Joint Relational Database Generation via Graph-Conditional Diffusion Models
Authors: Mohamed Amine Ketata
David Lüdke
Leo Schwinn
Stephan Günnemann
Venue: NeurIPS 2025 poster ()
Keywords: relational databases
tabular data
generative models
diffusion models
graph neural networks
graph generation
Primary area: deep_learning
TL;DR: We propose the first non-autoregressive generative model for relational databases based on a graph view of the database, and we achieve state-of-the-art results on real-world RDBs.

Abstract: Building generative models for relational databases (RDBs) is important for many applications, such as privacy-preserving data release and augmenting real datasets. However, most prior works either focus on single-table generation or adapt single-table models to the multi-table setting by relying on autoregressive factorizations and sequential generation. These approaches limit parallelism, restrict flexibility in downstream applications, and compound errors due to commonly made conditional independence assumptions. In this paper, we propose a fundamentally different approach: jointly modeling all tables in an RDB without imposing any table order. By using a natural graph representation of RDBs, we propose the Graph-Conditional Relational Diffusion Model (GRDM), which leverages a graph neural network to jointly denoise row attributes and capture complex inter-table dependencies. Extensive experiments on six real-world RDBs demonstrate that our approach substantially outperforms autoregressive baselines in modeling multi-hop inter-table correlations and achieves state-of-the-art performance on single-table fidelity metrics. Our code is available at https://github.com/ketatam/rdb-diffusion.

Joint Relational Database Generation via Graph-Conditional Diffusion Models
Mohamed Amine Ketata, David Lüdke, Leo Schwinn, Stephan Günnemann
School of Computation, Information and Technology & Munich Data Science Institute Technical University of Munich, Germany Correspondence to: a. k.eetta@@um..de
Abstract
Building generative models for relational databases (RDBs) is important for many applications, such as However, most prior works either focus on single-table generation or adapt single- table models the multi-table setting by relying autoregressive factorizations and sequential generation. These approaches limit parallelism, restrict flexibility downstream applications, and compound errors due commonly made conditional independence assumptions. In this paper, we propose a fundamentally different ap- proach: jointly modeling all tables in RDB without imposing any table order. By using natural graph representation of RDBs, we propose the Graph-Conditions Relational Diffusion Model (GRDM), which leverages graph neural network to jointly denoise row attributes and capture complex inter-table dependencies. Extensive experiments on six real-world RDBs demonstrate that our approach sub- stantially outperforms autoregressive baselines in modeling multi-hop inter-table correlations and achieves state-of-the-ee performance on single-table fidelity mer- rics. Our code is available at https:://// /ggtttb........ com
data release and augmenting real datasets.
Introduction
Relational databases (RDBs), which organize data into multiple interlinked tables, are the most widely used data management system, estimated to store over 70% of the world's structured data [. RDBs are used in var- ious domains, including healthcare, finance, education, and e-commerce 2 However, increasing legal and ethical concerns around data privacy have led to strict regulations that limit access to data containing personal sensitive information. While these safeguards protect individuals and organizations, they can hinder the de- velopment of data-driven technologies that benefit from rich, structured data, thereby slowing scientific progress.
Autoregressive
Joint (Ours)
Tullin
Stap
Table
Tablu
Synthetic data generation has emerged promising approach to mitigate this problem by training genera- tive models on private datasets and releasing synthetic samples that preserve key statistical properties without disclosing sensitive information [4. (). In addition, syn- thetic data can enhance fairness, facilitate data augmen- tation, and support robust downstream analysis 6 7
Limiting table order Sequential gen ginenation Simplifying assumations
A+No table order
+ Paralllll generation
ooittt modeling rg of rows
Figure 1: Comparison of autoregressive and joint relational database generation.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Despite recent advances in synthetic data generation for single-table settings 8 12 multi-table relational database generation remains relatively underexplored. Notably, current approaches share common limitation: they impose fixed order on the tables and model their distribution autore- gressively [13]16] (ef. Figure [ left). This sequential generation procedure introduces several key limitations. It constrains downstream tasks, such missing data imputation, prevents parallel sampling, and often struggles with capturing complex long-range dependencies between tables. In addition, prior methods often adopt strong conditional independence assumptions that facilitate learning but sacrifice fidelity or generality 15].
We discuss further related work in Appendix A
In this paper, we propose a fundamentally different approach: we jointly model all tables in relational database without relying on a specific table ordering (cf. Figure rightt and propose the first non-autoregressive generative model for RDBs. To achieve this, we adopt graph-based based representation of RDBs [7], where nodes correspond to individual rows, and edges encode primary- foreign key relationships. By framing RDB generation
graph generation problem, we leverage random graph generation methods to first sample graph that preserves node degree distributions of the real graph, then jointly generate all node features using our novel Traph-Conditional Relational Diffusion Model (GRDM). Our main contributions are the following:
We introduce new framework that models the joint distribution over tables an RDB, removing the need for a predefined table ordering or autoregressive factorization.
We propose GRDM, the first non-autoregressive generative model for RDBs. It uses a graph-based representation and jointly generates all row attributes.
We demonstrate that GRDM significantly outperforms autoregressive baselines across six real-world RDBs, especially in capturing longrrange dependencies between tables.
2
Background
Relational Databases. A Relational Database (RDB) 18 RR C) (ef. Figure 2 top) is defined by a set of m interconnected tables, called relations, R = ( L R R that specifies the connections between them. Specifically, tuple (Rehidd Rparent) belongs to L if rows from Ryhitd contain references to rows from R Rporens, forming a parent-child
R(m)} and database schema
relationship. Each table (() represents one entity type and consists of n. rows (r(() ,..,, T...,, where each row is an instance of this entity and consists of three components 17
(2)
Primary key pr: a unique identifier for each row within a table.
Foreign keys K,, C {pr r' E R' and (R, R') L): set of primary keys of rows in other tables, that row r points to. We refer to these rows as r's parents.
Attributes w,: set of columns (features) describing this specifie instance. In this work, focus on databases with categorical and numerical attributes, which include ordinal, date, time, discrete, and continuous data.
Synthetic Relational Database Generation. The goal of synthetic RDB generation is learn a parameterized distribution pe as p(R) to sample a synthetic database R that adheres to the schema and preserves the statistical properties of R. The challenges of this learning problem are twofold:
(i) the complex distributions of columns with heterogeneous multi-modal features (column shapes),
(ii) the complex correlations between columns from the same table (intra-table trends), as well columns from different tables (inter-table trends) which can be directly (one-hop) or indirectly (multi-hop) connected by primary-foreign key links.
Autoregressive Models for RDB Generation. To learn these distributions and capture their correlations, existing generative models for RDBs rely on a specific table order, often the topological order induced by the schema L, to autoregressively factorize the joint distribution 13 16
p(R) = (
R(m) = =II ((())
{R())
< <+))
'Some works [6 19 generate all primary and foreign keys in first step, then apply similar autoregressive factorization the attributes (non-key columns).

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While this factorization can, in principle, model the true joint distribution, its inherent sequential nature introduces two key limitations. First, fixing the table order limits downstream applications. e.g., missing value imputation, because each table is conditioned only on its predecessors and cannot incorporate information from tables later in the order. Second, the factorization forces the generation runtime scale linearly with the number of tables and prevents parallelization across tables.
Moreover, learning the required conditional probability distributions is practically challenging. particularly for large RDBs with high-dimensional attributes and complex multi-hop inter-table dependencies. To make the problem tractable, prior work focused on RDBs with only two tables [[6] or resorted to simplifying assumptions such as Markov-like conditional independence assumptions of tables or individual rows given their parents [13) Such assumptions are particularly problematic in this autoregressive setting, where small errors can accumulate across generation steps, causing compounding errors and loss of coherence.
Joint Modeling of All Tables with Graph-Conditional Diffusion Models
In this work, we introduce the first non non-autoregressive generative model for RDBs by directly modeling the joint distribution of tables in R. Specifically, by avoiding any table ordering and leveraging the relational structure of the database to jointly model its rows, our approach imposes fewer assumptions and enables models that are more expressive, flexible, and parallelizable. We structure this section as follows: we introduce how to faithfully model RDBs as graphs in Section [.1] motivate our two-step graph generation procedure Section 3.2 describe how we generate the graph structure in Section 3.3 present novel diffusion model, which is the main contribution of this paper, in Section 3.4 and finally, discuss the special case of dimension tables in RDBs in Section 3.5
3.1 Relational Databases as Attributed Heterogeneous Directed Graphs
The relational structure of an RDB is naturally represented as directed heterogeneous graph that elegantly models its different components. A key advantage of adopting this graph- view is that we can leverage the vast literature on graph theory and graph machine learning to process relational data. This idea has gained traction in recent representation learning work 20 21 and has also been explored in generative modeling [16] 19), though existing approaches remain autoregressive. While there are several ways to represent an RDB graph [2T) we adopt the simple and intuitive formulation from Fey et al. [17), modeling an RDB (R, L) heterogeneous graph in which each row is a node and edges are defined by primary- foreign key links (cf. Figure (egureBbotomm))
Formally, we define the graph as G = (V,E,X)) with node set V representing the rows, edge set & representing the frimary-foreign key connections, and feature set X' representing the attributes. First, map each resulting in heterogeneous graph. The full node
Tabular
Customers
Products
Iraid
Gmeer
R() to node of type i,
Suthig
Relational Database
Calcerta
Prseat
Ris
set is given by V U p(o), where each y(i) {u(r) | R contains all nodes of type and v(r) denotes the node corresponding row and r(v) its inverse. Second, we define the edge set as
300
9530
253
13:6
314
15:4
E = ((11,
x V/ Po. E K., }
Graph
where P. Pr(v) and K., Kr)v)) shorthand for accessing a node's primary and foreign keys.
Each edge represents a primary-foreige key relation- ship between two rows. We assign each edge a type (i.j), based on the types of its endpoints v1 and U2. Since edges are ordered pairs, the result- ing graph is directed-a property that is essential reconstructing the original RDB from graph representation. Finally, the feature set is defined as X' = (æx, U V), where F., = Er(() contains the non-key attributes of the corresponding row. Thus,
Figure Tabular and graph representations of relational databases. We use different colours and different arrow shapes to depict different node and edge types, respectively.
3

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X' contains all columns in R that are not primary or foreign keys, while the primary and foreign keys are fully encoded in the (featureless) graph structure (V,
Notably, this graph construction is a faithful representation of the original RDB in the sense that one can reconstruct the original RDB R from the graph 9 up to permutations of the primary keys (thus also of foreign keys). One concrete procedure for this reconstruction is provided in Appendix [ In the following, we use G and R interchangeably refer to the graph and RDB representation of the same object. With this, our target data distribution becomes:
(RR) = p(G) = P(,,,
3.2
On the Factorization of p(V,
To find an efficient and scalable parametrization for p(V, X'), we examine its possible factorizations. First, note that V is implicitly defined by either X or E. Hence, we can reduce the problem to three viable factorizations: (A) one-shot; p(V, E (B) featurex-thhen-edger, ())E)) 2), or (C) figes-then-featuree p(V. ()X(()
To illustrate the scalability trade-offs among these factorizations, consider an RDB with two tables containing and m rows, respectively. Note that for factorizations (A) and (B), the edges generated along with or after the features identifying each node. Thus, they need to consider all m possible edges, which is computationally prohibitive for large and m, and increasingly so for databases with multiple tables. In contrast, factorization (C) generates the edges before the features, which allows to sidestep this combinatorial explosion by exploiting the exchangeability of nodes and simply parameterizing the number of nodes and their per edge-type degree for efficient random graph generation similar Xu al. [6]. Specifically, factorizing the distribution
p R)) = p(9)
corresponds to a two-step process. First, E)V,E) defines the structure of the attribute-free graph, equivalent to generating the primary and foreign key columns of the RDB. Second, p(X(V, E) generates the node features of the graph, or equivalently, the row attributes of the RDB. Note that this can be attained by jointly modeling each node with its neighborhood defined by the graph structure, allowing us to scale to very large databases. In the following, we build up on this factorization: Section 3.3 presents simple and effective algorithm for modeling p(V. E), followed by our main contribution, graph-conditional diffusion framework for modeling p( VV &), in Section 3.4
3.3 Node Degree-Preserving Random Graph Generation
The featureless graph EV,E) is directed, heterogeneous, and m-partite graph exist only between nodes of different types, corresponding to rows from different tables. We model pVV, E) using simple random graph generation algorithm that preserves the node degree distributions observed in the real graph [22]. We define a node's 's indegree and outdegree for a given edge type as the number of incoming and outgoing edges of that type. respectively. Outdegrees are constant per edge type, since each table has a fixed number of foreign key columns. Thus, our goal is to preserve the indegree distribution for each edge type.
The learning phase computes the empirical indegree distributions per edge type from the real graph. Sampling begins with root nodes-those with no outgoing edges. We set the number of root nodes to match the real graph, ensuring a similar overall graph size, but the graph size can also be scaled by a multiplier (see Appendix [E.4 for the corresponding experiment). For each root node, we sample its children according to the learned indegree distributions and recursively repeat this process for the sampled child nodes. When node has multiple parents, this procedure initially produces multiple copies for it. To correct this while preserving the overall indegree distribution, we perform a random matching merge duplicate nodes across these sets, before sampling their indegree sequence.
3.4
Graph-Conditional Relational Diffusion Model (GRDM)
Given the graph structure (V, E), we are now interested in generating the features X. Since this is now a feature generation task for a fixed graph structure, an interesting analogy to keep in mind image generation. An image can be viewed as a graph with simple 2D grid structure, where each pixel represents node with a 3 3-dimensional feature vector of color channels. In our case,

