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.
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Acknowledgement
We thank the reviewers for their valuable suggestions. This work is supported by Natural Science Foundation of China (Grant Nos. 61532011, 61672311) and National Key Basic Research Program (2015CB358700).
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Appendix
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:
then there exists no randomization function \(f\in \mathcal {F}\) with the product-level monotonic property of:
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:
and this can be equivalently written as:
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:
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:
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:
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).
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Xiao, L., Min, Z., Yongfeng, Z. (2017). Joint Factorizational Topic Models for Cross-City Recommendation. In: Chen, L., Jensen, C., Shahabi, C., Yang, X., Lian, X. (eds) Web and Big Data. APWeb-WAIM 2017. Lecture Notes in Computer Science(), vol 10366. Springer, Cham. https://doi.org/10.1007/978-3-319-63579-8_45
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