Ordinal Non-negative Matrix Factorization for Recommendation Olivier Gouvert, Thomas Oberlin, Cédric Févotte
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Olivier Gouvert, Thomas Oberlin, Cédric Févotte. Ordinal Non-negative Matrix Factorization for Recommendation. International Conference on Machine Learning (ICML), 2020, Vienna (virtual), Austria. hal-03049397
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Olivier Gouvert 1 Thomas Oberlin 2 Cédric Févotte 1
Abstract 2011) which aims to predict future interactions between users and items in order to make recommendations. For We introduce a new non-negative matrix factor- this purpose, PF is often applied to a binarized version of ization (NMF) method for ordinal data, called the data, i.e., Y ∈ {0, 1}U×I , containing only the infor- OrdNMF. Ordinal data are categorical data which mation that a user is interacting with an item or not. A exhibit a natural ordering between the categories. variant of PF, called Bernoulli-Poisson factorization (Be- In particular, they can be found in recommender PoF) (Acharya et al., 2015), has been proposed to explicitly systems, either with explicit data (such as ratings) model binary data. However, for both PF and BePoF, the or implicit data (such as quantized play counts). binarization stage induces a loss of information, since the OrdNMF is a probabilistic latent factor model that value associated to an interaction is removed. Although generalizes Bernoulli-Poisson factorization (Be- several attempts in the literature tried to directly model raw PoF) and Poisson factorization (PF) applied to data, both for explicit (Hernandez-Lobato et al., 2014) and binarized data. Contrary to these methods, Ord- implicit data (Basbug & Engelhardt, 2016; Zhou, 2017; NMF circumvents binarization and can exploit a Gouvert et al., 2019), this remains a challenging problem. more informative representation of the data. We design an efficient variational algorithm based on In an attempt to keep as much information as possible, we a suitable model augmentation and related to vari- propose in this paper to consider ordinal rather than binary ational PF. In particular, our algorithm preserves data. Ordinal data (Stevens, 1946) are nominal/categorical the scalability of PF and can be applied to huge data which exhibit a natural ordering (for example: cold sparse datasets. We report recommendation ex- ≺ warm ≺ hot). This type of data is encountered in rec- periments on explicit and implicit datasets, and ommender systems with explicit data such as ratings. It show that OrdNMF outperforms BePoF and PF can also be created by quantizing implicit data such as play applied to binarized data. counts. Such a pre-processing remains softer than binariza- tion and stays closer to the raw data, as soon as the number of classes is chosen big enough. In this paper, without loss 1. Introduction of generality, we will work with ordinal data belonging to {0,...,V }. Note that for this type of data, the notion of Collaborative filtering (CF) is a popular recommendation distance between the different classes is not defined. For technique based only on the feedbacks of users on items. example, this implies that the mean is not adapted to these These feedbacks can be stored into a matrix Y of size data, unlike the median. U × I, where U and I are the number of users and items respectively. Matrix factorization (MF) methods (Hu et al., There are two naive ways to process ordinal data. The first 2008; Koren et al., 2009; Ma et al., 2011) aim to approxi- one consists in