# TableDC: Deep Clustering for Tabular Data

Hafiz Tayyab Rauf  0000-0002-1515-3187 Department of Computer Science,
University of ManchesterManchesterUK.M13 9PL<EMAIL_ADDRESS>,  Andre Freitas  0000-0002-4430-4837 Department of Computer Science,
University of Manchester
Manchester, UK.
IDIAP Research Institute
Martigny, Switzerland.M13 9PL<EMAIL_ADDRESS>and  Norman W.
Paton  0000-0003-2008-6617 Department of Computer Science, University of
ManchesterManchesterUK,M13 9PL<EMAIL_ADDRESS>
###### Abstract.

Deep clustering (DC), a fusion of deep representation learning and clustering,
has recently demonstrated positive results in data science, particularly text
processing and computer vision. However, joint optimization of feature
learning and data distribution in the multi-dimensional space is domain-
specific, so existing DC methods struggle to generalize to other application
domains (such as data integration and cleaning). In data management tasks,
where high-density embeddings and overlapping clusters dominate, a data
management-specific DC algorithm should be able to interact better with the
data properties for supporting data cleaning and integration tasks. This paper
presents a deep clustering algorithm for tabular data (TableDC) that reflects
the properties of data management applications, particularly schema inference,
entity resolution, and domain discovery. To address overlapping clusters,
TableDC integrates Mahalanobis distance, which considers variance and
correlation within the data, offering a similarity method suitable for tables,
rows, or columns in high-dimensional latent spaces. TableDC provides
flexibility for the final clustering assignment and shows higher tolerance to
outliers through its heavy-tailed Cauchy distribution as the similarity
kernel. The proposed similarity measure is particularly beneficial where the
embeddings of raw data are densely packed and exhibit high degrees of overlap.
Data cleaning tasks may involve a large number of clusters, which affects the
scalability of existing DC methods. TableDC’s self-supervised module
efficiently learns data embeddings with a large number of clusters compared to
existing benchmarks, which scale in quadratic time. We evaluated TableDC with
several existing DC, Standard Clustering (SC), and state-of-the-art bespoke
methods over benchmark datasets. TableDC consistently outperforms existing DC,
SC, and bespoke methods.

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##  1\. Introduction

Deep Clustering (DC) combines the unsupervised grouping of related data items
with the learning of clustering-friendly representations for the data to be
clustered. DC uses deep neural network architectures with unsupervised
clustering mechanisms to effectively classify multi-dimensional, unlabeled
data. Early implementations primarily used autoencoders for dimensionality
reduction, where the embeddings in the latent space were clustered using
standard clustering (SC) algorithms such as K-means (Yang et al., 2017).
Advancements in DC have led to more sophisticated architectures, based on
variational autoencoders (Dilokthanakul et al., 2016), Generative Adversarial
Networks (Mukherjee et al., 2019), and self-supervised learning paradigms
(Alwassel et al., 2020). Recent DC methods focus on integrating representation
learning and cluster assignment within end-to-end trainable models using
graph-based neural networks (Liu et al., 2022; Tu et al., 2021; Bo et al.,
2020) to capture relational structures within data.

DC has been employed successfully in several domains, particularly image
processing (Zhou et al., 2022, 2022; Qian, 2023), which includes downstream
tasks such as segmentation and object detection (Zhou et al., 2022),
clustering of in and out-of-distribution noise in corrupted images (Albert et
al., 2022), and unsupervised image alignment and clustering (Jin et al., 2023;
Qian, 2023). Further typical applications of DC include community and anomaly
detection (Emadi and Mazinani, 2018; Liu et al., 2020) and medical
applications (Kart et al., 2021). These DC proposals excel in determining
complex patterns within images and graphs, extracting features, and
subsequently clustering similar images and communities based on those
patterns. The successful application of DC in specific domains has led to the
development of more specialized algorithms that reflect specific properties of
the domain features, even if the algorithms are not explicitly domain-
specific. For example, in (Chen et al., 2022a), the authors use domain-
specific 3D geometric consistency to guide the learning of 2D image
representation and solve several 2D-image-based downstream tasks.

In data management, and in particular data integration and cleaning, a variety
of important problems make use of clustering, including schema inference
(Kellou-Menouer and Kedad, 2015; Bonifati et al., 2022), entity resolution
(Papadakis et al., 2020; Costa et al., 2010; Hassanzadeh et al., 2009) and
domain discovery (Piai et al., 2023; Ota et al., 2020). We have investigated
the application of existing DC algorithms to these three problems, with
positive results; DC algorithms consistently outperform established SC
algorithms for a variety of initial data representations for all of tables,
columns and rows (Rauf et al., 2024). However, existing DC algorithms do not
fully exhibit the properties required to address the clustering of latent
representations of table objects, required to support the transfer to data
cleaning and integration tasks. In our previous work (Rauf et al., 2024), we
observed that existing DC algorithms often fall short when applied to data
integration tasks, where the nature of the data is fundamentally different
from that in image processing applications, with tables, rows or columns being
compared in the latent space instead of images (Rauf et al., 2024).

In this paper, we propose a new deep clustering algorithm for tabular data
(TableDC) that reflects the properties of table embeddings and data management
applications, specifically: (i) Data representations are typically dense,
where features are closely packed together. As the dimensionality increases,
it becomes more challenging for the models to interpret the relationships
between features. High density in data reduces the differences in features and
makes it difficult to differentiate between clusters effectively. The
closeness of data points in the latent space leads to ambiguity, where a data
point could reasonably belong to multiple clusters. (ii) Tabular data contains
higher semantic and syntactic variability levels than image data, leading to
increased ambiguity and cluster overlap. Tables, columns and rows commonly
involve the composition of different types of semantic objects (e.g., named
entities, categories, numbers), each in their own way subject to ambiguity,
leading to potentially overlapping embeddings when projected to high-
dimensional latent spaces. In contrast, while complex, image data is based on
a single data modality, with the expectation of smoother transfer of visual
features across domains, naturally conforming to a better cluster separation.
(iii) In a data management setting, data can have a large number of clusters
compared to existing DC applications. In our previous work (Rauf et al.,
2024), we observed that existing DC algorithms struggle with scalability when
the data has a large number of clusters. This affects the DC method’s
performance in terms of the high probability of misclassification, a higher
computational burden, and cluster instability, where small changes in data and
model parameters lead to significant changes in cluster assignments.

The contributions of this paper are:

  1. (1)

The identification of features of data management applications that challenge
the assumptions that underpin existing DC algorithms.

  2. (2)

The proposal of a new DC algorithm, TableDC, that builds on the analysis at
(1) to provide an algorithm that: (i) Maintains a balance between cluster
compactness and separation during cluster center initialization to better
assist TableDC training with robust cluster shapes. (ii) Integrates
Mahalanobis distance, a covariance-aware distance that considers the data’s
variance and covariance to handle dense features, unlike Euclidean distance,
which treats all dimensions equally. (iii) Offers flexibility in overlapping
clusters through its heavy-tailed Cauchy distribution as a similarity kernel,
especially when the cluster boundaries are ambiguous. (iv) Scales in quasi-
linear time under significant increases in the number of clusters.

  3. (3)

The experimental evaluation of TableDC from (2) with both state-of-the-art SC
and DC algorithms and recent bespoke algorithms on real-world datasets for
schema inference, entity resolution and domain discovery, which shows that
TableDC significantly outperforms existing techniques.

##  2\. Related Work

This section reviews related work in areas of relevance to the development of
a deep clustering algorithm for data management applications, specifically:
(i) existing DC algorithms, their features and applications ; and (ii)
clustering in data management, focusing on the three applications on which we
evaluate techniques and representations for tables, columns and rows.

###  2.1. Existing DC algorithms

The basic DC architecture consists of two phases: (i) representation learning
and (ii) clustering. The joint optimization of both phases helps each to
improve the algorithm’s performance iteratively. A precursor to (i) and (ii)
is (iii) cluster initialization, which contributes to the algorithm’s
performance by providing the initial data shape. With better cluster
initialization, a DC algorithm is less likely to get stuck in local minima,
and requires fewer converging iterations.

