# Multi-Resolution Diffusion for Privacy-Sensitive Recommender Systems

Derek Lilienthal  San Jose State University1 Washington Sq, San Jose, CA
95192San JoseUnited States<EMAIL_ADDRESS>,  Paul Mello  San Jose
State University1 Washington Sq, San Jose, CA 95192San JoseUnited States
<EMAIL_ADDRESS>,  Magdalini Eirinaki  San Jose State University1
Washington Sq, San Jose, CA 95192San JoseUnited States
<EMAIL_ADDRESS>and  Stas Tiomkin  San Jose State University1
Washington Sq, San Jose, CA 95192San JoseUnited States<EMAIL_ADDRESS>
(20 November 2023)

###### Abstract.

While recommender systems have become an integral component of the Web
experience, their heavy reliance on user data raises privacy and security
concerns. Substituting user data with synthetic data can address these
concerns, but accurately replicating these real-world datasets has been a
notoriously challenging problem. Recent advancements in generative AI have
demonstrated the impressive capabilities of diffusion models in generating
realistic data across various domains. In this work we introduce a Score-based
Diffusion Recommendation Module (SDRM), which captures the intricate patterns
of real-world datasets required for training highly accurate recommender
systems. SDRM allows for the generation of synthetic data that can replace
existing datasets to preserve user privacy, or augment existing datasets to
address excessive data sparsity. Our method outperforms competing baselines
such as generative adversarial networks, variational autoencoders, and
recently proposed diffusion models in synthesizing various datasets to replace
or augment the original data by an average improvement of 4.30% in Recall@$k$
and 4.65% in NDCG@$k$.

Diffusion Models, Recommender Systems, Synthetic Data, Data Privacy

This work has been submitted to the IEEE for possible publication. Copyright
may be transferred without notice, after which this version may no longer be
accessible.

##  1\. Introduction

Recommender systems have become a ubiquitous part of the Web experience, with
applications in multiple domains. Such systems are trained on user data that
comes in the form of explicit ratings or implicitly inferred preferences for
items as well as various other sources of information including demographics,
item metadata, context data, social network connections, etc. (Aggarwal et
al., 2016). Inevitably, this heavy reliance on user data makes privacy one of
the major challenges for recommender systems (Paraschakis, 2016; Milano et
al., 2020). Breaches of user privacy and even rights’ violations may occur in
various stages of a recommender system’s life cycle, including inferences that
can be made from the output of the recommendation process (Friedman et al.,
2015; Paraschakis, 2016; Milano et al., 2020). Research into privacy-
preserving algorithms and anonymization of user data has made significant
progress, but a balance between accurately capturing complex user preferences
and data privacy remains challenging (Liu et al., 2022). Even when data
privacy is not the primary concern, there are other needs which can be
addressed by synthetic datasets for recommender systems. One of the main
issues of recommender systems is data sparsity (Idrissi and Zellou, 2020). A
dataset is considered sparse when a very small percentage of total user-item
preferences are available. In cases where a high number of missing values
exists, a synthetic dataset can be used to augment the original dataset by
increasing the total number of data points, ideally improving the prediction
accuracy of the recommender system.

Generative AI has become increasingly popular due to its ability to generate
realistic content across various modalities (GM et al., 2020; Bond-Taylor et
al., 2022). Generative models such as Generative Adversarial Networks (GANs)
(Goodfellow et al., 2014, 2020), Variational Autoencoders (VAEs) (Kingma and
Welling, 2014), and Diffusion models (DMs) (Ho et al., 2020) have been shown
to model complex data distributions extremely well, and have thus been
employed in various domains, including recommender systems. Most of prior work
employs GANs or VAEs as part of the collaborative filtering process (i.e. to
model user-item interactions), with the objective of improving predictions
rather than generating synthetic data (Wang et al., 2019b; Liang et al., 2018;
Askari et al., 2021; Chae et al., 2018; Wang et al., 2019a; Ma et al., 2019;
Sachdeva et al., 2019). DMs, a recent addition to the generative modeling
family, have emerged as a prominent framework for data synthesis and have been
shown to produce high-quality samples outperforming GAN architectures
(Dhariwal and Nichol, 2021). Only recently have DMs been applied to
recommendation systems for collaborative filtering (Walker et al., 2022; Wang
et al., 2023) and sequential predictions (Li et al., 2023; Du et al., 2023)
with very promising results.

In this work, we propose a Score-based Diffusion Recommendation Module (SDRM)
which allows the generation of synthetic data in the form of user-item
interactions. The artificial datasets capture the intricate patterns of the
real-world datasets that SDRM uses as input, and can thus be used to augment
or completely replace them. We utilize a VAE encoding scheme to map inputs to
a Gaussian distribution and then to a latent space where a diffusion model
transforms the Gaussian input to a novel Gaussian output for the VAE to
decode. Through this architecture, we generate arbitrarily many synthetic
samples which accurately capture the input distribution allowing for a partial
or total replacement of the input data. Through thorough experimental
evaluation, our approach has been proven to be more effective than other
methods that utilize GANs (Xu et al., 2019), VAEs (Xu et al., 2019; Liang et
al., 2018), and DMs (Walker et al., 2022; Wang et al., 2023) in creating
datasets of varying sizes and densities. It shows an average improvement of
6.81% in Recall and 7.73% in NDCG for augmented datasets, and 1.42% in Recall
and 1.98% in NDCG for synthetic datasets. This performance boost is consistent
across four distinct datasets and three different recommendation
algorithms111We have made our code available at https://github.com/Multi-
resolution-diffusion-recommender/SDRM.

##  2\. Related Work

Recently, generative models have been used in recommender systems, with most
of the prior work involving GANs and VAEs. GAN-based techniques have been used
to leverage adversarial training to optimize the generator’s ability to
augment user interactions (Gao et al., 2021; Wang et al., 2019b). The
application of GANs to enhance user-item data for training recommender models
has gained significant attention, with methods emerging to augment user
preferences for collaborative filtering (Chae et al., 2018; Wang et al.,
2019b, a), and sequential predictions (Bharadhwaj et al., 2018; Zhao et al.,
2018). Various works on incorporating GANs and reinforcement learning to
simulate user behavior have also been explored (Chen et al., 2019; Bai et al.,
2019).

Many works have been proposed to leverage VAEs in recommendation systems to
predict user ratings (Liang et al., 2018; Ma et al., 2019; Sachdeva et al.,
2019; Askari et al., 2021) by using their ability to compress data into a
Gaussian distribution using the re-parameterization trick (Kingma and Welling,
2014) and applying variational inference to reconstruct the original data. To
take advantage of VAE’s abilities to reconstruct data from a Gaussian
distribution, we propose pretraining a VAE to map data to a low-dimensional
Gaussian representation for which we apply diffusion on their latent space.
Similar work has been proposed using this method (Vahdat et al., 2021; Wang et
al., 2023), but trained the VAE and diffusion model together, whereas we train
them individually to take advantage of the variational inference ability of
VAEs.

There have been very few works incorporating diffusion into recommendation
systems. Existing research primarily focuses on predicting complete user
preferences (Walker et al., 2022; Wang et al., 2023) or making sequential
predictions (Li et al., 2023; Du et al., 2023). Walker et al. proposed CODIGEM
(Walker et al., 2022), a DDPM-inspired architecture that stacks autoencoders
to denoise the corrupted user-item preferences. Very recently, Wang et al.
proposed DiffRec (Wang et al., 2023), a model that trains a denoising
diffusion model by incorporating an array of MultiVAE encoders to map user
data to a low-dimensional space and reconstructs the input back to its
original high-dimensional form. While DiffRec and CODIGEM show promise as a
method to predict user ratings, they are not suitable as a data generator and
do not fully take advantage of diffusion’s ability to generate realistic data.
We make this claim for a few reasons: first, both DiffRec’s and CODIGEM’s
training approach implements early stopping, which prevents the denoising
model from fully learning the entire resolution of the latent data during the
denoising process (Ho et al., 2020); and second, CODIGEM and DiffRec train for
3 and 10 timesteps $T$ respectively, while training for larger $T$ (such as
1000 steps or more), has been shown to capture more detailed information, due
to increased reconstruction time to learn imperceptible details of the data,
as demonstrated in (Ho et al., 2020; Nichol and Dhariwal, 2021).