applying classification methods. The scale mate the feedback matrix Y by a low-rank structure WHT of the ordering relation which links the different categories U×K is then ignored. The second way considers these data as where W ∈ R+ corresponds to user preferences and I×K real values in order to apply regression models. By doing H ∈ R to item attributes. + this, it artificially creates a distance between the different Poisson factorization (PF) (Canny, 2004; Cemgil, 2009; categories. These two naive methods do not fully consider Gopalan et al., 2015) is a non-negative matrix factorization the specificity of ordinal data, since they remove or add in- (NMF) model (Lee & Seung, 1999; 2001; Févotte & Idier, formation to the data. Threshold models (McCullagh, 1980; Verwaeren et al., 2012) are popular ordinal data processing 1IRIT, Université de Toulouse, CNRS, France 2ISAE- SUPAERO, Université de Toulouse, France. Correspondence to: methods that alleviate this issue. They assume that the data Olivier Gouvert
The rest of the paper is organized as follows. In Section2, yui = Gb(xui). (3) Ordinal Non-negative Matrix Factorization for Recommendation
• Probit function. A common choice for the additive noise 10 2 v = Gb(x) is εui ∼ N (0, σ ) (Chu & Ghahramani, 2005; Paquet et al., 8 2012; Hernandez-Lobato et al., 2014). In that case the link −1 function Fε is the probit function. Inference can be carried 6 out with an EM algorithm based on the latent variable xui. v 4 • Other choices like log-log or cauchit functions have also been considered (Agresti & Kateri, 2011). The sur- 2 vey (Ananth & Kleinbaum, 1997) recaps some of these 0 choices. 100 102 104 106 x 2.2. Bernoulli-Poisson Factorization (BePoF) Figure 2. Example of a quantization function x Gb(x). 7→ In this section, we present BePoF (Acharya et al., 2015) which is a variant of PF for binary data (not directly related The goal of MF models for ordinal data is therefore to jointly to the CLMs introduced above). It employs a model aug- infer the latent variables W and H as well as the sequence mentation trick for inference that we will use in our own of thresholds b. algorithm presented in Section4. The Poisson distribution can easily be “augmented” to fit Cumulative distribution function. The c.d.f. associated binary data yui ∈ {0, 1}. Indeed, it suffices to introduce a with the random variable yui in Eqs. (2)-(3) can be calcu- thresholding operation that binarizes the data. The corre- lated as follows: sponding generative hierarchical model is therefore given by: P[yui ≤ v|λui] = P[Gb(xui) ≤ v|λui] (4) n ∼ T , = P[λui + εui < bv] (5) ui Poisson([WH ]ui) (11) y 1 n > , = P[εui < bv − λui] (6) ui = [ ui 0] (12)
= Fε (bv − λui) . (7) where nui ∈ N is a latent variable and 1 is the indicator func- U×I tion. We denote by N ∈ N the matrix such that [N]ui = It follows that the function v 7→ P[yui ≤ v|λui] is increas- nui. This variable can easily be marginalized by noting that T ing since the sequence of thresholds is itself increasing. T −[WH ]ui P[yui = 0] = Poisson(0|[WH ]ui) = e . We Moreover, the probability mass function (p.m.f.) associated obtain: to the ordinal data can be written as: T −[WH ]ui yui ∼ Bern(1 − e ) (13) P[y = v|λ ] = P[y ≤ v|λ ] − P[y ≤ v − 1|λ ] ui ui ui ui ui ui where Bern refers to the Bernoulli distribution. The condi- = Fε (bv − λui) − Fε (bv−1 − λui) . (8) tional distribution of the latent variable nui is given by: ( δ , if y = 0, Some examples. If the c.d.f. is strictly increasing, we can 0 ui nui|yui ∼ T (14) rewrite Eq. (7) as: ZTP([WH ]ui), if yui = 1.
