon$ is the target accuracy).
\end{remark}

\subsection{Efficient optimization by dropping convexity}
The main computational cost in each iteration of the \alg~algorithm lies in
computing the SVD decomposition of $\bW_k$. An alternative cheaper way to  solving  the optimization problem in Eq.~(\ref{eqn-primal-social-final-1}) is as follows. For a fixed rank of the target parameter matrix $\bW$, say $k$, one can decompose it as $\bW = \bU \bV$. From the equivalence relation between trace norm and the Frobenius of its components in decomposition,
$$ \|\bW\|_*  = \min_{\substack{\bU \in \mathbb{R}^{n \times k}, \bV \in \mathbb{R}^{m \times k}\\ \bW = \bU \bV^{\top} }} \frac{1}{2} \left( \|\bU\|_{\rm{F}}^2 + \|\bV\|_{\rm{F}}^2 \right)$$

we can write the objective in terms of these low rank components as:

\begin{equation}\label{eqn-primal-social-final-2}
\begin{aligned}
\min_{\bU \in \mathbb{R}^{n \times k}, \bV \in \mathbb{R}^{m \times k}}    \frac{\lambda}{2} \left( \|\bU\|_{\rm{F}}^2 + \|\bV\|_{\rm{F}}^2 \right) + \mathcal{L}(\bU\bV),
\end{aligned}
\end{equation}

These factored optimization problem does not have the explicit trace norm regularization. However, the new formulation is non-convex and potentially subject to stationary points that are not globally optimal. However, despite its non-convexity, the formulation in~Eq.~(\ref{eqn-primal-social-final-2})  is competitive  as compared to trace-norm minimization, while scalability is much better. In particular, the objective is not jointly convex in both $\bU$ and $\bV$ but it is convex in each of them fixing the other one. Therefore, to find a local solution one can  stick to  the standard gradient descent method to find a solution in an iterative manner as follows:
\begin{align*}
& \bU_{t+1} \leftarrow (1 - \lambda \eta_t) \bU_t - \eta_t \nabla_{\bU} \mathcal{L}\large{|}_{\bU = \bU_t, \bV = \bV_t}, \\
& \bV_{t+1} \leftarrow (1 - \lambda \eta_t) \bV_t - \eta_t \nabla_{\bV} \mathcal{L}\large{|}_{\bU = \bU_t, \bV = \bV_t}. \\
\end{align*}

\begin{remark} It is remarkable that  the large number of users or items  may cause computational  problems in solving the optimization problem  using GD method. The reason is essentially the fact that computing the gradient at each iteration requires to go through all the users and compute the gradient for pair of items. To alleviate this problem one can utilize stochastic gradient  method~\cite{nemirovski-2009} to solve the optimization problem. The main idea is to choose a fixed subset of pairs for gradient computation instead of all pairs  at \textit{each} iteration or a sample a user at random for gradient computation instead of including all users.  We note that this strategy generates unbiased estimates of the true gradient and makes each iteration of algorithm computationally more efficient compared to the full gradient counterpart.
\end{remark}

\begin{algorithm}[t]
\center \caption{\texttt{\alg+}}
\begin{algorithmic}[1] \label{alg:taco+}
     \STATE \textbf{input:}
 $\gamma, \lambda \in \mathbb{R}_{+}$: the regularization parameters,  $\{\eta_t\}_{t \geq 1}$: the sequence of scalar step sizes, and $\mtx{S} \in \mathbb{R}^{n \times n}$: the similarity matrix of usets
	\STATE Initialize $\bW_0 \in \mathbb{R}^{n \times d}$
	\STATE Choose an appropriate step size
	\FOR{$t = 1, \ldots, T$}
		\STATE Compute the sub-gradient of $\mtx{G}_t \in \partial \mathcal{L}(\bW_{t})$ using Eq.~(\ref{eqn-grad-u})
		\STATE Compute $\widehat{\mtx{G}}_t = \mtx{G}_t + 2 \gamma \mtx{X}\mtx{X}^{\top} \mtx{W} \mtx{L}$
		\STATE $[\mtx{U}_t, \mtx{\Sigma}_t, \mtx{V}_t] \leftarrow \texttt{SVD}(\bW_{t-1} - \frac{1}{\eta_{t-1}}\widehat{\mtx{G}}_t))$
		\STATE $\bW_t \leftarrow \mtx{U}_t \left[ \mtx{\Sigma} - \frac{\lambda}{\eta_{t-1}} \mtx{I}\right]_{+} \mtx{V}_t^{\top}$
	 \ENDFOR
    \STATE  \textbf{output:}
\end{algorithmic}
\end{algorithm}

\section{Regularization by Exploiting Similarity of Users}\label{sec:taco-users}

In many recommender systems, in addition to handling cold-start items, it would be beneficial to make high quality recommendations to new users.  The new user cold start issue represents a serious problem in recommender systems as it can lead to the loss of new users who decide to stop using the system due to the lack of accuracy in the recommendations received in that first stage in which they have not yet cast a significant number of ratings with which to feed the recommender system's collaborative filtering core.

The above formulation  estimates a parameter vector separately for each user regardless of potential similarities across users. In this section we generalize the proposed ranking model to the setting where the similarity information between users is available.  Let $\bS \in \mathbb{R}^{n \times n}$ be the similarity matrix of users which is inferred from the side information about users such as  social relations between them or explicit users' features.  Let $\bD$ be the diagonal matrix with $D_{ii} = \sum_{j = 1}^{n}{W_{ij}}$, and  $\bL = \bD- \bS$ be the Laplacian matrix. It is natural to require the similar users in the matrix $\bS$ have similarity evaluation  on rating the items~\cite{belkin2006manifold}.  Thus, the new regularization on the parameters of users $\bW$ using the similarity matrix $\bS$ can be achieved by minimizing:
\begin{equation*}\label{}
\begin{aligned}
& \frac{1}{2} \sum_{i = 1}^{m}{ \left( \sum_{j,k = 1}^{n}{S_{jk} \left(\dd{\bw_j}{\bx_i} - \dd{\bw_k}{\bx_i} \right)^2} \right)} \\
& = \frac{1}{2} \sum_{i = 1}^{m} \left( \bx_i^{\top} \bw_j ( \sum_{k}^{}{S_{jk}} ) \bw_j^{\top} \bx_i  - \sum_{j,k}^{}{\bx_i^{\top} \bw_j S_{jk} \bw_{k}^{\top} \bx_i}\right)\\
& = \sum_{i}^{}{ \left( \sum_{j}^{}{\bx_i^{\top} \bw_j S_{jj} \bw_j^{\top} \bx_i}  - \sum_{j,k}^{}{\bx_i^{\top} \bw_j S_{ij} \bw_k^{\top} \bx_i} \right)}\\
& = \sum_{i}^{}{\bx_i^{\top} \left( \bW \bD \bW^{\top} - \bW \bS \bW^{\top}\right)\bx_i}\\
& = \sum_{i}^{}{\bx_i^{\top} \bW \bL \bW^{\top}  \bx_i} =  \text{\texttt{tr}} \left( \bX^{\top} \bW \bL \bW^{\top} \bX\right).
\end{aligned}
\end{equation*}
By plugging the above regularization term in the~Eq.~(\ref{eqn-primal-social-final-1}), we  obtain the following optimization problem:
\begin{equation*}\label{}
\begin{aligned}
\mathcal{F}(\bW) =  \lambda \|\bW\|_{*} +  \mathcal{L}(\bW) + \gamma \;\text{\texttt{tr}} \left( \bX^{\top} \bW \bL \bW^{\top} \bX\right),
\end{aligned}
\end{equation*}

The above optimization problem  can also be solved using the optimization procedure discussed before. The only difference is the gradient computation step which now has an extra term due the introduction  of user's similarity  regularization. Specifically, we have the new gradient $\widehat{\mtx{G}}_t$ computed as:

$$\widehat{\mtx{G}}_t = \mtx{G}_t + 2 \gamma \mtx{X}\mtx{X}^{\top} \mtx{W} \mtx{L},$$
which replaces the $\mtx{G}_t$ in Algorithm~\ref{alg:taco}. The resulting algorithm, dubbed $\alg+$, is detailed in Algorithm~\ref{alg:taco+}.

