Vol-3194/paper41

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id	Vol-3194/paper41
wikidataid	Q117344879→Q117344879
title	Generating Synthetic Discrete Datasets with Machine Learning
pdfUrl	https://ceur-ws.org/Vol-3194/paper41.pdf
dblpUrl	https://dblp.org/rec/conf/sebd/0001RRSS22
volume	Vol-3194→Vol-3194
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Generating Synthetic Discrete Datasets with Machine Learning

Generating Synthetic Discrete Datasets with Machine
Learning
(Discussion Paper)

Giuseppe Manco1 , Ettore Ritacco1 , Antonino Rullo2 , Domenico Saccà2 and
Edoardo Serra3
1
  Institute for High Performance Computing and Networking (ICAR) of the Italian National Research Council (CNR), v. P.
Bucci 8/9C, 87036 Rende (CS), Italy
2
  DIMES Department, University of Calabria, Rende, CS, 87036, Italy
3
  Computer Science Department, Boise State University, Boise, ID 83725, USA


                                         Abstract
                                         The real data are not always available/accessible/sufficient or in many cases they are incomplete and
                                         lacking in semantic content necessary to the definition of optimization processes. In this paper we
                                         discuss about the synthetic data generation under two different perspectives. The core common idea is
                                         to analyze a limited set of real data to learn the main patterns that characterize them and exploit this
                                         knowledge to generate brand new data. The first perspective is constraint-based generation and consists
                                         in generating a synthetic dataset satisfying given support constraints on the real frequent patterns. The
                                         second one is based on probabilistic generative modeling and considers the synthetic generation as a
                                         sampling process from a parametric distribution learned on the real data, typically encoded as a neural
                                         network (e.g. Variational Autoencoders, Generative Adversarial Networks).

                                         Keywords
                                         Synthetic dataset, Data generation, Inverse Frequent Itemset Mining, Constraints-based models, Varia-
                                         tional Autoencoder, Generative Adversarial Networks, Generative models


           This paper is an extended abstract of [1].


1. Introduction
Emerging “Big Data” platforms and applications call for the invention of novel data analysis
techniques that are capable to effectively and efficiently handle large amount of data [2]. There
is therefore an increasing need to use real-life datasets for data-driven experiments but the
scarcity of significant datasets is a critical issue for research papers [3]. Synthetic data generation
can help in this, by reproducing the internal mechanisms and dependencies that justify the
occurrence of some specific pieces of information, and hence being able to replicate them
stochastically. In this paper we focus on the problem of generating high-dimensional discrete
data. Generating such data is a challenge because of the combination of both high dimensionality
and discrete components, which may result in a complex structural domain with lots of variety
and irregularities, not necessarily smooth. Throughout the paper, we shall study approaches to
data generation which rely on the idea that each point in the real domain can be mapped into a

SEBD 2022: The 30th Italian Symposium on Advanced Database Systems, June 19-22, 2022, Tirrenia (PI), Italy
                                       © 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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    Workshop
    Proceedings
                  http://ceur-ws.org
                  ISSN 1613-0073
                                       CEUR Workshop Proceedings (CEUR-WS.org)
�Table 1
Transactional Dataset end Frequent Itemsets.

                      𝑎   𝑏   𝑐    𝑑      𝜎𝒟
                      1   1   1    0      40    𝑆𝑖    𝑎    𝑏    𝑐   𝑑         𝜎 𝑆𝑖
                      0   1   1    1      40    𝑆1    1    1    0   0         100
                      1   1   0    0      60    𝑆2    0    1    1   0         100
                      0   1   1    0      20    𝑆3    0    0    1   1          50
                      1   0   1    1      10
                                                (𝑏) Frequent Itemsets with threshold 50
                          (𝑎) Dataset 𝒟



suitable latent space and vice versa. These mappings guarantee a consistency with regards to
the original dataset. At the same time, the manifold in the latent space summarizes the main
characteristics of the data, that can hence be injected into the synthesized data in a controlled
way. We first discuss two main approaches, that are the Inverse Frequent Itemset Mining (IFM)
and probabilistic generative modeling (PGM); finally, a comparison of the results obtained with
some of the presented algorithms are provided.