−1 where ZTP refers to the zero-truncated Poisson distribution Fε (P[yui ≤ v|λui]) = bv − λui. (9) and δ0 to the Dirac distribution located in 0. The latent Hence the name of CLM, since the factorization model is re- variable N can be useful to design Gibbs or variational lated to the c.d.f. of the ordinal data through a link function inference (VI) algorithms for binary PF (Acharya et al., −1 2015) and we will employ a similar trick in Section4. Fε : [0, 1] → R. Various choices of noise (equivalently, of link function F −1) have been considered in the literature. ε Remark. The generative model presented in Eq. (13) We present some of these choices in what follows. T is in the form yui ∼ Bern(G([WH ]ui)) where G : T • Logit function. The use of the logit function was first R (or R+) → [0, 1]. When [WH ]ui ∈ R, the function proposed in (Walker & Duncan, 1967). This model was pop- G can be the inverse of the probit (Consonni & Marin, ularized and renamed as "proportional odds model" by (Mc- 2007) or of the logit function for example. They are special Cullagh, 1980). The model can be rewritten as: cases of the model presented in Section 2.1 with V = 1. Mean-parametrized Bernoulli MF models have also been P[yui ≤ v|λui] logit P[yui ≤ v|λui] = log = bv − λui considered (Lumbreras et al., 2018). They correspond to P[yui > v|λui] G = Id and require additional constraints on the latent (10) T factors W and H in order to satisfy [WH ]ui ∈ [0, 1]. Ordinal Non-negative Matrix Factorization for Recommendation
3. Ordinal NMF (OrdNMF) 1.0 α = 0.5 1.0 α = 0.5 α = 1.0 α = 1.0 0.8 α = 2.0 0.8 α = 2.0 In this section, we introduce OrdNMF which is a NMF
) 0.6 ) 0.6 1 1 − − λ λ ( ( model pecially designed for ordinal data. A difference with ε ε F 0.4 F 0.4 Section 2.1 is that we impose that both matrices W and T 0.2 0.2 H are non-negative. Thus, we now have [WH ]ui ∈ R+ T 0.0 0.0 instead of [WH ] ∈ R. We denote λ = w h so 0 1 2 3 4 5 0 1 2 3 4 5 ui uik uk ik λ λ P T that λui = k λuik = [WH ]ui. −1 Figure 3. Functions λ Fε(λ ) for gamma (left) and inverse- 3.1. Quantization of the Non-negative Numbers gamma (right) noises. 7→ Our model works on the same principle as OrdMF (see Section 2.1) and seeks to quantize the non-negative real γ(α,x) R x α−1 −t by Fε(x) = Γ(α) where γ(α, x) = 0 t e dt is the line R+. For this, we introduce the increasing sequence lower incomplete gamma function. If α = 1, we recover an of thresholds b given by b−1 = 0 < b0 < ··· < bV −1 < exponential noise εui ∼ Exp(1) whose c.d.f. is Fε(x) = bV = +∞ (the thresholds are here non-negative). Moreover, 1 − e−x. we define the quantization function Gb : R+ → {0,...,V } 1 like in Eq. (1) but with support R . • Inverse-gamma (IG) noise: εui ∼ IG(α, 1). The + −1 c.d.f. is given by F (x) = Γ(α,x ) where Γ(α, x) = As compared to Section 2.1, we now assume a non-negative ε Γ(α) R ∞ tα−1e−t t is the upper incomplete gamma function. multiplicative noise on xui. This ensures the non-negativity x d −1/x of xui and it seems well suited for modeling over-dispersion, If α = 1, we obtain the c.d.f. Fε(x) = e . a common feature of recommendation data. Let εui be a non- • Any increasing function Fε : R+ → [0, 1] defines a non- negative random variable with c.d.f. Fε, we thus propose negative random variable which can be used in OrdNMF. the following generative model:
xui = λui · εui, (15) 3.2. OrdNMF with IG Noise (IG-OrdNMF) yui = Gb(xui). (16) In the rest of the paper, we focus on the special case where εui is a multiplicative IG noise with shape parameter α = 1, 2 Like before, our goal is to jointly infer the latent variables i.e., εui ∼ IG(1, 1) . We use the acronym IG-OrdNMF for W and H as well as the sequence of the thresholds b. In our this particular instance of OrdNMF. model, the c.d.f. associated to the ordinal random variable −1 For convenience we write θv = bv . The sequence θ corre- yui becomes: sponds to the inverse of the thresholds and is therefore de- P[yui ≤ v|λui] = P[Gb(xui) ≤ v|λui] (17) creasing, i.e., θ−1 = +∞ > θ0 > ··· > θV −1 > θV = 0.