\section{Experiments}\label{sec:results}

In this section, we conduct exhaustive experiments to demonstrate the merits and advantages of the proposed algorithm. We conduct our experiments on three well-known datasets MovieLens, Amazon and CiteULike. We investigate how the proposed \alg~performs in comparison to the state-of-the-art methods. In the following, first we introduce the datasets that we use in our experiments and then the metrics that we employ to evaluate the results, followed by our detailed experimental results on the real datasets.

In the following subsections, we intend to answer these key questions:
\begin{itemize}
\item \textbf{Ranking versus rating:} How  does  learning optimization for a ranking based loss function  affect  the  performance of recommending versus the square loss function?

\item \textbf{Employing missing ratings:} How does employing the missing ratings could help in making more accurate recommendations?

\item \textbf{Dealing with cold-start items:} How  does  the  proposed  algorithm, with incorporating side information of items, perform in comparison to the  state-of-the-art algorithms to deal with cold-start items?

\end{itemize}

\subsection{Datasets}~We use the following well known datasets to evaluate the performance of \alg{}:\\

\begin{itemize}
\item \textbf{ML-IMDB.} We used ML-IMDB{} which is a dataset extracted from the IMDB and the MovieLens $1$M  datasets by mapping the MovieLens and IMDB and collecting the movies that have plots and keywords. The rating values are 10 discrete numbers ranging from $1$ to $10$ and the rating were made binary by treating all the ratings greater than $5$ as $+1$ and below $5$ as $-1$.  \\
\item \textbf{Amazon.} We used the dataset of best-selling books and their ratings in Amazon. Each book has a one or two paragraphs of textual description, which has been used to have a set of features of the books. Ratings can be integers numbers from $1$ to $5$. The ratings were also made binary by treating all the ratings greater or equal to $3$ as $+1$ and below $3$ as $-1$.  \\
\item \textbf{CiteULike.} It is an online free service for managing and discovering scholarly references. Users can add those articles that they are interested in to their libraries. Collected articles in a user's library will be considered as relevant items for that user. This dataset does not have explicit irrelevant items and was chosen to illustrate the effect of considering missing data while only having relevant itmes.\\
\end{itemize}

For all above datasets, the description about the items were tokenized and after removing the stop words, the rest of the words were stemmed. Then those words that have been appeared in less than $20$ items and more that $20\%$ of the items were also removed ~\cite{sharmafeature2015b}. At the end, the TF-IDF was applied on the remaining words and the TF-IDF scores represented the features of the items. The statistics of the datasets are given in Table \ref{statisticsOfRealDatasets}. As it is shown in Table \ref{statisticsOfRealDatasets}, all these three datasets have high dimensional feature space.

\begin{table*}
\centering
\caption{Statistics of real datasets used in our experiments.}
\label{statisticsOfRealDatasets}
\begin{tabular} { l || c || c || c }
Statistics & ML-IMDB{} & Amazon{} & CiteULike{} \\
\hline \hline
Number of users    & 2,113     & 13,097     &   3,272  \\
\hline
Number of items    & 8,645    & 11,077  & 21,508\\
\hline
Number of ratings  & 739,973  & 175,612  &  180,622\\
\hline
Number of features   & 8,744  &  5,766  &  6,359\\
\hline
Average number of ratings by users   &  350.2 &  13.4 &  55.2\\
\hline
Average number of ratings for items  & 85.6  & 13.4  & 55.2\\
\hline
Density  & 4.05\%  & 0.12\%  & 0.13\% \\
\hline
\end{tabular}
\end{table*}

\subsection{Metrics}~
We adopt the widely used metrics, Discounted Cumulative Gain at $n$ and Recall at $n$, for assessing the performance of our and baseline algorithms. For each user $u$, given an item  $i$, let $s_k$ be the relevance score of the item ranked at position $k$, where $s_k = 1$ if the item is relevant to the user $u$ and $s_k = 0$ otherwise. Now, given the list of top-$n$ item recommendations for user $u$, Discounted Cumulative Gain at $n$, is defined as:
$$
{\rm{DCG_u}}@n = s_1 + \sum_{k=2}^{n}{\frac{s_k}{\log_2 (k)}}
$$
If we divide the ${\rm{DCG_u}}@n$ by its maximum value, we get the ${\rm{NDCG_u}}@n$ value.
Given the list of top-$n$ item recommendations for each user $u$, Recall at $n$ will count the number of relevant items appeared in the recommendation list divided by the size of the list. Recall at $n$ is defined as:
$$
{\rm{REC_u}}@n =  \frac{| \{\text{relevant items to $u$}\} \cap \{\text{top-$n$ recommended items}\} |}{|\{\text{top-$n$ recommended items}\}|}
$$
DCG$@n$, ${\rm{NDCG_u}}@n$ and REC$@n$ will be computed for each user and then will be averaged over all users.

\subsection{Methodology}~
Given the partially observed rating matrix, we transformed the observed ratings of all datasets from a multi-level relevance scale to a two-level scale $(+1, -1)$ while $0$ is considered for unobserved ratings.
We randomly selected $60\%$ of the observed ratings for training and $20\%$ for validation set and consider the remaining $20\%$ of the ratings as our test set. To better evaluate the results, we performed a 3-fold-cross validation and reported the average value for our results.

\subsection{Baseline Algorithms}~The proposed~\alg~algorithm is compared to the following algorithms:

\begin{itemize}
\item \textbf{Matrix Factorization (MF) \cite{srebro2004maximum}:} Is a matrix completion method that factorizes the incomplete observed matrix and completes the matrix using the unveiled  latent features.

\item \textbf{Matrix Factorization with Side Information (KPMF)~\cite{zhou2012kernelized}:}
Is a matrix completion based algorithm, which incorporates external side information of the users or items into the matrix factorization process. It imposes a Gaussian Process prior over all rows of the matrix, and the learned model explicitly captures the underlying correlation among the rows.

\item \textbf{Decoupled Completion and Transduction (DCT)~\cite{barjasteh2015cold}:} Is a matrix factorization based algorithm that decouples the completion and transduction stages and exploits the similarity information among users and items to complete the (rating) matrix.

\item \textbf{Feature Based Factorized Bilinear Similarity Model (FBS) ~\cite{sharmafeature2015b}:} This algorithm uses bilinear model to capture pairwise dependencies between the features. In particular, this model accounts for the interactions between the different item features.

\item \textbf{Collaborative User-specific Feature-based Similarity Models (CUFSM):} By using the history of ratings for users, it learns personalized user model across the dataset. This method is one of the best performing collaborative latent factor based model~\cite{elbadrawy2015user}.
 \item \textbf{Regression based Latent Factor Model (RLF):\footnote{The implementation of this method is available in LibFM library~\cite{rendle2012factorization}.}} This method incorporates the features of items in factorization process by transforming the features to the latent space using linear regression~\cite{agarwal2009regression}. If the learning method is Markov Chain Monte Carlo, we name it RLF-MCMC.
 \item \textbf{Cosine Similarity Based Recommender (CSR):}  Using the similarity between features of items, the preference score of a user on an item will be estimated.
 \end{itemize}

We would like to mention that as baseline  algorithms we only consider state-of-the art methods that are able to exploit the side information about items.

