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HHMF: hidden hierarchical matrix factorization for recommender systems

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Abstract

Matrix factorization (MF) is one of the most powerful techniques used in recommender systems. MF models the (user, item) interactions behind historical explicit or implicit ratings. Standard MF does not capture the hierarchical structural correlations, such as publisher and advertiser in advertisement recommender systems, or the taxonomy (e.g., tracks, albums, artists, genres) in music recommender systems. There are a few hierarchical MF approaches, but they require the hierarchical structures to be known beforehand. In this paper, we propose a Hidden Hierarchical Matrix Factorization (HHMF) technique, which learns the hidden hierarchical structure from the user-item rating records. HHMF does not require the prior knowledge of hierarchical structure; hence, as opposed to existing hierarchical MF methods, HHMF can be applied when this information is either explicit or implicit. According to our extensive experiments, HHMF outperforms existing methods, demonstrating that the discovery of latent hierarchical structures indeed improves the quality of recommendation.

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Notes

  1. http://www.amazon.com.

  2. http://www.netflix.com.

  3. http://www.yelp.com.

  4. http://www.tripadvisor.com.

  5. http://www.goodreads.com.

  6. http://www.netflixprize.com.

  7. Users in a trust network can indicate whether he/she thinks one review with a rating is ‘useful’. One trust statement forms an edge with the trust rating as edge weight between two users. Epinions (http://www.epinions.com) is an exemplified recommender systems based on trust network.

  8. \(Q^{(l+1)}=P^{(l+1)}=0\).

  9. http://www.librec.net/datasets.html.

  10. http://www.kaggle.com/c/yelp-recsys-2013.

  11. http://lihui.info/data/dianping.html.

  12. https://grouplens.org/datasets/movielens.

  13. https://www.netflixprize.com.

  14. https://github.com/yushuai/FISM.

  15. http://www.public.asu.edu/~swang187/codes/HSR.zip.

  16. https://github.com/RuidongZ/Deep_Matrix_Factorization_Models.

  17. https://github.com/hexiangnan/neural_collaborative_filtering.

  18. https://github.com/hexiangnan/neural_factorization_machine.

  19. https://software.intel.com/en-us/mkl.

  20. https://www.geforce.com/hardware/technology/cuda.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 61803249) and the Shanghai Sailing Program (Grant No. 18YF1407700).

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Authors and Affiliations

  1. Fujian Key Laboratory of Sensing and Computing for Smart City, School of Information Science and Engineering, Xiamen University, Xiamen, Fujian, China

    Hui Li

  2. Department of Computer Science, The University of Hong Kong, Pok Fu Lam, Hong Kong

    Yu Liu, Yuqiu Qian & David W. Cheung

  3. Department of Computer Science and Engineering, University of Ioannina, Ioannina, Epirus, Greece

    Nikos Mamoulis

  4. School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai, China

    Wenting Tu

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  1. Hui Li
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  2. Yu Liu
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  3. Yuqiu Qian
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  4. Nikos Mamoulis
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  5. Wenting Tu
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Appendix

Appendix

1.1 Calculation of Eq. 3

$$\begin{aligned} \begin{aligned}&Pr(Q, P, \theta , \mu , \phi , \kappa | R, \varXi )\\&\quad = Pr(Q, P | R, \varXi )\cdot Pr(\theta , \mu , \phi , \kappa | R, \varXi )\\&\quad = Pr(Q, P | R, \varXi )\cdot Pr(\theta , \mu , \phi , \kappa | R)\\&\quad = Pr(Q, P | R, \varXi )\cdot Pr(\theta , \kappa | R)\cdot Pr(\mu , \phi | R)\\&\quad = \frac{Pr(Q, P | R, \varXi )\cdot Pr(R | \theta , \kappa ) \cdot Pr(R | \mu , \phi ) \cdot Pr(\theta , \kappa )\cdot Pr(\mu , \phi )}{Pr(R)\cdot Pr(R)}\\&\quad \propto Pr(Q, P | R, \varXi )\cdot Pr(R | \theta , \kappa ) \cdot Pr(R | \mu , \phi ). \end{aligned} \end{aligned}$$