= P[λui · εui < bv] (18) Moreover, we denote by ∆ the positive sequence of decre- ments defined by ∆v = θv−1 − θv for v ∈ {1,...,V }. We bv PV = P εui < (19) have θv = l=v+1 ∆l and, in particular, θV −1 = ∆V . λui bv = Fε . (20) Interpretation. In IG-OrdNMF model, the c.d.f. associ- λui ated with an ordinal data yui is given by:
−λuiθv Therefore, we can deduce that the p.m.f. is given by: P[yui ≤ v|λui] = e , (23)
−λuiθv P[yui = v|λui] or P[yui > v|λui] = 1 − e , (24) = P[yui ≤ v|λui] − P[yui ≤ v − 1|λui] (21) with v ∈ {0,...,V }. Therefore, BePoF (see Section 2.2) is bv bv−1 a particular case of IG-OrdNMF with V = 1 and θ = 1. = F − F . (22) 0 ε λ ε λ ui ui This formulation allows for a new interpretation of IG- OrdNMF. As a matter of fact, the event {y > v} is a Various functions F can be used which determine the exact ui ε binary random variable which follows a Bernoulli distribu- nature of the multiplicative noise. Figure3 displays the tion: {y > v} ∼ Bern(1−e−λuiθv ). Then, we can see IG- function λ 7→ F (λ−1) for the examples considered next. ui ε OrdNMF as the aggregation of V dependent BePoF models 1 • Gamma noise: εui ∼ Gamma(α, 1). The c.d.f. is given for different thresholds of binarization v ∈ {0,...,V − 1}.
1The rate parameter β is fixed to 1 because of a scale invariance 2The expectation of a IG variable is not defined for α 1, ≤ with λui. however the model is still well-defined. Ordinal Non-negative Matrix Factorization for Recommendation
Probability mass function. The p.m.f. of an observation Table 1. Variational distributions for IG-OrdNMF. is given by:
−λ θ Var. Distribution e ui 0 , for v = 0, −λuiθv −λuiθv−1 ˜ P[yui = v|λui] = e − e , for 1 ≤ v < V, C q(cui|nui) = Mult cui; nui, φui 1 − e−λuiθV −1 , for v = V. ( δ0, if yui = 0 (25) N q(nui) = ZTP(nui;Λui∆yui ), if yui > 0 W ˜W Then, the log-likelihood of λui can be written as: W q(wuk) = Gamma wuk;α ˜uk, βuk H H ( H q(hik) = Gamma hik;α ˜ , β˜ −λuiθ0, if v = 0, ik ik log P[yui = v|λui] = −λui∆v −λuiθv + log(1 − e ), else.
(26) λuik vector with entries . Therefore, for ordinal data yui ∈ λui {1,...,V }, we obtain the following joint log-likelihood: This expression brings up a linear term in λui and a non- linear term of the form x 7→ log(1 − e−x), similar to the log p(yui, nui, cui|λui) = −λuiθy −1 (29) function used in Section 2.2. ui X + nui log ∆y + (cuik log λuik − log cuik!) , Moreover, the expectation of the observations is well- ui k defined and given by : ∗ X s.t. nui ∈ N and nui = cuik. V −1 k X −λuiθv E(yui|λui) = V − e . (27) v=0 Joint log-likelihood of IG-OrdNMF. We denote by Z = Note that, in the context of ordinal data processing, the {N, C, W, H} the set of latent variables of the augmented expectation is not a good statistic since it implicitly implies model. Moreover, we define Tv such that: a notion of distance between classes. However, this quantity ( θ0, if v = 0, will be useful to build lists of recommendations. Indeed, the Tv = (30) θv−1, if v > 0. function λui 7→ E(yui|λui) is increasing. Thus, the higher the λ = [WHT ] , the higher (in expectation) the level ui ui The joint log-likelihood of IG-OrdNMF is therefore given of interaction between the user and the item. by: X h 4. Bayesian Inference log p(Y, N, C|W, H) = nui log ∆yui ui We impose a gamma prior on the entries of both ma- X i W W + (cuik log λuik − log cuik!) − λuiTy . (31) trices W and H, i.e., wuk ∼ Gamma(α , βu ) and ui H H k hik ∼ Gamma(α , βi ). Gamma prior is known to induce sparsity which is a desirable property in NMF methods. It is important to note that nui = 0 and cui = 0K when 4.1. Augmented Model yui = 0. Consequently, the variables N and C are partially