\subsection{\alg{} vs. rating}
Many different algorithms are trying to provide recommendations to users such that the predicted rating values be very close to the actual rates that users would provide. These algorithms try to minimize the error between the predicted values and actual rating values by minimizing Mean Squared Error (MAE) or Root Mean Square Error (RMSE) or etc.  \\
Then, due to the fact that users tend to only care about the top of their recommendation list, predicting a ranking of items of interest instead of ratings became the main focus of recent works \cite{christakopoulou2015collaborative}. In this section we compare the results of \alg{} with those state-of-the-art algorithms that try to predict ratings for unrated items. Among the state-of-the-art algorithms, we chose a diverse set of algorithms, which are Matrix Factorization, Matrix Factorization with Side Information, Decoupled Completion and Transduction and Regression Based Latent Factor Model. Table \ref{ResultsOfRankingVsRating} shows the NDCG value of top 10 items of recommendation list. It shows that \alg{} outperformed all other rating prediction based algorithms in terms of NDCG measure. The results confirm the effectiveness of providing the ranks of items rather than their ratings.

\begin{table*}
\centering
\caption{Results of comparing rankings methods versus ratings methods on  ML-IMDB{}. $\lambda$ is regularization parameter, $h$ is dimension of latent features, $T$ is the number of iterations, $\eta $ is learning rate and $\sigma$ is the standard deviation.}
\label{ResultsOfRankingVsRating}
\begin{tabular}{ l | c | c  }
\multirow{2}{*}{\textbf{Algorithms}} &\multirow{2}{*}{\textbf{Hyperparameters}}& \multirow{2}{*}{\textbf{ NDCG$@10$}} \\
 &  & \\
           \hline\hline

  MF     &$\eta=0.003, h=10,  \sigma = 0.4$&  0.0947 \\ \cline{1-3}

 KPMF &$\eta=0.003, h=10,  \sigma = 0.4$ & 0.1005 \\ \cline{1-3}

 DCT &$      h  = 10$& 0.0095 \\ \cline{1-3}

 RLF-MCMC & $ T=100, \sigma= 0.1 $ & 0.0248    \\ \cline{1-3}

 \alg{} &$\lambda = 0.55, h =10, T=100$ & \textbf{0.265}    \\ \cline{1-3} \hline \hline

\end{tabular}
\end{table*}

\subsection{Robustness to not missing  at random ratings}
In this section we compare the effect of incorporating the unobserved ratings in our learning in comparison with excluding them from our learning. Most of the methods in the literature ignore the unobserved ratings and train their model only base on observed ratings. By incorporating the unrated items in ranking, our method can limit the bias caused by learning solely based on the observed ratings and consequently deals with the not missing at random issue of ratings. Table \ref{ResultsOfNotMissingAtRandom} shows results of comparing these two scenarios for \alg{} on ML-IMDB{}. In order to see the difference between these two scenarios, we considered 70\% of the ratings for training and 30\% for test to have more ground truth for our testing. Table \ref{ResultsOfNotMissingAtRandom} shows the NDCG@5, 10,15 and 20 for both scenarios and it shows that incorporating the unobserved ratings causes to improve the accuracy of recommendation list. Hence, the NDCG values for top 5, 10, 15 and 20 items improved when unrated items were included as part of the training process.

\begin{table*}
\centering
\caption{Results of employing missing ratings versus ignoring them on  ML-IMDB{}. $\lambda=0.6$ is regularization parameter, $h=10$ is dimension of latent features, $T=100$ is the number of iterations. }
\label{ResultsOfNotMissingAtRandom}
\begin{tabular}{ c | c | c  | c | c  }
\multirow{2}{*}{\textbf{Algorithm: \alg{}}} & \multirow{2}{*}{\textbf{ NDCG@5}}& \multirow{2}{*}{\textbf{ NDCG@10}}& \multirow{2}{*}{\textbf{ NDCG@15}}& \multirow{2}{*}{\textbf{ NDCG@20}} \\
 &  & && \\
           \hline\hline

Observed ratings     &1.1690   &   2.2218   & 2.8362  &  3.2849      \\ \cline{1-5}
Observed + missing ratings  &\textbf{1.1794}   &   \textbf{2.2405}   & \textbf{2.8585}  &  \textbf{3.3096}  \\ \hline \hline

\end{tabular}
\end{table*}

\subsection{Dealing with cold-start items}

We now turn to  evaluating the effectiveness of \alg{} for cold-start recommendation.
To do so, we randomly selected $60\%$ of the items as our training items and $20\%$ for validation set and considered the remaining $20\%$ of the items as our test set.
In this scenario, baseline algorithms that are used for comparison are CSR,  FBS, CUFSM and RLF. For the experiments, we used ML-IMDB{}, Amazon{} and CiteULike{} datasets. Table \ref{ResultsOfColdStartScenario} shows the measurement results of applying mentioned algorithms on these datasets. For each test, the parameters' values producing the best ranking on the validation set  were selected to be used and reported. As it can be seen from the results in Table~\ref{ResultsOfColdStartScenario}, the proposed ~\alg~ algorithm outperformed all other baseline algorithms and provided a recommendations with higher quality in comparison to other methods. We can also see from the results of Table~\ref{ResultsOfColdStartScenario} that for the ML-IMDB dataset, the improvement in terms of REC@10 is significant compared to other datasets. Since the density of this dataset is much higher than other two datasets, this observation indicates that our method is more effective in utilizing side information compared to other methods.
These results demonstrate the effectiveness of \alg{} in comparison with other state-of-the-art algorithms. \alg{} was able to outperform other state-of-the-art algorithms by considering the missing data and focusing on top of the recommendation list for cold-start items.

\begin{table*}
\centering
\caption{Results on cold-start items. $\lambda$, $\mu_1$ and $\beta$ are regularization parameters, $h$ is dimension of latent features, $l$ is the number of similarity functions and $T$ is the number of iterations.}
\label{ResultsOfColdStartScenario}
\begin{tabular}{c | c | c | c | c   }
&\multirow{2}{*}{\textbf{Algorithms}} &\multirow{2}{*}{\textbf{Hyperparameters}}& \multirow{2}{*}{\textbf{ DCG$@10$}}&\multirow{2}{*}{\textbf{REC$@10$}}  \\
  &&  & &  \\
           \hline\hline

\multirow{5}{*}{\rotatebox{90}{\textbf{ML-IMDB{}}}}&  CSR  &---&  0.1282 & 0.0525 \\ \cline{2-5}

& RLF &$h = 15$ & 0.0455 & 0.0155 \\ \cline{2-5}

& CUFSM &$l=1, \mu_1=0.005$& 0.2160 & 0.0937\\ \cline{2-5}

& FBS &$\lambda=0.01, \beta=0.1, h=5$ & 0.2270 & 0.0964  \\ \cline{2-5}

& \alg{} &$\lambda = 0.6, h =10, T=200$ & \textbf{0.2731}  & \textbf{0.2127}  \\ \cline{1-5} \hline \hline

\multirow{5}{*}{\rotatebox{90}{\textbf{Amazon{}}}}&  CSR  &---&  0.0228 & 0.1205\\ \cline{2-5}

