On the Effectiveness of Low-rank Approximations for Collaborative Filtering compared to Neural Networks

TL;DR

This paper compares low-rank matrix factorization and deep neural networks for collaborative filtering, finding that low-rank methods outperform neural networks due to better suitability for the task.

Contribution

The study clarifies why low-rank approximations are more effective than neural networks in pure collaborative filtering tasks without additional features.

Findings

Low-rank methods outperform neural networks in collaborative filtering.

Deep neural networks' universal approximation impairs latent vector determination.

Low-rank approximations are better suited for pure collaborative filtering tasks.

Abstract

Even in times of deep learning, low-rank approximations by factorizing a matrix into user and item latent factors continue to be a method of choice for collaborative filtering tasks due to their great performance. While deep learning based approaches excel in hybrid recommender tasks where additional features for items, users or even context are available, their flexibility seems to rather impair the performance compared to low-rank approximations for pure collaborative filtering tasks where no additional features are used. Recent works propose hybrid models combining low-rank approximations and traditional deep neural architectures with promising results but fail to explain why neural networks alone are unsuitable for this task. In this work, we revisit the model and intuition behind low-rank approximation to point out its suitability for collaborative filtering tasks. In several experiments we compare the performance and behavior of models based on a deep neural network and low-rank approximation to examine the reasons for the low effectiveness of traditional deep neural networks. We conclude that the universal approximation capabilities of traditional deep neural networks severely impair the determination of suitable latent vectors, leading to a worse performance compared to low-rank approximations.

Tables

Table 1. Table 1. Correlation statistics for user and item views in an implicit and explicit feedback scenario.

	Implicit		Explicit
	user	item	user	item
$n$	610	4275	610	6278
$mean$	0.8246	0.7256	0.8987	0.8156
${\sigma }_{\rho }$	0.1011	0.1275	0.0446	0.098
${\rho }_{min}$	0.5304	0.112	0.6589	0.1534
${\rho }_{0.25}$	0.7552	0.6505	0.8683	0.7752
${\rho }_{0.5}$	0.8425	0.7472	0.9034	0.8433
${\rho }_{0.75}$	0.9061	0.8207	0.9318	0.8827
${\rho }_{max}$	0.9806	0.9681	0.9867	0.9710

Table 2. Table 2. Comparison between our best DNN and LRA models with respect to different strategies for user-item latent vectors, e.g. p ​ r ​ e ​ t ​ r ​ a ​ i ​ n ​ e ​ d 𝑝 𝑟 𝑒 𝑡 𝑟 𝑎 𝑖 𝑛 𝑒 𝑑 pretrained and/or f ​ i ​ x ​ e ​ d 𝑓 𝑖 𝑥 𝑒 𝑑 fixed , and concatenation [ ⋅ , ⋅ ] ⋅ ⋅ [\cdot,\cdot] , resp. Hadamard product ⊙ direct-product \odot , in terms of MRR, MAP@10 and AUC.

		MRR	MAP@10	AUC
$M{F}_{best}$		0.0436	0.0706	0.9211
$[{𝐞}_i^u,{𝐞}_j^v]$	$DNN$	0.0369	0.0627	0.8920
	$DN{N}_{pretrained}$	0.0444	0.0738	0.9195
	$DN{N}_{pretrained}^{{}^{fixed}}$	0.0449	0.0706	0.9138
${𝐞}_i^u\odot {𝐞}_j^v$	$DNN$	0.0445	0.0716	0.9157
	$DN{N}_{pretrained}$	0.0473	0.0753	0.9241
	$DN{N}_{pretrained}^{{}^{fixed}}$	0.0449	0.0713	0.9216

