## Consistent Collaborative Filtering via Tensor Decomposition

### Shiwen Zhao, Guillermo Sapiro

Changes Since Last Submission: We revised the last sentence of Section 4.3 by adding more explanation to what values $\tau_i$ can take: > In SAD we estimate the value of $T$ from data. As discussed in Section 1, we encourage the values of entries in $T$ to be 1 unless there > is strong evidence from the data suggesting otherwise. This is achieved by adding an $l_1$ ​regularization centered around 1 to the > entries in $T$ independently. We rewrote the paragraph that introduced (8) to explain how $\tau$s were defined > SAD extends the original equations (2) and (3) by introducing a new non-negative vector $\tau_i$ for every item (a column in $T$ in Equation (7)). We can re-write Equation (6) as follows for every item pair $i$ and $j$: We made changes to the revised manuscript by adding a paragraph above Section 4.4 on Pg 6. > The $l_1$ regularization has another side effect. In Equation 8, multiplying any constant $c$ to $\eta_{hi}$ > and $1/c$ to $\tau_{hi}$ results in the same objective function, causing $H$ and $T$ unidentifiable. > The additional $l_1$ regularization around 1 mitigates the identifiability problem by discouraging any > constant multiplication that moves $\tau_{hj}$ away from 1, making the joint objective function identifiable > between $H$ and $T$. We added more explanation in the revised manuscript on Pg 10 when describing the two methods for recommendation in real data application. > SAD faces some unconventional challenges (hence opportunities) in producing a ranking when violation of transitivity exists. Consider three items $I$, $j$ and $t$. With transitivity, if user has $i \succ j$ and $j \succ t$, then $I \succ t$ must hold. However, SAD can result in a scenario in which $t \succ i$, forming a preference cycle among the three items, in which case no ranking can be inferred. We propose two methods using pairwise comparisons for SAD in the evaluation. We also added Figure 1, Algorithm 1 and missing captions in existing figures and table in revised manuscript. More details can be found in our responses to reviewers. All the changes in the revised manuscript are highlighted by yellow color. > further remarks We added revisions to reflect our responses to further remarks from reviewer Kngd. The revisions are highlighted in green. > Decision by Action Editors We made more revisions based on Action Editors final comments, by discussing the experiments we conducted, the two datasets referenced, and future extensions (in discussion section).