& RLF &$h = 30$ & 0.0076 & 0.0394 \\ \cline{2-5}

& CUFSM &$l=1, \mu_1=0.25$& 0.0282 & 0.1376  \\ \cline{2-5}

& FBS &$\lambda=0.1, \beta=1, h=1$ & 0.0284 & 0.1392 \\ \cline{2-5}

& \alg{} &$\lambda = 0.6, h =10, T=200$ & \textbf{0.1195} & \textbf{0.1683} \\ \cline{1-5} \hline \hline

\multirow{5}{*}{\rotatebox{90}{\textbf{CiteULike{}}}}&  CSR  &---& 0.0684 & 0.1791  \\ \cline{2-5}

& RLF &$h = 75$ & 0.0424 & 0.0874  \\ \cline{2-5}

& CUFSM &$l=1, \mu_1=0.25$& 0.0791 & 0.2017\\ \cline{2-5}

& FBS &$\lambda=0.25, \beta=10, h=5$ & 0.0792 & 0.2026  \\ \cline{2-5}

& \alg{} &$\lambda = 0.6, h =10, T=200$ &  \textbf{0.0920} &  \textbf{0.2243}  \\ \cline{1-5} \hline \hline
\end{tabular}
\end{table*}

\section{Conclusions}\label{sec:conclusions}

In this paper we introduced a   {semi-supervised collaborative ranking} model  by leveraging side information about {both observed and missing ratings} in collaboratively learning the ranking model.  In the learned model, unrated items are conservatively pushed after the relevant  and before the relevant items in the ranked list of items for each individual user. This crucial difference greatly boosts the performance and limits the bias caused by learning only from sparse non-random observed ratings. The proposed algorithm is compared with seven baseline algorithms on three real world datasets that demonstrated the effectiveness of  proposed algorithm in addressing cold-start problem and mitigating the data sparsity problem, while being robust to sampling of missing ratings.

This work leaves few interesting directions as future work. First, we would like to investigate the performance of the proposed \alg~algorithm when side information about users is also available using the graph regularization idea discussed in Section~\ref{sec:taco-users}. Second, we would like to empirically evaluate the performance of the optimization method derived by dropping the convexity in future. Also, we have largely ignored the case of differentiable smooth surrogate convex loss functions in this work and it would be interesting to consider smooth alternatives and apply the accelerated optimization methods for faster convergence. Moreover,  the scalability analysis of of proposed algorithm on large datasets using stochastic optimization methods is also worthy of investigation. Finally, we believe  there are   still many open questions relating to non-random nature of rating information in many real applications.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


Hi there! I made peanut butter cookies recently and thought I should share this recipe with my blog readers. I always liked to eat peanuts. In India when I was a little girl we would go to a speciality shop. In Marathi, the shop is called as Bhadabhunja (bha-duh-bhun-ja) shop. They only use to sell salted roasted peanuts, roasted chickpeas, rice puffs, wheat puffs, sorghum puff, rice cereals, etc. There was one such shop in the neighborhood where my grandparents lived. That shop is still there and run by the owner's future generations. Everytime I visit India I make sure to get some goodies from the shop especially roasted, salted peanuts.
After moving in US I encounter peanuts in different forms than I used to in India. I started liking peanut butter and then peanut butter cookies. I experimented with different peanut butter cookie recipes and finally made my own. I have been using this recipe from last 10 years. I am sure you will enjoy this recipes as well. Give it a try!
Sift all the dry ingredients (flour, baking powder, baking soda, nutmeg, and salt) together. Keep it aside.
In a food processor bowl add butter, peanut butter, brown sugar and regular sugar. Cream everything together to a smooth mixture. As you will start creaming the mixture it will become pale in color and will become fluffy.
Add the beaten eggs one at a time in the creamed butter and sugar mixture.
After mixing in the eggs, add the dry ingredients to the butter-sugar mixture in 3 batches. Make sure not to overmix the batter.
Fold in the chopped peanuts in the cookie batter.
Spoon 1 tablespoon of the mixture into your palm and roll the dough and place it on the baking sheet.
Flatten the rolled dough with the tines of a fork, making a crosshatch pattern on the cookies. Sprinkle sugar on top of the cookies.
Bake the cookies until golden brown around the edges about 10 to 12 minutes (depending on your oven) switching the position of the sheets halfway through baking. Transfer to racks to cool. Repeat with the remaining dough.
Instead of using combination of whole wheat and all purpose flour, you can use 2 1/2 cups of plain all purpose flour or whole wheat flour.
For Peanut butter – chocolate chip cookies : Replace 1/2 cup of chopped peanuts by 1/2 cup of semi-sweet chocolate chips.
if you like your cookies sweeter then roll the dough ball in granulated sugar before baking.


I am working professional and I want to make friends so that we can travel together. My hobbies are travelling and hanging with frends.
I want girl who is understandable, loves to travel and share our feelings. She should also caring and jolly nature.


Mamta Kanvinde received the Master of Data Science in 2016 and to this day, still uses notes from coursework to assist in her job tasks. She received a Master of Science in Computer Science in Mumbai, India before enrolling in the data science program at Illinois Tech. She was intrigued by the wide range of course study offered within curriculum, particularly in math.


Hello, and welcome to the first real lesson in real time system development.
and firstly the most basic concept, the task structure.
on a processor, so that the timeline guarantees for workload is never violated.
We quantify the workload into discrete pieces which we refer to as a task.
of those boxes illustrate the time it takes to execute the code inside the task.
To illustrate at what point in time such a task can be executed, we use a timeline.
represents the world clock time of the system execution.
it is placed on the timeline based on the decision of the scheduler.
we will refer to as a C-style function containing C code.
when a task is scheduled on the timeline.
and a delay in the end to determine the period of the task.
over again based on the loop structure.
which is an instance of a task.
And a new job is recreated every loop period.
We indicate a job with a letter J followed by two numbers.
The first number tells us which task the job belongs to, and in this case it is 1.
And the second number tells us which instance of a job is currently scheduled.
So here you see it goes from 1 to n.
which will release a new job every x time units.
can start to execute as the release time of the task.
which will request access to the CPU.
we must learn a couple of parameters.
The three most important ones are P, e, and D.
It is the time interval between releases of jobs in a task.
E is execution time of the task, simply speaking, how long the box is.
The longer the execution time, the more CPU resources are needed.
it cannot handle all the tasks within the time limit.
Speaking of time limit, the parameter D tells us the deadline of a task.
This tells us the latest time at which a task must be completed.
The deadline can either be defined as absolute or relative.
An absolute deadline means that the deadline is always relative to time 0.
A relative deadline, again, is with respect to the previous release of a job.
it is assumed that the relative deadline is equal to the task period.
Let's now have a look at how a system can be scheduled.
So we have a timeline without any tasks scheduled.
periodic tasks called T1, T2, and T3.
start to be generated immediately when the system starts up.
when a new job is generated from T1, the deadline is always 4 steps ahead.
it must wait until this time before jobs can start to be generated.
be completed 1 step before the next job is being released.
T3 is released at time 5, and it has a period of 5 and a deadline of 2.
when it has been released.
The execution time of the tasks are illustrated by the length of those boxes.
and T3 has an execution time of 0.5.
And we can start by looking at the release times of the tasks.
and T3 is released at time 5.
We also put out the deadlines for the first job in each task.
and D3 is at time 7.
So we can make an example schedule here with the first T1 jobs.
And the first job of T1 is clearly not violating the deadline, D1.
and its deadline is still okay.
it does not violate the deadline either.
So this here was an example of a successful schedule.
A periodic task is formally defined in the order period, execution time and deadline.
But we also have some more terms to define.
then the feasibility interval of the first job is between 0 and 4.
which is a time shift of the first job in the task.
then it must be released earliest at time 1.
And the phase is denoted with the Greek letter phi.
it can be expressed in a range from r- to r+.
is decreased, and you basically don't know when you can schedule a task exactly.
Let's define some periodic tasks.
We have T1, T2, and T3.
this algorithm is able to schedule all the jobs between 0 and 60.
So here we start with T1, T2, T3, T1, T2, T3, and so on.
this exact release pattern will start to repeat after time 60.
this will look exactly like a copy from time 0 to 60.
hyperperiod H, and is, in fact, the smallest multiple of all the periods.
keep all the deadlines forever and not only the first x time steps.
the least common multiple of the individual periods of all the tasks.
T2 with a period 4, and T3 with a period 10, then we calculate the following.
find the smallest common multiple.
For a task with a period of 3, we have 3,6,9,12, and so on.
For period 4, we have 4,8,12,16, and so on.
And for 10 we have 10,20,30, and so on.
functionality in a real time system.
periodic tasks are usually defined by period, execution time, and deadline.
Thus a task must never violate a deadline.