Equations

$$\begin{array}[]{c}\hat{r}_{ij}:=\mathbf{e}_{i}^{u}\cdot\mathbf{e}_{j}^{v}+b_{i}^{u}+b_{j}^{v},\mathbf{e}_{i}^{u},\mathbf{e}_{j}^{v}\in\mathbb{R}^{p},p\in\mathbb{N},b_{i}^{u},b_{j}^{v}\in\mathbb{R},\end{array}$$

$$-\sum _S\left[r_{ij}^{\prime }\cdot \log (\sigma ({\hat{r}}_{ij}))+(1-r_{ij}^{\prime })\cdot \log (1-\sigma ({\hat{r}}_{ij}))\right],$$

$$f_{\Theta ,U,V}(u_i,v_j)=g(h_u(u_i),h_v(v_j)\mid U,V)\mid \Theta ),$$

$$\mathrm{cov}(R,\sum _{k=1}^p{\alpha }_k{E}_k^u{E}_k^v)>0,$$

Full text

Recommender Systems; Neural Networks; Collaborative Filtering

††ccs: Information systems Collaborative filtering††ccs: Computing methodologies Neural networks††ccs: Computing methodologies Factorization methods

1. Introduction

Since the Netflix prize in 2009, variants of low-rank approximation (LRA) have been and still are among the most popular approaches to collaborative filtering (CF) problems despite the advent of Deep Learning (DL). In many other domains, e.g. computer vision, image and speech recognition, classical methods used in those domains were significantly surpassed by neural networks and thus great progress was made. Although neural network based recommender systems have become widespread and shown significant performance gains by exploiting content, contextual and sequential patterns, they prove insufficient in case of CF. Feature engineering and extraction capabilities of DL seem to be of no use and the sparsity of user-item interactions surely is an impeding factor. This is generally acknowledged by the community and neural networks are often combined with LRA approaches(He et al., 2017; Guo et al., 2013). Despite these works, we found no proper study comparing LRAs with neural networks especially with respect to the determined latent vectors by these methods. Our work remedies this by contributing an intuition for LRAs based on expected covariances between latent features and interactions. Moreover, we evaluate in several experiments the suitability of the latent vectors obtained by low-rank approximations compared to neural networks.

1.1. Problem Formulation

Given users $u_{i}$ with $i=1,\ldots,m$ and items $v_{j}$ with $j=1,\ldots,n$ from a single domain, we denote with a scalar$r_{ij}$ the interaction of user $u_{i}$ with item $v_{j}$. In order to express the user’s preference for an item, we assume $r_{ij}<r_{ik}$ if $u_{i}$ liked $v_{k}$ more than $v_{i}$ and $r_{ij}=r_{ik}$ if $u_{i}$ is indifferent between $v_{i}$ and $v_{k}$. In implicit feedback scenarios, we often have $r_{ij}=1$ for positive feedback, whereas in explicit feedback scenarios $r_{ij}$ takes a numerical rating. We denote the interaction matrix of all users and items with $R=\{r_{ij}\}$.

1.2. Low-Rank Approximations

LRAs exploit the fact that rows and columns of $R$ are highly correlated due to redundancies in the underlying ratings, e.g. similarly acting users or similarly rated items. This allows a robust approximation by a lower-rank matrix $\hat{R}=\{\hat{r}_{ij}\}$ (Ricci et al., 2015; Aggarwal, 2016). We have

[TABLE]

where $\mathbf{e}_{i}^{u}$ and $\mathbf{e}_{j}^{v}$ are elements of a joint latent space of dimension $p\ll\min\{m,n\}$ denoted as user or item* latent factors* in a LRA context** or more generally* latent vectors*. Their inner product models the user-item interaction. The bias terms $b_{i}^{u}$ and $b_{j}^{v}$ capture interaction-independent effects like users systematically rating lower than others or items that are more popular than others. The actual approximation depends on the problem setting. For explicit feedback, where $r_{ij}\in\{-1,1\}$ or $r_{ij}\in\{0,1\}$, a pointwise approach is often applied and for instance the binary cross-entropy loss, i.e.

[TABLE]

where $r_{ij}^{\prime}=\max\{0,r_{ij}\}$, $\sigma$ is the sigmoid function and $S:=\{(i,j\}|r_{ij}\>\mathrm{is\>known}\}$, is optimized. In case of implicit positive feedback, the de facto standard is Bayesian Personalized Ranking (BPR). This pairwise approach maximizes the probability that an item $v_{j}$ with observed interaction $r_{ij}$ of user $i$ is ranked higher than an item$v_{k}$ with no observed interaction $r_{ik}$ with an item $k$, i.e. $p(v_{j}>_{u_{i}}v_{k}\mid\mathbf{e}_{i}^{u}\mathbf{e}_{j}^{v},\mathbf{e}_{k}^{v},b_{i}^{u},b_{j}^{v})$ (Rendle et al., 2009). The model parameters are determined with the help of (stochastic) gradient descent or alternating least squares methods (Hu et al., 2008; Koren, 2008).