In contrast to DiffRec and CODIGEM, our model, SDRM, benefits from a simpler
architecture and trains over a larger amount of $T$ steps without early
stopping utilizing an ablated training objective which focuses on generating
user-item interaction data for enhancing, or completely replacing the original
datasets.

##  3\. Background

Diffusion Models (DMs) have emerged as a versatile class of generative models
applicable to various applications. DMs consist of two processes, forward and
reverse (Sohl-Dickstein et al., 2015). The forward process incrementally adds
Gaussian noise to the initial input $x_{0}$ towards the standard Gaussian
noise $x_{T}\sim N(0,\mathbf{I})$ using a noise scheduler $\beta_{t}$ for $T$
time steps. This forward process is defined conditional distribution of
$x_{t+1}$ given $x_{t}$ variable $x_{t}$: $q(x_{t+1}\mid
x_{t})=\mathcal{N}\left(x_{t};\mu_{t}=\sqrt{1-\beta_{t}}x_{t-1},\Sigma_{t}=\beta_{t}\boldsymbol{I}\right)$
(Xiao et al., 2022; Luo, 2022).

The subsequent reverse process gradually reconstructs the input, $x_{0}$, from
the Gaussian noise, $x_{T}$, where the $t$-th reconstruction step is given by
the conditional Gaussian distribution with trainable parameters, $\theta$:
$p_{\theta}(x_{t}|x_{t+1}):=\mathcal{N}(x_{t};\mu_{\theta}(x_{t+1},t),\Sigma_{\theta}(x_{t+1},t))$.
The reverse process parameters, $\theta$, are optimized to maximize the
evidence lower bound (ELBO) on the original data distribution (Ho et al.,
2020). The ELBO in DM can be represented by the mean square error between the
original data, which is gradually corrupted by noise in the forward process,
and the reconstructed data, from the reverse process. This enables the
efficient training of DMs and high-quality data synthesis (Ho et al., 2020;
Luo, 2022; Kingma et al., 2021; Kingma and Gao, 2023).

Generative models, such as diffusion, can be trained utilizing various
training objectives (Chefer et al., 2023; Zhang and Yu, 2023), including a
score-based approach (Vincent, 2011; Song and Ermon, 2019; Song et al., 2020;
Batzolis et al., 2021; Luo, 2022; Dockhorn et al., 2021; De Bortoli et al.,
2022; Voleti et al., 2022; Gnaneshwar et al., 2022; Sun et al., 2022).
Compared to the ELBO, the unbiased nature of score-based objectives naturally
lends itself to recommender systems. We leverage this score-based objective in
our work to capture the intricacies of user-item interactions.

##  4\. Score-Based Diffusion Recommender Method

Figure 1. SDRM training and sampling

We propose SDRM (Score-based Diffusion Recommender Module), a module designed
to generate artificial user-item preferences to augment or replace the dataset
it is trained on.

Diffusion models are capable of modeling complex distributions of arbitrarily
many dimensions, but suffer from intractability when datasets have sparse
features. As a result, reducing the overall dimensions before the diffusion
process can help with modeling accurate reconstructions over all possible
items of the dataset. For this reason, our proposed method employs a VAE to
map data into a lower-dimensional Gaussian distribution and enables more
efficient training for SDRM.

Generating recommendation data poses a unique problem due to the challenge of
training a model on datasets containing tens or hundreds of thousands of
potential user preferences. Recommendation data is highly sparse and skewed
(Aggarwal et al., 2016), making it necessary to reduce the overall dimensions
before the diffusion process. Through this method, diffusion can learn to
reconstruct all possible items of the dataset. For this reason, our proposed
method takes advantage of VAE’s ability to map data into a lower-dimensional
Gaussian distribution and enables more efficient training for SDRM.

As shown in Figure 1, SDRM uses a pre-trained encoder to map user ratings into
a smaller representation, followed by a Multi-Layer Perceptron (MLP) denoising
model, and finally, a decoder to decompress the latent vectors into user
ratings. We train SDRM in two steps: first, we pretrain a multinomial
likelihood variational autoencoder (MultiVAE) on user ratings, and then we
train SDRM with the encoder of the MultiVAE. To generate user ratings, we
sample from a normal Gaussian distribution, denoise it using SDRM, and pass
the denoised latent vectors into the VAE decoder to reconstruct the data into
the same dimensions as the original dataset.

We propose two sampling variations for SDRM: full-resolution sampling
(F-SDRM), and multi-resolution sampling (M-SDRM). F-SDRM denoises pure
Gaussian noise starting at timestep $T\rightarrow 0$ while M-SDRM starts at a
random timestep $t$ and denoises to $0$. Because the latent $z$ vector of the
pretrained encoder maps to a Gaussian distribution, SDRM learns to denoise
from a Gaussian distribution $z$ to a Gaussian distribution $\hat{z}$. By
starting at a random timestep $t$ instead of the full timestep $T$ and
denoising until we reach timestep $0$, we reconstruct Gaussian noise at
different resolutions into user data for M-SDRM.

###  4.1. SDRM objective

We propose the following score-based objective for training diffusion in
recommender systems.

(1)
$\small\ell(z_{t})=\frac{||\bigl{(}s_{\theta}(\hat{z}_{t})-s_{\theta}(z_{t})\bigr{)}-\Delta_{\theta}(z_{t})||_{2}^{2}+||s_{\theta}(z_{t})-\Delta_{\theta}(z_{t})||_{2}^{2}}{||\Delta_{\theta}(z_{t})||_{2}^{2}}$

where $t$ is the discrete time step, uniformly sampled from the range
$[1,\dots,T]$, $z_{t}$ is the corresponding latent representation of the input
data point, (e.g., user-item interactions), encoded by the VAE, and
$\hat{z}_{t}=z_{t}+\nu$ with $\nu\sim\mathcal{N}(0,\sigma_{\nu}\mathbf{I})$ is
its perturbed version by the Gaussian noise, $\eta$. $s_{\theta}(\cdot)$ is
the score function in the diffusion model with the parameters $\theta$.
$\epsilon_{\theta}(z_{t})$ is the prediction noise at the time $t$ in the
diffusion model. Finally,
$\Delta_{\theta}(z_{t})\doteq\epsilon_{\theta}(z_{t})-z_{t}$ quantifies the
difference between the prediction noise, $\epsilon_{\theta}(z_{t})$, and the
latent representation, $z_{t}$. The denominator,
$||\Delta_{\theta}(z_{t})||_{2}^{2}$, serves as a normalizing factor to
stabilize the loss function. We show, through experimentation on various
datasets, that this score-based objective is more effective than standard
training objectives applied in other diffusion models, such as CODIGEM (Walker
et al., 2022) and DiffRec (Wang et al., 2023).

###  4.2. SDRM Training and Sampling

We train MultiVAE (Liang et al., 2018) on the original dataset, using the
evidence lower-bound objective defined by

(2)
$\small\mathcal{L}_{u}(\theta,\phi)=\mathbb{E}_{q_{\phi}(z_{u}|x_{u})}[\log
p_{\theta}(x_{u}|z_{u})]-\beta\cdot KL(q_{\phi}(z_{u}|x_{u})\|p(z_{u}))$

where $\log p_{\theta}(x_{u}|z_{u})$ is the Gaussian log-likelihood for user
$u$, $q_{\phi}$ is the approximating variational distribution (inference
model), and $\beta$ is a heuristic annealing parameter. Next, we utilize the
MultiVAE encoder to compress user-item preferences into a smaller latent
representation for training SDRM with the loss in Equation (1). We prevent
gradient updates on the VAE model while training SDRM, ensuring SDRM purely
learns to transform the VAE latent variable $z$ to another latent variable
$\hat{z}$.

For sampling data with F-SDRM, we follow the same sampling procedure as Ho et
al. (Ho et al., 2020) and map $z_{0}$ to the decoder of the MultiVAE. We
follow the same process for M-SDRM, but start the denoising from a random
$t\in T$ for each sample. Additional details of the training and sampling
procedure are included in Algorithms 1 and 2.