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we can think about p(X(VV)) as generating a very large "image", with more complex structure and heterogeneous node features. Inspired by the success of diffusion models image generation and other domains [6]. and given their unprecedented capacity to model joint distributions, we propose to model V E) using diffusion model. Our model can be viewed as generalization of image diffusion models to graphs with arbitrary structures and node features. In this section, we
use ( to denote the original data X' as is common in the diffusion models literature 23
3.4.1 Forward Process
Diffusion models 27 23 are latent variable models that define a sequence of latent variables , x²(T) of the same dimensionality as x² through a fixed Markov chain, called the forward process diffusion process, that gradually adds noise to data:
T
q( ( V.E)
While this formulation is very general and allows defining complex forward processes, we choose to add noise independently to all nodes at each timestep & (analogous to how pixels are independently noised in image diffusion models), which will enable efficient training of our model:
q(x10)22 1) V.E)
II q (( (e("-1))) VEV
where (1) is the feature vector of node v at diffusion timestep L. In this work, we assume that Ty consists of numerical and categorical features. A common way model categorical tabular data is multinomial diffusion (28) 8). however, it suffers significant performance overheads and we found it unstable for high-cardinality categorical variables. Therefore, we follow Pang al. 115 and map categorical variables to continuous space through label encoding (See Appendix C.2
for details) and apply Gaussian diffusion (DDPM (23 the unified space: q(ax!" [o! (o((-)))
N( -IVIIBBaa (t-1) 1,,1), where BI...., Br define the variance schedule.
An important consequence of this choice of the forward process, i.e., adding Gaussian noise inde-
pendently to all nodes, is that we can sample !! of an individual node at any timestep t directly from the clean data at time O, which is key for efficient training of the model. Let = 1 - Br and
ā IT ag then q( (+) (2 V) q(>(") (0))) ~N(x Vāra!" (1 a,)
3.4.2 Reverse Process
The generative model is defined another Markov chain with learned transitions; the reverse process,
T
po ()) 8)
II po po V.E).
(5)
This formulation of the reverse process also offers significant modeling flexibility; to predict the less noisy state 2 (t-1) of node v, the model can leverage the graph structure (V,E) and the features of all other nodes at time f, xt) However, doing so is not scalable for large graphs, but also not necessary because most of the nodes will be irrelevant. Therefore, we propose to condition the denoising of node at each timestep on the local neighborhood encoding the relations of the database, which achieves our goal of jointly modeling the nodes faithfully to the RDB structure. This is analogous to how image diffusion models based on convolutional neural networks (CNNs) condition the denoising of pixel on its neighboring pixels 23
To allow a node to interact not only with its parents, but also with its children, for the diffusion model, we treat the graph as undirected by adding reverse edges all existing edges in G. Formally, let G = (Vr, & x(i) denote the subgraph from 9 (now undirected), centered at node v. containing all nodes in the K-hop neighborhood around v, with features at diffusion time t.
Concretely, for K = 1, the nodes in G. correspond to all rows in the database that are referenced by r(v) and those that reference it. Note that we also bound the worst-case space and time
5

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complexity by sampling a fixed-size set of neighbors proposed by Hamilton et al. (29 Formally, we model the following factorization of the reverse process (Equation 5
p(X(O)V)))
II p(a)")) and (0( V.E) DEV
II po
(0) v.K)
(6)
-EV
crucial property of our reverse process formulation the following: due to the multi-step nature of diffusion models, the denoising of node is not restricted to only conditioning on its K-hop neighborhood, but can extend to further away nodes. To see this, let us consider two consecutive timesteps and identify which nodes in (x+++)) influence x (t-1) From Equation 6 we see that from
I + 1, only nodes in x influence _e- (t-1) However, from L + + L, all nodes in 2, are in turn influenced by nodes in their respective K -hoop neighborhood time + 1, which will indirectly influence x (?-1) These nodes form the subgraph _0,2k and by induction on the number of diffusion steps, we can show that _!!) is influenced by ) at time L + N, meaning that our model can in principle model interactions between node pairs that are ((T hops away in the graph during the whole reverse diffusion process. Empirically, this is demonstrated by model, which uses K = and effectively captures higher-order correlations (See Section for detailed experiments).
In the case of Gaussian diffusion, the reverse process is also Gaussian: p(x!") = ((x I,
and po ( (z=!-"))) x). = -N (e V. t). of where μe is a neural network that learns to approximate the less noisy state of node given its neighhorhood at time I, and or is hyperparameter.
3.4.3 Training and Sampling
Training. Recall that our end goal is to learn a parametrized diffusion model py for the distribution (( po can be learned by minimizing the model's negative log-likelihood (NLL) of the data, - log Po (x( V, E). However, the NLL is intractable to compute; we therefore optimize its variational bound 27
(0:2)) (V log po (2 (V,) < E. log (( V.
I(O)-
(Ø) can be equivalently expressed in terms of conditional forward process posteriors, which are tractable to compute (see Appendix B for detailed derivation). Furthermore, we follow Ho et al.
23 in parametrizing μ4g as ( (. vor I vT f (t) t)), where co predicts the
B.
noise added time i, and using simplified training objective (details are provided in Appendix B leading to the following objective that we use to train our model:
- (a (g )
Lsimple (0) (V) T | V'EV,,,>
(8)
where U samples an element from its input set uniformly at random. Our model, eg, is trained by minimizing Lsimpic(0) using stochastic gradient descent, where each iteration works as follows: node v is sampled uniformly at random and is noised along with its K-hop neighborhood to the same time step f, which is also sampled uniformly at random. Given the noisy subgraph g!)) Co learns to predict the noise vector e, that was added to e (but not the noise vectors added to the other nodes in its neighborhood). Algorithm T shows the detailed training procedure.
Sampling.
To sample from our model, we initialize all node features in the graph with Gaussian noise, x (2) ~ NO, I) V V. Then, each time step / TT, T..... 1, all nodes are jointly denoised before moving to the next step as follows:
x (t-1)
1
x (8)
B.
EB (o oizo VuEV,
Vor
VI
where z. NO I) ift 1 else 2, = O. The full sampling procedure is shown in Algorithm 2
nice property of our sampling procedure that at each time step, the denoising step (Equation can be run fully in parallel across all nodes in the graph, allowing us to scale to very large graphs. This is in contrast to prior works [55 19 that specify an order on the tables and can only start with generating one table, once all its predecessors have already been generated.

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3.4.4 Model Architecture
The denoising model Eg is a learnable function that maps a graph G centered at node and timestep t to a vector of the same dimension as T.. Formally, Eg : G x R + R where G is the set of Pesssag--aasiing Graph Neural Networks IMP-GNNS 30 B1 to parameterize Eg. Let (G, t) E G x denote the input to Eş with g = (V, x)). We define initial node embeddings as h (ar 1) V v V. One iteration of heterogeneous message passing consists of updating each node's 's embedding at iteration I, h based its neighbors, to get updated embeddings h2t where nodes and edges of different types are treated differently. In Appendix C.4 we present general formulation of heterogeneous MP-GNNs, In this work, we use the heterogeneous version of the GraphSAGE model [29] 32) with sum- nam-based neighbor aggregation. After L iterations of message passing, we obtain a set of deep node embeddings (he Jeyy We set I to be the same as the number of hops K used to select G. Finally, to predict the noise vector &y for the target node v, we further pass the final embedding of node e through node type-specific multi-layer perceptron (MLP) 33 é. MLP() which is the final output of Eq, where is the type of node
graphs. In this work, we use heterogeneous
3.5 Dimension Tables
In RDBs, some tables have a unique property: they have relatively few rows compared to other tables, and each row represents a unique real-world entity. An example such a table a COUNTRIES table that stores the different countries in the world. This specific type of table is well-known in the RDB literature and called dimension tables [34 (7), which contrasts with so-called fact tables. As discussed in [17). dimension tables store contextual information, macro statistics, and/or immutable properties of real-world entities, which makes these tables relatively small. In contrast, fact tables typically store interactions between entities, e.g., transactions ratings. Since entities interact frequently, fact tables have the majority of rows in an RDB.
In our generative modeling setting, dimension tables present a unique challenge and require special care. Since rows of dimension tables represent unique entities in the real world, there is no clear notion of generalization, and it is unclear how to generate synthetic dimension tables. Therefore, we propose to view dimension tables as metadata, on which we condition the generation of all other tables. This is reminiscent of how possible categories of a categorical random variable not generated but rather encoded into the generative model. In fact, one can join a dimension table with table that references it and get new categorical column (in the joined table) whose possible categories are the entities in the dimension table.
To condition our generative process on dimension tables, we simply avoid adding noise to them and keep them fixed at time f during training and sampling of the diffusion model. In the aforemen- tioned example of the COUNTRIES table, this modified process would only use the information about the countries themselves from the real RDB, while, for example, the information about customers' nationalities is encoded in the edges of the graph and is still stochastically sampled from our model. In our experiments, we define dimension tables as tables that contain at least one categorical column, where each possible category appears only once in that table.
Experiments
We evaluate our model's performance in synthetic relational database generation, using single-table and multi-table fidelity metrics, including long-range dependency metrics introduced in [15]. We present a comparison of our proposed approach to different sfate-of-the- baselines on six worl-world databases, followed by detailed analysis and an ablation study for our model. In Appendix L we evaluate other aspects, including privacy preservation, missing value imputation, and database size extrapolation.
4.1 Experimental Setup
Real-world Databases. We use six real-world relational databases in our experiments. Five were used the evaluation setup in [5]: Berka (35), Instacart 05 (36), Movie Lens [37 38), CCS 37],
and California [39]. In addition, we use the RelBench-FI database [40] from RelBench 41
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recently introduced benchmark for relational deep learning 177. nine tables-maen -ooe than any of the previous five RDBs and includes relatively high number of numerical and categorical columns, compared to other databases in RelBench. 'he used databases vary in the number of tables, size, maximum depth, number of inter-table connections, and feature complexity. We provide more details on these databases in Appendix [.11 Note that Berka, Instacart 05, and RelBench-FI exhibit the most complex inter-table correlations with up to 3-hop interactions.
We chose FI because it contains
Baselines, We compare our model to four relational database generation methods from the literature. SDV [13] is a statistical method tailored for RDB synthesis based on Gaussian Copulas. CPavaDDPM 15 is a state-of-the-an diffusion- based model that leverages cluster labels and classifier glassifier-guided diffusion models to generate child table conditioned on its parent. In addition, we adopt two synthesis pipelines used by Pang et al. [5]] to provide additional insights: SingleTabll and Denorm. SingleTTable generates foreign keys based on the real group size distribution, i.e., the distribution of row counts that have the same foreign key, similarly to our random graph generation a algorithm. However, it learns and generates each table individually and independently of all other tables. It can be seen as a version of our model with the number of hops K set to 0. Denorm is based on the idea of joining tables first, it learns and generates the joined table of every connected pair, and then splits it. For both these baselines, we use the same
tabular diffusion backbone as our model 23 For all DDPM-based basclines, we use the same hyperparameters for model architecture and training as our model for fair comparison. For SDV, we used the default hyperparameters as in [15]. For our model, we set K as we found this setting to achieve a good -effiiiy-fficiinccy tradeoff, but better fidelity metrics could be achieved with K > 1 at the cost of more compute, which we leave for future investigation.
the context of our experiments, we use "GRDM" to refer to our overall model, i.e., the graph generation component and the diffusion component. More implementation details are provided in Appendix [D.2]
Evaluation Metrics, To evaluate the quality of the synthetic data in terms of fidelity, we follow [15] and report the following metries implemented in the SDMetrics package [42]. 1) Cardinality compares the distribution of the number of children rows that each parent row has, between the real and synthetic data. We compute this metric for each parent-child pair and report the average across all pairs. 2) Column Shapes compares the marginal distributions of individual columns between the real and synthetic data. We compute this metric for every column of each table and report its average. 3) Mutra-Table Trends measures the correlation between pair of columns in the same table and computes the similarity in correlation between the real and synthetic data. We report the average across all column pairs of each table. 4) Inter-Table Trends (k-hop) is similar to the previous metric but measures the correlation between column pairs from tables at distance k, e.g., 1-hop for columns from parent-child pairs. For each possible & 2 1, report the average across all column pairs that are k hops away from each other. This is the most challenging and most interesting metric because it considers interactions between different tables.
For each of these metrics, we use the Kolmogorov-Sninn Smirnov (KS) statistic and the Total Variation (TV) distance to compare distributions of numerical and categorical values, respectively. To compare correlations of column pairs, we use the Pearson correlation coefficient for numerical values and the contingency table for categorical values. All metrics are normalized to lie between 0 (least fidelity) and 100 (highest fidelity). Detailed descriptions of these metric computations are in Appendix D.3
4.2 Evaluation Results
Table I reports the fidelity metrics described in Section across six databases for all methods, with standard deviations over three random seeds. GRDM consistently outperforms all baselines on Inter-Table Trends across all databases, particularly in capturing multi-hop correlations for the most complex RDBs. On Berka, which has the most complex schema with up to 3-hop dependencies, GRDM surpasses the best baseline by 14.14% on 3-hop correlations and 11.3% on 2-hop correlations. On Instacart 05 and RelBench-F1, GRDM improves over the best baseline by more than 5.7 times and 12.62% on 2 2-hop correlations, respectively. For 1 l-hop correlations, GRDM achieves an average gain of 15% compared to the best baselines over all databases. These improvements highlight the effectiveness of our joint modeling approach over the autoregressive methods used by other baselines (with the exception of SingleTable, which treats tables independently). Notably, the fact that our model, which uses neighborhood size K = 1, achieves very good multi-hop correlation metrics
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Table 1: Comparison of the fidelity metrics described in Section 4.1 The best method in each metric highlighted, and we report the relative improvement of GRDM compared to the closest baseline. Databases are ordered by complexity. SDV does not support RDBs with more than 5 tables.
SDV
SingleTable
Denorm
ClavaDDPM
GRDM (Ours)
Improvement (%)
Berka CARDINALITY
97.06 sea0
96.06
1:165
96.75 1:20
99.65 =0s
+2.67
COLUMN SHAPES
94.58
83.28
94.60
96.84 a0ts
+2.37
INTRA-TABLE TRENDS
5 tables
91.72 +
72.12 em
90.53 -100
98.23 -oa
+7.10
INTER-TABLE TRENDS (1-H10P)
81.77 a:1
55.77
83.79
91.41 =um
49.09
INTER-TABLE TRENDS (2-HOP)
78. e.30
57.68 +3
85.87 +5.7
96.57 -oa1
+11.30
INTER-TABLE TRENDS (3-HOP)
75.56 4:0s
55.59
1:14
80.98
#3/2
92.43 =wp
+14.14
Instacart 05
CARDINALITY
94.73 4.6.4
94.98 k.0.4
94.91 1::00
99.96 -not
+5.24
COLUMN SHAPES
89.30 +000
71.83
90.18 +0.8
98.39 -0os
49.10
INTRA-TABLE TRENDS
5 tables
99.70
88.74
99.68
97.59
-2.12
INTER-TABLE TRENDS (1-HOP)
66.93 +oor
62.58 6o0
75.84 6.30
88.49 -03
+16.68
INTER-TABLE TRENDS (2-H1OP)
16.22
0.00
14.40
92.95 =wm
+4473.06
RelBench-F1
CARDINALITY
94.94 1:106
93.39 4:1.6
94.04 1.300
98.44 4.0.s
+3.69
COLUMN SHAPES
asus
89.25
95.19
95.71
-0.23
INTRA-TABLE TRENDS
5 tables
95.80 annz
90.30
1.066
94.71 a00
95.00 a.:1
-0.84
INTER-TABLE TRENDS (1-HOP)
79.61 +8.6
65.60
88.19
90.85 -015
+3.02
INTER-TABLE TRENDS (2-HOP)
74.10
62.21
83.17
93.67
+12.62
Movie Lens
CARDINALITY COLUMN SHAPES
98.90 stus
98.87 1:1
98.79
98.80 2:000
-0.19
5 tables
99.19
78.03
99.11
97.22
-1.99
TATRA-TABLE TRENDS
98.56 ant
57.33 a.0.
98.52 1oc
95.25 a.0
-3.36
INTER-TABLE TRENDS (1-HOP)
92.72 suD
77.15
91.31
+1.75
CCS
CARDINALITY
74.36 $:40
99.37 +:.10
26.70 6:m0
98.96 6.70
99.79 -013
+0.42
COLUMN SHAPES
69.04
95.20
79.29
92.64
97.30
+2.21
INTRA-TABIE TRENDS
94.84 1:00
98.96 +:00
86.60 k.
97.76 +120
94.82 +:100
.4.18
INTER-TABLE TRENDS (1-HOP)
21.74 2
51.62
6:22
57.77
2:00
72.65 =...
85.38 =xuss
+17.52
California
CARDINALITY
71.45 1.00
99.89 k.0
99.87 come
98.99 dute
99.96 Lnot
+0.07
COLUMN SHAPES
72.32
99.51 +0o
94.99 too
98.76
98.75 +:000
-0.76
INTRA-TABLE TRENDS
50.23
98.69
94.17
97.65
97.35
-1.36
INTER-TABLE TRENDS (1-HOP)
54.89
8.00
92.96 exo
87.24
95.34 +0-8
96.16 -035
+0.86
confirms that long-range dependencies can be effectively captured through the diffusion process, as a node's receptive field increases with each diffusion step (see Section 3.4.2) for detailed discussion on this point). On single-table metrics, GRDM performs comparably to the best methods and even outperforms them on some databases.
4.3 Ablation Studies
Table 2: Comparison of our default setting from Table I with using the real graph structure instead of sampling it and with sequential version of our model.
To gain further insights into model and understand the performance of its different components, conduct two ablation studies that analyze the capac- ity of random graph generation algorithm (Section (.3) and the im- portance of the joint modeling aspect. For brevity, we only report results on the three databases with the most com- plex schema, Berka, Insiacart 05, and RelBench- shown in Table 2
GRDM
Real Graph
Sequential
Berka CARDINALITY
99.65 -055
100.0 +.m
99.65 -as
COLUMN SHAPES
96.84
96.
38.97
INTRA-TABLE TRENDS
98.23 -03a
98.19 cm
52.10 -53
INTER-TABLE
TABLL TRENDS (1-HOF)
91.41 201
91.50 2
28.53 =3:4
INTER "TABLE TRENDS (2-HOP)
95.57
95.90
52.01
INTER-TABLE TRENDS (3-HOP)
92.43 20.s
94.45 1.000
49.15 24.34
Instacart 05
CARDINALITY
100.0 aum
99.96
COLUMN SHAPES
98.39
98.84 aus
67.51
INTRA-TABLE TRENDS
97.59
20.0
98.55
89.33
=2277
INTER-TABLE TRENDS TRENDS (1-H1P)
88.49
91.80
63.32
Real Graph Structure. To better understand the modeling capacity of our graph generation algorithm de- scribed in Section 3.3, we perform an ablation study by replacing the graph sampled using this algorithm with the
INTER -TARE' TRENDS (2-HOP)
92.95
-0.30
92.85
4.6.0
13..41 -361
RelBench-FF
CARDINALITY
-034
+mm 96.
98.44 -:33
COLUMN SHAPES
95.71 =0.6
94.88
INTRA-TAHL TRENDS
95.00 -a
95.84
4:06
95.01 -035
-TEEE TRENDS (1-HOP)
90.85
93.12 a0.00
90.82
INTER TABLE TRENDS (2 HOP)
93.67 -033
96.10
4:300
87.73 .0.5