Regarding (i), the most widely used architecture is the basic Auto-encoder
(AE) (Yang et al., 2017), where the encoder function $f_{e}$ encodes an input
representation $x_{i}$ into a latent representation given by
$h_{i}=f_{e}(x_{i})=\frac{1}{1+e^{-(Wx_{i}+b_{i})}}$. Similarly, the decoder
function $f_{d}$ decodes $h_{i}$ back into the reconstructed input
$x^{\prime}_{i}=f_{d}(h_{i})$. Here, $W$ and $b_{i}$ represent the weights and
biases of the neural network, respectively. The optimization function of AE
with $N$ samples is defined as
$f_{min}=\min\frac{1}{N}\sum_{i=1}^{N}\|x_{i}-x^{\prime}_{i}\|^{2}$. Many
recent DC proposals use generative models in representation learning,
especially Variational Auto Encoders (VAEs), which combine an AE architecture
with probabilistic graphical models. In VAEs, the encoder maps the input data
into a probabilistic distribution over the latent space while the decoder
reconstructs the data from the latent probabilistic representation. Some
recent proposals with VAEs in representation learning are DGG (Yang et al.,
2019), LTVAE (Li et al., 2019), VaDE (Jiang et al., 2017) and MFCVAE (Falck et
al., 2021).

To benefit from the local and global structure of data in representation
learning, several recent DC proposals combine basic AE learning with a Graph
Convolutional Network (GCN) to assist the model in learning the structural
information. Some recent examples are DCRN (Liu et al., 2022), DFCN (Tu et
al., 2021) and SDCN (Bo et al., 2020). Subspace representation learning is
another recent technique that identifies subspaces that optimize cluster
separation, unlike AEs and VAEs, which require additional clustering
mechanisms. Subspace learning seeks to identify and represent underlying
subspaces within high-dimensional data, while AEs and VAEs focus on
reconstructing the input data as accurately as possible. The latest DC methods
based on Subspace representation learning are EDESC (Cai et al., 2022), DMCAG
(Cui et al., 2023), SAGSC (Fettal et al., 2023) and OMSC (Chen et al., 2022b).

DC is inherently more complex for the tabular data embeddings obtained by
Large Language Models (LLMs) than widely used sparse image or textual datasets
like CIFAR-10 (Krizhevsky et al., 2009), CIFAR-100 (Krizhevsky et al., 2009),
STL-10 (Coates et al., 2011), ImageNet (Russakovsky et al., 2015), USPS (LeCun
et al., 1990) and Reuters (Lewis et al., 2004), primarily due to data
structure and feature representation differences. Image datasets include
spatial and visual coherence that DC methods can better differentiate, even
when the resolution is low or the number of classes is high. DC has been used
widely on complex and varied datasets unlike ImageNet, where, despite the vast
diversity, the visual patterns (such as extracted from different textures,
shapes, objects, and backgrounds) retain a level of consistency

On the other hand, the primary issue with data management datasets is the high
density and significant overlap in the feature space of their embeddings;
specifically, tabular data embeddings represent a complex relationship among
different features, including categorical and numerical data.

In relation to (ii), clustering, the aim of existing work on DC is to use the
low dimensional representation from the representation learning module and
obtain clustering assignment probabilities for soft clustering. In the related
work, several clustering mechanisms are employed to jointly work with
representation learning for optimal clustering. The most widely used one is
self-supervised clustering (Dizaji et al., 2017; Guo et al., 2017; Peng et
al., 2017; Xie et al., 2016; Bo et al., 2020; Tu et al., 2021), which aims to
group data points based on learned representations without explicit labels;
the approach uses two distributions: the cluster assignment distribution $Q$
representing the probability that a data instance belongs to a particular
cluster, and an auxiliary distribution $P$ which is a modified version of $Q$
that emphasizes more confident cluster assignments and minimizing the
Kullback-Leibler (KL) divergence between $Q$ and $P$. $Q$ is calculated using
the distance between the data point $x_{i}$ and each cluster centroid
$\mu_{k}$. The closer $x_{i}$ is to $\mu_{k}$, the higher the probability $Q$.
Self-supervised clustering focuses on high-confidence instances due to the
squaring of probabilities in $P$ and preventing cluster dominance by
normalizing the probabilities, ensuring balanced clustering assignments.

Pseudo-labeling is another clustering technique recently utilized in DC (Niu
et al., 2022; Zhou et al., 2022), where the model generates labels that are
then used to fine-tune the model, effectively creating a feedback loop where
the model iteratively refines its clustering assignment. The recent success of
contrastive learning has led to its use in DC (Sun et al., 2023; Zhang et al.,
2023; Liu et al., 2023; Ma and Kim, 2022), where the data points from low
dimensional representations are paired in a way that the same cluster
(positive pairs) is contrasted with the different clusters (negative pairs).
Further improvements in contrastive deep clustering include the contrast
comparison between instance-to-instance, instance-to-cluster, and cluster-to-
cluster (Zhou et al., 2022).

TableDC employs a self-supervised clustering approach by comparing the
encodings of tables, rows or columns within a latent space. The selection of
an appropriate distance function is critical in this context, as it determines
how the distances between each table, row or column and the cluster centroids
in the latent space are measured. A simple distance metric like Euclidean
distance is beneficial when dealing with data where features exhibit limited
variance and correlation, as in image vectors or sparse data matrices (Bo et
al., 2020; Tu et al., 2021). However, a more sophisticated approach is
required in dense embeddings, where the data points show significant feature
correlation. Here, the Mahalanobis distance is particularly advantageous
(Mahalanobis, 2018). Its ability to account for feature correlation enables
the model to discern even subtle differences among clusters, mainly when a
cluster shows high similarity across multiple features. Mahalanobis distance
is particularly relevant when comparing tables, columns or rows, as it allows
for a more nuanced interpretation and grouping of data points based on their
inherent relationships and similarities.

In relation to (iii), most existing works use K-means to initialize clusters,
whether self-supervised or subspace clustering such as EDESC (Cai et al.,
2022), DCRN (Liu et al., 2022), DFCN (Tu et al., 2021) and SDCN (Bo et al.,
2020). The quality of the initial clusters significantly affects the final
clusters. To our knowledge, there is no related work on different cluster
initialization approaches in the DC environment. To address this, we compare
different cluster initialization approaches in an ablation study in Section
5.1.

###  2.2. Clustering in data management

Clustering is important in data management for several data integration and
cleaning tasks, including those discussed here.

Schema Inference identifies the raw data’s types, relationships and attributes
to infer an overarching schema (Kellou-Menouer et al., 2022). Schema inference
has been explored to infer a schema that fully describes some closely related
datasets (Baazizi et al., 2019; Bex et al., 2007) or to summarise structures
that appear in more diverse datasets. A common first step in the latter case
is the identification of recurring structural patterns, which can be used as
types in an inferred schema. In this setting, a variety of SC algorithms have
been used to group similar structures (e.g., (Bonifati et al., 2022; Kellou-
Menouer and Kedad, 2015; Tsuboi and Suzuki, 2019)).

Entity Resolution is the well-explored problem of identifying two or more
records that represent the same real-world object (Christophides et al.,
2021). Several proposals use clustering, especially for entity matching and
duplicate detection, for which empirical comparisons have been carried out
(Hassanzadeh et al., 2009; Costa et al., 2010; Saeedi et al., 2018; Draisbach
et al., 2020; Hassanzadeh et al., 2009). Although many entity resolution
proposals focus on pairwise comparisons, some proposals incorporate clustering
because the transitive closure of pairwise similarity may not always lead to
robust clusters (Fisher et al., 2015; Mandilaras et al., 2021).

Domain Discovery involves identifying columns that draw values from the same
collection of values. Most existing works have used bespoke algorithms (Li et
al., 2017; Ota et al., 2020; Piai et al., 2023). However, we cast domain
discovery as a clustering problem, where the goal is to cluster a set of
columns that share semantic types. RaF-STD (Piai et al., 2023) uses clustering
to discover semantic types for heterogeneous sources. Similarly, D4 (Ota et
al., 2020) provides a column-based clustering approach to discovering local
and strong domains from a set of columns, while handling the challenge of
incomplete columns. Building on language models, Starmie (Fan et al., 2023)
provides column clustering based on the learned column representation through
contrastive learning.