1.2 Calculation of Eq. 7

$$\begin{aligned}&Pr\left( \left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{\left( \ell \right) }\right\} ^{l}_{\ell =1}\,|\,\left\{ R^{(\ell )}\right\} _{\ell =1}^{l}, \left\{ S^{(\ell )}\right\} _{\ell }^{l-1}, \left\{ V^{(\ell )}\right\} _{\ell =1}^{l-1}, \varXi \right) \\&\quad =\, \frac{Pr\left( \left\{ R^{(\ell )}\right\} _{\ell =1}^{l}, \left\{ S^{(\ell )}\right\} _{\ell }^{l-1}, \left\{ V^{(\ell )}\right\} _{\ell =1}^{l-1}, \varXi \,|\,\left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}\right) \cdot \Pr \left( \left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}\right) }{Pr(\left\{ R^{(\ell )}\right\} _{\ell =1}^{l}, \left\{ S^{(\ell )}\right\} _{\ell }^{l-1}, \left\{ V^{(\ell )}\right\} _{\ell =1}^{l-1}, \varXi )} \\&\quad \propto \, Pr\left( \left\{ R^{(\ell )}\right\} _{\ell =1}^{l}\,|\,\left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}, \sigma _R^2\right) \\&\qquad \cdot Pr\left( \left\{ S^{(\ell -1)}\right\} _{\ell =1}^{l}\,|\,\left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}, \sigma _S^2\right) \\&Pr\left( \left\{ V^{(\ell -1)}\right\} _{\ell =1}^{l}\,|\,\left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}, \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}, \sigma _V^2\right) \\&\qquad \cdot \Pr \left( \left\{ Q^{(\ell )}\right\} ^{l}_{\ell =1}\right) \cdot \Pr \left( \left\{ P^{(\ell )}\right\} ^{l}_{\ell =1}\right) \\&\quad =\, \prod _{\ell =1}^{l}\prod _{r_{ui}\in R^{(\ell )}}{\mathcal {N}}\left( \langle Q_{u}^{(\ell )},P_{i}^{(\ell )}\rangle , \sigma _{R^{(\ell )}}^2\right) \cdot \prod _{\ell =1}^{l-1}\prod _{s_{ui}\in S^{(\ell )}}{\mathcal {N}}\left( \langle Q_{u}^{(\ell +1)},P_{i}^{(\ell )}\rangle , \sigma _{S^{(\ell )}}^2\right) \\&\qquad \prod _{\ell =1}^{l-1}\prod _{v_{ui}\in V^{(\ell )}}{\mathcal {N}}\left( \langle Q_{i}^{(\ell )},P_{j}^{(\ell +1)}\rangle , \sigma _{V^{(\ell )}}^2\right) \cdot \prod _{\ell =1}^{l}\prod _{u=1}^{m^{(\ell )}}\phi ^{(\ell +1)}_{e_u}{\mathcal {N}}\left( Q^{(\ell +1)},\sigma _{Q^{(\ell +1)}}^2\mathrm {I}\right) \\&\qquad \prod _{\ell =1}^{l}\prod _{i=1}^{n^{(\ell )}}\kappa ^{(\ell +1)}_{t_i}{\mathcal {N}}\left( P^{(\ell +1)},\sigma _{P^{(\ell +1)}}^2\mathrm {I}\right) . \end{aligned}$$

1.3 Gradients used in Eq. 9

$$\begin{aligned} \frac{\partial {\mathcal {L}}_{R^{(\ell )}}}{Q_{u}^{(\ell )}}= & {} \nabla _q\left( R^{(\ell )}_{u\cdot },\ell ,\ell \right) -{\mathcal {R}}\left( Q^{(\ell )}_u\right) \end{aligned}$$
(16)
$$\begin{aligned} \frac{\partial {\mathcal {L}}_{R^{(\ell )}}}{P_{i}^{(\ell )}}= & {} \nabla _p\left( R^{(\ell )}_{\cdot i},\ell ,\ell \right) -{\mathcal {R}}\left( P^{(\ell )}_i\right) \end{aligned}$$
(17)
$$\begin{aligned} \frac{\partial {\mathcal {L}}_{S^{(\ell -1)}}}{Q_{u}^{(\ell )}}= & {} \nabla _q\left( S^{(\ell -1)}_{u\cdot },\ell ,\ell -1\right) -{\mathcal {R}}\left( Q^{(\ell )}_u\right) \end{aligned}$$
(18)
$$\begin{aligned} \frac{\partial {\mathcal {L}}_{S^{(\ell -1)}}}{P_{i}^{(\ell -1)}}= & {} \nabla _p\left( S^{(\ell -1)}_{\cdot i},\ell ,\ell -1\right) -{\mathcal {R}}\left( P^{(\ell -1)}_i\right) \end{aligned}$$
(19)
$$\begin{aligned} \frac{\partial {\mathcal {L}}_{V^{(\ell -1)}}}{Q_{u}^{(\ell -1)}}= & {} \nabla _q\left( V^{(\ell -1)}_{u \cdot },\ell -1,\ell \right) -{\mathcal {R}}\left( Q^{(\ell -1)}_u\right) \end{aligned}$$
(20)
$$\begin{aligned} \frac{\partial {\mathcal {L}}_{V^{(\ell -1)}}}{P_{i}^{(\ell )}}= & {} \nabla _p\left( V^{(\ell -1)}_{\cdot i},\ell -1,\ell \right) -{\mathcal {R}}\left( P^{(\ell )}_i\right) \end{aligned}$$
(21)