---

ground-truth graph from the RDB and using the same diffusion model from the previous experiments to generate the attributes. The results are shown in the second column of T Using the real graph achieves a perfect relational structure fidelity (ef. Cardinality metric) and provides upper-bound on the performance achievable by any graph generative model. The results show that our model achieves close performance to the real graph setting, confirming that our graph generation procedure can effectively capture the relational structure of the RDBs, with the small gap indicating that there is room for improvement by improving graph generation procedure, which we leave for future research. This gap smaller or even nonexistent on the simpler databases, as shown in Table the Appendix.
Effeet of Joint Modeling. To isolate the impact of our joint modeling approach, we implemented an autoregressive variant of model that retains all other components (c.g., graph representa- tion, GNN architecture). Instead of modeling pl p(((VEE) directly, it factorizes the distribution as
I p(x()) (X((') | < i)),, where x² denotes features of node type i. As shown in the third column of Table this version performs significantly worse than our proposed approach, not only on inter-table correlations, but also on single-table metrics. This aligns with observations from [55, which introduced latent cluster variables to mitigate the difficulty of learning conditional distributions with noisy and sigh-dimensional conditing space. Note that the sequential generation is significantly
slower than with the joint model. Concretely, on Berka, the autoregressive model was more than 2.5 times slower than its joint counterpart.
5 Conclusion
We introduced GRDM, the first non non-autoregressive generative model for RDBs. GRDM represents RDBs as graphs by mapping rows to nodes and grimary-forcigh key links to edges. It first generates graph structure using features-i.e., row attributes-vin a novel diffusion model that conditions each node's denoising on its K-hopp neighborhood. This enables modeling long-range dependencies due to the iterative nature of diffusion models. GRDM consistently outperformed baselines on six real-world RDBs, particularly in capturing long-range correlations.
random graph model, then jointly generates all node
Limitations and Future Work. This work lays the foundation for scalable and expressive gen- erative models for RDBs, and opens several avenues for future research. First, while our graph generation method is simple and effective, more tailored approaches could better capture the structure of relational data. Second, although we use a straightforward graph representation, alternatives such as modeling certain tables (e.g., transactions or reviews) as attributed edges-may be more appropriate for specific tasks. Third, while this work focuses on unconditional generation, GRDM can be extended to support downstream tasks and serve as a foundational model for relational data. Finally, equipping GRDM with differential privacy guarantees could further increase practical adoption, especially in sensitive domains.
Broader Impact
This paper introduces impacts of our work include enabling more accurate and powerful synthetic data generation, which promises tackle issues with privacy-preserving data sharing and advancing scientific research. However, such models can be used to generate fake content that can be misused. However, we strongly believe that the benefits significantly outweigh the small chance of misuse.
approach for generating synthetic relational databases. Some positive
Acknowledgments
This project was funded by SAP SE. It is also supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research. We thank Pedro Henrique Martins, Maximilian Schambach, Sirine Ayadi, and Tim Beyer for their valuable feedback.
10

---

References
[1] DB-Engines. DBMS popularity broken down by database model, 2023. Available: https
//db- wangginne..com/e//rakiinngg
[2] Alistair EW Johnson, Tom J Pollard, Lu Shen, Li-wei H Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Authony Celi, and Roger G Mark. Mimic-iii, a freely accessible critical care database. Scientific data, 3(1):1-9, 2016.
[3] PubMed. National Center for Biotechnology Information, U.S. National Library of Medicine, 1996. Available: httpss:/www, ncbi. nlm. mih. gov/pubmed/
[4] Aldren Gonzales, Guruprabha Guruswamy, and Scott R Smith. Synthetic data health care: narrative review. PLOS Digital Health, 2(1):e0000082, 2023.
[5] Vamsi K Potluru, Daniel Borrajo, Andrea Coletta. Niccolò Dalmasso, Yousef El-Laham, Eliza- beth Fons, Mohsen Ghassemi, Sriram Gopalakrishnan, Vikesh Gosai, Elconora Kreačić, et al. Synthetic data applications in finance. arXiv preprint arXiv:2401.0008) 2023.
[6] Joao Fonseca and Fernando Bacao. Tabular and latent space synthetic data generation: a literature review. Journal of Big Data, 10(1):115, 2023.
[7] Boris Van Breugel and Mihaela Van der Schaar. Beyond privacy: Navigating the opportunities and challenges of synthetic data. arXiv preprint arXiv:2304.03722, 2023.
[8] Akim Kotelnikov, Dmitry Baranchuk, Ivan Rubachev, and Artem Babenko. Tabddpm: Mod- elling tabular data with diffusion models. In International Conference Machine Learning, pages 17569. PMLR, 2023.
[9] Tennison Liu, Zhaozhi Qian, Jeroen Berrevoets, and Mihaela van der Schaar. Goggle: Generative modelling for tabular data by learning relational structure. In The Eleventh International Conference on Learning Representations, 2023.
[10] Hengrui Zhang, Jiani Zhang, Balasubramaniam Srinivasan, Zhengyuan Shen, Xiao Qin, Christos Faloutsos, Huzefa Rangwala, and George Karypis. Mixed-type tabular data synthesis with score-based diffusion in latent space. arXiv preprint arXiv:2310.09633 2023.
[11] Juntong Shi, Minkai Xu, Harper Hua, Hengrui Zhang, Stefano Ermon, and Jure Leskovec. Tabd- iff: multi-modal diffusion model for tabular data generation. arXiv preprint arXiv:2410.20626, 2024.
[12] Xi Fang, Weijie Xu, Fiona Anting Tan, Jiani Zhang, Ziqing Hu, Yanjun Qi, Scott Nickleach, Diego Socolinsky, Srinivasan Sengamedu, and Christos Faloutsos. Large language models (Imms on tabular data: Prediction, generation, and understanding-a survey. arXiv preprint a402v:2402.1 17944, 2024.
[13] Neha Patki, Roy Wedge, and Kalyan Veeramachaneni. The synthetic data vault. In 2016 IEEE international conference on data science and advanced analytics (DSAA), pages 399-410, IEEE, 2016.
[14] Kuntai Cai, Xiaokui Xiao, and Graham Cormode. Privlava: synthesizing relational data with foreign keys under differential privacy. Proceedings of the ACM on Management of Data, 1(2): 1-25, 2023.
[15] Wei Pang, Masoumeh Shatieinejad, Lucy Liu, Stephanie Hazlewood, and Xi He. Clavaddpm: Multi-relational data synthesis with cluster-guided diffusion models. Advances in Neural Information Processing Systems, 37:83521-83544 2024.
[16) Kai Xu, Georgi Ganev, Emile Joubert, Rees Davison, Olivier Van Acker, and Luke Robinson. Synthetic data generation of many-to-many datasets via random graph generation. In The Eleventh International Conference Learning Representations, 2022.
[17] Matthias Fey, Weihua Hu, Kexin Huang, Jan Eric Lenssen, Rishabh Ranjan, Joshua Robinson, Rex Ying, Jiaxuan You, and Jure Leskovec. Relational deep learning: Graph representation learning relational databases. arXiv preprint arXiv:2312.0446 2023.
11