2. Inverse Frequent Itemset Mining-based Generative Models
Let ℐ be a finite domain of 𝑛 elements, also called items. Any subset 𝐼 ⊆ ℐ is an itemset over
ℐ, also called a transaction. Let 𝒰ℐ denote the set of all itemsets on ℐ; then, |𝒰ℐ | = 2𝑛 . A
(transactional) database 𝒟 is a set of tuples [𝑘, 𝐼], where 𝑘 is the key and 𝐼 is an itemset. The size
|𝒟| of 𝒟 is the total number of its itemsets, i.e., transactions. A transactional database 𝒟 is very
often represented as a bag of itemsets, i.e, the keys are omitted so that tuples are simply itemsets
and may therefore occur duplicated – in this case 𝒟 is also called a transactional dataset. In
the paper we shall also represent an itemset 𝐼 ∈ 𝒟 by its one-hot encoding x , that is a binary
vector of size 𝑛 (the number of items in ℐ) such that its 𝑖-th position 𝑥𝑖 = 1 if the 𝑖-th item in ℐ
is in 𝐼, 0 otherwise. Consequently, 𝒳 = {x 1 , . . . , x 𝜂 }, where 𝜂 = |𝒟|, is the one-hot encoding
of the whole dataset 𝒟. For each itemset 𝐼 ∈ 𝒟, there exist two important measures: (i) the
number of duplicates of 𝐼, denoted as 𝛿 𝒟 (𝐼), that is the number of occurrences of 𝐼 in 𝒟, and (ii)
the support of 𝐼, denoted as 𝜎 𝒟 (𝐼), that
                                        ∑︀ is the sum of the number of duplicates of each itemset
𝐽 in 𝒟 containing 𝐼, i.e., 𝜎 𝒟 (𝐼) = 𝐽∈𝒟∧𝐼⊆𝐽 𝛿 𝒟 (𝐽) . A dataset 𝒟 can be represented in a
succinct format as a set of pairs (𝐼, 𝜎 𝒟 (𝐼)). Given ℐ = {𝑎, 𝑏, 𝑐, 𝑑}, an example of dataset in the
succinct, one-hot format is shown in Table 1-𝑎. We say that 𝐼 is a frequent (resp., infrequent)
itemset in 𝒟 if its support is greater than or equal to (resp., less than) a given threshold. A
classical data mining task over transaction datasets is to detect the set of the frequent/infrequent
itemsets, and a rich literature deals with this topic [4, 5, 6] Given the threshold 50, the frequent
itemsets for the dataset of Table 1-𝑎 are listed in Table 1-𝑏. The perspective of the frequent
itemset mining problem has been later inverted as follows: given a set of itemsets together with
their frequency constraints the goal is to compute, if any, a transaction dataset satisfying the
above constraints. This new problem is called the inverse frequent itemset mining problem (IFM)
algorithms [7], and has been later investigated also in privacy preserving contexts [8]. Given
a set ℐ of items, the IFM problem consists in finding a dataset 𝒟 that satisfies given support
�Table 2
Examples of Feasible Datasets with size 170 for an IFM instance.

             𝑎    𝑏   𝑐   𝑑   𝜎𝒟          𝑎    𝑏   𝑐   𝑑   𝜎𝒟          𝑎    𝑏   𝑐   𝑑    𝜎𝒟
             1    1   1   0   70          1    1   1   0   40          1    1   1   1    30
             0    1   1   1   10          0    1   1   1   40          1    1   1   0    30
             1    1   0   0   30          1    1   0   0   60          0    1   1   1    20
             0    1   1   0   20          0    1   1   0   20          1    1   0   0    40
             0    0   1   1   40          0    0   1   1   10          0    1   1   0    50
                 (𝑎) Dataset 𝒟1               (𝑏) Dataset 𝒟2               (𝑐) Dataset 𝒟3