##  3\. Proposed Model

We propose a clustering algorithm, TableDC, that benefits from representation
learning in an end-to-end framework to perform several tabular data
integration tasks, particularly schema inference, entity resolution, and
domain discovery. TableDC considers features of embeddings explicitly in the
latent space during its training, such as preserving correlation among highly
dense data features through the Mahalanobis distance measure and using a
heavy-tailed similarity kernel that provides the flexibility of cluster
assignments when the cluster boundaries are ambiguous.

The representation learning architecture of TableDC is based on self-
supervised learning (Bo et al., 2020). The self-supervised module adopts
Mahalanobis distance (Mahalanobis, 2018) between data points and their
centroids as it naturally accounts for the variance of different dimensions
and is less sensitive to noise, ensuring that no particular feature influences
the distance measure due to its scale. Unlike learning a Euclidean distance in
a self-supervised module (Bo et al., 2020), which assumes that all dimensions
of the data are orthogonal for the clustering, the Mahalanobis distance shows
some degree of correlation between different dimensions by weighting the
importance of all dimensions, which could be different based on their
semantics.

TableDC integrates an autoencoder for representation learning and a self-
supervised module to optimize representation for clustering. An autoencoder
consists of two main parts: the encoder and the decoder. The encoder function
compresses the input $x$ (an embedding matrix of tables, rows or columns) into
a latent-space representation. Mathematically, it can be represented as:

(1) $h=f(W_{e}\times x+b_{e})$

where $W_{e}$ is the weight matrix, $b_{e}$ is the bias term, and $f$ is an
activation function for the encoder, such as sigmoid or ReLU (Nair and Hinton,
2010). The result, $h$, is the encoded representation of the input $x$. The
decoder function works to reconstruct $x$ from the internal representation. It
aims to map the encoded data back from the latent space to the original space.
The decoder can be represented as:

(2) $\hat{x}=g(W_{d}\times h+b_{d})$

where $h$ is the encoded representation from the encoder, $W_{d}$ is the
weight matrix, $b_{d}$ is the bias term, and $g$ is an activation function for
the decoder. The result, $\hat{x}$, is the reconstructed representation of the
original input $x$. An overview of the TableDC framework is presented in
Figure 1.

Figure 1. Overview of TableDC framework. An autoencoder takes an embedding
matrix as input, representing a set of rows for entity resolution, a set of
columns for domain discovery, or a set of tables for schema inference. z is
the latent representation obtained from the autoencoder, and c is initialized
using Birch. In the self-supervised module, the m distribution is calculated
from q by taking the Mahalanobis distance between z and c and setting the
Cauchy distribution as a kernel to measure the similarity. p is the target
distribution, which is also calculated from q.

Assume we have a latent vector with $n$ data points
$z={z_{1},z_{2},\dots,z_{n}}$, where each $z$ represents a row, column or
table depending on the data integration problem. We want to cluster the data
points into $\mathbb{K}$ clusters with centers $c={c_{1},c_{2},\dots,c_{k}}$.
We create a covariance matrix $\Sigma$ to compute the Mahalanobis distance
between the data points $z={z_{1},z_{2},\dots,z_{n}}$ and the cluster centers
$c={c_{1},c_{2},\dots,c_{k}}$. Since we have noisy data (for example, columns
with missing instances, similar tables with different unit measurements, and
duplicate rows that affect the embeddings), the variance between different
dimensions misleads the importance of the features. We aim to minimize the
noise effect when calculating the covariance by scaling the identity matrix.
The scaled identity matrix ensures that all features do not have the same
importance to avoid the Euclidean effect and also handles multicollinearity in
the data when the covariance matrix is singular. In the case of overlapping
clusters, it avoids noisy correlations. The covariance matrix is created by
scaling the identity matrix with a scale factor $\delta=0.01$ to adjust the
strictness of distance between data points:

(3) $\Sigma=\delta\cdot I,$

where $I$ is the identity matrix. We calculate the Cholesky decomposition of
the covariance matrix $\Sigma$ to simplify its inversion. The Cholesky
decomposition is a decomposition of a Hermitian, positive-definite matrix into
the product of a lower triangular matrix $L$ and its conjugate transpose
$L^{T}$. In our case, since the covariance matrix $\Sigma$ is real and
symmetric, the Cholesky decomposition can be represented as:

(4) $C=LL^{T},$

where $C$ represents the Cholesky decomposition of the covariance matrix
$\Sigma$, $L$ is the lower triangular matrix with real and positive diagonal
entries, and $L^{T}$ is the transpose of $L$. The matrix $L$ is computed using
the Cholesky decomposition algorithm.

The Cholesky decomposition is used to solve a linear system to obtain the
Mahalanobis distance, as it provides an efficient way to compute the inverse
of the covariance matrix:

(5) $\Sigma^{-1}=(C)^{-1}=L^{-T}L^{-1}.$

By using the Cholesky decomposition, we can efficiently solve the linear
system required to compute the Mahalanobis distance between the latent
representations and the cluster centers.

(6)
$D_{\text{M}}(z_{i},c_{j})=\sqrt{(z_{i}-c_{j})^{\text{T}}\Sigma^{-1}(z_{i}-c_{j})},$

where $\Sigma$ is the covariance matrix.

The Student’s t-distribution (van der Maaten and Hinton, 2008) is a common
choice for existing DC applications as a kernel to measure the similarity
between data points and their cluster centers. However, due to its large
degree of freedom parameter, the Student’s t-distribution shows less tolerance
to outliers and overlapping clusters; it becomes closer to a normal
distribution. In data integration, where the data is more prone to overlaps,
i.e., the same magnitude with different semantics, we integrate the Cauchy
distribution (Cauchy et al., 1847) with undefined mean and variance, making it
robust to outliers, as its shape is unaffected by them. We calculate the soft
assignments $q$ for each data point to each cluster using the Cauchy
distribution. Soft assignments $q$ can be calculated using the Mahalanobis
distance:

(7) $q_{ij}=\frac{1}{1+\frac{D_{\text{M}}(z_{i},c_{j})^{2}}{\gamma^{2}}},$

where $\gamma$ is a hyperparameter for the Cauchy distribution. We then
normalize $q$ to ensure that the sum of the soft assignments for each data
point is 1:

(8) $q=\frac{q}{\sum_{j=1}^{k}q_{ij}+\epsilon},$

where $\epsilon$ is a small value added to prevent division by zero. We then
compute the predicted probabilities by applying the softmax function to $q$:

(9) $m=\frac{e^{q}}{\sum_{j=1}^{k}e^{q_{ij}}}.$

###  3.1. Loss Function

Since $m$ acts as clustering probabilities and can be further optimized, we
integrate KL-divergence to refine the clustering probabilities $m$ to be more
accurate and representative of the underlying data. KL divergence provides a
flexible way to guide the model toward the expected clustering assignments. We
use the following objective function to optimize the $m$.

(10)
$ce_{loss}=\text{KL}_{\text{div}}(p||m)=\sum_{i=1}^{n}\sum_{j=1}^{k}p_{ij}\cdot\log\frac{p_{ij}}{m_{ij}},$
(11) $p_{ij}=\frac{q_{ij}^{2}}{\sum_{i=1}^{n}q_{ij}},$

$ce_{loss}$ is clustering loss, where $p_{ij}$ is the target distribution,
calculated by adjusting the soft assignments $q$. This step aims to make the
soft assignments more robust during the clustering process. We use the
following reconstruction loss ($re_{loss}$) to maintain the inherent structure
and essential features in a compressed form.