Patio sliding doors have always been a popular choice amongst consumers, and it's easy to see why. With their smooth sliding operating system, opening and closing of the door panes is effortless, while its stylish looks stay looking good for longer.
'Choices' UPVC patio doors look great and are compatible with all of our ranges. They slide open and close effortlessly, they are available as two, three or four panels and featuring toughened safety glass as standard.
Available in a range of colours including Light Oak, Mahogany or Cherry woodgrains, the stainless steel lock hooks and keeps guarantees a high grade locking system. In addition, anti-lift blocks help prevent the door from being lifted out of its frame by working as a secure unit.


I stumbled across their album entitled "Into The Black", and I was hooked immediately. I could not find myself to skip past a single song. Each track changed it up with a different sound to enjoy, while still giving that feeling that hooks you in the first place. The sound this band delivers makes it hard to believe that they are a new group. Music is what I am most critical about, but I cannot find anything to complain about here, and I am sure that this is one local band that will not remain local for long. Staunton could soon be a household band name.
You can find more about Staunton on their Facebook Page by clicking "HERE", or check out some music at http://staunton.band/.


SELLER CAN CARRY! *GREAT INVESTOR OPPORTUNITY!* If you have been searching high and low for the perfect starter home for your family that has no HOA and lots of space, look no further! This clean move in ready home is perfectly located on an oversized corner lot with RV Gate! You'll feel right at home in this cozy 2 bedroom, 2 full bathroom home with upgraded tile throughout, and neutral colored paint. Each bedroom has its own Walk-In Closet plus a ''Bonus'' room in the garage that is ideal for additional storage. Don't let this dream pass you by – schedule your private showing today!


On this episode of HOT ROD Garage, Tony Angelo and Lucky Costa revive one of the most popular HRG builds ever, the Crown Hick. This do-everything shop truck is really a 2007 Crown Victoria Police Interceptor dressed up as a 1966 F-100. The Hick has been out of commission since it dominated the Roadkill truck shootout, leaving it battered and broken. The guys set out to get the truck back on the road, and make it better than ever. A Gearstar 4R75W replaces the trashed cop gearbox, a set of ADTR.net adjustable coilover suspension enhances the handling, and a bunch of fabricated bracing makes the whole truck more stable! At the end of the episode, the Crown Hick is king once again!


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Q: Why is $\lvert x:\lvert T(g)(x)\rvert>\alpha \rvert\le\alpha^{-p}\lvert\lvert T(g)\rvert\rvert^p_{L^p}$
Why is $\lvert x:\lvert T(g)(x)\rvert>\alpha \rvert\le\alpha^{-p}\lvert\lvert T(g)\rvert\rvert^p_{L^p}$

Is this a consequence of Chebyshev inequality, Does it not matter how $T$ is defined (here $T$ is an operator given by concolution with some function), Is it always true that $\lvert\lvert T(g)\rvert\rvert=\sup\limits_{\lvert x\rvert=1}\frac{\lvert T(g)(x)\rvert}{\lvert x \rvert}=\sup\limits_{\lvert x\rvert=1}\lvert T(g)(x)\rvert$
or should it be $\lvert g(x)\rvert$ instead of $\lvert x \rvert$ in the denominator ?
What if $\lvert g(x)\rvert=1$ is never reached ? Is it $0$ then ?

A: I'm not sure what $T$ is from the context of your post, but for a measurable function $f$ we have that
$$
\vert \{ \vert f \vert > \alpha\} \vert = \int_{\{\vert f \vert > \alpha\}} 1 \le \int \frac{\vert f\vert^p}{\alpha^p}
$$
for all $p < \infty$.

redpajama_set_name: RedPajamaStackExchange


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10% of Facebook's users are not human - including Mark Zuckerberg's dog Beast who has 1.5 MILLION followers
Facebook currently has more than one billion 'monthly active users'
At least 100 million of these users are pets, brands and companies
Biggest growth in membership is in India, Brazil, Russia and Middle East
Much slower growth in Western Europe and North America
Half of the population of the UK and the U.S. are Facebook members
Investors fear that young people are losing interest in the site
By Emma Innes
Dogs, cats, horses, toasters: You name it, it has a Facebook page.
Facebook now has one billion users - but ten per cent of them are not human, according to new research.
At least 100 million of Facebook's 'monthly active users' are pets, brands and companies - and even Facebook founder Mark Zuckeberg's dog Beast has a page.
Mark Zuckerberg with his dog Beast (who has his own page on the social network) at Facebook HQ. At least 100 million of Facebook's 'monthly active users' are pets, brands and companies, it has been revealed
According to eMarketer, an internet market research company, only 889.3 million of the one billion users are real people.
Quartz reports that Facebook claims it currently has 1.11 billion users but says that it will not actually reach this number of human users until 2014.
Currently the biggest growth in Facebook use is in India, Brazil, Russia, the Middle East and Africa.
There are 50 BILLION apps for that: Apple's App Store hits... Which ear are you? Researchers find how the brain decides...
Each of these regions is expected to see a 30 per cent increase in Facebook members this year.
In contrast, the U.S., UK and Western Europe are seeing a much slower growth in Facebook membership.
FACEBOOK INVESTOR FEARS
Facebook's investors are concerned teenagers are starting to lose interest in the site.
They worry people are developing 'Facebook fatigue' so are moving to sites such as Twitter instead.
The Financial Times reports that investors fear young people no longer think Facebook is 'cool' and that they are deserting it as their parents join the site.
The concerns are based on figures which show that 40 per cent of young Facebook users in the U.S. say that they expect to use the site less this year than they have in the past.
It is believed that the biggest challenge facing the social media site is gaining a foothold in the mobile market.
However, currently half of the population of the U.S. and the UK actively use Facebook but many of the users are less active than they were in the past.
In Western Europe as a whole only 37 per cent of people actively use the social networking site because people in France, Italy and Germany are less enthusiastic.
eMarketer also predicts that Facebook use will grow from 42.6 per cent of internet users now to 54.7 per cent in 2017.
While 46.6 per cent of people in North America and more than 30 per cent of people in Latin America and Western Europe will use Facebook at least once a month this year, the Middle East, Africa and Asia-Pacific will have much lower penetration rates, which brings down the worldwide percentage of the population using Facebook to 15.4 per cent.
eMarketer estimates that 63 per cent of social network users worldwide will access Facebook at least once a month in 2013.
Facebook founder Mark Zuckerberg's dog, Beast, has its own page on the site - with 1.5m followers
In recent years, it has been common for users in countries such as China, Japan, South Korea and Russia to use local social networking sites as opposed to Facebook, which gives the site room for growth.
By 2017, Facebook users will account for 69.5 per cent of social network users worldwide.
Facebook is about to celebrate the end of its first year as a public company but there are concerns among investors that teenagers are losing interest in it.
Facebook now has one billion users but ten per cent of them are not human - only 889.3 million of the one billion users are real people
Some investors are worried that young people are developing 'Facebook fatigue' which is causing them to move to other social networking sites such as Twitter.
The Financial Times reports that investors believe teenagers no longer see Facebook as 'cool' and that the arrival of their parents on the site is causing them to defect.
Ten per cent of Facebook users are not human including Mark Zuckerberg's dog Beast