constraints on some itemsets 𝑆𝑖 on ℐ — the set of such itemsets is denoted by 𝑆. The support
constraints are represented as follows: ∀𝑆𝑖 ∈ 𝑆 : 𝜎𝑚𝑖𝑛     𝑖    ≤ 𝜎 𝒟 (𝑆𝑖 ) ≤ 𝜎𝑚𝑎𝑥𝑖   , where 𝜎 𝒟 (𝑆𝑖 ) is
the sum of all number of duplicates of itemsets in 𝒟 containing 𝐼. As an example, consider
ℐ = {𝑎, 𝑏, 𝑐, 𝑑}, 𝑆 = {{𝑎, 𝑏}, {𝑏, 𝑐}, {𝑐, 𝑑}} and the support constraints represented in Table
1-𝑏 – in this example minimal and maximal supports coincide. The itemsets 𝑆1 = {𝑎, 𝑏} and
𝑆2 = {𝑏, 𝑐} must occur in exactly 100 transactions (possibly as their sub-transactions) whereas
the itemset 𝑆3 = {𝑐, 𝑑} must occur in exactly 50 transactions. It is also required that the dataset
size (i.e., the total number of transactions) be 170. The dataset 𝐷1 shown in Table 2-𝑎 is feasible
as it satisfies all constraints: 𝑆1 is satisfied by the transactions {𝑎, 𝑏, 𝑐} and {𝑎, 𝑏}, 𝑆2 by the
transactions {𝑎, 𝑏, 𝑐}, {𝑏, 𝑐, 𝑑} and {𝑏, 𝑐} and 𝑆3 by the transactions {𝑏, 𝑐, 𝑑} and {𝑐, 𝑑}.
   Let 𝑆 ′ be the set of all itemsets that are neither in 𝑆 nor subsets of some itemset in 𝑆. In
the example, 𝑆 ′ consists of {𝑎, 𝑏, 𝑐, 𝑑}, {𝑎, 𝑏, 𝑐}, {𝑎, 𝑏, 𝑑}, {𝑎, 𝑐, 𝑑}, {𝑏, 𝑐, 𝑑}, {𝑎, 𝑐}, {𝑎, 𝑑} and
{𝑏, 𝑑}. IFM does not enforce any constraint on the itemsets in 𝑆 ′ and, therefore, it may happen
that 𝒟 contains additional (and, in some cases, unsuspected or even undesired) frequent itemsets.
In the dataset 𝒟1 of Table 2-𝑎, the itemset {𝑎, 𝑏, 𝑐} is in 𝑆 ′ but it turns out to be frequent with a
support of 70. To remove the anomaly, Guzzo et al. [9] have proposed an alternative formulation,
called IFM𝑆 , that requires that only itemsets in 𝑆 can be included as transactions in 𝒟 and,
therefore, no unexpected frequent itemsets may eventually occur. Obviously, the decision
complexity of this problem is lower as it is NP-complete. Despite the complexity improvement,
the IFM𝑆 formulation has a severe drawback: it is too restrictive in excluding any transaction
besides the ones in 𝑆 as confirmed by the fact that no feasible dataset exists for our running
example. To weaken the tight restrictions of IFM𝑆 , Guzzo et al. [10] proposed a new formulation
of the problem, called IFM with infrequency support constraints (IFMI for short), which admits
transactions in 𝑆 ′ to be in a feasible dataset if their supports are below a given threshold 𝜎 ′ . By
the anti-monotonicity property, the number of infrequency support constraints can be reduced
by applying them only to a subset of 𝑆 ′ consisting of its minimal (inclusion-wise) elements. This
subset, denoted by 𝐵𝑆 ′ , is called the negative border and coincides with the set of all minimal
transversals of the hypergraph 𝐸     ¯ = {ℐ ∖ 𝐼 : 𝐼 ∈ 𝑆} (as defined in [11]). In the example,
𝐵𝑆 ′ = {{𝑎, 𝑐}, {𝑎, 𝑑}, {𝑏, 𝑑}} and the dataset 𝒟2 in Table 2-𝑏 is a feasible dataset for IFMI for
𝜎 ′ = 40. In fact, all infrequency support constraints on 𝐵𝑆 ′ are satisfied as the supports of
{𝑎, 𝑐}, {𝑎, 𝑑}, {𝑏, 𝑑} are respectively 40, 0 and 40. Another possibility to enforce infrequency
constraints is to fix a duplicate threshold 𝛿 ′ so that an itemset in 𝑆 ′ is admitted as transaction
�in a feasible dataset if its number of occurrences is at most 𝛿 ′ . This formulation has been given
in [12] with the name of IFM with infrequency duplicate constraints (IFMD for short). Observe
that duplicate constraints are less restrictive than infrequency constraints in the sense that
some itemset 𝐼 in 𝐵𝑆 ′ may happen to be eventually frequent as it may inherit the supports of
several itemsets in 𝑆 ′ with duplicates below the threshold. For instance, given the threshold
𝛿 ′ = 30, the dataset 𝒟3 in Table 2-𝑐 is a feasible dataset for IFMD . However, the supports of
{𝑎, 𝑐}, {𝑎, 𝑑}, {𝑏, 𝑑} are respectively 60, 30 and 50, thus {𝑎, 𝑐} and {𝑏, 𝑑} are frequent.