observed and the inference take advantage of the sparsity of As described in Section 3.2 , the log-likelihood for ordinal the observed matrix Y. data such that v ∈ {1,...,V } brings up a non-linear term −x log(1 − e ) which is not conjugate with the gamma dis- 4.2. Variational Inference tribution, making the inference complicated. To solve this issue, we use the trick presented in Section 2.2 by augment- The posterior distribution p(Z|Y) is intractable. We use ing our model with the latent variable: VI to approximate this distribution by a simpler variational distribution q. Here, we assume that q belongs to the mean- ( δ0, if yui = 0, field family and can be written in the following factorized nui|yui, λui ∼ (28) form: ZTP(λui∆yui ), if yui > 0. Y Y Y q(Z) = q(n , c ) q(w ) q(h ). (32) Moreover, as commonly done in the PF setting (Cemgil, ui ui uk ik ui uk ik 2009; Gopalan et al., 2015), we augment our model with the latent variable cui|nui, λui ∼ Mult(nui, φui), where Note that the variables N and C remains coupled. We use Mult is the multinomial distribution and φui is a probability a coordinate-ascent VI (CAVI) algorithm to optimize the Ordinal Non-negative Matrix Factorization for Recommendation parameters of q. The variational distributions are described Algorithm 1 CAVI for IG-OrdNMF. in Table1. The associated update rules are summarized in Data: Matrix Y Algorithm1. Result: Variational distribution q and thresholds θ 1 Initialization of variational parameters and thresholds θ; repeat Approximation and link with PF. Algorithm1 can be 2 foreach couple (u, i) such as yui > 0 do 3 simplified by assuming that q(nui) = δ1 if yui > 0. This Λuik = exp (Eq(log wuk) + Eq(log hik)); −x P amounts to replacing the non-linear term log(1 − e ) by Λui = k Λuik; Λui∆yui log x in Eq. (26). However, this approximation will produce E (n ) = −Λ ∆ ; q ui ui yui −x 1−e similar results only if x is very small, since log(1 − e ) = Λuik Eq(cuik) = Eq(nui) ; log x+o(x). In practice, this can only be verified a posteriori Λui 4 end by observing that Eq(nui) ≈ 1. 5 foreach user u ∈ {1,...,U} do W W P As mentioned above, BePoF is a special case of IG-OrdNMF 6 α˜uk = α + i Eq(cuik); W W for V = 1 and θ0 = 1. Thus, we can notice that PF algo- ˜ P βuk = βu + i Tyui Eq(hik); rithm applied to binary data is an approximation of BePoF 7 end algorithm for q(nui) = δ1 if yui = 1. 8 foreach item i ∈ {1,...,I} do H H P 9 α˜ik = α + u Eq(cuik); 4.3. Thresholds Estimation ˜H H P βik = βi + u Tyui Eq(wuk); A key element of threshold models is the learning of thresh- 10 end olds (corresponding here to θ parameters). For this, we 11 Update of thresholds: Eq. (35) and Eq. (36); W H use a VBEM algorithm. It aims to maximize the term Update of rate parameters βu and βi ; q, θ Eq(log p(Y, Z; θ)), w.r.t. the variables θ, which is given Calculate ELBO( ); by: 12 until ELBO converge;
Eq(log p(Y, Z; θ)) = (33) Thus, we obtain the following update rules: h i X P 1 y l E n Eq(nui) log ∆yui − Eq(λui)Tyui + cst, ui [ ui = ] q( ui) ∆l = P , ∀l ∈ {1,...,V }, (35) ui ui 1[yui ≤ l]Eq(λui) s.t. θ > θ > ··· > θ > θ = 0. V 0 1 V −1 V X θv = ∆l, ∀v ∈ {0,...,V − 1}. (36) l=v+1 Note that both terms Tv (defined in Eq. (30)) and ∆v = θv−1 − θv > 0 depend on the sequence θ. Algorithm1 scales with the number of non-zero values in the observation matrix Y. The complexity of OrdNMF is of the same order of magnitude as BePoF and PF. The Decrements optimization. We choose to work on the only difference with these algorithms in terms of computa- decrement sequence ∆ rather than on the threshold se- tional complexity is the update of the thresholds (Line 11 of quence θ. Indeed, by doing