redpajama_set_name: RedPajamaCommonCrawl


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General Mills recalls some 5-pound bags of flour over salmonella fears
General Mills is recalling some five-pound bags of its Gold Medal unbleached flour over fears of salmonella contamination.
NEW YORK — General Mills is recalling some five-pound bags of its Gold Medal unbleached flour over fears of salmonella contamination.
"This recall is being issued out of an abundance of care as General Mills has not received any direct consumer reports of confirmed illnesses related to this product," the company said in a statement.
The recall is for Gold Medal unbleached flour with a better-if-used-by date of April 20, 2020. The company said those who have the affected product should dispose it.
"Food safety is our top priority, and though we have not had any confirmed illnesses, we are voluntarily recalling this specific lot of Gold Medal unbleached flour to prevent potential illnesses," said Jim Murphy, president of General Mills Meals and Baking Division.
Salmonella causes serious and sometimes fatal infections, especially in young children, the elderly and people with weakened immune systems.
Symptoms include fever, nausea, diarrhea and abdominal pain.


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Home ED Originals Picture Story Victorian Era Fashion: Then And Now
Victorian Era Fashion: Then And Now
The Victorian Era may be long over, but the influence of Victorian Era fashion can still be seen in some of the trends followed across the world.
Mayank Dixit
Our clothes define who we are. But who or what defines those same clothes.
Would you be surprised if I told you that majority of our dressing styles and trends are inspired by the Victorian Era fashion?
This was the same time period when finally the cloth manufacturing process started shifting towards power looms and sewing machines appeared on the scene.
Victorian fashion usually comprises of the series of fashion changes introduced during the reign of Queen Victoria, a period lasting from 1837 to 1901. The changes introduced for men and women were almost introduced around the same time.
CHANGES IN DRESSING STYLE OF MEN AND WOMEN
The radical changes in women's clothing were more centered around celebrating their slim waists and how to make them appear more lovely. These included loads of things such as corsets, petticoats, boots, lace etc.
On the other hand, men's fashion was centered around costumes that made them look formal, sober and elegant at their workplace and otherwise. Men's clothing usually involved wearing long coats, hats, vests, and blazers.
Boots– A comparison between the Victorian era boot design and the boots worn in Modern times. The design of the shoelaces has more or less remained the same over time.
Hair- The hairstyle of women in Victorian times involved curls and braids, whose reflection can be seen even know.
Victorian Jewellery– The Victorian Jewellery designs still act as a source of inspiration for modern day rings and necklaces.
ALSO READ: Bizarre Kind Of Phobias You Never Knew Existed
Skirt- The Victorian skirt was worn in combination with the corset. Modern-day Victorian skirt designs are somewhat similar to those worn in the 19th century, sans corsets.
Men's Hats– The hats worn by men in the 19th century Britain have not seen major design changes in today's time. Although still in circulation, they are not fairly popular.
Vests- Modern-era vests have a less complicated design than those worn in Victorian times. But they are still worn as an intricate part of a 3 piece suit.
Coats- The long coats of the 19th century were slowly replaced by waist-length coats and blazers. But the design of the coats has not seen drastic changes.
Our clothes derive heavy inspiration from Victorian Era fashion. The clothes reflected the societal setup of that time, which was fairly formal and involved class.
The Victorian vibe can still be felt even now. The details of clothing of that time combined with the comfort of today's time form a perfect combination, and this is the reason for Victorian items still being popular.
Image Credits- Google
Sources: Wikipedia, Historic UK
YOU'D ALSO LIKE TO READ:
http://edtimes.in/icse-passing-marks-lowered-35-mean-students/
19th century fashion style of britain
Victorian clothes
victorian fashion inspiration
victorian fashion now
victorian fashion today
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Ayesha thakur November 30, 2017 At 7:28 PM
Really great article for a fashionista like me :)
Mayank Dixit November 30, 2017 At 11:03 PM
Thanks a lot:)
Terry March 17, 2019 At 8:11 AM
I'm pretty sure that the skirt example (actually a dress) is not from the Victorian era but from the 1700's. Also, women wore petticoats for hundreds of years before the Victorian era. Corsets were worn quite a long time before the Victorian era, also.
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Know Dixie Hollins HS Class of 1996 graduates that are NOT on this List? Help us Update the 1996 Class List by adding missing names.
More 1996 alumni from Dixie Hollins HS have posted profiles on Classmates.com®. Click here to register for free at Classmates.com® and view other 1996 alumni.
The Dixie Hollins High class of '96 alumni are listed below. These are former students from Dixie Hollins High in St Petersburg, FL who graduated in 1996. Alumni listings below have either been searched for or they registered as members of this directory.