(12) $re_{loss}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x}_{i})^{2},$

where $\bar{x}$ is the reconstructed input. $ce_{loss}$ and $re_{loss}$ work
together to achieve effective clustering while maintaining the quality of the
data reconstruction. The total loss is given by:

(13) $loss=\alpha\cdot ce_{loss}+re_{loss},$

where $\alpha$ is a weighting factor that balances the contribution of the
$ce_{loss}$ and the reconstruction loss ($re_{loss}$) to the total loss.
Combining $ce_{loss}$ and $re_{loss}$ enables autoencoder learning to generate
meaningful latent representations that can be effectively clustered while
preserving the structure and information in the input data. We show the
pseudo-code for training of TableDC in Algorithm 1.

Algorithm 1 Pseudo-code for training of TableDC

0: $x$: representation of rows, columns or tables; $\mathbb{K}$: number of
clusters; $maxiter$: maximum number of iterations

0: m: clustering probabilities

1: Initialize $W_{e},b_{e},W_{d},b_{d}$ by pre-training autoencoder

2: Initialize cluster centers $c$ using Birch algorithm

3: for $i=1$ to $\text{length}(maxiter)$ do

4: Generate latent representation $z$ from autoencoder using (1)

5: Compute inverse of covariance matrix $\Sigma^{-1}$ using Cholesky
decomposition from (5)

6: Calculate Mahalanobis distance $D_{\text{M}}$ between $z$ and $c$ using (6)

7: Apply Cauchy distribution on $D_{\text{M}}$ to get soft assignments $q$
from (7)

8: Normalize $q$ and apply softmax to get predicted probabilities $m$

9: Calculate target distribution $p$ using (11)

10: Calculate $ce_{loss}$ and $re_{loss}$ from (10) and (12) respectively

11: Back propagate and update parameters in TableDC

12: end for

13: return Updated $m$

The reconstruction loss, $re_{loss}$ (Equation 12), of TableDC is consistently
reducing over epochs during the training (see ablation study in Section 5.2),
which shows that TableDC is learning a meaningful representation of the data,
capturing essential features and structures. In $ce_{loss}$ (Equation 10), m
is a softmax-ed q consistently diverging from the target distribution p (see
ablation study in Section 5.2). After applying softmax, q becomes a
probability distribution m over clusters. Each row of the probability matrix
has a sum of 1, representing the probability for each point to belong to each
cluster. The softmax function emphasizes the significant values in a
probability matrix m while suppressing smaller values, making TableDC soft
assignments sharper. The choice of the hyperparameter $\alpha$ allows for
fine-tuning the trade-off between clustering performance and reconstruction
quality. We use $\alpha$ = 0.9 (Equation 13), so TableDC gives more importance
to the clustering loss, $ce_{loss}$ (Equation 10) than $re_{loss}$. We
determined the weight of $\alpha=0.9$ for the $ce_{loss}$ through empirical
evaluation.

The presence of noise or exact probabilities towards more than one cluster can
help provoke the model towards more confident cluster assignments despite
having inherent ambiguities. p is artificially derived from q by re-
normalizing and making high-probability assignments higher and diminishing
low-probability assignments. However, the high divergence of m from the
sharped p shows that TableDC is more confident with the actual soft
assignments q becoming sharper over time. Moreover, the data points in target
p need more intra-cluster distance instead of high cohesion. A consistent
increase in divergence of m from q indicates that TableDC’s internal
clustering, as represented by m, is moving in a direction that better aligns
with the true data structure. p is an auxiliary target to guide the clustering
process and does not represent the ground truth; in the presence of an
overlapping cluster, the divergence from p is not inherently poor. Considering
this phenomenon, we can consider the KL-divergence as a maximization problem
where we prefer m to have a maximum divergence from p and can be defined as a
sequence of KL-divergences over $N$ iterations: $\\{D_{KL}(p\parallel
m^{1}),D_{KL}(p\parallel m^{2}),\ldots,D_{KL}(p\parallel m^{N})\\}$ where
$D_{KL}(p\parallel m^{k})$ represents the KL-divergence between $p$ and $m$ at
the $k^{th}$ iteration. The KL divergence is increases over $N$ if:
$D_{KL}(p\parallel m^{k+1})>D_{KL}(p\parallel m^{k})$ for all $k$ in
$\\{1,2,\ldots,$N-1$\\}$.

###  3.2. Initialisation

In the training phase of DC algorithms, the quality of the initial centroids
can directly affect the quality of the final clusters. Poor initialization can
lead to poorly formed clusters, such as unevenly sized, too small, or too
large. Existing DC methods (Bo et al., 2020; Liu et al., 2022; Tu et al.,
2021) use K-means to initialize cluster centers c. However, in data
integration problems dealing with highly dense and overlapping embedding
representation, K-means may not be a suitable initialization (see ablation
study in Section 5) for the following reasons: (i) In densely packed and
overlapping data dimensions, the Euclidean distance between c and z becomes
uniform, which leads to poor cluster initialization. (ii) In a dense data
space, many sub-optimal solutions exist, and K-means is not optimal enough to
avoid local minima. (iii) K-means initialization is based on hard assignment,
where the centers are randomly initialized. However, dense data with
overlapping clusters requires soft assignments based on probabilities. K-means
initialization is often considered suitable in existing DC proposals due to
the nature of the data, which is mainly sparse or semi-dense, particularly in
image representations. The Euclidean distance between different points becomes
more distinct in a sparse data space, resulting in well-separated clusters.
This distinct separation enhances the effectiveness of K-means initialization
in these contexts, facilitating the clustering process by providing a clear
starting point for the algorithm.

In TableDC, we integrate the Birch algorithm (Zhang et al., 1996) to
initialize the clusters. Birch is a traditional hierarchical clustering
algorithm that handles large and noisy databases. Birch uses a Clustering
Feature tree (CF-tree) to process each data point. A CF-tree is a set of
nodes, and each node consists of a triplet that efficiently stores information
on data points in a compressed format to the cluster instead of processing a
complete dataset. The triplet includes the number of data points per cluster,
squared, and linear sum. In a dense space, each node of the CF-tree represents
aggregated properties of each data point instead of the distance between data
points, which enables Birch initialization to avoid close proximity and high
overlaps. Since Birch is hierarchical, the multiple levels of the CF-tree can
effectively capture the different granularities of clusters even if the
overlaps exist on one level. We compare different cluster center
initialization techniques in the ablation study (see Section 5.1). The pseudo-
code for the initialization of c using Birch is given in Algorithm 2).

Algorithm 2 Pseudo-code for initialization of c using Birch

0: z: data points; K: number of clusters; T: threshold; B: branching factor
(the maximum number of child nodes); L: The maximum number of entries in a
leaf node

0: c: cluster centers and cluster assignments for each z

1: Create an empty CF-tree with T, B, and L.

2: Initialize Birch clustering with K

3: for each z in the dataset do

4: Traverse the CF-tree to find a suitable corresponding leaf node

5: Update the CF-tree in the path from the root to the last leaf node at each
level

6: if the value of leaf node $>$ T then

7: Split the leaf node

8: end if

9: end for

10: Compute clusters k and assign a cluster index to each z

11: for each k do

12: Compute c as a mean of z assigned to k

13: Add c to the list of cluster centers

14: end for

15: return $c_{i}=c$

##  4\. Evaluation

This section evaluates TableDC using existing benchmarks with a variety of
different embedding approaches, in comparison with other clustering algorithms
and bespoke solutions.

###  4.1. Experimental setup

####  4.1.1. Datasets

TableDC is evaluated on six benchmark datasets widely adopted for data
integration problems, specifically schema inference, domain discovery and
entity resolution. A brief description and statistical detail of each data set
are given below and in Table 1.

  * •

T2D Entity-Level Gold standard (T2D) (Ritze and Bizer, 2017) is a widely used
collection of web tables with mappings to their corresponding DBpedia class.
Regarding schema inference, T2D can be used to align the schema of web tables
(structure and format) with the schema used in DBpedia. In T2D, we excluded
all tables categorized under the DBpedia class ’Thing’. This is because the
’Thing’ category is mapped to over 50% of the web tables, leading to a
significant risk of data imbalance.

  * •

Table Union Search (TUS) benchmark (Nargesian et al., 2018), refers to finding
unionable tables for a given query table. However, for clustering application,
the problem can be redefined as each cluster shares unionable tables.

