Joint Factorizational Topic Models for Cross-City Recommendation

Abstract

The research of personalized recommendation techniques today has mostly parted into two mainstream directions, namely, the factorization-based approaches and topic models. Practically, they aim to benefit from the numerical ratings and textual reviews, correspondingly, which compose two major information sources in various real-world systems, including Amazon, Yelp, eBay, Netflix, and many others.

However, although the two approaches are supposed to be correlated for their same goal of accurate recommendation, there still lacks a clear theoretical understanding of how their objective functions can be mathematically bridged to leverage the numerical ratings and textual reviews collectively, and why such a bridge is intuitively reasonable to match up their learning procedures for the rating prediction and top-N recommendation tasks, respectively.

In this work, we exposit with mathematical analysis that, the vector-level randomization functions to harmonize the optimization objectives of factorizational and topic models unfortunately do not exist at all, although they are usually pre-assumed and intuitively designed in the literature.

Fortunately, we also point out that one can simply avoid the seeking of such a randomization function by optimizing a Joint Factorizational Topic (JFT) model directly. We further apply our JFT model to the cross-city Point of Interest (POI) recommendation tasks for performance validation, which is an extremely difficult task for its inherent cold-start nature. Experimental results on real-world datasets verified the appealing performance of our approach against previous methods with pre-assumed randomization functions in terms of both rating prediction and top-N recommendation tasks.

Appendix

Let $\gamma$ denote arbitrary vectors with length K , and $\mathcal {F}$ is the set of all randomization functions $f:\mathbb {R}^K\rightarrow \mathbb {R}^K$ satisfying:

$$\begin{aligned} \left\{ \begin{aligned}&0 \le f(\gamma )_{i} \le 1, \Vert f(\gamma )\Vert _1 = 1\\&\gamma _{i}< \gamma _{j} \rightarrow f(\gamma )_{i} < f(\gamma )_{j} \end{aligned} \right. ,\forall \gamma \in \mathbb {R}^{K},1\le i,j \le K \end{aligned}$$

(20)

then there exists no randomization function $f\in \mathcal {F}$ with the product-level monotonic property of:

$$\begin{aligned} \gamma _{1}\cdot \gamma _{2}<\gamma _{3}\cdot \gamma _{4}\rightarrow f(\gamma _{1})\cdot f(\gamma _{2})<f(\gamma _{3})\cdot f(\gamma _{4}),~\forall \gamma _1,\gamma _2,\gamma _3,\gamma _4 \end{aligned}$$

(21)

Proof: Suppose there exists a randomization function $f\in \mathcal {F}$ that meets Eq. (21). Let $t>1$, and let $\alpha$ and $\beta$ be vectors with $\alpha \cdot \beta >0$, then we have $t\alpha \cdot \beta >\alpha \cdot \beta$. By applying the property of product-level monotonic in Eq. (21) we have:

$$\begin{aligned} f(t\alpha )\cdot f(\beta )>f(\alpha )\cdot f(\beta ) \end{aligned}$$

(22)

and this can be equivalently written as:

$$\begin{aligned} \big (f(t\alpha )-f(\alpha )\big )\cdot f(\beta )>0 \end{aligned}$$

(23)

Let $\varDelta \doteq f(t\alpha ) - f(\alpha )$, and according to the definition of randomization function in Eq. (20), we know that $\sum _{k}f(t\alpha )_{k}= \sum _{k}f(\alpha )_{k} = 1$, thus we have:

$$\begin{aligned} \sum \nolimits _{k}\varDelta _{k} = \sum \nolimits _{k}\big (f(t\alpha ) - f(\alpha )\big )_{k} = 0 \end{aligned}$$

(24)

According to Eq. (23) we know that $\varDelta \ne \mathbf 0$. Let $\mathcal {P}$ denote the indices of all positive elements in vector $\varDelta$, and $\mathcal {N}$ denote the indices of negative elements. We have:

$$\begin{aligned} \sum \nolimits _{k\in \mathcal {P}}\varDelta _{k}+\sum \nolimits _{k\in \mathcal {N}}\varDelta _{k}=0 \end{aligned}$$

(25)

As Eq. (23) holds for any $\beta$ with $\alpha \cdot \beta >0$, without loss of generally, let $\beta$ be a vector where $\beta _{k\in \mathcal {P}}=0$ and $\beta _{k\in \mathcal {N}}=1$. According to the vector-level monotonic property in Eq. (20) and the fact that $0<1$, we have $f(\beta )_{k\in \mathcal {P}}<f(\beta )_{k\in \mathcal {N}}$ and $0\le f(\beta )_{k\in \mathcal {P}\cup \mathcal {N}}\le 1$. Combined with Eq. (25), we further obtain the following:

$$\begin{aligned} \varDelta \cdot f(\beta )=\sum _{k\in \mathcal {P}}\varDelta _{k}f(\beta )_k+\sum _{k\in \mathcal {N}}\varDelta _{k}f(\beta )_k<0 \end{aligned}$$

(26)

which is a direct contradiction with Eq. (23). As a result, there exists no randomization function $f\in \mathcal {F}$ that satisfies the product-level monotonic property in Eq. (21).