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Home Posts tagged "cash"
Tag: cash
Edgewater's $13 million civic center, partly funded by cannabis cash, opens Sunday
Stoner November 17, 2018 November 17, 2018 Marijuana News Cannabis, cash, Center", Civic, Edgewater's, Funded, million, Opens, partly, Sunday	0
EDGEWATER — This weekend, thanks to its bustling marijuana trade, this tiny city on the western shore of Sloan's Lake will forge a new civic identity with the opening of a $ 13 million municipal complex complete with library, police station, fitness center and city offices. Without $ 3 million…
With marijuana banking legislation stalled, cannabusinesses awash with cash, vulnerable to violent crime
Stoner February 10, 2018 February 10, 2018 Marijuana News awash, Banking, Cannabusinesses, cash, Crime, Legislation, Marijuana, stalled, Violent', vulnerable	0
Published: Feb 10, 2018, 2:05 pm • Updated: Feb 10, 2018, 2:05 pm By Stuart Leavenworth, McClatchy Washington Bureau WASHINGTON — Stung by robberies in California, Colorado, Washington and other states, the cannabis industry is pressing Congress to change federal banking laws so that its retailers no longer have to…
Cops seized $22 million in marijuana cash from a Miami hydroponics shop owner. He won't lose it all
Stoner February 9, 2018 February 9, 2018 Marijuana News cash, Cops, from, Hydroponics, lose, Marijuana, Miami, million, owner, seized, shop, Won't	0
Published: Feb 9, 2018, 1:51 pm • Updated: Feb 9, 2018, 1:52 pm By David Ovalle, Miami Herald MIAMI — The Miami Lakes man found with $ 22 million in suspected marijuana cash stuffed in orange buckets inside his house pleaded guilty Wednesday — but he won't be losing all…
Elderly California Couple Arrested Again With Cannabis "Gifts" and Duffel Bag Full of Cash
Stoner January 11, 2018 January 11, 2018 Marijuana News Again, arrested, California, Cannabis, cash, couple, duffel, Elderly, Full, Gifts	0
Elderly California Couple Arrested Again With Cannabis "Gifts" and Duffel Bag Full of Cash The same elderly couple from California arrested last month for transporting 60 pounds of cannabis which they claimed to be given away as Christmas gifts were caught again in Nebraska. However, this time they were suspected…
A Maryland bank is quietly solving the marijuana industry's cash problem
Stoner January 3, 2018 January 3, 2018 Marijuana News bank, cash, Industry's, Marijuana, Maryland, problem, Quietly, solving	0
Published: Jan 3, 2018, 9:33 am • Updated: Jan 3, 2018, 9:35 am By Aaron Gregg, The Washington Post Most banks refuse to open accounts for cannabis-related businesses even in states where pot is legal, citing federal laws that outlaw the drug and consider it on par with cocaine and…
Six people pose as cops to rob a marijuana grow of 500 plants, cash and a truck
Stoner November 10, 2017 November 10, 2017 Marijuana News cash, Cops, Grow, Marijuana, People, plants, Pose, truck	0
Published: Nov 10, 2017, 8:50 am • Updated: Nov 10, 2017, 8:50 am By Tribune News Service, The Cannabist Staff One of at least six people who allegedly posed as law enforcement to tie up and rob victims at a Calaveras County marijuana grow last month was arrested after a…
California proposes coalition of marijuana states — and armored cars for tax cash
Stoner November 7, 2017 November 7, 2017 Marijuana News armored, California, car's, cash, coalition, Marijuana, Proposes, States	0
Published: Nov 7, 2017, 11:49 am • Updated: Nov 7, 2017, 1:02 pm By Michael R. Blood, The Associated Press LOS ANGELES (AP) — California should use armored cars to transport hundreds of millions of dollars in cash tax payments expected next year with the state's legal marijuana market, the…
Can Nevada casinos cash in on cannabis industry without breaking federal law?
Stoner September 21, 2017 September 21, 2017 Marijuana News Breaking, Cannabis, cash, Casinos, Federal, Industry, Nevada, Without	0
Published: Sep 21, 2017, 7:59 am • Updated: Sep 21, 2017, 8:36 am By Regina Garcia Cano, The Associated Press LAS VEGAS — Nevada Gov. Brian Sandoval on Wednesday ordered a committee to make policy recommendations on the interactions between the state's long-established gambling industry and a budding, yet exceedingly…
Jeff Sessions wants police to seize more cash from Americans
Stoner July 19, 2017 July 19, 2017 Marijuana News Americans, cash, from, Jeff, more, Police, Seize, Sessions, Wants	0
Published: Jul 18, 2017, 8:19 am • Updated: Jul 18, 2017, 6:26 pm By Christopher Ingraham, The Washington Post Attorney General Jeff Sessions on Monday said he'd be issuing a new directive this week aimed at increasing police seizures of cash and property. "We hope to issue this week a…
U.S. Postal Service refuses to deliver box of tax cash from Alaskan pot shop
Stoner April 29, 2017 April 29, 2017 Marijuana News Alaskan, cash, deliver, from, Postal, Refuses, service, shop, U.S.	0
Published: Apr 27, 2017, 8:59 am • Updated: Apr 27, 2017, 10:24 am By The Associated Press JUNEAU, Alaska — An Alaska marijuana retailer's tax payment has been sent back to the business after the U.S. Postal Service refused to ship the money. Rainforest Farms had recently tried to send…


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HomeArtists
Ialreadyforgotwhoiam's track "The Thrill" is dope level hypnotizing
TMV 0 June 7, 2021 8:58 pm
Ialreadyforgotwhoiam's ingenuity is revealed in his hit track "The Thrill". Two minutes into this track, you will be lost in another world of a "feel-good reality" created by the artist.
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Hello music earthlings, I am back, and it is so good to be back. Have you listened to the tracks of Ialreadyforgotwhoiam? If you have not, then, I came back to do you a favor. When you have listened to his songs especially his latest albums', you will have a strong compulsion to thank me. I have been out of this world since I started listening to Ialreadyforgotwhoiam's songs particularly The Thrill. So, therefore allow me to guide you through this newfound melodious journey.
Firstly, you must agree with me that that uniqueness is class and if you doubt, Michael Jackson was unique, his music is class, Daft Punk was unique and their music is class. Ialreadyforgotwhoiam is an artist who has honed his musical skills, dynamism, and talent to join the "class" group. He has released two albums so far and the latest of them is titled "Life is Such a Bother". Amongst 11 tracks, "The Thrill" is the opening track of the album "Life is Such a Bother".
The Thrill
The Thrill is absolutely dope! What a way to start an album whose title is drowned in so much melancholy. Honestly, Ialreadyforgotwhoiam is not just an enigma but a cunning and crafty artist who just wants to blow your mind out of the depths of this world. The album title says "Life is Such a Bother" but The Thrill leaves you with the option of whether to be happy or sad. While you are still contemplating what to feel in the song, 2minutes in, you are already lost in another world of a "feel-good reality". Trust me! I have been there before, you will be forced to put The Thrill on repeat, thanks to Ialreadyforgotwhoiam ingenuity.
The track starts with a soft piano syncopation after which it was overshadowed by a melodious distorted synth, deep bass, and intense kick. It also features fragments of motor racing before veering off into a delightful lo-fi ambiance of tranquility. The rhythmic sequence of synth beat continues soon after the melodious episode of car racing sounds and tranquil beats. Overall, The Thrill is a synth track, it could be classified into the music genres of synth-pop, electronic dance music, and electro-pop. Click here to listen to the song
Simply put, Ialreadyforgotwhoiam is a sheet of the book of Daft Punk. His range of music is so wide and just like Daft Punk, he is not seeking personal fame or glory, his profile and music stage name speaks volumes of that. Ialreadyforgotwhoiam just wants to do good music and so far his music has attracted thousands of listeners, which says a lot for someone who keeps a low profile personality. Ialreadyforgotwhoiam has released two albums so far. His first album was released in 2020 and is titled The Rainy Morning.
The latest album from Ialreadyforgotwhoiam which has been buzzing since its release in 2021 is titled "Life is such a Bother". The album comprises 11 dope tracks of different music varieties. Although the album title may sound sad and depressing, the message the artist wishes to portray is for earthlings to take a break and see another reality from this album.
Give us your feedback on the song, album, or artist.
Artists 577 EDM 15 EDM pop 5 electronic pop 8 synth pop 2
MadLeo's new track "Reminisce" is on fire
Sofia Evangelina' "Endure", exudes great pleasure


Below are sites that I have found that provide information on the lastest news regarding the publishing industry. I will add other sites as I find theme and have made a review of the information furnished.
A site to get up-to-date publishing news and articles on the publishing industry.


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Kiszka szwedzka – kiełbasa ziemniaczana, tradycyjne danie charakterystyczne dla okolic Wałcza.

Kiszka szwedzka wpisana została na Listę produktów tradycyjnych Ministerstwa Rolnictwa i Rozwoju Wsi 6 marca 2012. Wolno ją wytwarzać wyłącznie na terenie gminy Wałcz.

Historia wsi Szwecja sięga roku 1590, kiedy to teren zaludniono osadnikami z Pomorza (20 pańszczyźnianych chłopów narodowości niemieckiej). Głównym składnikiem diety ich potomków były, po rozpowszechnieniu się tej rośliny w Europie, ziemniaki. We wsi funkcjonowała duża gorzelnia, przerabiająca ten surowiec, który stał się również podstawą tutejszego wyżywienia. Po II wojnie światowej tradycja przetrwała (gorzelnia nadal funkcjonowała). Tradycję przejęli osadnicy z innych części Polski.