1.3. Neural Network Approaches

In recent years, DL based recommenders have become widespread in academia and industry (Zhang et al., 2018). Leveraging flexible, non-linear models for representation learning and sequence modeling has proven highly beneficial, especially in content-based and hybrid settings. However, some works also use neural networks, e.g. multi-layer perceptrons (MLP), autoencoders, and convolutional neural networks, purely for CF (Wang et al., 2015; He et al., 2017, 2018; Song et al., 2018). This invites to research their effectiveness compared to LRAs.

A general MLP-based model, illustrating the basic structure of neural collaborative filtering networks (NCFN), is

[TABLE]

where $f$ is a mapping from a user-item tuple $(u_{i},v_{j})$ into $\mathbb{R}$. It is composed of functions $h_{u},h_{v}$ that transforms user and item indices into their joint latent space and a MLP $g$ with parameters $\Theta$ which maps the concatenation, outer product or Hadamard product of these latent vectors into $\mathbb{R}$ modeling the user-item interaction (He et al., 2017, 2018; Song et al., 2018). Since neural networks are universal function approximators, they are theoretically capable to derive these interactions when the concatenation of user and item latent vectors are provided as inputs. Analogously to LRAs, NCFNs can be trained in an explicit context using the binary cross-entropy (1) or in an implicit context with BPR. The parameters are inferred by using backpropagation for loss minimization and hence weight adaption by means of (stochastic) gradient descent.

2. Covariances and low-rank approximations

We want to establish the connection between LRAs and the covariances of latent vectors and interactions to give a novel intuition behind the LRA model. In case of a purely CF task, $u_{i}$ as well as $v_{j}$ are just entities without any observable features. Thus we assume the existence of latent item features $l_{k}$ with $k=1,\ldots,p$ which describe the items since we assume a single domain. Consequently, we have for each item $v_{j}$ a latent vector $\mathbf{e}_{j}^{v}\in\mathbb{R}^{p}$ where $e_{jk}^{v}$ defines how strong the latent feature $l_{k}$ is prevalent in $v_{j}$. Analogously, we have for each user $u_{i}$ a latent vector $\mathbf{e}_{i}^{u}\in\mathbb{R}^{p}$ where $e_{ik}^{u}$ represents the strength of the preference for $l_{k}$.

Under these assumptions, we can conclude that if user $u_{i}$ has a preference for $l_{k}$ this will also be reflected in the items the user has interacted with. With $R_{i}^{u}$ we denote the random variable for the interactions of user $u_{i}$ having realizations $r_{ij}$ and analogously the random variable for the prevalence of the feature $k$ in items with $E_{k}^{v}$ and realizations $e_{jk}^{v}$. We can now formalize the joint variability of $R_{i}^{u}$ and the prevalence of $l_{k}$ in the items, $u_{i}$ has interacted with, as $\mathrm{cov}(R_{i}^{u},E_{k}^{v})>0$. Following the same reasoning but from an item’s perspective, we can argue that the preference of users for $l_{k}$ should be reflected in their interactions with an item $v_{j}$ having a strong prevalence of $l_{k}$ and thus we have that $\mathrm{cov}(R_{j}^{v},E_{k}^{u})>0$. If we now interpret $r_{ij}$ as realizations of a random variable $R$, we can express these relationships jointly as $\mathrm{cov}(R,E_{k}^{u}E_{k}^{v})>0$ due to the fact that $E_{k}^{u}$ and $E_{k}^{v}$ can be assumed independent.