---

[18] Edgar F Codd. A relational model of data for large shared data banks. Communications of the ACM, 13(6)3377-37,, 1970.
[19] Valter Hudovemik. Relational data generation with graph neural networks and latent diffusion models. In NeurlPS 2024 Third Table Representation Learning Workshop, 2024.
[20] Lukáš Zahradník, Jan Neumann, and Gustav Šír. A deep learning blueprint for relational databases. In NeurIPS 2023 Second Table Representation Learning Workshop, 2023.
[21] Minjie Wang, Quan Gan, David Wipf, Zhenkun Cai, Ning Li, Jianheng Tang, Yanlin Zhang, Zizhao Zhang, Zunyao Mao, Yakun Song, et al. 4dbinfer: A 4d benchmarking toolbox for graph-centric predictive modeling relational dbs. arXiv preprint arXiv:2404. /8209, 2024.
[22] Michael Molloy and Bruce Reed. A critical point for random graphs with given degree sequence. Random structures & algorithms, 6(2-3):161-180, 1995.
[23] Jonathan Ho, Ajay Jain, and Picter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
[24] Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances neural information processing systems, 34:8780-8794 2021.
[25] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily L Denton, Kamyar Ghasemipour, Raphael Gontijo Lopes, Burcu Karagol Ayan, Tim Salimans, et al. Photorealistic text-to-image diffusion models with deep language understanding. Advances in neural information processing systems, 35:36479-36499 2022.
[26] Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution. arXiv preprini 1:XXiv2310016833 2023.
[27] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsuper- vised learning using nonequilibrium thermodynamics. In International conference on machine learning, pages
pmlr, 2015.
[28] Emiel Hoogeboom, Didrik Nielsen, Priyank Jaini, Patrick Forré, and Max Welling, Argmax flows and multinomial diffusion: Learning categorical distributions. Advances in neural information processing systems, 14:12454-12265, 2021.
[29] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning large graphs. Advances in neural information processing systems, 30, 2017.
[30] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. International conference on machine learning, pages 1263-1272. PMLR, 2017.
[31] Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne Van Den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional networks. In The semantic web: 15th international conference, ESWC 2018, Heraklion, Crete, Greece, June 3-7, 2018, proceedings 15, pages 593-607. Springer, 2018.
[32] Matthias Fey and Jan Eric Lenssen. Fast graph representation learning with pytorch geometric. arXiv preprint arXiv: 1903,02428, 2019.
[33] Yury Gorishniy, Ivan Rubachev, Valentin Khrulkov, and Artem Babenko. Revisiting deep learning models for tabular data. Advances in neural information processing systems, 34: 18932-18943, 2021.
[34] Hector Garcia-Moling Database systems: the complete book. Pearson Education India, 2008.
[35] Petr Berka et al. Guide to the financial data set. PKDD2000 discovery challenge, 2000.
[36] jeremy stanley, Meg Risdal, sharathrao, and Will Cukierski. Instacart mar- ket. basket analysis, 2017. URL https: comp//ttmptiiiins//////
Instacart-marker wareett--erreetbeeeeee----- masket-analyee
12

---

[37] Jan Motl and Oliver Schulte. The ctu prague relational learning repository. arXiv preprint arXiv:1511.03086, 2015.
[38] Oliver Schulte, Zhensong Qian, Arthur E Kirkpatrick, Xiaoqian Yin, and Yan Sun. Fast learning of relational dependency networks, Machine Learning, 103:377-406, 2016.
[39] MP Center. Integrated public microdata series, international: Version 7.3 [data set]. minneapolis, mn: Ipums, 2020.
[40] Open source formula database. htw.b///p//ggtttuu...o///////e//
[41] Joshua Robinson, Rishabh Ranjan, Weihua Hu, Kexin Huang, Jiaqi Han, Alejandro Dobles, Matthias Fey, Jan Eric Lenssen, Yiwen Yuan, Zecheng Zhang, al. Relbench: A benchmark for deep learning on relational databases. Advances in Neural Information Processing Systems, 37:21330-21333 2024.
[42] Synthetic Data Metrics. DataCebo, Inc., 12 2024.
URL https://docs.. sdv dev
sdmetrics/ Version 0.18.0.
[43] Jim Young, Patrick Graham, and Richard Penny. Using bayesian networks to create synthetic data. Journal of Official Statistics, (5(4))549-567, 2009.
[44] Lei Xu, Maria Skoularidou, Alfredo Cuesta-Infante, and Kalyan Veeramachaneni. Modeling tabular data using conditional gan. Advances in neural information processing systems, 32, 2019.
[45] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. Communications of the ACM, 63(1 (1(11))139-14, 2020.
[46] Thomas N Kipf and Max Welling. Variational graph auto-encoders. arXiv preprint arXiv:1611.07308, 2016.
[47] Aleksandar Bojchevski, Oleksandr Shchur, Daniel Züggner, and Stephan Günnemann. Netgan: Generating graphs via random walks. In International conference machine learning, pages 610-619, PMLR, 2018
[48] Mufei Li, Eleonora Kreačić, Vamsi K Potluru, and Pan Li. Graphmaker: Can diffusion models generate large attributed graphs? arXiv preprint arX1v.2310.13833, 2023.
[49] Zilong Zhao, Aditya Kunar, Robert Birke, and Lydia Chen. Ctab-gan: Effective table data synthesizing. In Asian conference machine learning, pages 97-112. PMLR, 2021.
[50] Andreas Lugmayr, Martin Danelljan, Andres Romero, Fisher Yu, Radu Timofte, and Luc Van Gool. Repaint: Inpainting using denoising diffusion probabilistic models. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 1161111477,
2022.
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A Related Work
While the focus of this work on relational database generation, our method draws on ideas from various fields, which we briefly review in this section. As RDBs are composed of multiple tables, tabular data generation methods are a major building block of RDB generative models. We start by reviewing both lines of work. Then, we review some recent efforts to apply graph machine learning to RDBs, and finally discuss existing works for general graph generation.
Tabular data generation. For single-table generation, early work 43 applied Bayesian networks to model the distribution of tabular data with graphical models. CTGAN 44
is a seminal work that applied Generative Adversarial Networks (GANs) [45 to tabular data generation. Recently, with the rise of diffusion models, more diffusion-
models have been proposed to model tabular data. Notably, TabDDPM [8] was the first to apply denoising diffusion probabilistic models to the tabular data domain. TabSyn [T]] adopts the powerful idea of latent diffusion using a transformer- based variational autoencoder. More recently, TabDiff [I] proposes learning feature-specific node schedules and applying masked diffusion to model categorical variables.
Relational Database Generation. The Synthetic Data Vault (SDV) [3] laid the foundation of this field. It uses the Gaussian copula process to model every parent-child relationship. LrivLava [4] applies the framework of differential privacy generate RDBs. It uses graphical models with latent variables to capture the inter-table correlations. ClavaDDPM [3] is a recent diffusion- model that introduced cluster-based latent variables and uses classifier dassiffer-guided diffusion models generate the rows a child table conditioned on its parent. Xu et al. [6] and Hudovemik [19] adopt graph view of the RDB and model the autoregressive factorization of the tables' distribution. Note that all prior methods for RDB generation also follow similar autoregressive factorization.
Relational Deep Learning. Recently, the idea of representing RDBs graphs and applying graph machine learning methods to process them has gained traction by the introduction of Relational Deep Learning (RDL) [17 and the release of benchmark of real-world databases (RelBench) [41. While we use a similar graph representation, the sole focus of these works is on predictive tasks on RDBs, e.g., to predict user churn future purchases. In contrast, we are interested in generative modeling of RDBs, which is a more general and challenging problem.
Graph generation. By framing RDB generation graph generation problem, our work draws on a large literature on graph generation. Early work in this direction builds on the classic problem of finding graphs with a given degree sequence (22. Closest our setup is the setting of large graph generation, typically applied to social networks or citation networks. GAE and VGAE [46]
extend autoencoders and variational autoencoders the the graph setting. NetGAN 47 applies GANs for graph structure generation by sequentially generating random walks. Li et al. [48] introduces a diffusion model for large graph generation. However, all these works focus on general graphs like social networks, which typically have complex graph structure but rather simple features. In contrast, in our case, the graphs induced by relational databases exhibit a rather simple graph structure with specific properties (like m-partite) but very complex attributes with heterogeneous and multi-modal features. This distinguishes our work from these prior works and presents new challenges.
14

---

B Detailed Loss Derivation
We provide detailed derivation for the NLL loss following 27 2X For notation brevity, we drop the conditioning on V.E in the following:
log Po (2)
Pe (o((()))))
log
(((()))))
((10)
po ( () x(0)
En
log log
Σ
log (() ((1-1)))111 11)-
po (a (1-1) 1 o.
Bg
logp(z!! ())
log
ạ (1)
DEV
1=1 NEV
19 (1)
(a
Eg
log logp!a)! (?))
log
Σ log
q( 2
a(<0! @2) (x0
NEV
t=2 NEV
DEV
(a
g
(ax!
Eg
log logp(z.!?) (T)
log
g( (t-1) a0v (1)
x
(x!!" lo!
NEV
t= NEV
(x (0) (9(kk))
log
(xx 1
(0))
w V
pla!")
po (x (t-1)
g
Eg
log
log
log Po (a. (0) 10
q(xb ( (0)))
(T)
(x x (1))
(2)
VEV
vEV
NE
En
Σ DKL (a(a!?") 2c! (1lp(c
Σ Σ
DEL (g(aft-1) (x) (0) Ilppl (ap 10
VE
t= vEV
log po (z (1)
(
(0
const, + Eg
Σ ΣΣDn DKL (9(0)- x=!)) 20(1)
\ppo Mlpo (((0))))()))
log pe (a: (((
1
L(0)
The advantage of this reformulation is that the forward process posteriors needed to compute the loss tractable when conditioned x: (0),
(= (0!))) = (ox 1); (1) 20), B.I), gxx((-1) 2z))
(10)
with
i. (,) 20!) Vor
1
(xr {) VI- a₁
Br
. N.00
(11)
and
1 - 0z-1
3r =
Br
(12)
1 - ōi
15

---

The term Le (t-11)
which is KL divergence between two Gaussians, can now be written as:
E
Me (x)
+ const.
2
(13)
B.
LEg
2 (x!!) Ev) VI α4 ) no (0
,u)
+ const.
2
Var
This suggests new parametrization of n4o, which predicts the noise instead:
ED
Eo(C
(14)
Vou
āt
With this, L is follows:
B3 T By 20 a,)
c
c0(g())
v.K 0/1²]
(15)
Ho et al. 23 found that training works even better with simplified objective that ignores the weighting term in L (t-1) and treats _ similarly to all ("11)
||co ca (g..x: )1²)
ssimple ( ()
(16)
Sampling from 9 requires sampling set of noise vectors, one for each node in v's neighborhood: (e, I) E Vu,k), and computing the noisy features in Gl.kk- The final loss can be written as:
Lsimple (0) = -. veV..t |6,-- - 66 (g 1) 11]
(17)
C Additional Method Details
C.1 Database Reconstruction from its Graph Representation
presented the graph representation of the RDB that for generation. An important property of this representation is that it is invertible, i.e., one can reconstruct the RDB from its graph representation after generating the graph. One way perform this reconstruction is as follows:
In Section 3.1
For each node type i, assign unique primary key Pe E (1,...,,,] ni) to each node e of type i. Nodes of the same type should have different primary keys.
2. For each edge (v1, U2) E E, add primary key Pe, to the set of foreign keys Ke,,
3. For each node type i, construct table R(0) by stacking rows of the form (pe, Ko, for every node 100)
C.2 Gaussian Diffusion for Categorical Variables
In Section 3 we discussed that our diffusion model applies Gaussian diffusion both categorical
and numerical features by first mapping categorical variables to continuous space through label encoding. Specifically, given a categorical feature with distinct values {cI,..., Cx,). we apply label encoding scheme E (ci...., Cn {O...., 11-1 that maps each unique category to unique integer value. Note that this process is invertible, allowing to map generated values back to the categorical space.
16

---

C.3 Training and Sampling Algorithms
Algorithm Training
1: repeat
2:
Sample e ~ Uniform(V)
3: (Ve,k, &, &e.
k) &
+ (1LECTx(1; D,,
K KC-hop neighborhood around node
4: Sample t
(1tformm(.....111 ,T)
5:
Sample {E. ~ NO, 1) d' E Vo.k
X (Vāra (0) +II d,e o Vo
6:
E (V, x(th
7:
é,
8: Take gradient descent step on Volle, E.11? until converged
9:
Algorithm 2 Sampling
1: ((T) (w!!)) (T') (O, I) VEV]
2: for t = 1 do
3: Z + [ze N(O, I) ift clse 2, = =0
u uEVv V}
4: (e
are vI-ae En (oox)l)) + ozz [VEV]
(t)
Be
Voe
5: end for
6: return (0)
C.4 Heterogeneous Message-Passing Graph Neural Networks
A common way process graph-structured data with deep learning models is using Message-Passing Graph Neural Networks (MP-GNNs). Since we are dealing with heterogeneous graphs, we leverage heterogeneous message passing formulation, which supports wide range of GNN architectures. Let (G, 1) E G x IR denote the input to 6g with G = (V, E, 2). We define initial node embeddings ( I) V V. One iteration of message passing consists of computing updated node embeddings [h[+]] Jvey from the embeddings at the previous iteration { Juey.
h
Given node U, let (bv) denote its type the graph and let E, denote the of distinct edge types adjacent One iteration (or layer) of heterogeneous message passing updates v's embedding h! as follows. First, distinct message is computed for each edge type:
Ve E E., mi,e
1+1 = g[ (h.,)) w E (c(v)))
where (v) is the e-specific neighborhood of node Then, these messages are combined into single unified message:
m ++1
[(fo(m(t))
1+1 e E.D.
which is used to update the node embedding:
h I+1 Solu) (hi, ml11).
Pe. Je and fo(e) are arbitrary differentiable functions with learnable parameters and (.) is permutation-invarian set aggregator such as mean, max, etc.. Different GNN architectures can be obtained through specific choices of these functions.
17