Pierwotnie, przed wojną, kiszkę wytwarzano ze startych, surowych ziemniaków, kaszy jęczmiennej, soli, ryżu, cynamonu i majeranku. Obecnie wykorzystuje się do jej wytwarzania ziemniaki surowe i gotowane, surowy oraz wędzony boczek, cebulę, kaszę mannę, jajka i przyprawy. Całością nadziewa się jelita i parzy około 10 minut. W czerwcu każdego roku odbywa się w Szwecji festyn Świętojanki, na którym istnieje możliwość degustacji kiszki. Kiszkę spożywać można zarówno na ciepło, jak i na zimno

Zobacz też
 kiszka ziemniaczana – tradycyjna potrawa kuchni białoruskiej i kuchni podlaskiej
 dzionie rakowskie – tradycyjna potrawa świętokrzyska

Przypisy

Kuchnia pomorska
Kuchnia polska
Kuchnia niemiecka
Polskie produkty tradycyjne

Potrawy z ziemniaków

redpajama_set_name: RedPajamaWikipedia


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BlizzCon 2016: Allen Adham, co-founder of Blizzard Entertainment, rejoins company
The co-founder came out of a 12-year retirement.
David Craddock
During this morning's BlizzCon 2016 keynote, Blizzard president and CEO Mike Morhaime announced that Allen Adham, his fellow co-founder, has come out of retirement to rejoin the company.
Allen Adham was the driving force behind the founding of Blizzard in 1991. A lifelong gamer, he got his start when a mutual friend, Interplay founder Brian Fargo, gave Allen's then-nascent Blizzard company contracts to develop ports of console games.
When the Blizzard staff got hooked on Westwood's Dune II RTS, Adham assigned programmer Patrick Wyatt the task of breaking ground on WarCraft, Blizzard's take on the genre. The team grew from there, and WarCraft: Orcs and Humans launched in 1994.
Adham maintained a strong leadership role at the company after it was acquired by Davidson & Associates, an "edutainment" corporation known for creating the Math Blaster series of games on Apple II and IBM PCs.
Particulars of Adham's involvement are not known as of press time. He was president before his departure, the position occupied by his co-founder and college buddy, Mike Morhaime.
Long Reads Editor
David L. Craddock writes fiction, nonfiction, and grocery lists. He is the author of the Stay Awhile and Listen series, and the Gairden Chronicles series of fantasy novels for young adults. Outside of writing, he enjoys playing Mario, Zelda, and Dark Souls games, and will be happy to discuss at length the myriad reasons why Dark Souls 2 is the best in the series. Follow him online at davidlcraddock.com and @davidlcraddock.
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David Craddock posted a new article, BlizzCon 2016: Allen Adham, co-founder of Blizzard Entertainment, rejoins company


[You can read my first review here] How do I keep myself away from the yummy diner goodness of Kaye's Kitchen? I simply can't. Even though I was going to eat a big meal later that day, I snagged whatever time I had to see Duc to get breakfast at Kaye's. No regrets.
I only got a side of hash browns since I knew that I would be eating something big later. I swear - they have perfected the art of making hash browns. Crispy on the outside and soft (but not too soft) on the inside, the hash browns are superb and not too badly priced either. Duc's order of biscuits and gravy was pretty good as well but not the best I've had. Then again, I've been to the South where biscuits and gravy are a staple.
The French toast was divine however. I could smell the cinnamon sweetness when it was served. We didn't even need syrup because all the flavors were well-packed into the pieces of toast. Talking about hitting the breakfast spot! Another meal well-done Kaye's. I shall be making you my regular for any breakfasts out.
Quips & Anecdotes - Happy Birthday Blog!


It's demolition derby time with this Disney.Pixar Cars 3 edition of the Trouble game! Lightning McQueen has been challenged by the Crazy 8 Racers to compete in a demolition derby at Thunder Hollow Speedway. He needs a great team to help him go full throttle around the muddy track to win. Players bump opponents back to the pits, move through speed zones, get stuck in pileups, and get help by rolling Miss Fritter and having her in their corner. The first player to get all 4 of their cars home wins! Game includes the iconic Pop-O-Matic dice roller, and features a die with pictures that can change a player's move.
Includes gameboard, Pop-O-Matic dice roller, 16 car pawns (four of each color), Miss Fritter figure, and instructions.
" WARNING: CHOKING HAZARD-Small parts. Not for children under 3 years.


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AMD's Ryzen 7000 mobile CPUs feature up to 16 cores and 5.4GHz speeds
It's going to be a great year for powerful laptops.
Devindra Hardawar
·Senior Editor
January 4, 2023, 10:30 p.m. ·3 min read
AMD is bringing out some big guns to take on Intel's powerful HX laptop CPUs, and once again, they're also called "HX." The company's new Ryzen 9 7945HX processor is its most premium mobile offering, with 16 cores and 32 threads, as well as speeds between 2.5GHz and 5.4GHz. It's joined by an array of other Ryzen 7000 CPUs unveiled at CES (AMD also announced low-end chipsin September), which will power everything from budget laptops to extreme gaming and creator machines. The big takeaway this year? There will be a Ryzen laptop chip for practically every need.
It makes sense for AMD to focus even more on its HX chips this year. Intel unveiled its high-powered lineup last year, and this week it revealed that there will be even more HX processors in 2023, led by the 24-core i9-13980HX. Like its previous chips, Intel is using a hybrid design which combines Performance and Efficient cores. So while Intel's core count is higher, its chips may end up performing similarly to AMD's, which utilize full-speed cores across the board.
AMD Ryzen 7000 chips
The Ryzen HX CPUs will run at 55-watts and above (also like Intel's), meaning they'll focus more on raw power than battery life. AMD claims the 7945HX is 78 percent faster than the previous top-end Ryzen 6900HX in Cinebench's multithreaded benchmark, and 18 percent faster in the single threaded test. Compared to Intel's 12900HX, its high-end chip from last year, the 7945HX is 169 percent faster in Handbrake encoding and 75 percent faster with Blender rendering. You can expect to see the 7945HX in some of the premiere gaming laptops this year, like the Alienware m16 and m18, as well as the latest ASUS ROG Strix.
For a better balance between size and power, there's also the new Ryzen 7040 HS series chips, running between 35W and 45W. AMD says these chips are aimed at thin enthusiasts, and you can expect to see them in laptops this March. The Ryzen 9 7940HS is the pinnacle of this lineup, featuring 8 cores, 16 threads and up to 5.2 GHz boost speeds.
The 7040 chips will also feature Ryzen AI, an integrated AI engine that can handle tasks like making background blurs during video calls, or sorting through your photo library. AMD says Ryzen AI can multitask up to 4 different AI workloads (or combine those capabilities on one task), and it's reportedly 50 percent more power efficient than Apple's M2. Intel is also planning to introduce a VPU AI accelerator in some of its 13th-gen chips — previously, we've only seen these accelerators on ARM chips, like the Microsoft SQ3 on the Surface Pro 9.
When it comes to true ultraportables, AMD has the U-series Ryzen 7030 and 7035 lineups, which range from 15W to 35W chips. Those include the Ryzen 3 7330U at the low end, featuring 6 cores and 12 threads, to the 8 core/16 thread Ryzen 7 7730U. (Curiously, Ryzen 3 7335U is just a quad-core chip, but it can use almost twice as much power as the 7330U.)
ChatGPT Could Spell an "AI Revolution" for These Canadian Stocks
ChatGPT indicates a breakthrough in AI that could bolster stocks like Shopify (TSX:SHOP). The post ChatGPT Could Spell an "AI Revolution" for These Canadian Stocks appeared first on The Motley Fool Canada.
Short phone video played in court could put Alex Murdaugh at scene of 2 murders
The video found on Paul's cellphone could be clearest evidence placing his father, Alex Murdaugh, at the scene of the murders shortly before his wife and son were killed, despite his claims to have been somewhere else at the time.
Samsung Galaxy S23 Ultra hands-on: A 200MP camera is the biggest update
The first flagship phone of 2023 is here.


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