  * •

MusicBrainz (Saeedi et al., 2017) dataset is derived from authentic records of
songs sourced from the MusicBrainz database. It combines the data from five
distinct sources, where approximately 50% of the original song records are
duplicated across two to five sources.

  * •

Geographic Settlements (GeoSet) (Saeedi et al., 2017) represents a collection
of geographical real-world entities from four different sources (Geonames,
Freebase, DBpedia, and NYTimes).

  * •

Di2KG (Camera and Monitor)111http://di2kg.inf.uniroma3.it/datasets.html
contains a set of cameras and monitors web records scraped from multiple
e-commerce web pages. Both datasets are highly heterogeneous. For example,
specification lcd display from www.price-hunt.com and monitor from
www.cambuy.com.au semantically represent the same domain despite having
different syntactic structures.

We use T2D and TUS for schema inference, MusicBrainz and GeoSet for entity
resolution, and Di2KG (Camera and Monitor) datasets for domain discovery.

Table 1. Statistics of benchmark datasets used to evaluate TableDC for schema inference, entity resolution and domain discovery. |  | Dataset | Instances | Clusters
---|---|---|---
Tables | web tables | 429 | 26
| TUS | 4248 | 37
Rows | Music Brainz | 2002 | 684
| GeoSet | 3021 | 786
Columns | Camera | 19036 | 56
| Monitor | 34481 | 81

####  4.1.2. Benchmark Methods

We compared TableDC222https://github.com/hafizrauf/TableDC with three SC and
five DC clustering methods. The selected SC methods represent different
paradigms. Similarly, the chosen DC methods represent different approaches to
representation learning (self-supervised and subspace). A short description of
each benchmark method is given below:

  * •

K-means (Hartigan and Wong, 1979) is a classical clustering algorithm; it
assigns instances to the nearest centroid to create clusters. Centroids are
updated based on the assigned points.

  * •

DBSCAN (Ester et al., 1996) identifies distinct clusters based on the data
density, considering the number of neighboring points within a specified
radius. DBSCAN is particularly good with data having variation in densities.

  * •

Birch (Zhang et al., 1996) is a hierarchical clustering algorithm built on a
CF-tree to summarize the information (on each tree node) needed to cluster
instances.

  * •

SHGP (Yang et al., 2022) is a self-supervised method that shares an attention-
aggregation mechanism between two modules, Att-LPA  to produce pseudo-labels
and Att-HGNN, to learn object embeddings. Both modules effectively improve
each other to optimize embeddings.

  * •

DCRN (Liu et al., 2022) learns representations through AE and GCN by reducing
the information correlation to improve the discriminative property of the
resulting features.

  * •

DFCN (Tu et al., 2021) focuses on the fusion of representation learning of AE
and graph neural networks. It works on a dynamic fusion mechanism to refine
the graph neural network structure to better integrate with AE representation
learning.

  * •

EDESC (Cai et al., 2022) EDESC is a subspace representation-based learning
method that iteratively refines the bases of the subspace derived from deep
representations.

  * •

SDCN (Bo et al., 2020) effectively utilizes the structural information in the
AE representation learning. The dual self-supervised mechanism effectively
updates the model parameters to produce clustering-specific representations.

Table 2. Comparing schema inference clustering results (TableDC vs. existing DC methods). The best results are shown in bold. Representations marked with $*$ are obtained on instance-level data and $TT^{*}$ refers to TabTransformer. Method | TUS | Web tables
---|---|---
| SBERT | FastText | $TT^{*}$ | $SBERT^{*}$ | SBERT | USE | $TT^{*}$ | $SBERT^{*}$
| ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC
K-means | 0.73 | 0.79 | 0.63 | 0.69 | 0.22 | 0.35 | 0.46 | 0.56 | 0.27 | 0.45 | 0.19 | 0.40 | -0.013 | 0.27 | 0.32 | 0.51
DBSCAN | 0.17 | 0.47 | 0.01 | 0.25 | 0.02 | 0.26 | 0.09 | 0.38 | 0.0 | 0.29 | 0.0 | 0.29 | -0.007 | 0.26 | 0.0 | 0.29
Birch | 0.22 | 0.40 | 0.0 | 0.20 | 0.18 | 0.35 | 0.54 | 0.58 | 0.33 | 0.49 | 0.22 | 0.40 | -0.013 | 0.27 | 0.46 | 0.61
SHGP | 0.61 | 0.72 | 0.43 | 0.56 | 0.05 | 0.19 | 0.13 | 0.29 | 0.08 | 0.29 | 0.07 | 0.28 | -0.0031 | 0.27 | 0.09 | 0.30
DCRN | 0.50 | 0.59 | $\infty$ | $\infty$ | 0.018 | 0.24 | 0.15 | 0.26 | 0.07 | 0.25 | 0.03 | 0.25 | -0.02 | 0.25 | 0.17 | 0.36
DFCN | 0.55 | 0.65 | 0.43 | 0.52 | 0.20 | 0.37 | 0.34 | 0.43 | 0.24 | 0.40 | 0.19 | 0.33 | -0.0006 | 0.24 | 0.37 | 0.48
EDESC | 0.28 | 0.35 | 0.22 | 0.32 | 0.14 | 0.30 | 0.36 | 0.43 | 0.26 | 0.44 | 0.22 | 0.37 | 0.08 | 0.25 | -0.04 | 0.26
SDCN | 0.60 | 0.57 | 0.53 | 0.47 | 0.25 | 0.40 | 0.45 | 0.49 | 0.20 | 0.34 | 0.08 | 0.33 | 0.0 | 0.29 | 0.30 | 0.42
TableDC | 0.88 | 0.87 | 0.73 | 0.76 | 0.24 | 0.37 | 0.63 | 0.66 | 0.62 | 0.65 | 0.25 | 0.42 | -0.006 | 0.26 | 0.61 | 0.65

####  4.1.3. Representations

Each data integration task has an associated embedding model for representing
tables, rows or columns which is integrated into the DC method for clustering.
We use six different embedding models, each of which is specific to one or
more data integration tasks; details are given below:

Schema Inference: We use pre-trained embeddings to get the representations of
raw data. We categorized these into two categories. When we consider headers,
we use SBERT (Reimers and Gurevych, 2019), FastText (Grave et al., 2018) and
USE (Cer et al., 2018); with instances, we use TabTransformer and SBERT. SBERT
uses siamese and triplet network structures to extract semantically meaningful
embeddings from sentences. FasText is specialized in extracting morphological
information and considers subwords, representing each word as a bag of
character n-grams. Like SBERT, USE is also a sentence encoder specializing in
capturing the semantic meanings of sentences. Considering the instances along
with the header is a more tricky task and requires more robust transformers.
We fine-tuned TabTransformer (Huang et al., 2020) on both datasets to benefit
the categorical and contentious features in the dataset. We also encoded
instance data with SBERT; we convert each row of the table into a sentence
sequence, where each row is represented as a sequence of its cell values
appended with [SEP] token for SBERT to separate segments and maintain the row
boundaries in the text. [SEP] carries the table structure within the BERT
model in the encoded form.

Entity Resolution: We train EmbDi (Cappuzzo et al., 2020) on both datasets to
embed rows and used pre-trained SBERT. EmbDi is a graph-based representation
method based on tripartite connectivity within a graph of table elements to
extract structural information. A tripartite connection contains a column node
(attribute representation), a value node (unique value representation), and a
row node (unique tuple token).

Domain Discovery: Similar to schema inference, we embed columns in two
categories, with and without column headers. We used SBERT and T5 (Raffel et
al., 2020) pre-trained embeddings for both purposes. When considering
instances, we embed column values using SBERT and T5 and aggregate the
individual embeddings to get the final column embeddings. T5 is a widely used
language model developed by Google for information retrieval tasks and
generating embeddings, specifically capturing semantic information within
data.

###  4.2. Metrics

We adopted two widely used standard clustering evaluation metrics, Accuracy
(ACC) (Yang et al., 2010) and Adjusted Rand Score (ARI) (Wu et al., 2019) to
test the performance of TableDC and existing benchmark methods. ARI ranges
from -1 to 1, where 1 indicates perfect matching, 0 indicates random labeling
and negative values indicate worse than random labeling. ACC ranges from 0 to
1.