---

D Experimental Details
D.I Datasets
TABLES
FOREIGN KEY PAIRS
DEPTH
TOTAL # AFTRIBUTES
# RowS IN LARGEST TABLE
DIMENSION TABLES
Berka
8
8
41
1,056, 320
Intecart 05
6
12
1,616,311 315
RelBench-Fi
996,
CCS
5
4
11
383, 282
Califoriia
2
15
1, 1,690,668
Table 3: Database Details
D.2
Implementation Details
D.2.1 GRDM hyperparameters and training
Architecture. For the GNN, we use the heterogeneous version of the GraphSAGE model 29 with the number of layers set the number hops K from the diffussin model. In all our experiments, we set K = 1. We use sum-based aggregation and a hidden dimension of 256 for the GNN.
In addition the GNN, GRDM contains MLP for each node type. These MLPs have the same architecture as those used in [15] and follow the TabDDPM implementation [8]. We use layers of sizes 512, 1024, 1024, 1024, 1024, 512, if the corresponding table has more than 10, 000 rows, and use smaller MLP with layers of sizes 512, 1024, 1024, 512 otherwise. This, in total, leads to smaller number of MLP parameters than the models used in [15], which we use our baselines. All input and output dimensions of the GNN and MLP are adapted to the data dimensions. We also incorporate timestep information by encoding timesteps into sinusoidal embeddings, which are then added to the model's input.
Diffusion hyperparameters. We follow cosine scheduler for the noise schedule.
[5] and set diffusion timesteps 2000 and use
Training hyperparemters. We also follow 15 and use the A optimizer with learning rate 6c-4 and weight decay 1e-5. We use 100, 000 training steps for California and 200, 000 all other databases. We use batch size of 4096 on all databases.
D.2.2 Baselines
For all diffusion baselines, we use the same set of hyperparameters, including diffusion timesteps. MLP architecture, and training from [15]. In particular, the only difference to our model is that the baselines use larger MIPss, which gives them an advantage over our model in terms of the total number of parameters.
For SDV, the default setting of their HMASynthesizer, which uses Gaussian Copula synthesizer.
D.2.3 Hardware
All experiments are run using a single NVIDIA A100-PCIE-4OGB GPU.
D.3
Detailed Metrics Computation
First, we define the distribution similarity measures that are used across the different metrics in this work.
For numerical variables, we use the complement of the Kolmogorov-Smimon (KS) statistic. The KS statistic between two numerical distributions is defined as the maximum difference between their respective cumulative density functions (CDFs).
KS sup Freat()) - Faym (x))
x
18

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where Freat and Feym denote the CDFs of the real and synthetic variables, respectively. We always report the scaled complement (1 KS) * 100 such that a higher score means better quality.
For categorical variables, we use the complement of the Total Variation (TV) distance. The TV distance between two categorical distributions compares their probabilities follows:
TV
Σ R S 2
1
where Q is the set of all categories, Rw and S. are the real and synthetic probabilities for a category w E Q. We always report the scaled complement (1 TV) 100 such that a higher score means better quality.
We now present how each metric is computed in detail.
Cardinality. For each parent-child pair in the database, we compute the cardinality cach parent row, i... the number of children that each parent row has. This defines numerical distribution for the real and synthetic data, for which we compute the complement of the KS statistic. The cardinality score is then the average across all parent-child pairs in the database.
Column Shapes. For each column in every table of the database, we measure the similarity between the column's 's marginal distribution in the real and synthetic databases. For numerical values, we use the complement the KS statistic, while for categorical values, we use the complement of the TV distance. The Column Shapes metric is the average across all columns in the database.
Table Trends. For each pair of columns within the same table, we compute different correla- tion metrics depending on whether the columns are numerical or categorical (for a pair numerical and categorical columns, as categorical). The Intra-Table Trends is the average of the correlation scores across all column pairs from the same table in the database.
first discretize the numerical column into bins, then treat both columns
For pair of numerical columns, A and B, we compute the Pearson correlation coefficient defined as
Coc( A, B)
PA.B
where Cov is the covariance, σA and are the standard deviations of A and B, respectively. We compute this metrie for both the real and synthetic pair, yielding RA,B and SA.B, respectively. We
report the normalized score 1 -A.B-RAA
100
For pair of categorical columns, A and B, we compute the normalized contingency table, which consists of the proportion of rows that have each combination of categories in A and B, i.e., how often each pair of categories co-occur in the same row. We compute this matrix both the real and synthetic pairs, yielding RA,B and S SAB respectively. We then compute the difference between them using the TV distance as follows:
1
score
Σ aEΩA BE
Σ
(S,.b(a,3) RA, B)),
where SA and Qg the set of possible categories of columns A and B, respectively. We again report the normalized score (1 score) 100.
Inter-Table Trends (k-hop). This metric is similar to the Column Pair Trends, but instead of computing the correlation on column pairs from the same table, computes it on column pairs from tables that are within & hops from each other. For example, 1-hop considers all parent-child pairs, 2-hop considers a column some table and its correlation with columns from its parent's parent or child's child, etc. For every possible hop k we report the average across all possible pairs. In practice, this implemented by denormalizing the pair of tables into single one and then computing the Column Pair Trends metric.
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E
Additional Experiments
E.1 Privacy Sanity Check
In order to evaluate the privacy preservation of our model and to test if any memorization is happening. we follow prior work [8] [5] and compute the distance to closest record (DCR) [49 between the generated and the training database. Specifically, for each synthetic sample, distance to the real records. We report the mean DCR, i.e., the average of these distances over all generated samples. Since different tables have different feature scales and ranges, also report the DCR values from the holdout set to the training set, which serve as "ground-truth generalizations"
get the minimum L1
We use four tables from different databases that contain sensitive information (but have been anonymized for these public benchmarks), such as census data containing household and indi- vidual information, sensitive financial information from a bank, and order information from an c-commerce platform. The results, provided Table 4 show that our model yields higher DCR values than the baseline ClavaDDPM (i.e., our model is more private) across all tables and it even yields higher DCR values than the holdout set on 3 out of 4 tables, while being close on the 4th one. These results suggest that model is able to generate new records from the underlying distribution and is not simply memorizing the training database. Note that DCR values need to be considered along with fidelity metrics (see Table T., since, e.g., random noise can yield high DCR values.
Table 4: Mean DCR values selected tables.
California (Household)
California (Individual)
Berka (Transaction)
Instacart 05 (Order)
MEAN DCR HOLDOUT SET MEAN DCR CLAVADDPM
0.094
0.206
0.012
0.012
0.104
0.177
0.019
0.007
MEAN DCR - GRDM (OURS)
0.119
0.259
0.028
0.009
E.2 Missing Value Imputation
To assess the performance of GRDM in the task of missing value imputation, we design a new experiment on the California database, which consists of two tables: parent table and child table. First, we split the database into a training set and a holdout set based on the parent table, and use the training set to train the diffusion model in the same way presented in the main text. Next, perform experiments on the holdout set by conditioning the generation on specific sections and using the same pre-trained model to inpaint the missing sections without any modification to the pre-trained model nor to the sampling procedure [50]. We report the metrics comparing the generated database with the holdout database:
Table 5
We use three different strategies to select the missing parts of the
1. Masking entire tables (parent or child table).
2. Masking entire rows from both tables with different rates.
3. Masking single cells (individual attributes of rows) from both tables with different rates.
The results show that GRDM consistently maintains good performance across the different masking settings, which again highlights the effectiveness of our proposed joint modeling approach capturing the complex distributions of RDBs. Note that the first column of Table 5 (Unconditional) is obtained by evaluating against the real test set, which explains the minor differences to the results reported in T
Table S: Results of missing value imputation experiments on the California dataset.
Unconditional
Parent Table
Child Table
50% Rows Missing
75% Rows Missing
50% Cells Missing
75% Cells
Missing
Missing
Missing
CARDINALITY
100.0
100.0
100.0
100.0
COLUMN SHAPES
99.03
99.26
99.37
99. 99.22
99.05
98.70
98.10
INTRA-TABLE TRENDS
97.75
98.65
98.56
98.36
97.91
96.34
94.44
97.34
97.17
97.20
95.96
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Table 6: Comparison of our default setting from Table L with using the real graph structure instead of sampling it on the databases missing from Table 2
GRDM
Real Graph
Movie Lens
CARDINALITY
98.80
20.30
100.0
2:00
COLUMN SHAPES
97.22 #0.48
97.31
+0.33
INTRA-TABLE TRENDS
95.25
10.35
95.50
1..55
1-HOP
94.34
+...9
94.71
10.50
CCS
CARDINALITY
99.79 10008
100.0 a:000
COLUMN SHAPES
97.30 1:030
97.08 1:0.1
INTRA-TABLE TRENDS
94.82 11.00
93 35 2:00
1-HOP
85.38
87 .87 #:270
California
CARDINALITY
99.96 +00.01
100.0
+:000
COLUMN SHAPES
98.75 #00.0
98.75 +:000
INTRA-TABLE TRENDS
97.35
#0000
97.36
#000
1-HOP
96.16 1.0.6
96.14 1.0.6
E.3 Detailed Evaluation of the Random Graph Generation Procedure
In this section, provide more fine- uine-grained evaluation of our graph generation algorithm deseribed in Section 3.3 Specifically,
Table 7: Higher-order structural properties. Car- dinality compares first-order node indegrees, Cor- relations compares the correlation between node indegrees from different edge types, and I-Hop, 2 2-Hop compare inter-table indegree correlations.
pare the graphs generated using this algorithm with the ground-truth graphs from the original RDBs beyond the Cardinality metric reported in the main text. The cardinality metric com- pares the node indegree distributions between the real and synthetic databases. In this experi- ment, we also compute metries for higher-order structural properties, such as the correlation be- tween node indegrees from different edge types (Correlations), e.g., the correlation between the number of actors in movie and the number of ratings it received. In addition, we report inter- table indegree correlations (I-Hop, 2-Hop), e.g.. the correlation between the number of stores in specific region and the number of purchases these stores get.
ClavaDDPM GRDM (Ours)
Berka CARDINALITY
96.75 20.36
99.65 -0.0
CORRELATIONS
93.07
=2,7
93.64
=0.88
1-HOP
98.35 20080
98.57 =0.30
2-HOP
97.87 0.00
99.17 =0.51
Instacart
CARDINALITY CORRELATIONS
94.91
1000
99.96 00.01
76.51
20.3
95.06 0.000
1-HOP
97.99
20.36
100.0 0.000
Movic Lens CARDINALITY
98.79
0.30
98.80 0.336
CORRELATIONS
97.02
0.57
94.56
=0.51
CCS
The results, presented in Table 7 show that, even though our graph generation algorithm only models first-order indegree distributions, it achieves good higher-order performance. This suggests that the used databases do not exhibit strong correlations between different types of relationships and that these relationships are, to large extent, independent, which our approach can capture very well (which was the main motivation behind our choice in the early stages of this project). However, more complex approaches may be required for more complex databases, which an interesting research direction, albeit orthogonal to the main focus of this work, namely the joint modeling of tables in RDBs.
CARDINALITY
98.96
0.79
99.79
00000
CORRELATIONS
98.00
95.73
E.4 Database Size Extrapolation
Our graph generation algorithm, presented in Section enables the generation of graphs of arbitrary sizes. We leverage this property to evaluate the performance of GRDM when increasing the database size. We use different multipliers of the original database size and generate databases with up to more than 20 million rows (10x multiplier) using the same diffusion model. For this experiment, we use smaller models than the ones used in the main text (specifically, smaller MI.ss) for efficiency purposes. The results, shown in Tables and 9 for two databases (California and MovieLens, respectively).
21