Using the bilinearity of the covariance, we have for all latent item features that

[TABLE]

where $\alpha_{k}>0$ weights the importance of $l_{k}$ with respect to the other latent features. Having derived this canonical condition allows us to justify many traditional methods for CF tasks. For instance classical matrix factorization based methods in an implicit feedback use-case, fulfill (2) by assuming equal importance of $l_{k},$ i.e. setting $\alpha_{k}=1$ for $k=1,\ldots,p$ and determining $e_{ik}^{u}$ and $e_{jk}^{v}$ such that $\hat{r}_{ij}=\sum_{k=1}^{p}e_{ik}^{u}e_{jk}^{v}=\mathbf{e}_{i}^{u}\cdot\mathbf{e}_{j}^{v}$.

3. Experiments

In the following section we provide empirical validation of the covariance intuition together with an extensive comparison between LRA and DL.111The source code is available at https://github.com/FlorianWilhelm/lrann.

3.1. Dataset

We use the MovieLens 100k dataset (Harper and Konstan, 2015) for our empirical study. The dataset contains $100,836$ ratings between $m=610$ users and $n=9,724$ items on a discrete rating scale with $r_{ij}\in\{0.5,1.0,...,5.0\}$. We provide an implicit and explicit interpretation of the rating data to analyze the results in both feedback scenarios.

For the* implicit feedback* scenario, we only keep all interactions rated equal and above each user’s mean rating labeled with $1$ and set all others to [math]. This yields a remainder of $54,732$ ratings. We use $BPR$ loss for training which also maximizes the AUC (Dhanjal et al., 2015). $BPR$ randomly samples negative feedback from the remaining unobserved items for each user. In the* explicit feedback* scenario, we binarize the original ratings using the users’ mean ratings as threshold resulting in labels $1$ and $-1$. We then use the binary cross-entropy (1) to fit the data to our models.

3.2. Methodology

3.2.1. Covariances and LRAs

In both scenarios, we fit latent vectors of size $p=32$ with minibatch gradient descent (batch size 128) for 15 training epochs using the Adam Optimizer (Kingma and Ba, 2014). We use a learning rate $\alpha=0.003$, exponential decay rates $\beta_{1}=0.9$ and $\beta_{2}=0.999$ and no regularization. Hence, we obtain latent vectors $\mathbf{e}_{i}^{u}$ and $\mathbf{e}_{j}^{v}$. We calculate the covariance $\mathrm{cov}(R_{i}^{v},E_{k}^{u})$ between the interactions $r_{ij}$ of a fixed user $i$ and latent vectors component $e_{jk}^{v}$ of the items she interacted or not interacted with. The covariances over varying $k$ are then correlated to the user’s latent vector using Pearson which we denote as $\rho_{i}^{u}:=\rho(E_{k}^{u},\mathrm{cov}(R_{i}^{u},E_{k}^{v}))$. Analogously, we define the correlation $\rho_{j}^{v}:=\rho(E_{k}^{v},\mathrm{cov}(R_{j}^{v},E_{k}^{u}))$ from the view of a fixed item and varying users that interacted or not interacted with it. Highly positive correlation provides empirical support for our theory from Section (2).

3.2.2. NCFN

In order to compare LRA and DL, we limit experiments to implicit feedback due to its greater abundance in real-world applications and competitive results by using the pairwise ranking approach BPR. Essentially, we examine to which extent network input modeling as well as pretraining latent vectors influence the capability of a DNN to reproduce or potentially outperform a strong LDA baseline in terms of accuracy as measured by the Mean Reciprocal Rank (MRR), Mean Average Precision at 10 (MAP@10) and Area Under receiver operating characteristic Curve (AUC). We explore different DNN architectures and distinguish between three pretraining strategies. $DNN$ refers to the first setting where the latent vectors are initialized randomly and constitute the parameter space to fit together with the network parameters. We augment this setting by initializing the latent vectors to those of our LDA baseline model. In this case we distinguish whether the latent vectors can still be adjusted or whether they stay fixed, yielding $DNN_{pretrained}$ and $DNN_{pretrained}^{fixed}$. Input modeling wise, we separate between feeding concatenated user-item latent vectors $[\mathbf{e}_{i}^{u},\mathbf{e}_{j}^{v}]$ and their Hadamard product $\mathbf{e}_{i}^{u}\odot\mathbf{e}_{j}^{v}$, similar to (He et al., 2017, 2018), into the network.