###  4.3. Parameter setting

TableDC consists of four AE layers and is trained using the Adam optimizer.
The loss function combines KL-divergence (to minimize the Mahalanobis distance
in self-supervised learning) and mean squared error loss for reconstruction.
The latent space size is fixed at 100. In order to be consistent with the
existing DC benchmarks, we pre-trained TableDC with AE on 30 epochs for schema
inference and domain discovery. Pretraining provides a good starting point for
the network parameters, avoiding random initialization and local minima during
the clustering phase, which leads to faster convergence. Since entity
resolution involves many clusters, the CF-tree in Birch initialization needs
more nodes with detailed cluster summaries, each representing a smaller and
more defined cluster. We have pre-trained TableDC on 100 epochs for entity
resolution. Training epochs for schema inference are fixed to 200, domain
discovery to 100, and entity resolution to 50. We run all existing benchmark
methods on the same number of epochs adopted for TableDC, excluding the
internal parameters for which we use the originally published values for each
existing method, such as the number and size of each layer. We tuned the
threshold T (the maximum range of radius of a cluster in the CF-tree) when
initiating the centroids in TableDC for those experiments where the number of
subclusters found on each CF-tree node is less to be able to summarize cluster
information. We adopted a grid search on threshold T and divided the number of
sub-clusters unless the CF-tree becomes stable. Some existing methods need
K-means for centroids initialization and clustering assignments. To avoid
extreme cases, we initialize 20 times and choose the best solution.

###  4.4. Schema inference clustering results

Schema inference as a clustering problem can be defined as the clustering of
tables that share a similar schema. Table 2 shows the clustering results of
schema inference on Table Union Search (TUS) and web tables datasets encoded
with several representations.

We observe the following: (i) TableDC obtained the best results over both
datasets when considering schema-level data (SBERT, FastText, USE) compared to
all benchmark methods, including SC. TableDC improves the clustering
performance (with SBERT) by 0.28 and 0.27 ARI and 0.30 and 0.15 ACC, compared
to SDCN and SHGP, respectively. (ii) TableDC exhibits good but not always best
performance on instance-level data (TabTransformer, SBERT) and was
outperformed by SDCN by a small margin (0.01 ARI and 0.03 ACC) on the TUS
dataset and by EDESC on the web tables dataset using TabTransformer. However,
TableDC outperformed the other methods with SBERT, improving clustering
performance by 0.31 average ARI and 0.23 average ACC on the TUS dataset. (iii)
TableDC handles overlapping clusters due to its heavy-tailed Cauchy
distribution as a similarity kernel. For example, TableDC assigns two tables
that share similar schemas, RadioStation.(Radio, Country, Language) and
Country.(Country, Language, Broadcasters) with a 0.79 cosine similarity score
in the representation space to two different clusters, compared to SDCN and
DFCN, which failed to handle high overlap and put them in one cluster, leading
to misclassification. (iv) TableDC provides effective distance measures in a
dense space through the Mahalnobis distance measure when calculating the
distance between data points. Two instances from different GT clusters are
closely packed in the latent space when the distance measured is Euclidean
compared to the Mahalbonis distance, which helps the model by placing both
instances apart. For example, the Mahalanobis distance between different
attributes of two tables Book.(rank, title, genre, creator, date, freq) and
Film.(rank, title, year, director, overall rank), is 0.99 compared to the
Euclidean distance, which is 0.71, indicating that during self-supervision in
TableDC, both instances are further apart despite having highly overlapping
values. Experimentally, TableDC correctly placed the two instances in
different clusters whereas EDESC and DCRN misclassified them into one cluster
due to a low Euclidean distance score. (v) TableDC focuses on semantic
similarity more than distributional frequency. For example, when considering
instances on TUS with SBERT, attributes (Day, Month) with different
frequencies (i.e., the frequency of each day) present in a column should be in
the same clusters, ignoring the synthetic properties or the count of a
particular day repeated in a column. However, SDCN and DCRN consider the
syntactic similarity and the frequency (Tuesday and Wednesday are the highest
counts in both tables) and incorrectly place the two tables in different
clusters, in contrast with TableDC.

Table 3. Comparing entity resolution clustering results (TableDC vs. existing DC methods). The best results are shown in bold. | Music Brainz | GeoSet
---|---|---
Method | SBERT | EmbDi | SBERT | EmbDi
| ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC
K-means | 0.40 | 0.68 | 0.39 | 0.64 | 0.57 | 0.83 | 0.56 | 0.71
DBSCAN | 0.0 | 0.002 | N/A | N/A | 0.0 | 0.001 | 0.0 | 0.001
Birch | 0.56 | 0.76 | 0.41 | 0.67 | 0.56 | 0.75 | 0.59 | 0.71
SHGP | 0.15 | 0.47 | 0.20 | 0.51 | 0.52 | 0.77 | 0.43 | 0.63
DFCN | 0.03 | 0.31 | 0.03 | 0.33 | 0.34 | 0.59 | 0.37 | 0.57
EDESC | -0.001 | 0.03 | 0.03 | 0.29 | 0.30 | 0.73 | 0.25 | 0.52
SDCN | 0.002 | 0.13 | 0.06 | 0.47 | 0.05 | 0.66 | 0.52 | 0.66
TableDC | 0.80 | 0.88 | 0.51 | 0.71 | 0.65 | 0.86 | 0.60 | 0.71
Table 4. Comparing domain discovery clustering results (TableDC vs. existing DC methods). The best results are shown in bold. Method | Camera | Monitor
---|---|---
| SBERT | $SBERT^{*}$ | $T5^{*}$ | SBERT | $SBERT^{*}$ | $T5^{*}$
| ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC | ARI | ACC
K-means | 0.74 | 0.70 | 0.49 | 0.56 | 0.49 | 0.56 | 0.59 | 0.58 | 0.54 | 0.53 | 0.52 | 0.53
DBSCAN | 0.73 | 0.69 | -0.005 | 0.25 | -0.005 | 0.25 | 0.27 | 0.50 | 0.007 | 0.22 | 0.006 | 0.22
Birch | 0.76 | 0.70 | 0.78 | 0.74 | 0.69 | 0.68 | 0.55 | 0.55 | 0.60 | 0.61 | 0.53 | 0.53
SHGP | 0.65 | 0.65 | 0.47 | 0.56 | 0.41 | 0.51 | 0.59 | 0.60 | 0.48 | 0.52 | 0.46 | 0.49
DCRN | 0.60 | 0.60 | 0.41 | 0.44 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A
DFCN | 0.77 | 0.72 | 0.64 | 0.68 | 0.57 | 0.61 | 0.59 | 0.59 | 0.53 | 0.54 | 0.50 | 0.50
EDESC | 0.41 | 0.48 | 0.57 | 0.59 | 0.46 | 0.54 | 0.32 | 0.42 | 0.48 | 0.50 | 0.46 | 0.49
SDCN | 0.68 | 0.67 | 0.68 | 0.67 | 0.66 | 0.63 | 0.47 | 0.52 | 0.52 | 0.54 | 0.55 | 0.54
TableDC | 0.80 | 0.72 | 0.82 | 0.75 | 0.77 | 0.74 | 0.64 | 0.65 | 0.63 | 0.61 | 0.58 | 0.57

###  4.5. Entity resolution clustering results

Entity resolution as a clustering problem can be defined as clustering
multiple records that refer to the same real-world entity. Table 3 shows the
clustering results of entity resolution on GeoSet and Music Brainz datasets
encoded with several representations.