so, the decreasing constraint Alg.1). of θ becomes a non-negativity constraint of ∆. Moreover, we obtain only terms in x and log x in the function to be 4.4. Posterior Predictive Expectation. maximized. Thus, the problem can be solved analytically. The posterior predictive expectation E(Y∗|Y) corresponds We can rewrite the term Tv w.r.t. the sequence ∆ by noting to the expectation of the distribution of new observations PV ∗ that: Tv = l=1 1[v ≤ l]∆l, ∀v ∈ {0,...,V }. Therefore, Y given previously observed data Y. This quantity allows the optimization problem presented in Eq. (33) amounts to us to create the list of recommendations for each user. We maximizing the following function: can approximate it by using the variational distribution q: Z E(Y∗|Y) ≈ E(Y∗|W, H)q(W)q(H)dWdH. V X X h W,H Eq(log p(Y, Z; ∆)) = 1[yui = l]Eq(nui) log ∆l (37) ui l=1 i Unfortunately, this expression is not tractable. But for rec- − 1[y ≤ l]E (λ )∆ + cst, s.t. ∆ ≥ 0. (34) ui q ui l ommendation we are only interested in ordering items w.r.t. Ordinal Non-negative Matrix Factorization for Recommendation
Table 2. Recommendation performance of OrdNMF using the MovieLens dataset. Bold: best NDCG score. R: raw data. B: binarized data.
NDCG @100 with threshold s Model Data K s = 1 s = 4 s = 6 s = 8 s = 10 OrdNMF R 150 0.444 0.444 0.439 0.414 0.353 BePoF B ( 1) 50 0.433 0.430 0.421 0.383 0.310 PF B (≥ 1) 100 0.431 0.428 0.418 0.380 0.306 BePoF B (≥ 8) 50 0.389 0.393 0.399 0.408 0.369 PF B (≥ 8) 150 0.386 0.389 0.395 0.403 0.365 ≥
Table 3. Recommendation performance of OrdNMF using the Taste Profile dataset. Bold: best NDCG or log-likelihood score. Q: quantized data. B: binarized data. R: raw data.
NDCG @100 with threshold s Model Data K s = 1 s = 3 s = 6 s = 11 s = 21 s = 51 log-lik OrdNMF Q 250 0.213 0.174 0.153 0.135 0.123 0.117 2.8 105 dcPF R 150 0.209 0.173 0.154 0.137 0.128 0.121 − 3.0 · 105 BePoF B ( 1) 250 0.210 0.170 0.149 0.131 0.120 0.115− N/A· PF B (≥ 1) 250 0.206 0.167 0.146 0.129 0.118 0.115 N/A ≥
∗ this quantity. The function λui 7→ E(yui|λui) being in- Be careful not to confuse the predefined quantization used to creasing, we can use instead of Eq. (37) the simpler score obtain ordinal data, with the quantization of the latent vari- T sui = [Eq(W)Eq(H) ]ui. able in OrdNMF model which is estimated during inference. Although we expect OrdNMF to recover a relevant scaling 5. Experimental Results between the categories, there is no reason to get the same quantization function that was used for pre-processing. 5.1. Experimental Set Up
Datasets. We report experimental results for two datasets Evaluation. Each dataset is split into a train set Ytrain and described below. a test set Ytest: the train set contains 80% of the non-zero • MovieLens (Harper & Konstan, 2015). This dataset con- values of the original dataset Y, the other values are set to tains the ratings of users on movies on a scale from 1 to the class 0; the test set contains the remaining 20%. All 10. These explicit feedbacks correspond to ordinal data. We the compared methods are trained on the train set and then consider that the class 0 corresponds to the absence of a evaluated on the test set. rating for a couple user-movie. The histogram of the ordinal First, we evaluate the recommendations with a ranking met- data is represented in blue on Figure4. We pre-process a ric. For each user, we propose a list of m = 100 items subset of the data as in (Liang et al., 2016), keeping only (movies or songs) ordered w.r.t. the prediction score pre- users and movies that have more than 20 interactions. We T sented in Section 4.4: sui = [Eq(W)Eq(H )]ui. The qual- obtain U = 20k users and I = 