3. Machine Learning-based Generative Models
The probabilistic approach to model transactional data (PGM - probabilistic generative modeling)
assumes that in a database 𝒟 the itemsets are modeled as stochastic events: that is, they are
sampled from an unknown true distribution P𝑟 . The analysis of the statistical distribution of the
stochastic events provides insights on the mathematical rules governing the generation process.
The problem hence becomes how to obtain a smooth and reliable estimate of P𝑟 .
   In general, it is convenient to use a parametric model to estimate P𝑟 when the constraints on
the shape of the distribution are known. By associating each observation x with a probability
measure 𝑃 (x |𝜃) ≡ 𝑃𝜃 (x ) where 𝜃 is the set of the distribution parameters, our problem
hence becomes to devise the optimal parameter set 𝜃 that guarantees a reliable approximation
P𝜃 ≈ P𝑟 that can emulate the sampling process x ∼ P𝑟 in a tractable and reliable way. A
clear advantage of a parametric approaches to data generation lies in the insights that it can
provide within the data generation process. They allow to detect the factors governing the
data, providing a meaningful explanation of complex phenomena. In this work, we are focusing
on two state-of-the-art probabilistic approaches: Variational Autoencoders and Generative
Adversarial Networks.
   A Variational Autoencoder (VAE) is an artificial neural architecture that combines tra-
ditional autoencoder architectures [13] with the concept of latent variable modeling [14].
Essentially, we can assume the existence of a a 𝐾-dimensional latent space 𝒵, that can be the
generation engine of the samples in 𝒳 . The transactions 𝒳 = {x 1 , . . . , x 𝜂 } can be modeled
through a chain dependency: (i) given a distribution 𝑃𝜃 (over the parameter set 𝜃) we can
sample z ∈ 𝒵, and (ii) given a z and another distribution 𝑃𝜑 (over the parameter set 𝜑) we
can sample x . According to the maximum likelihood principle, optimal values of 𝜃 and 𝜑 can
be found by trying to maximizing 𝑃 (𝒳 ), but, unfortunately, this optimization is typically an
intractable problem that requires the exploitation of heuristics.
   Variational inference [15] introduces a proposal normal distribution 𝑄𝜆 (z |x ) (over the
parameter set 𝜆), whose purpose is to approximate the true posterior 𝑃 (z |x ). Hence, a VAE
is devised by concatenating two neural networks: an “Encoder”, that maps an input x into
a latent variable z , exploiting 𝑄𝜆 (z |x ), and a “Decoder” that reconstructs x by applying
𝑃𝜑 (x |z ) to z . The gain function, to be optimized to learn the network parameters 𝜆 and
𝜑, is obtained by marginalizing the log-likelihood of 𝑃 (𝒳 ) over z and applying Jensen’s
inequality: log 𝑃 (x ) ≥ Ez ∼𝑄 [log 𝑃 (x |z )] − KL [𝑄(z |x )‖𝑃 (z )], where KL is the Kullback-
Leibler divergence.
   The Decoder can be opportunely exploit to solve the discrete data generation problem. In fact,
�                 Embedding plot                               Embedding plot                               Embedding plot
 4                                            4                                            4
 3                                            3                                            3
 2                                            2                                            2
 1                                            1                                            1
 0                                            0                                            0
 1                                            1                                            1
 2                                            2                                            2
 3                                            3                                            3
 4                                            4                                            4
     5   4   3   2      1    0    1   2   3       5   4   3   2      1    0    1   2   3       5   4   3   2      1    0    1   2   3

     (a) Real data mapping in 𝒵               (b) Synthetic data generation                (c) Synthetic data generation
                                                      using MVAE                            using Clustering-variant of
                                                                                                      MVAE

Figure 1: Illustration of the data generation process within VAE.