where

$$\begin{aligned} \nabla _q(R^{(\ell )}_{u\cdot },\ell _1,\ell _2)= & {} \frac{1}{\sigma _R^2}\sum _{r_{ui}\in R^{(\ell )}}(r_{ui}^{(\ell )}-\langle Q_{u}^{(\ell _1)},P_{i}^{(\ell _2)}\rangle )\cdot P_i^{(\ell )} \\ \nabla _p(R^{(\ell )}_{\cdot i},\ell _1,\ell _2)= & {} \frac{1}{\sigma _R^2}\sum _{r_{ui}\in R^{(\ell )}}(r_{ui}^{(\ell )}-\langle Q_{u}^{(\ell _1)},P_{i}^{(\ell _2)}\rangle )\cdot Q_u^{(\ell )} \\ \nabla _{q}(S_{u\cdot }^{(\ell -1)}, \ell , \ell -1)= & {} \frac{1}{\sigma _S^{2}}\sum _{r_{ui}\in R^{(\ell -1)}}\Big (s_{ui}^{(\ell -1)}-\langle Q_{u}^{(\ell )}, P_{i}^{(\ell -1)}\rangle \Big )\cdot P_{i}^{(\ell -1)} \\ \nabla _{p}(S_{\cdot i}^{(\ell -1)}, \ell , \ell -1)= & {} \frac{1}{\sigma _S^{2}}\sum _{r_{ui}\in R^{(\ell )}}\Big (s_{ui}^{(\ell -1)}-\langle Q_{u}^{(\ell )}, P_{i}^{(\ell -1)}\rangle \Big )\cdot Q_{u}^{(\ell )} \\ \nabla _{q}(V_{u\cdot }^{(\ell -1)}, \ell -1, \ell )= & {} \frac{1}{\sigma _V^{2}}\sum _{r_{ui}\in R^{(\ell )}}\Big (v_{ui}^{(\ell -1)}-\langle Q_{u}^{(\ell -1)}, P_{i}^{(\ell )}\rangle \Big )\cdot P_{i}^{(\ell )} \\ \nabla _{p}(V_{\cdot i}^{(\ell -1)}, \ell -1, \ell )= & {} \frac{1}{\sigma _V^{2}}\sum _{r_{ui}\in R^{(\ell -1)}}\Big (v_{ui}^{(\ell -1)}-\langle Q_{u}^{(\ell -1)}, P_{i}^{(\ell )}\rangle \Big )\cdot Q_{u}^{(\ell -1)} \\ {\mathcal {R}}(Q_{u}^{(\ell )})= & {} \frac{1}{\sigma _Q^2}\sum _{r_{ui}\in R^{(\ell )}}\phi _{(e_u,u)}^{(\ell +1)}\Big [\,| WQ _{u}^{(\ell )}|(Q_u^{(\ell )}-{\bar{Q}}^{(\ell -1)}_{ WQ _u^{(\ell )}})+(Q_{u}^{(\ell )}\\&-Q^{(\ell +1)}_{ FQ _u^{(\ell )}})\,\Big ] \\ {\mathcal {R}}(P_{i}^{(\ell )})= & {} \frac{1}{\sigma _P^2}\sum _{r_{ui}\in R^{(\ell )}}\kappa _{(t_i,i)}^{(\ell +1)}\Big [\,| WP _{i}^{(\ell )}|(P_i^{(\ell )}-{\bar{P}}^{(\ell -1)}_{ WP _i^{(\ell )}})+(P_{i}^{(\ell )}\\&-P^{(\ell +1)}_{ FP _i^{(\ell )}})\,\Big ] \end{aligned}$$

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Li, H., Liu, Y., Qian, Y. et al. HHMF: hidden hierarchical matrix factorization for recommender systems. Data Min Knowl Disc 33, 1548–1582 (2019). https://doi.org/10.1007/s10618-019-00632-4

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