12k movies. ity of these lists is then measured through the NDCG met- • Taste Profile (Bertin-Mahieux et al., 2011). This dataset, ric (Järvelin & Kekäläinen, 2002). The NDCG rewards provided by the Echo Nest, contains the play counts of users relevant items placed at the top of the list more strongly on a catalog of songs. As mentioned in the introduction, we than those placed at the end. We use the relevance definition choose to quantize these counts on a predefined scale in or- proposed in (Gouvert et al., 2019): der to obtain ordinal data. We arbitrarily select the following quantization thresholds: [1, 2, 5, 10, 20, 50, 100, 200, 500]. rel(u, i) = 1[ytest ≥ s]. (38) For example, the class labeled 6 corresponds to a listening ui counts between 21 and 50. As for MovieLens, the class 0 corresponds to users who have not listen to a song. The his- In other words, an item is considered as relevant if it belongs togram of the ordinal data are displayed in blue on Figure4. at least to the class s in the test set. The NDCG metric is We pre-process a subset of the data as before and obtain between 0 and 1, the higher the better. U k users and I k songs. = 16 = 12 Moreover, for the Taste Profile dataset, we calculate the log-likelihood of the non-zero entries on the test set, as it is Ordinal Non-negative Matrix Factorization for Recommendation
106 107 Truth (0.48) Truth (2.47) OrdNMF (0.48) 2.49 105 OrdNMF ( ) dcPF (0.47)
104 106
103
102 105 Occurence number Occurence number
101
100 104 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Class Class
Figure 4. PPC of the distribution of the classes in the Taste Profile dataset (left) and MovieLens dataset (right). The blue bars (Truth) represents the histogram of the classes in the train set. The colored bars represent the simulated histograms obtained from the different inferred OrdNMF or dcPF models. The percentages of non-zero values are written in parentheses. done in (Basbug & Engelhardt, 2016): on the NDCG scores. BePoF with data thresholded at 1 (≥ 1) perform well on small NDCG threshold s but has X test test ˆ ˆ LNZ = log p(yui |yui > 0, W, H), (39) poor performance after. On the contray, with data thresh- (u,i)∈Testset olded at 8 (≥ 8), BePoF achieves best performances for NDCG s = 10 but poor performances for small s. OrdNMF where ytest is the quantized data, wˆ = E[w ] and hˆ = ui uk uk ik does not exhibit such differences between NDCG scores E[h ] are the estimated latent factors. ik and benefits from the additional information brought by the ordinal classes. Nevertheless, we can note a small decrease Compared methods. We compare OrdNMF with three of the performance with s = 10 which is the hardest class other models: PF, BePoF and discrete compound PF to predict. (dcPF) (Basbug & Engelhardt, 2016) with a logarithmic ele- ment distribution as implemented in (Gouvert et al., 2019). Table3 displays the same kind of results for the Taste Profile Each model is applied either to raw data (R), quantized data dataset. Again, OrdNMF outperforms BePoF and PF which (Q) or binarized data (B). For the MovieLens dataset, two exploit less data information. OrdNMF is competitive with different binarizations are tested: one with a threshold at 1 dcPF which gives the best results for the highest thresholds (≥ 1) and one with a threshold at 8 (≥ 8). For the Taste Pro- s. However, OrdNMF presents a higher log-likelihood score file dataset, dcPF is applied to the count data (R) whereas than dcPF. Thus, OrdNMF seems better suited to predict the OrdNMF is applied to the quantized data (Q). feedback class of a user than dcPF. This observation is con- firmed by the posterior predictive