Algorithm 1 Training SDRM

1:Sample candidate hyperparameters $x^{*}$ from $\chi$

2:Initalize VAE and SDRM with hyperparameters $x^{*}$

3:while until converged do

4: for batch size $\in$ $x^{*}$ do

5: Train VAE:
$\scriptsize{\mathcal{L}_{u}(\theta,\phi)=\mathbb{E}_{q_{\phi}(z_{u}|x_{u})}[\log
p_{\theta}(x_{u}|z_{u})]-\beta\cdot KL(q_{\phi}(z_{u}|x_{u})\|p(z_{u}))}$

6: end for

7: if Recall@10 improves then

8: Continue

9: end if

10:end while

11:Freeze VAE encoder

12:for epochs $\in x^{*}$ do

13: Map user-item preferences with VAE encoder to $z$

14: Train SDRM:
$\scriptsize{\ell(z_{t})=\frac{||\bigl{(}s_{\theta}(\hat{z}_{t})-s_{\theta}(z_{t})\bigr{)}-\Delta_{\theta}(z_{t})||_{2}^{2}+||s_{\theta}(z_{t})-\Delta_{\theta}(z_{t})||_{2}^{2}}{||\Delta_{\theta}(z_{t})||_{2}^{2}}}$

15:end for

Algorithm 2 Sampling SDRM

1:Select $t$ number of denoising timestep from $[1,T]$

2:Initialize synthetic user-item preference vector with sampled Gaussian noise
$z_{T}\sim N(0,I)$

3:for $T$,…,$1$ do

4: Sample Gaussian noise: $z\sim N(0,I)$ if $t>1$, else $z=0$

5: Transform Gaussian noise to Gaussian noise:
$\hat{z}_{t-1}=\frac{1}{\sqrt{\alpha_{t}}}\left(\hat{z}_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}}}\epsilon_{\theta}(\hat{z}_{t},t)\right)+\sigma_{t}z$

6:end for

7:Reconstruct $\hat{z}$ using VAE decoder to $\hat{x}$

8:Apply piecewise conditional to $\hat{x}$ with threshold $\lambda$:
$f(\hat{x})=\begin{cases}1&\text{if }\hat{x}\geq\lambda\\\ 0&\text{if
}\hat{x}<\lambda\end{cases}$

##  5\. Evaluation

We evaluate SDRM by comparing how well the generated user-item data can
augment or replace the original data when used as input for any recommendation
algorithm. For this purpose, we demonstrate the ability of SDRM to synthesize
recommendation datasets by employing three recommendation algorithms, namely
SVD (Mnih and Salakhutdinov, 2007), MLP (Covington et al., 2016), and NeuMF
(He et al., 2017), and evaluate the generated top-$k$ recommendations in terms
of Recall@$k$ and ranking order, using NDCG@$k$. We compare our model’s two
variations, namely F-SDRM and M-SDRM, with six generative model baselines on
four diverse datasets.

###  5.1. Experiment Setting

####  5.1.1. Datasets

We employ four publicly available and commonly used datasets to evaluate SDRM:
Amazon Luxury Beauty (ALB) (Ni et al., 2019), Amazon Digital Music (ADM) (Ni
et al., 2019), MovieLens 100k (ML-100k) (Harper and Konstan, 2015), and
MovieLens 1M (ML-1M) (Harper and Konstan, 2015). We selected these datasets
because they encompass a wide range of items and vary in user/item/rating
cardinality and sparsity, as shown in Table 1. Moreover, their relatively
small size allowed us to conduct extensive, lengthy optimization runs. Each
dataset is pre-processed by converting ratings greater than three to 1 and
ratings three or less to 0. We also filtered out low-frequency users and
items, so only users who rated at least five items and items with at least
five ratings are included in the datasets.

Each original dataset was split into a training, testing, and validation set
using a 70:20:10 split on all the users. Additionally, a small portion of user
ratings in the testing set are needed as ground truth users in the training
dataset to evaluate the effectiveness of synthetic data. The ratings of those
users who are in the training and test set are masked during the evaluation
phase.

Dataset | #Users | #Items | #Ratings | Sparsity
---|---|---|---|---
Amazon Luxury Beauty | 1,344 | 729 | 15,359 | 98.43%
Amazon Digital Music | 10,621 | 8,582 | 108,509 | 99.88%
MovieLens 100k | 938 | 1,008 | 95,215 | 89.93%
MovieLens 1M | 3,125 | 6,034 | 994,338 | 94.73%
Table 1. Dataset Statistics

####  5.1.2. Baseline Generative Models

We evaluate the ability of SDRM to generate synthetic data against the
original dataset (representing the baseline before augmenting or substituting
data), and the following generative models:

  * •

CTGAN (Xu et al., 2019) (Conditional Tabular Generative Adversarial Network)
is a GAN designed for synthesizing tabular data.

  * •

TVAE (Xu et al., 2019) is a conditional Tabular Variational Auto-Encoder
designed for synthesizing tabular data, similar to CTGAN.

  * •

CODIGEM (Walker et al., 2022) is a diffusion-inspired collaborative filtering
model that utilizes stacking denoising autoencoders during the reverse
denoising process.

  * •

MultiVAE (Liang et al., 2018) is a Variational Autoencoder that uses a
multinomial likelihood loss objective and variational inference to model
implicit user data.

  * •

MultiVAE++ is the pre-trained multinomial variational autoencoder used in
SDRM, but its hyperparameters are optimized as shown in Table 2. This model
represents a true baseline to compare the increased performance of SDRM.

  * •

DiffRec (Wang et al., 2023) is a denoising diffusion probabilistic model used
for predicting whole user-item interactions.

####  5.1.3. Baseline Recommender Algorithms

We evaluate the effectiveness of the synthetic data using three state-of-the-
art recommendation algorithms: SVD (Singular Value Decomposition) (Mnih and
Salakhutdinov, 2007), a matrix factorization technique that approximates the
original matrix with lower-dimensional matrices representing latent factors;
MLP (Multilayer Perceptron) (Covington et al., 2016), a deep neural network
architecture incorporating user and item embeddings which feed into multiple
feed-forward layers to model and predict whole user-item preferences; and
NeuMF (He et al., 2017), that fuses matrix factorization and an MLP network to
predict single user-item preferences.

####  5.1.4. Evaluation Metrics

Similar to related work, we adopt two top-$k$ metrics, namely Recall@$k$ and
NDCG@$k$. Recall@$k$ is defined as the percentage of ground-truth positives
(items) that the system recommends as positive for a recommendation list of
size $k$. NDCG@$k$ (Normalized Discounted Cumulative Gain) is defined as the
ratio of the cumulative gain to its ideal value, which measures how close the
recommended top-$k$ ranking is to the ground truth.

####  5.1.5. Generating Synthetic Data

Generating data from CTGAN and TVAE was straightforward, as both approaches
use a Gumbel-softmax (Jang et al., 2017) to directly create categorical
features. To generate data from CODIGEM, MultiVAE, MultiVAE++, and DiffRec, we
first trained those models, then modified the code base to inject Gaussian
noise into the decoder of MultiVAE and denoised Gaussian noise with CODIGEM
and DiffRec. This process allows each baseline model to generate synthetic
samples starting from Gaussian noise. For each synthetic dataset, we generated
an equal number of samples to match the number of users in each training split
of the original dataset. We utilized a conditional piece-wise function to
transform the logit outputs from the generative baseline models and SDRM. We
used a threshold value $\lambda$ on the synthetic data, which matches the
value at the sparsity quantile of each dataset.