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highlight that the performance does not drop when increasing the database size. This can be explained by the of the GNN in the denoising model. which operates on local subgraphs that roughly invariant to the overall size of the RDB. Note that all RDBs were generated within less than 2 hours on a single GPU.
Table 8: Performance of GRDM the California database when increasing its size by different multipliers.
Size multiplier 0.5x (1.037M)
2x
Sx (10.381M)
10x (20.755M)
(total number of rows)
(2.076M)
(4.154M)
California CARDINALITY
99.89
99.96
99.96
99.99
99.97
COLUMN SHAPES
97.67
97.67
97.69
97.68
97.67
INTRA-TABLA TRENDS
95.12
95.
95.15
95.14
95.14
1-HOP
94.67
94.64
94.64
94.63
94.64
Table 9: Performance of GRDM on the MovieLens database when increasing its size by different multipliers.
Size multiplier
0.5x
1x
2x
5x (6.270M)
10x (12.414M)
(total number of rows)
(0.588M)
(1.221M)
(2.501M)
Movie Lens CARDINALITY COLUMN SHAPES
98.69
98.87
99.10
99.63
99.65
97.67
97.56
97.86
97.72
97.83
INTRA-TABLE TRENDS
95.38
94.99
95.58
95.34
95.42
1-HOP
95.17
95.35
95.57
95.62
95.63
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Reviews:
[{"review_id": "qRr20lSTfa", "reviewer": "Reviewer_gCht", "summary": "This paper proposes GRDM, a novel non-autoregressive model for synthetic relational database generation. GRDM represents databases as graphs and uses diffusion models to jointly generate all attributes. The method captures complex inter-table dependencies without requiring table ordering. Experiments show strong results across multiple real-world datasets.", "questions": "How would GRDM perform if some tables were missing or incomplete at test time?\n\nCan the graph generation be made differentiable to allow end-to-end training?\n\nDoes the performance drop when increasing database size?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n-- Nice work. It avoids autoregressive assumptions, allowing parallel generation and better scalability. \n-- Captures long-range dependencies via graph-based diffusion.\n\nWeaknesses:\n-- Graph generation step is quite simple and may limit structure realism. Consider more advanced generative graph models to improve relational structure fidelity.\n-- Categorical data is encoded as continuous using label encoding, which may harm semantics. Future work could explore discrete diffusion or hybrid encodings to better handle categorical variables.\n-- Unclear how well the model handles relational schema changes or dynamic schemas. Adding a discussion or experiments on generalization to modified schemas would improve the work.\n-- Missing related works: Check the papers by V. Hudovernik. Especially his RelDiff and benchmark against it", "was_revised": true, "final_justification": "Appreciate authors' response and have raised my score.\nThanks again for the hard work and well executed response.", "rebuttal": "We thank the reviewer for their valuable feedback. In the following, we address the reviewer's concerns in detail. We will include all the new experiments and discussions in the camera-ready version.\n\n---\n## [W1] Evaluation of the graph generation step\n\nTo better understand the modeling capacity of our graph generation algorithm, we perform a new ablation by replacing the sampled graph with the ground-truth graph and using the same diffusion model used in the paper to generate the attributes. The ground-truth graph achieves a perfect relational structure fidelity (cf. Cardinality metric) and provides an upper-bound on the performance achievable by graph generative models.\nThe results in the following table show that our model achieves **very similar performance to the ground-truth graph setting**, showing that **our graph generation procedure can effectively capture the relational structure of the RDBs**.\nWe also note that existing deep generative models for graphs are limited to graphs with a few thousand nodes, and scaling them to relational graphs with millions of nodes is a very interesting research direction.\n\n||Sampled Graph (Ours)|Ground-truth Graph\n|-|-|-|\n|**Berka**\n|Cardinality|99.70 ± 0.07|100.0 ± 0.00\n|Column Shapes|96.90 ± 0.07|97.00 ± 0.14\n|Intra-Table Trends|98.21 ± 0.05|98.24 ± 0.06\n|1-Hop|92.94 ± 0.06|93.16 ± 0.02\n|2-Hop|96.04 ± 0.20|96.67 ± 0.14\n|3-Hop|93.13 ± 0.47|94.47 ± 0.94\n|**Movie Lens**\n|Cardinality|98.80 ± 0.36|100.0 ± 0.00\n|Column Shapes|98.22 ± 0.05|98.25 ± 0.10\n|Intra-Table Trends|96.24 ± 0.24|96.31 ± 0.15\n|1-Hop|95.60 ± 0.17|94.19 ± 2.12\n|**California**\n|Cardinality|99.96 ± 0.01|100.0 ± 0.00\n|Column Shapes|99.15 ± 0.02|99.15 ± 0.01\n|Intra-Table Trends|98.00 ± 0.01 |98.02 ± 0.02\n|1-Hop|97.68 ± 0.01 |97.68 ± 0.02\n\n---\n## [W2] Discrete diffusion for categorical variables\nIn the early stages of the project, we experimented with multinomial diffusion [1] for categorical variables, which is based on uniform transitions between discrete states. However, we found it to **perform significantly worse than unified Gaussian diffusion**, as can be seen from the table below. This is likely because discrete diffusion models like uniform and masked diffusion completely **ignore the relationships between the different states** that can be important in, e.g., ordinal data, which is abundant in real-world RDBs (counts, discrete timesteps, ratings, etc.). On the other hand, Gaussian diffusion leverages these relationships by transitioning between more similar states with higher probability.\nWe also compare to concurrent work, RelDiff [2], which uses masked diffusion, and show that our model outperforms it on most datasets (see response to W4).\n\n||Multinomial + Gaussian diffusion|Unified Gaussian diffusion\n|-|-|-|\n|**California**\n|Column Shapes|83.27 ± 0.07|**99.51 ± 0.04**\n|Intra-Table Trends|79.27 ± 0.08|**98.69 ± 0.08**\n|**Berka**\n|Column Shapes|76.41 ± 2.21|**94.58 ± 0.01**\n|Intra-Table Trends|72.80 ± 1.23|**91.72 ± 0.23**\n\n---\n## [W3] Generalization to modified schemas\nWhile we are not aware of any other model that can handle schema changes, we discuss how our model can be adapted to handle different schemas:\n1. First, we need a collection of databases with different schemas to enable model generalization. We could use different databases and/or create different schemas of the same database using schema change operations.\n2. To enable training on different schemas, the generation process needs to be conditioned on the schema. For example, we could model the schema as a graph (nodes=tables, edges=table-level connections) and use an encoding of this graph as input to the generation process.\n3. Schema changes typically involve data type modifications, so we need a unified representation for different data types. We could use tabular foundation models pre-trained on large amounts of data to map different data types onto a unified latent space that can be used for generation.\n4. With these changes, we can apply our framework to jointly model the rows of any database in a unified space conditioned on its schema. After training on a variety of schemas, the model should be able to generalize to new schemas.\n\nDue to the limited time during the rebuttal, we leave this exciting direction for future research.\n\n---\n## [W4] Comparison to RelDiff and RGCLD\nUnfortunately, RelDiff [2] first appeared online 2 weeks after the NeurIPS full paper submission deadline, and at the time of writing, their code is still not available. Nonetheless, we provide a comparison to RelDiff on the datasets and metrics that are common to our evaluation setup. We also include RGCLD [3], which generates tables autoregressively. The results show that **GRDM significantly outperforms both models on the vast majority of the metrics**. \n\n||RGCLD|RelDiff|GRDM (Ours)\n|-|-|-|-|\n|**Berka**\n|Cardinality|**100.0**|**100.0**|99.70 ± 0.07\n|Column Shapes|79 ± 0.04|**97.72 ± 0.03**|96.90 ± 0.07\n|Intra-Table Trends|74 ± 0.03|**98.81 ± 0.02**|98.21 ± 0.05\n|1-Hop|81 ± 0.03|**96.88 ± 0.06**|92.94 ± 0.06\n|2-Hop|74 ± 0.03|95.79 ± 0.02|**96.04 ± 0.20**\n|3-Hop|65 ± 0.09|90.19 ± 0.22|**93.13 ± 0.47**\n|**Instacart 05**\n|Cardinality|N.A.|**100.0**|99.96 ± 0.01\n|Column Shapes|N.A.|96.85 ± 0.85|**98.87 ± 0.06**\n|Intra-Table Trends|N.A.|95.71 ± 0.42|**98.33 ± 0.04**\n|1-Hop|N.A.|85.83 ± 1.20|**92.03 ± 0.30**\n|2-Hop|N.A.|70.74 ± 0.14|**96.17 ± 0.15**\n|**RelBench-F1**\n|Cardinality|**100.0**|**100.0**|98.26 ± 0.12\n|Column Shapes|91 ± 0.03|94.89 ± 0.08|**97.28 ± 0.29**\n|Intra-Table Trends|92 ± 0.01|95.10 ± 0.12|**96.74 ± 0.36**\n|1-Hop|88 ± 0.03|93.46 ± 0.10|**93.74 ± 0.50**\n|2-Hop|88 ± 0.03|95.91 ± 0.03|**96.81 ± 0.28**\n|**MovieLens**\n|Cardinality|**100.0**|**100.0**|98.80 ± 0.36\n|Column Shapes|94 ± 0.05|96.91 ± 0.49|**98.22 ± 0.05**\n|Intra-Table Trends|94 ± 0.04|93.88 ± 0.87|**96.24 ± 0.24**\n|1-Hop|86 ± 0.09|94.84 ± 0.35|**95.60 ± 0.17**\n\n---\n## [Q1] Missing and incomplete tables at test time\nTo assess the performance of GRDM in the case of missing or incomplete tables at test time, we design a new experiment on the California database.\nFirst, we split the database into a train and a holdout set based on the parent table, and use the train set to train the diffusion model in the same way as presented in the paper. Next, we perform experiments on the holdout set by conditioning the generation on some sections and using the same pre-trained model to inpaint the missing sections without any modification [4]. We report the metrics comparing the generated and the holdout set. We use 3 different strategies to select the missing parts of the database:\n\n1. Masking entire tables.\n2. Masking entire rows from all tables with different rates.\n3. Masking single cells (individual attributes of rows) from all tables with different rates.\n\nThe results show that **GRDM consistently maintains good performance across the different conditional generation settings**.\n\n||Unconditional|Parent table is missing|Child table is missing|50% of rows are missing|75% of rows are missing|50% of cells are missing|75% of cells are missing\n|-|-|-|-|-|-|-|-|\n|Cardinality|99.78|100.0|100.0|100.0|100.0|100.0|100.0\n|Column Shapes|99.03|99.26|99.37|99.22|99.05|98.70|98.10\n|Intra-Table Trends|97.75|98.65|98.56|98.36|97.91|96.34|94.44\n|1-Hop|97.34|96.48|97.16|97.66|97.17|97.20|95.96\n\n---\n## [Q2] Differentiable graph generation\nYes, the graph generation can be made differentiable. For instance, we can use another diffusion model to generate the graph structure. However, under our proposed factorization $p(\\mathcal{G})=p(\\mathcal{V},\\mathcal{E})p(\\mathcal{X}|\\mathcal{V},\\mathcal{E})$, typically the graph generative model and the conditional feature generative model are trained separately, using teacher forcing for the conditional model, similarly to how autoregressive models are trained in practice.\nTo allow end-to-end training, we would need to directly model the joint $p(\\mathcal{G})=p(\\mathcal{V},\\mathcal{E},\\mathcal{X})$, which can pose computational challenges when scaling to very large graphs (see section 3.2 for a detailed discussion).\n\n---\n## [Q3] Performance when increasing database size\nOur graph generation algorithm allows us to generate graphs of arbitrary sizes. We leverage this property to test the performance of GRDM when increasing the database size. We use different multipliers of the original database size and generate databases with up to more than **20 million rows** (10x multiplier) using the same diffusion model. The results, shown below, highlight that **the performance does not drop when increasing the database size**. This can be explained by the use of our GNN in the denoising model, which operates on local subgraphs that are roughly invariant to the overall size of the RDB. Note that all RDBs were generated within less than 2 hours.\n\n|Size multiplier (number of rows)|0.5x (1.037M)|1x (2.076M)|2x (4.154M)|5x (10.381M)|10x (20.755M)\n|-|-|-|-|-|-|\n|**California**\n|Cardinality|99.89|99.96|99.96|99.99|99.97\n|Column Shapes|97.67|97.67|97.69|97.68|97.67\n|Intra-Table Trends|95.12|95.12|95.15|95.14|95.14\n|1-Hop|94.67|94.64|94.64|94.63|94.64\n\n|Size multiplier (number of rows)|0.5x (0.588M)|1x (1.221M)|2x (2.501M)|5x (6.270M)|10x (12.414M)\n|-|-|-|-|-|-|\n|**Movie Lens**\n|Cardinality|98.69|98.87|99.10|99.63|99.65\n|Column Shapes|97.67|97.56|97.86|97.72|97.83\n|Intra-Table Trends|95.38|94.99|95.58|95.34|95.42\n|1-Hop|95.17|95.35|95.57|95.62|95.63\n\n---\n## References\n[1] Kotelnikov, Akim, et al. Tabddpm: Modelling tabular data with diffusion models.\n\n[2] Hudovernik, Valter, et al. RelDiff: Relational Data Generative Modeling with Graph-Based Diffusion Models.\n\n[3] Hudovernik, Valter. Relational data generation with graph neural networks and latent diffusion models.\n\n[4] Lugmayr, Andreas, et al. Repaint: Inpainting using denoising diffusion probabilistic models.\n\n---\nWe hope our response can address the reviewer's concerns, and we are happy to answer any further questions during the discussion period."}, {"review_id": "z3MYzHI6JJ", "reviewer": "Reviewer_s1Pb", "summary": "This paper proposes Graph-Conditional Relational Diffusion Model (GRDM), a novel method that jointly modeling all tables in a relational database (RDB) without imposing any predefined order by adopting a natural graph representation of RDBs. GRDM leverages a graph neural network to simultaneously denoises row attributes and capture complex inter-table dependencies. Experimental results on 4 real-world graph datasets show some improvement against selected baseline methods (diffusion based), although for several cases the method underperforms SingleTable baseline.", "questions": "- In the ablation study, the performance gain from K=2 (2-hop) versus K=1 is very small. Can the authors explain why we need a multi-hop GNN in the final version? In other words, can we simply rely on diffusion to propagate information on the graph?\n- Are there ways to verify the claims of preserving the privacy of the training database?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strength**\n \n- The paper is moderately clear.\n- The experimental results show improvement on the majority of the tasks (especially inter-table metrics)\n- The proposed method is novel, where good modifications to a standard diffusion model for tabular data has been made, allowing it to generalize to multi-table settings.\n\n**Weakness**\n\n- While the paper claims data privacy as the major motivation for relational database generation, there is no analysis on how well the proposed method could preserve the privacy of the training dataset, making the effect of the proposed method questionable. This is especially concerning as the diffusion model may have simply memorized the training database, making the results concerning.