checks (PPC) presented For all models, we select the shape hyperparameters below. αW = αH = 0.3 among {0.1, 0.3, 1} (Gopalan et al., 2015). The number of latent factors is chosen among 5.3. Posterior Predictive Check (PPC) K ∈ {25, 50, 100, 150, 200, 350} for the best NDCG score with threshold s = 8 for the MovieLens dataset, and s = 1 A PPC consists of generating new data based on for the Taste Profile dataset. All the algorithms are run 5 the posterior predictive distribution p(Y∗, W, H|Y) ≈ times with random initializations and are stopped when the p(Y∗|W, H)q(W)q(H), and then compare the structure relative increment of the expected lower bound (ELBO) falls of the original data Y with the artificial data Y∗. Here, under τ = 10−5. The computer used for these experiments we focus on the distribution of the ordinal categories. Fig- was a MacBook Pro with an Intel Core i5 processor (2,9 ure4 presents the results of these PPCs for the Taste Profile GHz) and 16 Go RAM. All the Python codes are available dataset. The blue bars correspond to the empirical histogram on https://github.com/Oligou/OrdNMF. of the data (Ytrain), the red and orange bars correspond to the histograms of the simulated data obtained with OrdNMF 5.2. Prediction Results and dcPF respectively. While dcPF fails to model the very large values present in the data (from the class 7, which cor- Table2 displays the results for the MovieLens dataset. First, responds to values greater than 50 plays), OrdNMF seems we can compare BePoF with its approximation, i.e., PF ap- to precisely describe all the ordinal categories. This is also plied to binarized data. BePoF is slightly better than PF for the case on the MovieLens dataset too. Even if the empirical both binarizations, and requires less latent factors. Then, we histogram is here less regular, OrdNMF can adapt itself to observe that the choice of the binarization has a big impact Ordinal Non-negative Matrix Factorization for Recommendation all type of data through the inferred thresholds b. Chu, W. and Ghahramani, Z. Gaussian processes for ordinal regression. The Journal of Machine Learning Research, 6. Conclusion pp. 1019–1041, 2005. We developed a new probabilistic NMF framework to pro- Consonni, G. and Marin, J.-M. Mean-field variational ap- cess ordinal data. In particular, we presented IG-OrdNMF proximate Bayesian inference for latent variable models. which is an extension of BePoF and conducted experiments Computational Statistics & Data Analysis, pp. 790–798, on two different datasets. We show the ability of OrdNMF 2007. to process different kinds of ordinal data both explicit and Févotte, C. and Idier, J. Algorithms for nonnegative matrix implicit. This work opens up several exciting perspectives. factorization with the beta-divergence. Neural computa- As we described in Section 3.1, OrdNMF can be used for tion, pp. 2421–2456, 2011. different choices of multiplicative noise. It would be in- teresting to develop OrdNMF for the exponential noise in Gopalan, P., Hofman, J. M., and Blei, D. M. Scalable rec- a similar way than IG-OrdNMF. Finally, when applied to ommendation with hierarchical Poisson factorization. In implicit data, it would be of particular interest to learn the Proc. Conference on Uncertainty in Artificial Intelligence pre-processing during the factorization, in order to automat- (UAI), pp. 326–335, 2015. ically tune the level of pre-processing adapted to a given Gouvert, O., Oberlin, T., and Févotte, C. Recommendation dataset. This is yet left for future investigations. from raw data with adaptive compound Poisson factor- ization. In Proc. 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