(3) $f(x)=\begin{cases}1&\text{if }x\geq\lambda\\\ 0&\text{if
}x<\lambda\end{cases}$

####  5.1.6. Hyperparameter Search for SDRM

During the training procedure, we aim to find the set of hyperparameters
$x^{*}$ that yields the optimal $f(x)$ where a hyperparameter $x$ can take any
value in the search space of $\chi$ (Yang and Shami, 2020). We formalize the
search as $x^{*}=\arg\\!\max_{x\epsilon\chi}f(x)$, where we optimize
Recall@10. We utilize a TPE (Tree-structured Parzen Estimator) (Bergstra et
al., 2011) and a successive halving pruner (Li et al., 2020) to search over
the search space $\chi$ efficiently. We perform 5-fold cross-validation for
each set of hyperparameters used to train SDRM and report the average Recall
and NDCG results. The hyperparameter search space used during SDRM training is
shown in Table 2.

Table 2. Hyperparameter search space $\chi$ for SDRM Hyperparameters | Search Space
---|---
SDRM epochs | {5, 501, step=5}
SDRM learning rate | {1.e-6, 1.e-4, step=1.e-6}
Gaussian variance | {0.01, 1, step=0.1}
SDRM timesteps | {3, 200, step=5}
SDRM batch size | {30, 1000, step=10}
MLP hidden layers | {0, 5, step=1}
MLP latent neurons | {50, 1000, step=50}
MLP hidden layers | {0, 5, step=1}
VAE hidden layer neurons | {MLP latent neurons, 1000, step=50}
VAE learning rate: | {1.e-4, 1.e-2, step=1.e-2}
VAE batch size | {30, 1000, step=10}

To find the optimal set of hyperparameters for SDRM, we employed Bayesian
optimization search over the search space mentioned in Table 2 using the
Optuna framework (Akiba et al., 2019). Due to the observed variance in
performance between runs with the same hyperparameters, we ran each set of
candidate hyperparameters five times and averaged the results presented in
Tables 3 and 4. We ran each evaluation model (SVD, MLP, NeuMF) with the
benchmark datasets and SDRM between 30 to 300 trials. We computed these
results using RTX 3080, RTX 4090, T4, V100, and A10 GPUs over several weeks.

###  5.2. Training Procedure

We showcase the ability of diffusion to generate realistic user-item
preferences in two conditions: when the synthetic data is used to augment the
original training set, and when the synthetic data is used to replace the
original training set. For augmenting data, we combine the synthetic data with
the training portion of the original dataset and evaluate the results on the
holdout ratings from the test set. Similarly, for replacing the original data
with synthetic, we train each baseline recommender algorithm with the
synthetic data combined with 20% of ground truth user ratings from the test
set. This approach is necessary to show how synthetic data can enhance the
predictive performance on real users, as it is impractical to measure the
effectiveness of synthetic data in a recommendation model without
incorporating some actual user data. During the evaluation process, we predict
all possible items for each user in the test set and mask any items rated
during training to prevent data leakage in our results.

Each generative model was trained five times on each dataset and a synthetic
dataset was generated matching the same number of users in its respective
training set. Subsequently, each baseline recommender model was trained with
and without the original data. Similarly, SDRM was trained five times and each
run generated a synthetic dataset for the MultiVAE++ and SDRM using full
(F-SDRM) and multi-resolution (M-SDRM) sampling. This way, every MultiVAE++,
F-SDRM, and M-SDRM result was calculated from the same exact runs. We generate
top-$k$ recommendations for each user in the validation set and compare them
to the ground truth using Recall@$k$ and NDCG@$k$.

To account for the negative sampling needed for training NeuCF, we utilized
items with the lowest ratings from each user in the synthetic datasets. For
the actual users, we selected the items with a 0 rating.

Table 3. Overall performance comparison between baselines by training with
synthetic and the original dataset. The best results are in bold and the
second best are underlined. Average overall improvement: Recall 4.48%, NDCG
5.07%