we can sample as many z as we want, feed them to the Decoder and obtain brand new synthetic
data that is similar to the real one, if the VAE was properly trained. A simple generation approach,
called Multinomial VAE (MVAE), was proposed in [16], where x ∼ Multinomial (𝜋(z )), with
𝜋(z ) = softmax {exp [𝑃𝜑 (z )]} and z ∼ 𝒩 (0 , I 𝐾 ). A more sophisticated approach, that can
better observe the latent space of z and overcome the strong bias of considering only one
standard normal distribution, is implemented by clustering all the z ∼ 𝑄𝜆 (𝒳 ). The generation
of a synthetic x starts by choosing a cluster 𝑐, by exploiting a multinomial sampling according
to the clusters’ densities. Then, we can apply the MVAE sampling to pick up z ∼ 𝒩 (𝜇𝑐 , 𝜎 𝑐 ),
where 𝜇𝑐 and 𝜎 𝑐 are the mean and standard deviation within 𝑐. A comparison of these two
approaches is shown in Figure 1. Unfortunately, the approaches based on maximum likelihood
have been shown to suffer from over generalization [17], especially when data from P𝑟 is limited.
   Generative Adversarial Networks (GANs) [18] propose an alternative modeling which
departs from the maximum likelihood and instead focuses on an alternative optimization strategy.
In order to optimize the weights 𝜃 of a neural network, called Generator G, able to learn the
probability space P𝜃 , Adversarial Networks rely on an auxiliary classifier 𝐷, with weights 𝜑,
trained to discriminate between real and generated data. In practice, optimality can be achieved
when x ∼ P𝜃 is indistinguishable from x ∼ P𝑟 . The training process can be hence devised as a
competitive game, with the generator trying to produce realistic samples, starting from random
samples in the latent space 𝒵, and the classifier focusing on the detection of generated data,
where the objective function is min𝜃 max𝜑 Ex ∼P𝑟 [log 𝐷𝜑 (x )] + Ex ∼P𝜃 [log(1 − 𝐷𝜑 (x ))].
   It can be shown [18] that the adoption of this alternate optimization is equivalent to train 𝜃
to minimize the Jensen-Shannon divergence, that, differently from the approaches based on
maximum likelihood, has the objective of a complete adherence of P𝑟 and P𝜃 . In our context,
transactions are discrete vectors, with x ∈ {0, 1}𝑛 , thus backpropagation does not directly
apply to G and a workaround is needed. The simplest one is to admit a continuous relaxation of
the output of the generator. Just like with the variational autoencoder, the output of 𝐺 can be
modeled a multinomial probability (with no direct sampling channel), rather than a binary vector.
However, there is a major problem with this: the input of the discriminator would be a softmax
distribution from the generated transactions, and a binary vector for the real transactions. As
a result, the discriminator could easily tell them apart, with the result that the GAN would
get stuck in an equilibrium that is not good for the Generator. Different formulations of the
�Table 3
Originary experimental dataset.
                   Tuple ID   𝑖1   𝑖2   𝑖3   𝑖4    nd      itemset   frequent   support
                   𝑡1          1    0    1    1   210     𝑖1            y           640
                   𝑡2          0    1    0    1   140     𝑖2            y           550
                   𝑡3          1    1    1    0   110     𝑖3            y           450
                   𝑡4          1    0    0    0   100     𝑖4            y           500
                   𝑡5          1    1    0    0    90     𝑖1 , 𝑖2       y           270
                   𝑡6          0    0    0    1    80     𝑖1 , 𝑖3       y           380
                   𝑡7          0    1    1    0    70     𝑖1 , 𝑖4       y           280
                   𝑡8          1    1    0    1    70     𝑖3 , 𝑖4       n           210
                   𝑡9          0    1    0    0    70     𝑖2 , 𝑖3       n           180
                   𝑡10         1    0    1    0    60     𝑖2 , 𝑖4       n           210




adversarial training (based on Wasserstein distance [19], namely Wasserstein GANs or WGANs)
can partially mitigate this issue.