We observe the following: (i) The best results are with TableDC on both
datasets, outperforming all other methods by 0.63 average ARI and 0.53 average
ACC on Music Brainz and 0.31 average ARI and 0.24 average ACC on GeoSet using
SBERT. (ii) TableDC performed well with SBERT compared to EmbDi, delivering
good semantic similarity performance among different records of the same real-
world entity. TableDC is better with SBERT than EmbDi by 0.20 ARI and 0.17 ACC
with Music Brainz, while 0.05 ARI and 0.15 ACC with GeoSet. (iii) TableDC more
effectively adopted the semantic mapping by SBERT than SHGP and DFCN by
clustering contextually similar records. For example, two records of
geographical locations (name: Manchester (England), longitude: -2.23743,
latitude: 53.4809) from data.nytimes.com and (name: manchester united kingdom,
longitude: -2.24, latitude: 53.48) from rdf.freebase.com have a low syntactic
similarity (pairwise Euclidean distance) of 0.47 and high cosine similarity
(pairwise cosine distance) of 0.86. SHGP and DFCN misclassified both instances
and grouped them into different clusters compared to TableDC, which forms one
cluster with contextually similar records. (iv) TableDC with EmbDi on GeoSet
created fewer unary clusters (50) than SHGP (101) and EDESC (90), indicating
that TableDC effectively balances cluster purity and cluster size. A higher
ARI score with fewer unary clusters implies that TableDC is better at avoiding
unnecessary fragmentation of the rows into too many small clusters. (v) Like
schema inference, in entity resolution, existing clustering methods failed to
form clusters based on the context of a particular text. For example, two
different songs with some similar values of attributes are clustered together
by SDCN and SHGP with EmbDi, i.e. (title: Jabberwock, length: 292706, year:
2010, language: French) and (title: Bubble Star, length: 244346, year: 2004,
language: French). Both records share the same language and numeric values
with a similar range encoded by EmbDi despite having different titles. TableDC
managed to put them in different clusters based on the context of both records
despite them having several similar attribute values.

###  4.6. Domain discovery clustering results

Domain discovery as a clustering problem can be defined as a clustering of the
columns from a collection of datasets that refer to the same domain. Table 4
shows the clustering results of domain discovery on Camera and Monitor
datasets encoded with several representations.

We observe the following: (i) TableDC obtained the best results with SBERT and
T5 when considering column values of the Camera dataset outperforming other
methods by an average of 0.32 and 0.18 with SBERT and 0.30 and 0.20 with T5 on
ACC and ARI, respectively. (ii) TableDC also led on the Monitor dataset by an
average of 0.18 and 0.11 with SBERT and an average of 0.14 and 0.09 with T5 on
ARI and ACC, respectively. (iii) TableDC shows a superior capacity for
learning and interpreting the column’s context in the latent space compared to
other methods. TableDC’s ability to learn and utilize complex patterns shows
the better integration of the embeddings obtained using T5 on the Camera
dataset. For example, the cosine similarity of the T5 vectors of two columns
(image sensor: CMOS) and (optical sensor: CMOS) is 0.61, and TableDC learns
the context and assigns both columns to the same cluster, compared to SDCN and
DFCN, which incorrectly assigns them to different clusters. (iv) The
clustering performance of existing methods is affected by the length of
columns compared to TableDC, which handles the columns uniformly regardless of
the length. For example, two columns of the Camera dataset with different
lengths (camera color: silver gray/ black/ red/ silver) and (color: black) are
categorized in different clusters by TableDC and DCRN with SBERT; however,
TableDC overrides the column structure and correctly puts them together in the
same cluster. In the experimental evaluation, some experiments with DCRN have
not managed to scale on available resources and are reported as N/A.

###  4.7. Comparison with bespoke solutions

Previous experiments have compared TableDC with other SC and DC algorithms,
using the same representations. This section complements such results by
comparing TableDC with existing state-of-the-art solutions.

####  4.7.1. Schema Inference

Although there are many proposals for schema inference techniques, as surveyed
in (Cebiric et al., 2019; Kellou-Menouer et al., 2022), few of these are for
tabular data. Thus the comparators in this section use state-of-the-art
techniques for table similarity along with standard clustering algorithms.
Specifically for table similarity, we use:

  * •

$D^{3}L$ (Bogatu et al., 2020), which combines the results of several largely
syntactic methods for comparing column headers and values into an overall
table similarity score. $D^{3}L$ is used with the K-means for clustering
because it exhibits the best results compared to Birch and DBSCAN.

  * •

Starmie (Fan et al., 2023), which uses self-supervised contrastive learning to
fine tune ROBERTa (Liu et al., 2019) language models for column similarity;
table similarity is then supported by combining these column similarities.
Starmie is used with the connected component algorithm (Fan et al., 2023) as
in the original paper.

####  4.7.2. Entity Resolution

There are many proposals for entity resolution techniques, as surveyed in
(Christophides et al., 2021; Elmagarmid et al., 2007), though many of these
consider pairwise similarity rather than producing clusters. Thus we
specifically select a framework that can produce clusters of instances, namely
JedAI (Papadakis et al., 2020; Mandilaras et al., 2021), which provides
several workflows for entity resolution. Within JedAI, we use the schema
agnostic workflow, as TableDC is also schema agnostic; in essence, both
techniques can be used to look for duplicates in tables with different
structures. JedAI workflows can also be configured to support different
comparison metrics, and results are reported for Jaccard, Cosine and Dice.

####  4.7.3. Domain Discovery

There are relatively few fully unsupervised proposals for Domain Discovery.
Here we use one bespoke method, namely D4 (Ota et al., 2020), which identifies
domains by considering overlaps between column instances, and as in Schema
Inference we use Starmie (Fan et al., 2023) along with an SC algorithm. The
comparison with Starmie involves comparing a language model that has been
fine-tuned for similarity with a language model in TableDC that is fine-tuned
for clustering.

####  4.7.4. Observations

We compare two proposals for each problem that use techniques published in top
outlets (ICDE, PVLDB, Information Systems) since 2020. The results of the
comparison of TableDC with bespoke solutions are presented in Figure 2, which
reports the ARI and ACC for two datasets for each of schema inference, entity
resolution and domain discovery. We can observe that, for Schema inference,
TableDC consistently outperforms $D^{3}L$ and Starmie for web tables. For TUS,
$D^{3}L$ obtains 0.06 higher ACC than TableDC, while TableDC is still ahead
with 0.07 ARI. This indicates that TableDC is good at identifying the overall
clustering structure, but $D^{3}L$ is better at assigning the exact labels
correctly. TableDC shows its ability to achieve good results on entity
resolution compared to JedAI with different distance similarity measures.
JedAI with cosine distance is second best with 0.58 ARI and 0.69 ACC on the
Music Brainz dataset. Lastly, TableDC outperformed D4 (0.29 ARI and 0.27 ACC)
in column clustering due to its ability to learn contextualized column
embeddings effectively, providing more accurate clustering, with 0.77 ARI and
0.74 ACC on the Camera dataset. Overall, we can see that in the 12
experiments, TableDC provides the best results in 11 cases, thereby
demonstrating that, though generic, TableDC compares well with recent,
specialized state-of-the-art proposals.

ARI (TUS)ACC (TUS)ARI (web tables)ACC (web tables)ARI (GeoSet)ACC (GeoSet)ARI
(Music Brainz)ACC (Music Brainz)ARI (Camera)ACC (Camera)ARI (Monitor)ACC
(Monitor)

$D^{3}L$StarmieTableDC$0$$0.2$$0.4$$0.6$$0.8$$1$Scores (a) Schema Inference

CosineDiceTableDC$0$$0.2$$0.4$$0.6$$0.8$$1$Scores (b) Entity Resolution

D4StarmieTableDC$0$$0.2$$0.4$$0.6$$0.8$$1$Scores (c) Domain Discovery

Figure 2. TableDC vs. bespoke solutions for each problem. TableDC is
integrated with SBERT in (a) and (b) and T5 in (c). In (b), Jaccard and Cosine
are different similarity metrics of the JedAI framework (Papadakis et al.,
2020).

###  4.8. Scalability

We observe that existing DC proposals struggled to scale to large numbers of
clusters $\mathbb{K}$ (see Figure 3). To compare the scalability of different
DC methods, we used Music Brainz, which was scaled to large numbers of
clusters, up to $\mathbb{K}=2400$. The times are reported running on Nvidia
A100 GPU with 80GB GPU RAM.