Model | SVD
---|---
Dataset | ALB | ML-100k | ML-1M | ADM
Baseline | Recall@10 | NDCG@10 | Recall@10 | NDCG@10 | Recall@10 | NDCG@10 | Recall@10 | NDCG@10
Original |  0.3113 $\pm$ 0 |  0.287 $\pm$ 0 |  0.3716 $\pm$ 0 |  0.3973 $\pm$ 0 |  0.3769 $\pm$ 0 |  0.4035 $\pm$ 0 |  0.0624 $\pm$ 0 |  0.0427 $\pm$ 0
CTGAN |  0.2642 $\pm$ 0.018 |  0.2407 $\pm$ 0.014 |  0.3533 $\pm$ 0.006 |  0.3813 $\pm$ 0.004 |  0.3652 $\pm$ 0.003 |  0.3917 $\pm$ 0.004 |  0.0405 $\pm$ 0.002 |  0.0267 $\pm$ 0.001
TVAE |  0.3113 $\pm$ 0 |  0.2873 $\pm$ 0 |  0.3668 $\pm$ 0.003 |  0.3918 $\pm$ 0.003 |  0.3638 $\pm$ 0.004 |  0.3869 $\pm$ 0.003 |  0.0624 $\pm$ 0 |  0.0427 $\pm$ 0
CODIGEM |  0.3025 $\pm$ 0.003 |  0.2819 $\pm$ 0.002 |  0.3492 $\pm$ 0.006 |  0.3745 $\pm$ 0.01 |  0.3113 $\pm$ 0.004 |  0.3336 $\pm$ 0.004 |  0.049 $\pm$ 0.003 |  0.0301 $\pm$ 0.001
DiffRec |  0.2936 $\pm$ 0.003 |  0.2748 $\pm$ 0.003 |  0.37 $\pm$ 0.008 |  0.4021 $\pm$ 0.011 |  0.3598 $\pm$ 0.007 |  0.383 $\pm$ 0.008 |  0.0306 $\pm$ 0.002 |  0.0228 $\pm$ 0.003
MultiVAE |  0.3226 $\pm$ 0.007 |  0.2998 $\pm$ 0.007 |  0.3822 $\pm$ 0.006 |  0.4103 $\pm$ 0.007 |  0.3391 $\pm$ 0.006 |  0.3658 $\pm$ 0.007 |  0.0612 $\pm$ 0.004 |  0.0389 $\pm$ 0.002
MultiVAE++ |  0.3163 $\pm$ 0.014 |  0.2912 $\pm$ 0.007 |  0.3878 $\pm$ 0.006 |  0.4126 $\pm$ 0.007 |  0.3717 $\pm$ 0.002 |  0.3946 $\pm$ 0.003 |  0.0613 $\pm$ 0.003 |  0.0406 $\pm$ 0.002
F-SDRM |  0.325 $\pm$ 0.012 |  0.2988 $\pm$ 0.01 |  0.3924 $\pm$ 0.005 |  0.4181 $\pm$ 0.005 |  0.3722 $\pm$ 0.002 |  0.3977 $\pm$ 0.002 |  0.0623 $\pm$ 0.003 |  0.041 $\pm$ 0.001
M-SDRM |  0.3249 $\pm$ 0.004 |  0.2957 $\pm$ 0.005 |  0.3971 $\pm$ 0.004 |  0.417 $\pm$ 0.004 |  0.372 $\pm$ 0.002 |  0.3962 $\pm$ 0.003 |  0.0641 $\pm$ 0.004 |  0.0442 $\pm$ 0.002
Improvement | 0.74 % | -0.33 % | 2.39 % | 1.33 % | -1.26 % | -1.44 % | 2.72 % | 3.51 %
| MLP
Original |  0.3183 $\pm$ 0.006 |  0.3007 $\pm$ 0.004 |  0.3569 $\pm$ 0.007 |  0.3788 $\pm$ 0.004 |  0.3319 $\pm$ 0.006 |  0.3558 $\pm$ 0.005 |  0.0194 $\pm$ 0.001 |  0.0128 $\pm$ 0.001
CTGAN |  0.3188 $\pm$ 0.013 |  0.2987 $\pm$ 0.005 |  0.3567 $\pm$ 0.007 |  0.3781 $\pm$ 0.006 |  0.3228 $\pm$ 0.005 |  0.3463 $\pm$ 0.005 |  0.0208 $\pm$ 0.004 |  0.0135 $\pm$ 0.002
TVAE |  0.2438 $\pm$ 0.002 |  0.2431 $\pm$ 0.001 |  0.3437 $\pm$ 0.012 |  0.3622 $\pm$ 0.014 |  0.2949 $\pm$ 0.019 |  0.3186 $\pm$ 0.02 |  0.0184 $\pm$ 0.001 |  0.012 $\pm$ 0.001
CODIGEM |  0.3365 $\pm$ 0.014 |  0.3071 $\pm$ 0.008 |  0.3479 $\pm$ 0.01 |  0.3686 $\pm$ 0.013 |  0.3132 $\pm$ 0.007 |  0.3377 $\pm$ 0.006 |  0.0596 $\pm$ 0.003 |  0.0385 $\pm$ 0.002
DiffRec |  0.3283 $\pm$ 0.012 |  0.3013 $\pm$ 0.004 |  0.355 $\pm$ 0.011 |  0.3769 $\pm$ 0.008 |  0.3246 $\pm$ 0.001 |  0.3482 $\pm$ 0.002 |  0.0213 $\pm$ 0.002 |  0.0139 $\pm$ 0.001
MultiVAE |  0.3328 $\pm$ 0.009 |  0.3048 $\pm$ 0.006 |  0.053 $\pm$ 0.014 |  0.0615 $\pm$ 0.016 |  0.3588 $\pm$ 0.003 |  0.372 $\pm$ 0.006 |  0.0199 $\pm$ 0.003 |  0.0137 $\pm$ 0.001
MultiVAE++ |  0.3316 $\pm$ 0.008 |  0.3 $\pm$ 0.007 |  0.3901 $\pm$ 0.007 |  0.4101 $\pm$ 0.009 |  0.3528 $\pm$ 0.005 |  0.3763 $\pm$ 0.006 |  0.0756 $\pm$ 0.01 |  0.0489 $\pm$ 0.007
F-SDRM |  0.3246 $\pm$ 0.008 |  0.3004 $\pm$ 0.004 |  0.3839 $\pm$ 0.013 |  0.4055 $\pm$ 0.014 |  0.3595 $\pm$ 0.002 |  0.3845 $\pm$ 0.001 |  0.0146 $\pm$ 0.01 |  0.0095 $\pm$ 0.008
M-SDRM |  0.3343 $\pm$ 0.013 |  0.303 $\pm$ 0.007 |  0.3947 $\pm$ 0.011 |  0.419 $\pm$ 0.01 |  0.3591 $\pm$ 0.003 |  0.384 $\pm$ 0.003 |  0.0798 $\pm$ 0.0012 |  0.0523 $\pm$ 0.002
Improvement | -3.66 % | -1.35 % | 1.18 % | 2.17 % | 0.19 % | 2.18 % | 5.55 % | 6.95 %
| NeuMF
Original |  0.2399 $\pm$ 0.009 |  0.0818 $\pm$ 0.012 |  0.1006 $\pm$ 0.009 |  0.096 $\pm$ 0.005 |  0.0113 $\pm$ 0.001 |  0.0094 $\pm$ 0.001 |  0.0057 $\pm$ 0.001 |  0.0011 $\pm$ 0.001
CTGAN |  0.2444 $\pm$ 0.002 |  0.0877 $\pm$ 0.001 |  0.0889 $\pm$ 0.012 |  0.086 $\pm$ 0.013 |  0.0219 $\pm$ 0.003 |  0.0192 $\pm$ 0.004 |  0.007 $\pm$ 0.002 |  0.0014 $\pm$ 0.001
TVAE |  0.2439 $\pm$ 0.001 |  0.0874 $\pm$ 0.001 |  0.1258 $\pm$ 0.011 |  0.1282 $\pm$ 0.006 |  0.0126 $\pm$ 0.004 |  0.0105 $\pm$ 0.004 |  0.0053 $\pm$ 0.001 |  0.0013 $\pm$ 0.001
CODIGEM |  0.1709 $\pm$ 0.116 |  0.0477 $\pm$ 0.035 |  0.1015 $\pm$ 0.016 |  0.0839 $\pm$ 0.016 |  0.0642 $\pm$ 0.001 |  0.0601 $\pm$ 0.002 |  0.007 $\pm$ 0.002 |  0.0014 $\pm$ 0.001
DiffRec |  0.2621 $\pm$ 0.006 |  0.0923 $\pm$ 0.004 |  0.1147 $\pm$ 0.013 |  0.1093 $\pm$ 0.016 |  0.0203 $\pm$ 0.005 |  0.0178 $\pm$ 0.005 |  0.0138 $\pm$ 0.002 |  0.0035 $\pm$ 0.001
MultiVAE |  0.2475 $\pm$ 0.057 |  0.0826 $\pm$ 0.026 |  0.1253 $\pm$ 0.012 |  0.118 $\pm$ 0.014 |  0.0487 $\pm$ 0.003 |  0.0483 $\pm$ 0.002 |  0.0154 $\pm$ 0.004 |  0.0036 $\pm$ 0.001
MultiVAE++ |  0.2687 $\pm$ 0.009 |  0.0877 $\pm$ 0.01 |  0.2357 $\pm$ 0.003 |  0.1858 $\pm$ 0.004 |  0.1046 $\pm$ 0.005 |  0.0972 $\pm$ 0.005 |  0.0224 $\pm$ 0.007 |  0.005 $\pm$ 0.002
F-SDRM |  0.3225 $\pm$ 0.019 |  0.109 $\pm$ 0.01 |  0.232 $\pm$ 0.007 |  0.1891 $\pm$ 0.009 |  0.1026 $\pm$ 0.007 |  0.0935 $\pm$ 0.007 |  0.0234 $\pm$ 0.003 |  0.0054 $\pm$ 0.001
M-SDRM |  0.2953 $\pm$ 0.019 |  0.1026 $\pm$ 0.006 |  0.2425 $\pm$ 0.015 |  0.186 $\pm$ 0.008 |  0.1059 $\pm$ 0.014 |  0.099 $\pm$ 0.015 |  0.0273 $\pm$ 0.0059 |  0.006 $\pm$ 0.001
Improvement | 20.02 % | 24.28 % | 2.88 % | 1.77 % | 1.24 % | 1.85 % | 21.87 % | 20 %

Table 4. Overall performance comparison between baselines by training with
synthetic data. The best results are in bold and the second best are
underlined. Average overall improvement: Recall 2.08%, NDCG 0.88%