4. Comparative Analysis
In this section we compare the two approaches discussed in sections 2 and 3 by analyzing their
behavior on a set of controlled experiments. For these, we use a toy dataset, shown in Table 3,
comprising 4 items upon which 10 patterns are selected. The dataset used in the experiments is
hence built by replicating such patterns with some fixed frequencies. We use the following
approaches to generate the synthetic datasets:

    - IFM: IFM formulation with support ≥ 250.
    - IFM𝐼 : IFM formulation with infrequent itemsets’ support ≤ 210.
    - IFM𝐷 : IFM formulation with transaction duplicates ≤ 100.
    - IFM𝐷𝐼 : IFM𝐷 merged with IFM𝐼 .
    - IFM𝐷𝐼−5% : IFM𝐼 formulation imposing that each transaction has a number of duplicates
      that differ less than 5% from the number of duplicates in the original dataset.
    - VAE: Variational Auto Encoder generator.
    - VAE−{𝑡4 , 𝑡6 }: VAE without the generation of 𝑡4 and 𝑡6 transactions (as visible in Table 3)
      by means of sampling with rejection.
    - IFM𝐷𝐼−5% − {𝑡4 , 𝑡6 }: IFM𝐷𝐼−5% formulation imposing the number of duplicates for 𝑡4
      and 𝑡6 to be zero.

   The purpose of the experiments is to observe the reconstruction process in both methods
and compare the resulting reconstructed datasets. The comparison relies on the transactions
(𝑡1 . . . 𝑡10 in the table) as well as both simple items and item pairs. We evaluate the faithfulness
of the reconstruction in two respects: (1) whether the patterns are reproduced, and (2) whether
their frequencies are faithful. A simple∑︀      metric to measure the reconstruction accuracy is the
                                            |𝐷𝑖 −𝑂𝑖 |
discrepancy 𝒮, computed as 𝒮 =            𝑖∑︀
                                                      , where, for a pattern 𝑖 (either a transaction or an
                                              𝑖 𝑂𝑖
item pair), 𝐷𝑖 and 𝑂𝑖 represent the frequency of the pattern in the reconstructed and original
�Table 4
Discrepancy.
 Patterns                IFM    IFM𝐼     IFM𝐷    IFM𝐷𝐼    IFM𝐷𝐼−5%      VAE   VAE−{𝑡4 , 𝑡6 }   IFM𝐷𝐼−5% − {𝑡4 , 𝑡6 }
 Transactions         33.00%   12.00%   48.00%   50.00%       4.80%   7.00%        36.00%                  23.10%
 Items & item pairs    3.68%    0.82%    1.91%    0.82%       0.11%   3.02%        16.68%                   4.71%




Figure 2: Two-dimensional representation of the latent space


dataset, respectively. Table 4 reports the values of discrepancy, which are further detailed in
Figures 3-5.
   We first analyze the results of the reconstruction for a VAE-based generative model. Figure
3 shows a comparison with IFM𝐷𝐼−5% , the IFM-based formulation which provides the best
performances. The reconstruction provided by IFM𝐷𝐼−5% is extremely faithful, both on the
itemsets and the transactions. This because it enforces that the number of duplicates of each
transaction differs less than 5% from the number of duplicates in the original dataset. Figure 2
shows the details of how the original patterns are mapped into the latent generative space:
the leftmost picture shows the mapping of the original data, and the rightmost shows how
the generation results from a larger region of the latent space. The main advantage of the
approach based on generative modeling through latent variables is that the latter allows to
control the reconstruction process. By acting on the latter we can modify the characteristics of
the reconstructed space. For example, we see that transaction 𝑡11 (the only spurious transaction
generated by VAE) is placed in a specific region, denoted by a red star in the figure. Sampling
repeatedly from that region would allow us to change the overall distribution of the transactions
while still maintaining the itemset distribution. By contrast, the IFM-based approaches are in
general successful in maintaining the itemset distributions. However, they tend to produce a
higher noise with transactions unless not explicitly constrained by the IFM𝐷𝐼−5% formulation
(see Figure 4). This noise can in principle be considered an advantage in specific contexts where
a differentiation from the original dataset is required (e.g., due to privacy concerns).
   In principle, the adoption of IFM allows to implement a reconstruction "by design", by
choosing which itemsets to maintain or suppress. As an evidence, we report the cases of
IFM𝐷𝐼−5% − {𝑡4 , 𝑡6 } and VAE−{𝑡4 , 𝑡6 }, where transaction 𝑡4 and 𝑡6 are removed from the
generation phases. In fact, Figure 5 shows that IFM𝐷𝐼−5% − {𝑡4 , 𝑡6 } keeps the number of
�                             duplicates comparison IFM-VAE                                                                                        support comparison IFM-VAE
          250                                                                                                      700
                                                                                                                   600
          200
                                                                                                                   500
          150                                                                                                      400
          100                                                                                                      300
                                                                                                                   200
           50
                                                                                                                   100
            0                                                                                                        0
                 t1     t2   t3        t4   t5     t6     t7        t8        t9    t10 1001 1011 0010                       i1         i2         i3        i4       i1,i2      i1,i3      i1,i4   i3,i4    i2,i3    i2,i4