Most existing DC algorithms use GCN to create an adjacency matrix to capture
underlying relationships among different features. This leads to a sparse
adjacency matrix in scenarios where the representation is not dense. The
complexity of GCN-based representation learning, assuming a sparse adjacency
matrix, is proportional to the number of vertices and edges. We often deal
with dense representations with overlapping distances in data management
applications. For dense representations, the multiplication between an
adjacency matrix $\mathbb{A}$ (size $\mathbb{N}\times\mathbb{N}$) and a
feature matrix (size $\mathbb{N}\times\mathbb{D}$) becomes more
computationally intensive as $\mathbb{K}$ and $\mathbb{D}$ increase, leading
to greater complexity.

Figure 3. Scalability comparison of TableDC with existing DC approaches with
respect to the number of clusters $\mathbb{K}$. DFCN and DCRN are not included
in the comparison because we have not managed to run both on the available
hardware resources with a high number of clusters.

In contrast, TableDC does not rely on GCN to capture relationships but instead
uses Mahalanobis distance to determine correlations between different
features. TableDC employs an autoencoder with a linear relationship between
the number of data points $\mathbb{N}$ and the size of the hidden layer
$\mathbb{H}$. The self-supervised module, which calculates the differences
between cluster centroids from dense representations, maintains a manageable
computational load because the calculation is around the distance from each
data point to every cluster centroid. TableDC’s efficiency is optimized using
Cholesky decomposition to compute the Mahalanobis distance. This method
efficiently handles dense representations by decomposing the covariance
matrix, allowing for faster computation of distances (Golub, 2013).
Additionally, the integration of a Cauchy distribution kernel normalizes these
distances and applies softmax operations, ensuring that the computational
overhead remains low even as the number of clusters $\mathbb{K}$ increases.

##  5\. Ablation study

This section presents an ablation study to investigate the impact of three
components: (i) cluster initialization, (ii) loss function, and (iii)
different distance measures and similarity kernels on self-supervised
learning.

###  5.1. Cluster Initialization

We compare several cluster centroid initialization approaches with Birch
initialization to investigate how cluster initialization affects clustering
performance. We record ARI using different initialization methods to evaluate
the cluster quality. Figure 4 shows a bar chart comparison of TableDC trained
over different initialization methods, including K-means, which has been used
significantly in existing DC algorithms.

K-meansK-means++Bisecting K-MeansGMMBirch$0$$0.2$$0.4$$0.6$$0.8$$1$ARISchema
InferenceEntity ResolutionDomain Discovery Figure 4. Impact of different
cluster initialization methods on clustering performance. We used SBERT on web
tables for schema inference, EmbDi on GeoSet for entity resolution, and SBERT
on Camera for Domain Discovery.

We can observe from Figure 4 that TableDC with Birch initialization provides
the best results for all problems. The higher ARI score of TableDC with Birch
initialization indicates the accurate identification of homogeneous clusters
and clear separation between different clusters. K-means++ initialization
appeared as the second best in domain discovery. From the comparative
analysis, we also observed that with Birch initialization, the points are more
widely dispersed around the centroid, indicating enhanced cluster compactness
(how tightly the data points are grouped around the centroid of the cluster)
and homogeneity (the similarity of data points within the same cluster).

(a) $re_{loss}$ minimization

(b) divergence of $q$ from $p$

Figure 5. $re_{loss}$ and $\text{KL}_{\text{div}}$ (between q and p) comparative analysis of TableDC with benchmark methods for schema inference on web tables data.  Table 5. TableDC clustering results on different distance metrics and similarity kernels used in the self-supervised module. We used SBERT (schema only) on web tables and Monitor datasets for schema inference and domain discovery. We adopted SBERT (values only) for entity resolution on the MusicBrainz dataset. | Schema Inference | Entity Resolution | Domain Discovery
---|---|---|---
Distance | Euclidean | Cosine | Mahalanobis | Euclidean | Cosine | Mahalanobis | Euclidean | Cosine | Mahalanobis
ARI | 0.49 | 0.30 | 0.62 | 0.0014 | 0.0014 | 0.88 | 0.56 | 0.30 | 0.64
ACC | 0.62 | 0.51 | 0.65 | 0.0599 | 0.0300 | 0.87 | 0.57 | 0.41 | 0.65
Distribution | Student’s t | Normal | Cauchy | Student’s t | Normal | Cauchy | Student’s t | Normal | Cauchy
ARI | 0.52 | 0.45 | 0.62 | 0.37 | 0.0 | 0.88 | 0.54 | 0.56 | 0.64
ACC | 0.60 | 0.64 | 0.65 | 0.69 | 0.0025 | 0.87 | 0.06 | 0.0 | 0.65

###  5.2. Loss Function

To empirically assess the impact of loss functions in TableDC and existing
benchmarks for data management problems, we use schema inference and web table
data to plot (i) $re_{loss}$ and (ii) $\text{KL}_{\text{div}}$ (see Figure 5).
A well-trained AE with effective $re_{loss}$ minimization, leads to embeddings
where clusters are more distinct and separable, specifically in the cases of
high density and overlap in the original embeddings. We can observe from
Figure 5 that the $re_{loss}$ of TableDC is significantly reduced and more
consistent compared to the benchmark methods, which exhibit fluctuations in
loss during the learning of dense embeddings. For $\text{KL}_{\text{div}}$,
the relationship between p and q (see Section 3.1) is investigated to show if
q deviates from p. This divergence is necessary because the data points in the
target p require greater intra-cluster distance rather than high cohesion. We
only consider those benchmarks using p and q distribution and
$\text{KL}_{\text{div}}$ loss to align the comparison with TableDC. TableDC
shows a high divergence (see Figure 5) of q from p, which means that TableDC
is more confident with the actual soft assignments q becoming sharper over
time, and over highly packed p, the q needs to minimize the cohesiveness.

###  5.3. Self-supervision over different distance functions and similar
kernels.

We evaluate the impact of difference distance measures and similarity kernels
on the performance of TableDC, which utilizes a self-supervised module
employing the Mahalanobis distance and Cauchy distribution to manage densely
overlapping data effectively. We record the ARI and ACC (see Table 5) of
TableDC by keeping the Cauchy distribution as a similar kernel and replacing
the Mahalanobis distance with Euclidean (Bo et al., 2020) and Cosine distances
(Salton et al., 1975), each providing different geometric properties. The
Euclidean distance measures the straight-line distance between data points and
their centroids, and the Cosine distance evaluates the cosine of the angle
between vectors, emphasizing orientation over magnitude. Similarly, we kept
the Mahalanobis distance constant while varying the similarity kernels. We
replaced the Cauchy distribution with the Student’s t-distribution (Bo et al.,
2020) (with an additional degree of freedom parameter that adjusts the
distribution’s kurtosis.) and the Normal distribution (that provides a
standard Gaussian decay of similarity).

From Table 5, we can observe that the choice of Mahalanobis distance combined
with the Cauchy distribution appears optimal for all three problems. Euclidean
distance with a Cauchy distribution performs relatively better in schema
inference and domain discovery than cosine distance, showing that the
straight-line distance between points is more reasonable than the angle when
we have dense and overlapping clusters. With Student’s t distribution, TableDC
shows a high ARI for schema inference but significantly lower ACC than the
Normal distribution. The higher ARI suggests that it can effectively group
similar items into clusters, but the low ACC indicates that it failed to
handle overlapping clusters and mismatches predicted labels against the actual
labels.

##  6\. Conclusions

This paper proposes a deep clustering algorithm for data cleaning and
integration problems, and has demonstrated its applicability to schema
inference, entity resolution and domain discovery. TableDC utilizes the
Mahalonbis distance measure in its self-supervised module to optimize data
distribution during training. The strength of TableDC is its ability to
account for the covariance among different dimensions of the embeddings, often
ignored by the conventional Euclidean distance. This ensures that the distance
between data points remains consistent irrespective of the orientation of the
data in the latent space. Our method enhances cluster separation in scenarios
with overlapping features in the latent space by weighing dimensions based on
the degree of their inter-dependencies. The experimental evaluation shows that
TableDC consistently outperforms existing clustering algorithms and problem-
specific methods.

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