Model | SVD
---|---
Dataset | ALB | ML-100k | ML-1M | ADM
Baseline | Recall@10 | NDCG@10 | Recall@10 | NDCG@10 | Recall@10 | NDCG@10 | Recall@10 | NDCG@10
Original |  0.3113 $\pm$ 0 |  0.287 $\pm$ 0 |  0.3716 $\pm$ 0 |  0.3973 $\pm$ 0 |  0.3769 $\pm$ 0 |  0.4035 $\pm$ 0 |  0.0624 $\pm$ 0 |  0.0427 $\pm$ 0
CTGAN |  0.0654 $\pm$ 0.054 |  0.056 $\pm$ 0.054 |  0.2215 $\pm$ 0.011 |  0.225 $\pm$ 0.012 |  0.2623 $\pm$ 0.008 |  0.2769 $\pm$ 0.008 |  0.0152 $\pm$ 0.002 |  0.0095 $\pm$ 0.001
TVAE |  0.2521 $\pm$ 0.005 |  0.2503 $\pm$ 0.004 |  0.2641 $\pm$ 0.008 |  0.2766 $\pm$ 0.006 |  0.3082 $\pm$ 0.006 |  0.325 $\pm$ 0.008 |  0.0214 $\pm$ 0 |  0.0143 $\pm$ 0
CODIGEM |  0.3044 $\pm$ 0.008 |  0.2813 $\pm$ 0.007 |  0.2799 $\pm$ 0.004 |  0.286 $\pm$ 0.007 |  0.2624 $\pm$ 0.005 |  0.2774 $\pm$ 0.005 |  0.0481 $\pm$ 0.003 |  0.0295 $\pm$ 0.001
DiffRec |  0.2702 $\pm$ 0.013 |  0.2541 $\pm$ 0.02 |  0.3476 $\pm$ 0.006 |  0.3624 $\pm$ 0.009 |  0.3354 $\pm$ 0.011 |  0.3595 $\pm$ 0.013 |  0.0258 $\pm$ 0.004 |  0.0197 $\pm$ 0.003
MultiVAE |  0.3299 $\pm$ 0.011 |  0.3063 $\pm$ 0.008 |  0.3724 $\pm$ 0.008 |  0.3937 $\pm$ 0.007 |  0.3134 $\pm$ 0.011 |  0.3359 $\pm$ 0.012 |  0.0393 $\pm$ 0.005 |  0.0246 $\pm$ 0.004
MultiVAE++ |  0.3406 $\pm$ 0.003 |  0.3188 $\pm$ 0.001 |  0.3889 $\pm$ 0.008 |  0.414 $\pm$ 0.008 |  0.3704 $\pm$ 0.004 |  0.3935 $\pm$ 0.004 |  0.0617 $\pm$ 0.003 |  0.0412 $\pm$ 0.001
F-SDRM |  0.3471 $\pm$ 0.005 |  0.3229 $\pm$ 0.007 |  0.3931 $\pm$ 0.006 |  0.4176 $\pm$ 0.003 |  0.3707 $\pm$ 0.002 |  0.3944 $\pm$ 0.002 |  0.0628 $\pm$ 0.002 |  0.0428 $\pm$ 0.001
M-SDRM |  0.3384 $\pm$ 0.007 |  0.3174 $\pm$ 0.004 |  0.3946 $\pm$ 0.006 |  0.4176 $\pm$ 0.006 |  0.3703 $\pm$ 0.003 |  0.3946 $\pm$ 0.003 |  0.0622 $\pm$ 0.003 |  0.0423 $\pm$ 0.002
Improvement | 1.87% | 1.27% | 1.45% | 0.86% | -1.65% | -2.21% | 0.64% | 0.23%
Model | MLP
Original |  0.3183 $\pm$ 0.006 |  0.3007 $\pm$ 0.003 |  0.3569 $\pm$ 0.007 |  0.3788 $\pm$ 0.004 |  0.3319 $\pm$ 0.006 |  0.3558 $\pm$ 0.005 |  0.0194 $\pm$ 0.001 |  0.0128 $\pm$ 0.001
CTGAN |  0.2651 $\pm$ 0.008 |  0.2189 $\pm$ 0.021 |  0.2017 $\pm$ 0.016 |  0.2197 $\pm$ 0.011 |  0.2081 $\pm$ 0.001 |  0.2216 $\pm$ 0.003 |  0.0011 $\pm$ 0.001 |  0.0007 $\pm$ 0.001
TVAE |  0.2438 $\pm$ 0.001 |  0.2431 $\pm$ 0.001 |  0.1334 $\pm$ 0.007 |  0.1586 $\pm$ 0.006 |  0.1569 $\pm$ 0.017 |  0.1674 $\pm$ 0.015 |  0.0153 $\pm$ 0.003 |  0.0095 $\pm$ 0.001
CODIGEM |  0.1568 $\pm$ 0.095 |  0.1012 $\pm$ 0.061 |  0.1445 $\pm$ 0.019 |  0.1403 $\pm$ 0.018 |  0.1882 $\pm$ 0.006 |  0.187 $\pm$ 0.01 |  0.0249 $\pm$ 0.003 |  0.0173 $\pm$ 0.001
DiffRec |  0.3065 $\pm$ 0.005 |  0.2947 $\pm$ 0.003 |  0.2997 $\pm$ 0.016 |  0.3107 $\pm$ 0.014 |  0.2349 $\pm$ 0.033 |  0.2526 $\pm$ 0.034 |  0.0181 $\pm$ 0.002 |  0.0115 $\pm$ 0.001
MultiVAE |  0.311 $\pm$ 0.012 |  0.2925 $\pm$ 0.012 |  0.3359 $\pm$ 0.005 |  0.3465 $\pm$ 0.007 |  0.3385 $\pm$ 0.005 |  0.3633 $\pm$ 0.005 |  0.0172 $\pm$ 0.004 |  0.012 $\pm$ 0.002
MultiVAE++ |  0.3356 $\pm$ 0.011 |  0.3079 $\pm$ 0.011 |  0.3488 $\pm$ 0.01 |  0.3616 $\pm$ 0.007 |  0.3409 $\pm$ 0.004 |  0.3636 $\pm$ 0.004 |  0.071 $\pm$ 0.011 |  0.0492 $\pm$ 0.007
F-SDRM |  0.3375 $\pm$ 0.008 |  0.3103 $\pm$ 0.01 |  0.3601 $\pm$ 0.006 |  0.3754 $\pm$ 0.006 |  0.3451 $\pm$ 0.006 |  0.3693 $\pm$ 0.006 |  0.013 $\pm$ 0.004 |  0.0085 $\pm$ 0.003
M-SDRM |  0.3474 $\pm$ 0.009 |  0.3142 $\pm$ 0.005 |  0.3544 $\pm$ 0.006 |  0.3755 $\pm$ 0.007 |  0.3438 $\pm$ 0.005 |  0.3702 $\pm$ 0.004 |  0.0668 $\pm$ 0.005 |  0.0451 $\pm$ 0.004
Improvement | 3.51 % | 2.04 % | 0.89% | -0.87 % | 1.23% | 1.81 % | -6.28 % | -9.09%
Model | NeuCF
Original |  0.2399 $\pm$ 0.009 |  0.0818 $\pm$ 0.011 |  0.1006 $\pm$ 0.009 |  0.096 $\pm$ 0.005 |  0.0113 $\pm$ 0.001 |  0.0094 $\pm$ 0.001 |  0.0057 $\pm$ 0.001 |  0.0011 $\pm$ 0.001
CTGAN |  0.1786 $\pm$ 0.083 |  0.0405 $\pm$ 0.018 |  0.0532 $\pm$ 0.014 |  0.05 $\pm$ 0.019 |  0.0053 $\pm$ 0.001 |  0.0055 $\pm$ 0.001 |  0.0073 $\pm$ 0.002 |  0.0014 $\pm$ 0.001
TVAE |  0.0957 $\pm$ 0.034 |  0.0231 $\pm$ 0.011 |  0.1289 $\pm$ 0.011 |  0.1326 $\pm$ 0.008 |  0.0237 $\pm$ 0.005 |  0.023 $\pm$ 0.005 |  0.009 $\pm$ 0.008 |  0.0013 $\pm$ 0.001
CODIGEM |  0.1154 $\pm$ 0.108 |  0.0294 $\pm$ 0.032 |  0.0782 $\pm$ 0.014 |  0.0552 $\pm$ 0.012 |  0.0606 $\pm$ 0.006 |  0.0574 $\pm$ 0.005 |  0.0022 $\pm$ 0.001 |  0.0004 $\pm$ 0.001
DiffRec |  0.2613 $\pm$ 0.017 |  0.0881 $\pm$ 0.006 |  0.0726 $\pm$ 0.025 |  0.0629 $\pm$ 0.027 |  0.0143 $\pm$ 0.007 |  0.0134 $\pm$ 0.006 |  0.0168 $\pm$ 0.005 |  0.0031 $\pm$ 0.001
MultiVAE |  0.0834 $\pm$ 0.025 |  0.0196 $\pm$ 0.007 |  0.0951 $\pm$ 0.014 |  0.0894 $\pm$ 0.014 |  0.0198 $\pm$ 0.004 |  0.0195 $\pm$ 0.004 |  0.0174 $\pm$ 0.003 |  0.0042 $\pm$ 0.001
MultiVAE++ |  0.331 $\pm$ 0.01 |  0.1143 $\pm$ 0.002 |  0.2309 $\pm$ 0.013 |  0.1853 $\pm$ 0.009 |  0.126 $\pm$ 0.008 |  0.1154 $\pm$ 0.008 |  0.0259 $\pm$ 0.004 |  0.0059 $\pm$ 0.001
F-SDRM |  0.3339 $\pm$ 0.009 |  0.1137 $\pm$ 0.002 |  0.2421 $\pm$ 0.009 |  0.1913 $\pm$ 0.007 |  0.1265 $\pm$ 0.011 |  0.1146 $\pm$ 0.012 |  0.028 $\pm$ 0.014 |  0.0058 $\pm$ 0.002
M-SDRM |  0.332 $\pm$ 0.008 |  0.1141 $\pm$ 0.001 |  0.2303 $\pm$ 0.014 |  0.1814 $\pm$ 0.01 |  0.1296 $\pm$ 0.014 |  0.1212 $\pm$ 0.014 |  0.0297 $\pm$ 0.008 |  0.0064 $\pm$ 0.002
Improvement | 0.87 % | -0.17 % | 4.85 % | 3.23 % | 2.85 % | 5.02 % | 14.67 % | 8.47 %