For each of the resulting six combinations we explore neural networks with $L\in\{0,1,2,3\}$ hidden layers, using different activation functions $\{ReLU,ELU,\tanh,sigmoid\}$ arriving at 13 combinations. We choose $\alpha\in\{{0.001,0.003,0.01}\}$ and five different random initializations for the network parameter initialization. Each combination is trained for 20 epochs. We compute test set MRR after every epoch for early stopping.222Thus, with $2\times 3$ settings, $13$ neural network architectures, $3\times 5\times 20$ training epochs, we consider $23,400$ experiments. For sake of brevity of our study, we leave other goals of recommendations as diversity, serendipity for future work. We apply a 80/20 train-test split and keep it consistent across LRA and NCFN.333We perform a hyperparameter grid search to find the best and therefore most competitive LRA configuration using MRR as selection criterion.

3.3. Results and Discussion

3.3.1. Covariances and LRAs

Evaluating $\rho_{i}^{u},i\in I$ as well as $\rho_{j}^{v},j\in J$ for both feedback scenarios, we observe highly positive correlations with $\rho_{mean}^{u}=0.8246$ and $\rho_{mean}^{v}=0.7256$ (implicit feedback) and $\rho_{mean}^{u}=0.8987$ and $\rho_{mean}^{v}=0.8156$ (explicit feedback) as detailed in Table 1. Due to the fact that we only consider significant individual correlations and that many items have just few or no interactions, there remains just a fraction of items in each scenario. The distributions of these correlations are also shown in Figures 2 and 2. These results empirically support our derivation of LRA based on covariances of users’ preferences and items’ features manifested by user-item interactions.

3.3.2. NCFN

A traditional DNN with concatenation of user and item latent vectors could not match the performance of LRA in our experiments. This result is in line with Dziugaite and Roy (2015) who stated “Conceivably, a deep neural network could learn to approximate the element-wise product or even outperform it, but this was not the case in our experiments, which used gradient-descent techniques to learn the neural network weights.“ Despite the universal approximation capability of such a network, we need to explicitly model the inner product of latent user and item vectors to match LRA’s performance. We believe that the flexibility of a traditional DNN impedes the determination of proper latent vectors. This hypothesis is supported by our experiments that use pretrained latent vectors from LRA. In this case even a DNN with concatenation is on par with LRA, and even more so, if the pretrained latent vectors are fixed. The largest performance gain is achieved, however, by explicitly modeling the user-item interaction with the help of the Hadamard product resulting in a boost of 20% in MRR. Combining the fitted latent factors of the best LRA that already embed the underlying preference relations with the Hadamard product even outperforms LRA to a certain extent, i.e. 8.58% in $DNN_{pretrained}$. This can be interpreted as an adaption of $\alpha_{k}$ in (2) by the neural network which is more flexible as setting $\alpha_{k}=1$. All results are summarized in Figure 3 and Table 2.

4. Conclusion

This work contributes theoretical and empirical studies examining the effectiveness of LRA compared to DNNs. We showed that standard DNNs fail to approximate element-wise multiplications which is the cornerstone of LRA’s effectiveness according to our model derivation using covariances. Traditional DNNs perform significantly worse than LRAs for CF. However, when using proper initialization of the latent vectors from a pretrained LRA and potentially joining them using the Hadamard product, DNNs can outperform LRAs. These are important insights to consider when designing DNN based recommender systems that (partially) depend on collaborative signals. Our results are also supported by latest works that show surprising incapacities of neural networks. For example, Trask et al. (2018) propose a neural arithmetic logic unit (NALU) to alleviate the fact that DNNs fail to systematically abstract and to extrapolate from the provided training data. Lin et al. (2017) propose dedicated multiplication gates to enable DNNs solving a seemingly simple task.

For future work, we want to further deepen the understanding of DNNs for CF tasks compared to LRAs. We believe that the concurrent adaption of both, latent vector space and neural network parameters, leads to suboptimal configurations which is supported by the gained performance when we used pretrained latent vectors in our experiments. Eventually, a deeper understanding of these inner workings will result in more advanced DNN-based recommenders.