                                       original         VAE     IFM_DI 5%                                                                               original          VAE       IFM_DI 5%


Figure 3: Details of the discrepancies on the number of duplicates (left) and the itemsets support (right)
between VAE and IFM-based techniques.

                                                                                         duplicates IFM-based techniques
           250
           200
           150
           100
            50
            0
                       t1         t2              t3           t4                  t5           t6           t7           t8                 t9           t10             1001           0111         1011           0010
                                                                         original         IFM        IFM_I        IFM_D           IFM_DI           IFM_DI 5%

                                                                                           support IFM-based techniques
                      700

                      600

                      500

                      400

                      300

                      200

                      100

                       0
                                  i1               i2                    i3               i4            i1,i2             i1,i3               i1,i4               i3,i4             i2,i3            i2,i4
                                                                              original     IFM       IFM_I        IFM_D        IFM_DI         IFM_DI 5%


Figure 4: Details of the discrepancies on the number of duplicates (top) and the itemsets support
(bottom) between the original dataset and IFM-based techniques.


duplicates and the supports almost similar to the ones of the original dataset, while VAE−{𝑡4 , 𝑡6 }
changes many of them to remove the two transactions. The approach based on generative
modeling is in general more efficient. However, constraint-based generation is sensitive to the
frequency threshold and a suitable tuning can make these approaches comparable.
   To summarize, these experiments support an underlying intuition: Constraint-based genera-
tion allows more control on the expected outcome at the expense of a higher computational
cost, whereas probabilistic generative models provide more faithful reconstructions but are less
controllable. This essentially means that, without any further modeling artifact (that we do
not consider here), generative models are prone to fail in providing tailored reconstructions
where some patterns can be suppressed and new ones introduced. By contrast, contraint-based
generation is more suitable for reconstructions "by design".
�           duplicates comparison IFM-VAE w/o { t4,t6}                                    support comparison IFM-VAE w/o { t4,t6}
250                                                                          700
200
                                                                             500
150
100                                                                          300

 50
                                                                             100
  0
      t1      t2      t3       t5    t7       t8     t9    t10   1001 0011   -100   i1     i2    i3    i4   i1,i2   i1,i3   i1,i4   i3,i4   i2,i3   i2,i4
           original        VAE w/o { t4,t6}        IFM_DI w/o { t4,t6}              original     VAE w/o { t4,t6}      IFM_DI 5% w/o { t4,t6}



Figure 5: Details of the discrepancies on the number of duplicates (left) and the itemsets support (right)
between VAE and IFM-based techniques, without transactions 𝑡4 and 𝑡6 .


5. Conclusions
This paper has provided an overview about state-of-the-art approaches for synthetic transac-
tional data generation. A transaction has been modeled as a high-dimension sparse itemset,
that can be mapped into a binary vector, defined over the item space. The investigated algo-
rithms are (variants of the) Inverse Frequent Itemset Mining (IFM), and Probabilistic Generative
Models (PGMs). According to our analysis, the IFM approaches result to be extremely flexible
and understandable; they enable the control of the data generation procedure by the direct
identification of the discovery patterns to preserve. However, they proved to have extremely
onerous computational costs, making them not feasible in high-dimensional contexts. An
opposite conclusion has been obtained by analyzing PGMs: they are extremely fast and accurate,
but strongly lacking in control, flexibility and understandability. As future work, an interesting
research line is trying to investigate novel methodologies and techniques that are able to take
advantage of IFM and PGMs, by combining their strong points and mitigating their weakness.
Another promising research line is to apply the combination of the two approaches to NoSQL
applications by considering the extension of IFM that has been recently proposed in [12].


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