###  5.3. Results

We compare SDRM against the baseline generative models used during SDRM
training and present the average and standard deviation of Recall@$k$ and
NDCG@$k$ over five runs in Tables 3 and 4. We also report the overall
percentage improvement of SDRM over the best-performing baseline model for
top-$10$ generated recommendations. In Table 6, we report the average
improvement over MultiVAE++ in Recall@$k$ and NDCG@$k$, calculated over all
the recommender algorithm and dataset combinations for both the augmented and
the synthetic dataset training setups. We report additional improvements for
various top-$k$ ($k\in{1,3,5,10,20,50}$) over the MultiVAE++ baseline in the
appendix in Tables 7 and 8.

####  5.3.1. Training with augmented dataset

As shown in Table 3, F-SDRM and M-SDRM achieved the best or second-best
results in all Recall@10 and all but one NDCG@10 results. There are many runs
where MultiVAE++ placed second-best over either F-SDRM or M-SDRM, highlighting
that even doing diffusion over a Gaussian latent variable can boost the
performance. Overall, SDRM improves over the baseline generative methods by
4.48% for Recall@10 and 5.07% for NDCG@10 with the single best overall
improvement from baseline of 24.28% and 21.87% for Recall and NDCG,
respectively.

####  5.3.2. Replacing training data with the synthetic dataset

As shown in Table 4, when substituting the original datasets with synthetic
datasets, SDRM again consistently achieves the best or second best overall
results for each of the recommender models and averages an overall improvement
of 2.08% for Recall@10 and 0.88% for NDCG@10, achieving as high as 14.67% and
8.47% for Recall and NDCG, respectively, on the Amazon Digital Music dataset.
This demonstrates the potential of using SDRM as a method to substitute an
original dataset and achieve competitive or improving performance.

####  5.3.3. Comparing SDRM against MultiVAE

Since SDRM employs a MultiVAE for compressing and representing user-item data
as a smaller latent variable $z$, then learns to transform one Gaussian
representation into another, it raises a legitimate question whether the
additional computational effort for SDRM genuinely enhances performance
compared to using MultiVAE alone. To address this concern, we compared the
percentage increase of max Recall and NDCG value between M-SDRM and F-SDRM
(represented as SDRM in Figure 2) against MultiVAE++ from the same runs.

Figure 2. Baseline improvement of SDRM over MultiVAE++ across all datasets for
various sizes of top-$k$ recommendation lists. Augmented average improvement:
Recall@$k$: 6.81%, NDCG@$k$: 7.73%, Synthetic average improvement: Recall@$k$:
1.79%, NDCG@$k$: 1.56%, Combined average improvement: Recall@$k$: 4.30%,
NDCG@$k$: 4.65%

We observe that SDRM improves the overall Recall@$k$ and NDCG@$k$ scores
compared to using MultiVAE across different $k$ values. Our method achieves an
average 6.81% and 7.73% increase in Recall@$k$ and NDCG@$k$ when augmenting
the original data with synthetic and 1.79% and 1.56% increase in Recall@$k$
and NDCG@$k$. The scores shown in Figure 2 and Table 6 (located in the
appendix) are the average increase across each of the three baseline
recommender models and four datasets for each Recall@$k$ and NDCG@$k$. SDRM
has the greatest average performance increase in Recall for lower $k$ values
scores when used for augmentation and a higher average performance increase in
NDCG for larger $k$ scores when used as a substitution method. These findings
further demonstrate that using a score-based diffusion model between the
encoder and decoder of a MultiVAE architecture can significantly increase
performance.

###  5.4. Privacy-Preservation

It has been previously shown that diffusion models and variational
autoencoders can effectively reproduce training data even when sampled from a
Gaussian distribution (Carlini et al., 2023; Li et al., 2019). It has also
been observed that using a Gaussian distribution for generating synthetic data
does not guarantee the privacy of the data (Heller and Fetaya, 2023). However,
considering the high sparsity of recommendation data and the design of SDRM to
generate binary item-rating preferences, we posit that our approach does not
replicate the original training data, even when adhering to the same sparsity
constraints as the original data.

To confirm that the proposed model is privacy-sensitive, we present empirical
evidence that demonstrates that the overwhelming majority of SDRM-generated
data does not reproduce the training examples from each dataset, while
maintaining a similar distribution of rated items and user ratings as to the
original.

####  5.4.1. Assessing Similarity

To assess the difference between the synthetic and original data, we employ
the Jaccard similarity metric, which quantifies the similarity between two
sets by dividing the size of the intersection of the sets by the size of their
union.

(4)
$J(x,\hat{x})={{|x\cap\hat{x}|}\over{|x\cup\hat{x}|}}={{|x\cap\hat{x}|}\over{|x|+|\hat{x}|-|x\cap\hat{x}|}}$

We obtain a score for each pairwise comparison of each synthetic user
$\hat{x}$ to genuine user data $x$ using M-SDRM and the MultiVAE++ and present
the results in Table 5. We also explore the distribution of the items-per-user
and user-per-items cardinalities in the synthetic data compared to the
original dataset, as shown in Figure 3.

Table 5. Pairwise comparison using Jaccard similarity on original data vs synthetic data with M-SDRM and MultiVAE++ on Amazon Digital Music dataset Similarity | M-SDRM | Total % | MultiVAE++ | Total %
---|---|---|---|---
0.0 - 0.1 | 90,331,367 | 98.88% | 89,949,754 | 98.46%
0.1 - 0.2 | 989,059 | 1.08% | 1,365,015 | 1.49%
0.2 - 0.3 | 32,636 | ¡ 1% | 38,417 | ¡ 1%
0.3 - 0.4 | 1,842 | ¡ 1% | 1,796 | ¡ 1%
0.4 - 0.5 | 320 | ¡ 1% | 289 | ¡ 1%
0.5 - 0.6 | 77 | ¡ 1% | 64 | ¡ 1%
0.6 - 0.7 | 43 | ¡ 1% | 16 | ¡ 1%
0.7 - 0.8 | 12 | ¡ 1% | 8 | ¡ 1%
0.8 - 0.9 | 8 | ¡ 1% | 2 | ¡ 1%
0.9 - 1.0 | 0 | ¡ 1% | 3 | ¡ 1%
Figure 3. Distribution of the number of items and users for the Amazon Digital
Music dataset.

Even though each original dataset has been pre-processed to account for each
user having a minimum of 5 ratings per item, SDRM produces synthetic users
with less than five items (something that is not necessarily bad, given that
it better reflects real datasets that have such cold-start users and items).
Even so, SDRM still adheres to the natural long-tail distribution in
recommendation data, where a smaller number of users (items) account for the
majority of related item (user) ratings in the dataset, leading us to believe
SDRM accurately models intricate item-rating distributions in a recommendation
dataset.

####  5.4.2. Assessing Private Synthetic Data

Table 5 reveals that using M-SDRM to generate datasets of the same size and
sparsity as the original does not reproduce (copy) the training examples,
resulting in almost 99% dissimilarity. This significant difference is partly
illustrated in Figure 3, demonstrating that the synthetic dataset’s long-tail
distribution of rated items is less right-skewed than the original. Despite
this, the user-per-item distribution remains consistent between the synthetic
and original datasets, highlighting SDRM’s effectiveness in synthesizing data
while preserving preference distributions per item.

Overall, our analysis demonstrates that SDRM-generated data can balance the
ability to protect users in privacy-sensitive circumstances while still
offering the ability to train recommender system models with equal or better
performance than the original, something that true differential privacy
implementations inherently struggle with (Bagdasaryan et al., 2019).