\n- Following the first point, the overall setting assumes that a different diffusion model is trained for different database, preventing a trained model to make meaningful generalizations. Moreover, this setting reinforces the concern where the model may have simply memorized the training database. There is no metric to ensure that memorization is not happening. As a thought experiment, if a model takes in a database, add some minor noises to the values, the model may achieve very high evaluation scores under the current setting as well.\n- The method underperforms on intra-table metrics for 80% of the time compared to SingleTable baseline, it seems to sacrifice single table performance in some degree.\n- The exact definition and implementations of the MP-GNN is not clear.", "was_revised": true, "final_justification": "While I still find the evaluation on privacy as a weakness, I decided to increase my score to 4, given the limited time left and the author's argument that the generation task itself is challenging enough to outweigh privacy concerns.", "rebuttal": "We thank the reviewer for their valuable feedback. In the following, we address the reviewer's concerns in detail. We will include all the new experiments and discussions in the camera-ready version.\n\n---\n## [W1, W2, Q2] Privacy preservation and generalization\nIn order to evaluate the privacy preservation of our model and to test if any memorization is happening, we follow prior work [1,2] and compute the distance to closest record (DCR) [3] between the generated and the training database. Specifically, for each synthetic sample, we get the minimum L1 distance to the real records. We report the mean DCR, the average of these distances over all generated samples. Since different tables can have different feature scales and ranges, we also report the DCR values from the holdout set to the training set, which serve as \"ground-truth generalizations\".\n\nWe use 4 tables from different databases that contain sensitive information (but have been anonymized for these public benchmarks), such as census data containing household and individual information, sensitive financial information from a bank, and order information from an e-commerce platform. The results, provided in the table below, show that **our model yields higher DCR values than the baseline (i.e., is more private) across all tables** and it **even yields higher DCR values than the holdout set on 3 out of 4 tables**, while being close on the 4th one. These results suggest that **our model is able to generate new records from the underlying distribution and is not simply memorizing the training database**. Note that DCR values need to be considered along with fidelity metrics (see Table 1 in the paper), since, e.g., random noise can yield high DCR values.\n\n**Remark on W2:** While we use different diffusion models for different databases, each database contains multiple tables and potentially millions of rows, which are the central entity modeled by the same diffusion model. Note the contrast to single-table settings, where a different model is used for each table. While these single-table models showcased good levels of privacy protection and generalization [1,3], it is even more challenging for our model to memorize the training database because it models multiple tables at once.\n\n**Privacy metrics achieved by our model compared to the best baseline, and to the holdout set on 4 tables (the higher, the more private):**\n||California - Household|California - Individual|Berka - Transaction|Instacart 05 - Order|\n|-|-|-|-|-|\n|Mean DCR - Holdout set|0.094|0.206|0.012|**0.012**|\n|Mean DCR - ClavaDDPM|0.104|0.177|0.019|0.007|\n|Mean DCR - GRDM (Ours)|**0.119**|**0.259**|**0.028**|0.009|\n\n---\n## [W3] Comparison to SingleTable baseline\nWe designed our experiments such that all models share the same hyperparameters for the common components, including model architecture and size, diffusion steps, etc. In particular, the SingleTable baseline models tables separately and needs only to capture their individual distributions, while our model additionally needs to capture the correlations between different tables, given a similar parameter budget (the only difference is the GNN component, which constitutes approximately 5% of the total number of parameters).\n\nTo better compare these two models, we provide the average single-table and multi-table metrics, respectively, in a new table below. This comparison shows that **the relative drop in single-table metrics is negligible compared to the relative gain in multi-table metrics achieved by our model**. Also, when averaged across all the databases, our model outperforms SingleTable on both single-table and multi-table metrics.\n\nIn addition, we report the DCR values for the SingleTable baseline in the second table below (see response to [W1, W2, Q2] for the definition of this metric) and we can see that **the samples generated by SingleTable are consistently closer to the training data than our model, meaning that it generalizes less than our model**, which explains the higher single-table scores.\n\n**Comparison of average single-table and multi-table metrics between SingleTable and our model (the higher, the better):**\n||SingleTable|GRDM (Ours)|Improvement(%)|\n|-|-|-|-|\n|**Berka**|\n|average single-table metrics|93.15|**97.56**|+4.73|\n|average multi-table metrics|78.47|**94.04**|+19.84|\n|**Instacart 05**|\n|average single-table metrics|94.5|**98.60**|+4.34|\n|average multi-table metrics|41.58|**94.10**|+126.31|\n|**RelBench-F1**|\n|average single-table metrics|95.87|**97.01**|+1.19|\n|average multi-table metrics|76.85|**95.28**|+23.98|\n|**MovieLens**|\n|average single-table metrics|**98.88**|97.23|-1.67|\n|average multi-table metrics|92.72|**95.60**|+3.11|\n|**CCS**|\n|average single-table metrics|**97.08**|95.32|-1.81|\n|average multi-table metrics|51.62|**85.54**|+65.71|\n|**California**|\n|average single-table metrics|**99.10**|98.58|-0.52|\n|average multi-table metrics|92.96|**97.68**|+5.08|\n|**Average across all databases**|\n|average single-table metrics|96.43|**97.38**|+0.99|\n|average multi-table metrics|72.37|**93.71**|+29.49|\n\n**Comparison of the privacy metrics between SingleTable and our model (the higher, the more private):**\n||California - Household|California - Individual|Berka - Transaction|Instacart 05 - Order|\n|-|-|-|-|-|\n|Mean DCR - SingleTable|0.107|0.155|0.019|0.007|\n|Mean DCR - GRDM (Ours)|**0.119**|**0.259**|**0.028**|**0.009**|\n\n---\n## [W4] MP-GNN details\nIn Appendix D.4, we provide a general formulation of heterogeneous MP-GNNs. In the following, we provide the exact equations of the heterogeneous GraphSAGE model, which we used in our experiments, by plugging in the concrete choices of the different model components (e.g. choice of the aggregation function). Our implementation is based on the official PyTorch Geometric (PyG) implementation of the heterogeneous GraphSAGE model.\n\nWe describe one layer of this model, which maps node embeddings $\\\\{h_v^l \\\\} _{v \\in \\mathcal{V}}$ to $\\\\{h_v^{l+1} \\\\} _{v \\in \\mathcal{V}}$, where $l$ denotes the layer count. We describe the GNN update for a single node, and the same operations are applied to all nodes in parallel.\n1. **Compute a message for each edge type**. To accommodate the heterogeneity of the graph, a distinct message is computed for each edge type. For each edge type $e$, $\\mathcal{N}_ e (v)$ denotes the neighbors of $v$ that are connected via the edge type $e$, and $g_e$ is a neural network (typically an MLP) that is specific to this edge type. Then, the following message is computed for each edge type $e$: $$m_{v,e}^{l+1}=\\sum_{w\\in\\mathcal{N}_e(v)}g_e(h_w^l).$$\n2. **Combine the messages into a unified message**. Let $f_e$ be another MLP that processes messages of type $e$. Then, the combined message is computed as: $$m_v^{l+1}=\\sum_ef_e(m_{v,e}^{l+1}).$$\n3. **Update node embedding**. The unified message is used to update the node embedding. Again, to accommodate the heterogeneity of the graph, different MLPs are applied depending on the node's type, which we denote as $\\phi(v)$. The updated node embedding is: $$h_v^{l+1}=f_{\\phi(v)}(h_v^l,m_v^{l+1}).$$\n\nThese computations are run for all nodes, defining a single layer. Multiple such layers can be stacked to map initial node embeddings $\\\\{h_v^0 \\\\} _{v \\in \\mathcal{V}}$ to $\\\\{h_v^L \\\\} _{v \\in \\mathcal{V}}$, where $L$ is the total number of layers, with different neural networks for each layer. For each node $v$, $h_v^L$ contains information about the node $v$ and its $L$-hop neighborhood and can be used for downstream processing, e.g., type-specific denoising.\n\n---\n## [Q1] K=2 vs K=1 \nIndeed, the K=1 version achieves very strong performance and already outperforms all the baselines on the inter-table metrics, which confirms that we can safely rely on diffusion to propagate information on the graph (as long as we use K>0).\nThe multi-hop version can be seen as a simple way to further improve the performance at the cost of some additional compute.\n\n---\n## References\n[1] Kotelnikov, Akim, et al. Tabddpm: Modelling tabular data with diffusion models.\n\n[2] Pang, Wei, et al. Clavaddpm: Multi-relational data synthesis with cluster-guided diffusion models.\n\n[3] Zhao, Zilong, et al. Ctab-gan: Effective table data synthesizing.\n\n---\nWe hope our response can address the reviewer's concerns, and we are happy to answer any further questions during the discussion period."}, {"review_id": "KmvOpxdM57", "reviewer": "Reviewer_eXTm", "summary": "This paper addresses the problem of generating synthetic relational databases (RDBs). It identifies key limitations in prior autoregressive approaches, such as sequential generation and fixed table ordering, which can fail to capture complex inter-table dependencies. To overcome this, the authors propose the Graph-Conditional Relational Diffusion Model (GR-DDM), a novel two-stage framework. In the first stage, the model generates the high-level relational graph structure, defining the number of rows for each table and the primary-foreign key relationships between them. In the second stage, conditioned on this generated graph, a diffusion model is used to jointly and non-autoregressively generate the feature values for all rows across all tables. The authors claim this approach better models the complex joint distribution of RDBs and achieves state-of-the-art performance.", "questions": "The paper is very promising. My evaluation would be further improved if the authors could clarify the following points:\n\n- On the Richness of Graph Generation: Your node degree-preserving algorithm for graph generation is a clear and reasonable approach. However, have you considered if this is sufficient for capturing more complex, higher-order structural correlations in real-world databases (e.g., correlations between the counts of different relationships)? A brief discussion on the limitations of this structural generation approach would be valuable.\n\n- Scalability: How does GR-DDM scale with the complexity of the database, i.e., the number of tables, columns, and total rows? \n\n- Handling Hard Constraints: How does your framework enforce hard database constraints, such as unique primary keys, or ensure that foreign keys in one table correctly map to existing primary keys in another?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- Novel and Elegant Framework (Originality & Significance): The core contribution—a two-stage process that decouples graph structure generation from row content generation—is highly original and compelling. This non-autoregressive approach is a significant departure from prior work and has the potential to better capture holistic inter-table dependencies. The problem of high-fidelity RDB generation is very important, and this work presents a major potential advancement.\n\n- Use of Powerful Generative Models (Quality): The adoption of diffusion models for the content generation stage is a modern and powerful choice. Conditioning these state-of-the-art models on a relational graph structure is a technically sound and interesting research direction.\n\n- Clarity of Presentation: The paper's core idea is communicated very effectively. The abstract is clear, and figures provide an excellent and intuitive understanding of the proposed method and its advantages. The pipeline, including the statistical graph generation method, is well-explained.\n\nWeaknesses:\n\n- Potential Simplifications in Graph Structure Generation (Quality): The paper now clearly explains the graph generation algorithm, which preserves node indegree distributions from the real graph. While this is a reasonable and data-driven approach, it may be a simplification of more complex real-world graph structures. For instance, this method may not capture higher-order structural properties, such as correlations between different types of relationships for a given entity. A discussion on the limitations of only preserving first-order degree distributions would strengthen the paper.\n\n- Scalability Concerns (Quality): While the parallel generation of rows is an advantage, the overall scalability of the method is not discussed. Diffusion models can be computationally intensive, and conditioning on an entire, potentially massive, relational graph could pose significant memory and computational challenges. The paper would be stronger if it addressed how the framework scales to databases with a large number of tables or millions of rows.\n\n- Handling Constraints (Quality): Real-world databases are rich in discrete data and are governed by hard constraints (e.g., unique primary keys). The paper does not specify how the proposed diffusion model framework handles these fundamental characteristics of tabular data, which is a crucial aspect for practical usability.", "was_revised": true, "final_justification": "The response addressed my concerns of model scalability, therefore I will maintain the positive rating.", "rebuttal": "We thank the reviewer for their valuable feedback. In the following, we address the reviewer's concerns in detail. We will include all the new experiments and discussions in the camera-ready version.\n\n---\n## [W1, Q1] Richness of Graph Generation\nThe cardinality metric reported in the paper compares the node indegree distributions between the real and synthetic databases. As per the reviewer's request, we now also compare higher-order structural properties, such as the correlation between node indegrees from different edge types (**Correlations**), e.g., the correlation between the number of actors in a movie and the number of ratings it received. In addition, we report inter-table indegree correlations (**1-Hop, 2-Hop**), e.g., the correlation between the number of stores in a specific region and the number of purchases these stores get.\n\nThe results show that, even though our graph generation algorithm only models first-order indegree distributions, **it achieves good higher-order performance**. This suggests that the used databases do not exhibit strong correlations between different types of relationships and that these relationships are, to a large extent, independent, which our approach can capture very well (which was the main motivation behind our choice in the early stages of this project). However, more complex approaches may be required for more complex databases, which is an interesting research direction, albeit orthogonal to the main focus of this work, namely the joint modeling of tables in RDBs.\n\n**Higher-order structural properties. Cardinality compares first-order node indegree distributions, Correlations compares the correlation between node indegrees from different edge types, and 1-Hop, 2-Hop compare inter-table indegree correlations:**\n||ClavaDDPM|GRDM (Ours)|\n|-|-|-|\n|**Berka**|\n|Cardinality|96.75 ± 0.26|99.7 ± 0.07|\n|Correlations|93.07 ± 2.67|93.64 ± 0.18|\n|1-Hop|98.35 ± 0.08|98.57 ± 0.13|\n|2-Hop|97.87 ± 0.46|99.17 ± 0.51|\n|**Instacart 05**|\n|Cardinality|94.91 ± 1.50|99.96 ± 0.01|\n|Correlations|76.51 ± 0.34|95.06 ± 0.09|\n|1-Hop|97.99 ± 0.16|100.0 ± 0.00|\n|**Movie Lens**|\n|Cardinality|98.79 ± 0.13|98.8 ± 0.36|\n|Correlations|97.02 ± 0.37|94.56 ± 0.31|\n|**CCS**|\n|Cardinality|98.96 ± 0.79|99.79 ± 0.03|\n|Correlations|98.0 ± 0.25|95.73 ± 0.18|\n\nTo further evaluate whether our graph generation approach is sufficient to capture the complexity of the RDBs, we perform a new ablation by replacing the sampled graph with the ground-truth graph and using the same diffusion model to generate the attributes. The ground-truth graph achieves a perfect relational structure fidelity (cf. Cardinality metric) and provides an upper-bound on the performance achievable by graph generative models.\nThe results show that our model achieves **very similar performance to the ground-truth graph setting**, showing that **our graph generation procedure can effectively capture the relational structure of the RDBs**.\n\n||Sampled Graph (Ours)|Ground-truth Graph|\n|-|-|-|\n|**Berka**|||\n|Cardinality|99.70 ± 0.07|100.0 ± 0.00|\n|Column Shapes|96.90 ± 0.07|97.00 ± 0.14|\n|Intra-Table Trends|98.21 ± 0.05|98.24 ± 0.06|\n|1-Hop|92.94 ± 0.06|93.16 ± 0.02|\n|2-Hop|96.04 ± 0.20|96.67 ± 0.14|\n|3-Hop|93.13 ± 0.47|94.47 ± 0.94|\n|**Movie Lens**|||\n|Cardinality|98.80 ± 0.36|100.0 ± 0.00|\n|Column Shapes|98.22 ± 0.05|98.25 ± 0.10|\n|Intra-Table Trends|96.24 ± 0.24|96.31 ± 0.15|\n|1-Hop|95.60 ± 0.17|94.19 ± 2.12|\n|**California**|||\n|Cardinality|99.96 ± 0.01|100.0 ± 0.00|\n|Column Shapes|99.15 ± 0.02|99.15 ± 0.01|\n|Intra-Table Trends|98.00 ± 0.01 |98.02 ± 0.02|\n|1-Hop|97.68 ± 0.01 |97.68 ± 0.02|\n\n---\n## [W2, Q2] Scalability to larger databases\nThe databases used in our evaluation cover a wide range of sizes. The number of tables ranges from 2 to 9, the number of columns from 11 to 41, and the total number of rows goes up to 2 million (see Table 3 in the paper for details).\nTo evaluate the scalability of our model beyond the setting presented in the paper, we leverage the capability of our graph generation algorithm to generate graphs of arbitrary sizes. We use different multipliers of the original database size and generate databases with up to more than **20 million rows** (10x multiplier) using the same diffusion model.\nThe table below shows that **our model maintains its good performance across different scales** and that the runtime scales roughly linearly with the database size, **generating the largest database (> 20 million rows) in less than 2 hours on a single GPU**. Both these results can be explained by the use of our GNN in the denoising model, which operates on local subgraphs that are roughly invariant to the overall size of the RDB.\nNote that we used the same batch size for all scales, but the generation can be even faster with larger batch sizes, as the denoising of all rows is performed in parallel at each denoising step.\n\n|Size multiplier (number of rows)|0.5x (1.037M)|1x (2.076M)|2x (4.154M)|5x (10.381M)|10x (20.755M)|\n|-|-|-|-|-|-|\n|**California**|\n|Cardinality|99.89|99.96|99.96|99.99|99.97|\n|Column Shapes|97.67|97.67|97.69|97.68|97.67|\n|Intra-Table Trends|95.12|95.12|95.15|95.14|95.14|\n|1-Hop|94.67|94.64|94.64|94.63|94.64\n|**Runtime (min)**|**7**|**12**|**21**|**58**|**105**|\n\n|Size multiplier (number of rows)|0.5x (0.588M)|1x (1.221M)|2x (2.501M)|5x (6.270M)|10x (12.414M)|\n|-|-|-|-|-|-|\n|**Movie Lens**|\n|Cardinality|98.69|98.87|99.10|99.63|99.65|\n|Column Shapes|97.67|97.56|97.86|97.72|97.83|\n|Intra-Table Trends|95.38|94.99|95.58|95.34|95.42|\n|1-Hop|95.17|95.35|95.57|95.62|95.63|\n|**Runtime (min)**|**6**|**11**|**22**|**51**|**96**|\n\n---\n## [W3, Q3] Handling Hard Constraints\nIn our framework, the database constraints are enforced through our graph-based representation of the RDB. Specifically, each row of every table is represented as a unique node in the graph, with its type defined by the table. The identity of each node in the graph can be used as its primary key in the database, **ensuring the uniqueness of primary keys**. On the other hand, foreign keys are represented as edges: each foreign key in the RDB corresponds to a directed edge from the child row's node to the parent row's node. Since edges are defined between nodes of different types (m-partite graph), **this ensures that foreign keys in one table correctly map to existing primary keys in another**.\nOur diffusion model is conditioned on the graph structure and only denoises the features; therefore, all these constraints are maintained during the diffusion process. Finally, given a generated graph, we get the RDB representation as follows (see Appendix D.1 for more details):\n1. For each node type $i$, we assign a unique primary key $p_v \\in \\\\{1,\\dots,n_i \\\\}$ to each node $v$ of type $i$, where $n_i$ is the total number of nodes of type $i$. This ensures that nodes of the same type (i.e., from the same table) will have **unique primary keys**.\n2. For each directed edge $(v_{child}, v_{parent})$ in the graph, we add $v_{parent}$'s primary key to the set of $v_{child}$'s foreign keys. This ensures that **foreign keys correctly map to existing primary keys**.\n3. For each node type $i$, we construct the table $R^{(i)}$ as a collection of rows corresponding to all nodes of type $i$ in the graph, containing the primary keys, foreign keys, and the generated attributes.\n\n---\nWe hope our response can address the reviewer's concerns, and we are happy to answer any further questions during the discussion period."}, {"review_id": "ihQiIBJ0Cb", "reviewer": "Reviewer_FJA8", "summary": "The authors propose a generative model for relational tables represented as graphs, where rows are treated as vertices and connected via key–foreign key relationships. The proposed Graph-Conditional Relational Diffusion Model (GRDM) is built on graph neural networks and demonstrates good performance.", "questions": "Q1: Can the authors provide concrete evidence or real-world applications to demonstrate the significance of the problem being addressed?\n\nQ2: Can the authors clarify the use cases where augmenting real datasets is necessary? In practice, relational tables often contain large volumes of records.\n\nQ3: Regarding multi-modal features, can the authors specify which data types are considered in this work? In relational databases, data such as images and audio are typically stored as BLOBs and managed externally.\n\nQ4: Can the authors explain why key–foreign key relationships were the focus, and whether other types of relationships (e.g., many-to-many) were considered?", "limitations": "No.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nS1: Proposed techniques are methodologically sound.\n\nS2: Experimental results demonstrate that GRDM achieves good performance.\n\nS3: The paper reads well.\n\n\nWeaknesses:\n\nW1: The motivation for the problem is not sufficiently articulated.\n\nW2: The paper only considers key–foreign key relationships; other relationship types, such as many-to-many, are not discussed.\n\nW3: The limitations of the proposed approach are not directly discussed.", "was_revised": true, "final_justification": "I maintain a score of Borderline accept.", "rebuttal": "We thank the reviewer for their valuable feedback. In the following, we address the reviewer's concerns in detail. We will include all the new discussions in the camera-ready version.\n\n---\n## [W1, Q1] Motivation and real-world applications\nIn the following, we provide concrete real-world applications of the problem of synthetic relational database generation. Note that good generative models for RDBs are a prerequisite for all the following tasks.\n\n* **Privacy-preserving research and development**. Real relational databases typically contain sensitive information (e.g., personal data, financial records). Sharing them with third parties (e.g., LLM providers) or using them for product development can raise serious legal issues, e.g., on the basis of the European Union's General Data Protection Regulation (GDPR). Synthetic relational databases can be used to develop new features or valuable insights, without exposure of real user data. As a concrete example, the SyntheticMass project created a synthetic population of Massachusetts residents (based on EHR schemas) to support research and development without patient risk. For an evaluation of our model's privacy preservation capacity, we refer to our response to Reviewer s1Pb ([W1, W2, Q2] Privacy preservation and generalization).\n* **Data augmentation for machine learning models**. Many real-world machine learning tasks leverage relational databases (e.g., fraud detection, recommendation systems). Synthetic RDBs can help augment underrepresented groups to improve the fairness of the downstream models or simulate rare events to improve their robustness.\n* **Training foundation models on structured data**. There have been recent efforts to build foundation models for structured data [1,2]. However, training such models requires huge amounts of relational databases, which can be accelerated using generative models for RDBs.\n* **Adapting generative models for predictive tasks**. A pre-trained diffusion model for RDBs can be easily adapted to perform conditional generation without the need for re-training [3]. This allows us to leverage a single model for different downstream prediction tasks, such as regression or classification, without the need for training different models for each task, thereby saving cost and time. For concrete results on how our model can be adapted to such cases, we refer to our response to Reviewer gCht ([Q1] Missing and incomplete tables at test time)\n\nWe will update the motivation section in the camera-ready version to include these real-world applications.\n\n---\n## [W2, Q4] Many-to-many relationships\nWe confirm that many-to-many relationships were considered in our work. In fact, **4 out of the 6 used databases contain this type of relationship (Instacart 05, RelBench-F1, MovieLens, and CCS)**. As pointed out by the reviewer, relational databases cannot directly implement many-to-many relationships due to the fixed number of foreign keys a table can have. To overcome this, a so-called join table is typically introduced, which acts as an intermediary, storing the relationships between the two connected tables, essentially breaking down the many-to-many relationship into two one-to-many relationships.\nExamples of this type of relationship from our databases include ratings (from MovieLens), which connect multiple users with multiple movies, and transactions (from CCS), which connect multiple customers with multiple products.\nSince we represent databases as graphs, this implementation offers a unique advantage: the resulting graphs are simple (no multi-edges) and all the attributes are stored as node features instead of edge features (e.g., rating scores), allowing us to use standard GNN architectures out-of-the-box.\n\nWe refer to Table 1 in the paper for results on these databases. We will clarify this point in the camera-ready version.\n\n---\n## [W3] Limitations\nWe kindly point to the last paragraph in the paper (right after the conclusion) for a detailed discussion on the limitations of our approach and promising future research directions.\n\n---\n## [Q2] Augmenting real datasets\nReal datasets can be biased towards a specific population or only model common events. In such cases, using generative models to augment real datasets can improve the fairness of the dataset and improve downstream performance on rare edge cases (if, e.g., used to train downstream machine learning models).\nAs a concrete example, in the context of healthcare research, synthetic data can be generated to simulate rare diseases. Or in the case of recommendation systems, synthetic data can be used to improve the performance of downstream models on underrepresented groups.\n\n---\n## [Q3] Multi-modal features\nIn this work, we focus on categorical and numerical data types, which cover different data formats, such as ordinal data, enums, boolean, datetime, and any discrete or continuous data.\nFor high-dimensional data such as images, audio, or text, a natural idea would be to model low-dimensional latent representations for these data types as numerical features and directly apply our framework to generate them. Then, a suitable decoder can be applied to map back to the respective original data space.\n\n---\n## References \n\n[1] Wang, Yanbo, et al. Griffin: Towards a Graph-Centric Relational Database Foundation Model.\n\n[2] Fey, Matthias, et al. KumoRFM: A Foundation Model for In-Context Learning on Relational Data.\n\n[3] Lugmayr, Andreas, et al. Repaint: Inpainting using denoising diffusion probabilistic models.\n\n---\n\nWe hope our response can address the reviewer's concerns, and we are happy to answer any further questions during the discussion period."}]

Author rebuttal:

Meta-review: This paper studies the limitations of prior RDB generative models—focusing on single tables or using autoregressive sequential generation. The authors propose GRDM, a graph-conditional relational diffusion model that jointly models all RDB tables via graph neural networks to denoise rows and capture inter-table dependencies. The authors and reviewers had thorough rebuttal discussions, and finally, the reviewers converged to consistently accept this paper. After reading the paper & discussions, I think this paper is acceptable for publication in NeurIPS this year.
Decision: Accept (poster)