##  6\. Conclusion

In this work, we propose SDRM, a novel approach to generating synthetic data
through score-based diffusion modeling. We take advantage of the VAE framework
to capture compressed latent representations into a Gaussian distribution and
apply multi-resolution sampling to further improve distributional modeling for
recommendation systems. Through various ablation studies, we show that our
model is capable of modeling complex distributions and achieves an overall
average improvement of 4.30% in Recall@$k$ and 4.65% in NDCG@$k$ when applying
these novel samples to augment or replace existing datasets. We presented
evidence that our method produces data that is 99% dissimilar from the
original dataset while still retaining the characteristics of the original
data distribution. This work addresses the problem of dataset sparsity but
also creates a novel way for organizations to preserve the privacy of their
users’ data while still training robust recommender systems.

One limitation of our method is its dependence on a Multi-Layer Perceptron
(MLP) network for denoising Gaussian data, which generates synthetic user data
one sample at a time. This approach differs from diffusion models in image
processing, which employ segmentation models to address local pixel
relationships. Consequently, our method may overlook the opportunity to
generate user data in batches, capturing inherent clustering patterns between
users within the data. Additionally, our findings demonstrate the performance
primarily in smaller recommendation datasets. Due to resource limitations,
testing on such large-scale datasets was not feasible. The largest dataset we
experimented with (Amazon Digital Music) revealed the highest and lowest
performance of SDRM. This leaves room for further research to apply and
evaluate SDRM on larger datasets to determine whether SDRM can generalize to
large datasets.

Future work will integrate auxiliary and temporal data about items or users to
create conditioned synthetic data. This approach can help mitigate biases
against underrepresented populations in recommendation algorithms.

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##  7\. Additional Results Tables

Table 6. Baseline improvement over MultiVAE++ across all datasets. Augmented
average improvement: Recall@$k$: 6.81%, NDCG@$k$: 7.73%, Synthetic average
improvement: Recall@$k$: 1.79%, NDCG@$k$: 1.56%, Combined average improvement:
Recall@$k$ 4.30%, NDCG@$k$ 4.65%

Metrics | R@1 | R@3 | R@5 | R@10 | R@20 | R@50 | N@1 | N@3 | N@5 | N@10 | N@20 | N@50
---|---|---|---|---|---|---|---|---|---|---|---|---
Augmented | 12.7 % | 7.31 % | 5.96 % | 5.36 % | 4.5 % | 5.06 % | 12.7 % | 8.33 % | 7.69 % | 6.15 % | 6.01 % | 5.50 %
Only Synthetic | 1.04 % | -0.28 % | 1.87 % | 2.55 % | 2.46 % | 3.12 % | 0.97 % | 0.48 % | 1.55 % | 1.85 % | 2.02 % | 2.52 %

Table 7. SDRM augmented data percentage improvement over MultiVAE++ Datasets | ML-100k | ALB | ML-1M | ADM
---|---|---|---|---
Baseline Models | SVD | NeuMF | MLP | SVD | NeuMF | MLP | SVD | NeuMF | MLP | SVD | NeuMF | MLP
Recall@1 | 1.16 | 18.81 | 2.43 | 6.04 | 40.28 | 5.29 | 2.57 | 7.92 | 3.91 | 5.00 | 51.72 | 7.26
Recall@3 | 0.45 | 4.33 | 3.57 | 2.56 | 9.62 | 0.96 | 1.37 | 6.16 | 2.99 | 16.67 | 30.95 | 8.12
Recall@5 | 0.29 | 4.60 | 1.68 | 1.94 | 13.89 | -0.47 | 1.07 | 1.76 | 2.66 | 16.20 | 20.29 | 7.64
Recall@10 | 2.40 | 2.89 | 1.18 | 1.86 | 20.02 | 0.81 | 0.13 | 1.24 | 1.90 | 4.57 | 21.88 | 5.56
Recall@20 | 1.03 | -1.95 | 1.76 | 0.76 | 18.38 | 0.55 | 0.40 | 1.11 | 2.10 | 2.60 | 20.11 | 7.18
Recall@50 | 0.84 | 2.00 | 0.10 | 2.36 | 17.60 | -0.05 | -0.02 | -1.08 | 2.99 | 5.15 | 22.31 | 8.61
NDCG@1 | 1.16 | 18.81 | 2.43 | 6.04 | 40.28 | 5.29 | 2.57 | 7.92 | 3.91 | 5.00 | 51.72 | 7.26
NDCG@3 | 0.83 | 7.06 | 3.91 | 3.43 | 18.63 | 1.47 | 1.83 | 6.80 | 2.88 | 16.07 | 28.57 | 8.49
NDCG@5 | 0.70 | 5.27 | 2.65 | 2.80 | 22.31 | 0.84 | 1.49 | 3.60 | 2.64 | 15.89 | 25.93 | 8.20
NDCG@10 | 1.33 | 1.78 | 2.17 | 2.68 | 24.29 | 1.00 | 0.79 | 1.85 | 2.18 | 8.87 | 20.00 | 6.95
NDCG@20 | 0.62 | 2.82 | 1.83 | 2.33 | 22.52 | 0.80 | 0.71 | 2.25 | 2.18 | 7.18 | 21.43 | 7.48
NDCG@50 | 0.82 | 3.08 | 1.12 | 2.71 | 21.20 | 0.49 | 0.42 | 0.67 | 2.49 | 7.24 | 17.65 | 8.12
Table 8. SDRM synthetic training data percentage improvement over MultiVAE++ Datasets | ML-100k | ALB | ML-1M | ADM
---|---|---|---|---
Baseline Models | NeuMF | MLP | SVD | NeuMF | MLP | SVD | NeuMF | MLP | SVD | NeuMF | MLP | SVD
Recall@1 | -6.94 | 13.36 | 2.42 | 0.52 | 0.00 | 2.48 | 9.88 | 3.90 | 1.30 | -15.11 | -9.06 | 9.72
Recall@3 | 2.07 | 6.40 | -2.04 | -1.91 | 2.05 | -2.01 | 8.69 | 3.07 | 0.74 | -10.07 | -12.86 | 2.48
Recall@5 | 3.73 | 5.23 | 0.64 | -0.62 | 2.26 | -0.65 | 8.08 | 2.22 | 1.04 | 4.74 | -9.88 | 5.67
Recall@10 | 4.85 | 3.24 | 1.47 | 0.88 | 3.52 | 1.91 | 2.86 | 1.23 | 0.08 | 14.67 | -5.92 | 1.78
Recall@20 | 2.31 | -1.02 | 0.17 | -0.70 | 1.72 | 1.76 | 2.07 | 1.10 | 0.11 | 31.18 | -9.68 | 0.52
Recall@50 | 2.69 | 0.35 | 1.37 | -1.87 | 1.68 | 2.88 | 1.03 | 0.23 | -0.02 | 33.84 | -6.02 | 1.29
NDCG@1 | -6.94 | 13.36 | 2.42 | 0.52 | 0.00 | 2.48 | 9.88 | 3.90 | 1.30 | -15.91 | -9.06 | 9.72
NDCG@3 | 0.64 | 7.92 | -0.90 | -0.92 | 1.44 | -0.39 | 8.94 | 3.28 | 0.77 | -8.70 | -11.97 | 5.59
NDCG@5 | 2.08 | 5.47 | 0.58 | 0.00 | 1.48 | 0.19 | 8.53 | 2.71 | 0.88 | 1.30 | -10.92 | 6.27
NDCG@10 | 3.24 | 3.84 | 0.87 | -0.17 | 2.05 | 1.29 | 5.03 | 1.82 | 0.28 | 8.47 | -8.33 | 3.88
NDCG@20 | 2.75 | 2.00 | 0.27 | -0.65 | 1.36 | 1.14 | 3.79 | 1.55 | 0.27 | 19.57 | -9.80 | 1.93
NDCG@50 | 2.54 | 2.44 | 0.73 | -0.90 | 1.00 | 1.58 | 2.82 | 1.05 | 0.26 | 24.24 | -8.10 | 2.55