GRACE: LLM-Grounded Semantic Metric Spaces for Scalable Mixed-Data Clustering

# GRACE: LLM-Grounded Semantic Metric Spaces for Scalable Mixed-Data Clustering
Source: [https://arxiv.org/html/2608.07881](https://arxiv.org/html/2608.07881)
Zihua Yang1, Zhencheng Xie1, Junyang Chen2, Liang Xie3, Yiqun Zhang1, Mengke Li4, Yang Lu5 1Guangdong University of Technology,2Tsinghua University 3Peking University,4Shenzhen University,5Xiamen University \{yangzihua1, xiezhencheng\}@mails\.gdut\.edu\.cn, chenjuny25@mails\.tsinghua\.edu\.cn, liangxie@pku\.edu\.cn, yqzhang@gdut\.edu\.cn, mengkeli@szu\.edu\.cn, luyang@xmu\.edu\.cn

###### Abstract

Clustering mixed tabular data requires a unified metric space to bridge the inherent heterogeneity between continuous numerical measurements and discrete categorical symbols\. Traditionally, algorithms rely entirely on dataset\-internal statistics to estimate categorical relationships, which confines the learned metric to empirical co\-occurrences and ignores conceptually obvious yet statistically unobserved affinities\. Although LLMs offer external world knowledge, applying their text\-centric reasoning to highly abstract tabular concepts presents significant challenges\. Bridging this modality gap to construct a semantically complete metric typically requires embedding LLMs into iterative metric learning loops to dynamically optimize cross\-modality representations\. This incurs intractable computational overhead, forcing a compromise between semantic enrichment and scalability\. Therefore, we propose GRACE, an LLM\-grounded framework for scalable mixed\-data clustering\. GRACE shifts semantic acquisition to the attribute\-value level via a multi\-perspective LLM querying strategy, mapping heterogeneous values into knowledge\-informed descriptions\. Crucially, this one\-shot grounding extracts general\-purpose semantic representations that embed heterogeneous attributes into a unified space, decoupling expensive LLM invocation from iterative optimization\. Furthermore, GRACE cross\-validates these external semantics against dataset\-internal statistical evidence to ensure alignment with the dataset\-specific cluster structure\. Ultimately, GRACE matches the scalability of conventional statistics\-driven baselines while achieving superior clustering accuracy and conceptual interpretability over 11 competing methods\. The source code is available at[GRACE](https://github.com/develop-yang/GRACE-GRACE-A)\.

## IIntroduction

Tabular data with heterogeneous numerical and categorical attributes is ubiquitous in healthcare, finance, and e\-commerce\[[30](https://arxiv.org/html/2608.07881#bib.bib41),[33](https://arxiv.org/html/2608.07881#bib.bib42)\]\. In these domains, clustering supports label\-free organization of large\-scale records for exploratory analysis, cohort discovery, and data\-driven decision\-making\. Clustering such mixed data requires measuring pairwise similarity across all attributes\[[34](https://arxiv.org/html/2608.07881#bib.bib43),[32](https://arxiv.org/html/2608.07881#bib.bib44)\]\. For numerical attributes, value differences provide a readily defined distance, but categorical attributes take values from finite unordered sets and carry no intrinsic distance\[[24](https://arxiv.org/html/2608.07881#bib.bib45)\]\. Whethernurseis closer todoctoror toengineer, for example, cannot be judged from their symbolic forms alone\. Categorical distance is therefore a key component of mixed\-data similarity without a natural definition, and how well it is constructed governs clustering performance\[[31](https://arxiv.org/html/2608.07881#bib.bib46),[35](https://arxiv.org/html/2608.07881#bib.bib49)\]\.

To estimate categorical distances, one line of research defines distance functions directly in the original feature space, using statistical signals of varying granularity, from attribute\-wise entropy weighting and cross\-attribute dependency modeling\[[50](https://arxiv.org/html/2608.07881#bib.bib11),[51](https://arxiv.org/html/2608.07881#bib.bib12)\]to higher\-order value interaction graphs\[[42](https://arxiv.org/html/2608.07881#bib.bib1)\]\. These methods recover relationships that symbolic forms alone cannot reveal, but only when such relationships are reflected by co\-occurrence, dependency, or interaction patterns in the observed data\. Representation learning, by contrast, maps heterogeneous attributes into a shared continuous space where standard distance metrics, such as Euclidean or cosine distance, can be applied\[[43](https://arxiv.org/html/2608.07881#bib.bib2),[5](https://arxiv.org/html/2608.07881#bib.bib3)\], enabling unified treatment of numerical and categorical attributes\. Such embeddings capture dependencies that feature\-space metrics cannot express\[[37](https://arxiv.org/html/2608.07881#bib.bib48)\]\. However, their learning signal remains derived from dataset\-internal statistics\. As a result, the learned representations mainly reorganize empirical regularities already present in the observed data\[[12](https://arxiv.org/html/2608.07881#bib.bib47)\]\. Taken together, these methods share a common limitation: categorical relationships are inferred only from evidence internal to the dataset\. This reliance on internal statistics is restrictive because categorical values often denote real\-world concepts with readily available external semantics, and ignoring such prior knowledge can leave obvious conceptual affinities invisible to the learned metric\. This motivates the use of external semantic knowledge for mixed\-data clustering\.

To meet the demand for semantic knowledge beyond the dataset, the recent emergence of LLMs offers a natural source, as these models intrinsically encode world knowledge about real\-world concepts and their relationships\[[45](https://arxiv.org/html/2608.07881#bib.bib9),[6](https://arxiv.org/html/2608.07881#bib.bib50)\]\. Much like humans infer the closeness between occupations from domain knowledge rather than from their bare names, grounding attribute values with LLM knowledge lifts isolated symbols into a semantic metric space where conceptual affinities become geometrically structured\. Sparse symbol encoding derives distances from observed data patterns, leaving the induced metric space semantically incomplete\. As a result, conceptual proximity that is evident from world knowledge may not be faithfully translated into geometric proximity\. Once each value is expanded into descriptive text grounded in LLM world knowledge, the external semantics reshape the metric space, making domain\-level associations explicit and better aligned with the underlying clustering structure\.

The remaining challenge is how to turn this external semantic prior into a clustering\-ready metric space for mixed data\. Existing LLM\-based clustering mainly exploits LLMs in their native language space, where each instance is already a sentence or document and the model can directly reason about semantic similarity\[[47](https://arxiv.org/html/2608.07881#bib.bib7),[38](https://arxiv.org/html/2608.07881#bib.bib8)\]\. This text\-centric paradigm does not transfer naturally to mixed tabular data\. Since categorical values are isolated but semantically condensed concepts, whereas numerical values are computable measurements with their own metric structure, both are expected to be represented within a unified similarity space\. In principle, the semantic compression of categorical values makes LLM guidance especially valuable for metric construction\. However, repeatedly invoking an LLM during semantic grounding, neighborhood construction, or cluster refinement would introduce substantial latency and computational overhead, especially when pairwise or neighborhood structures scale with the data\. Therefore, the central obstacle to applying LLM knowledge to mixed\-data clustering is the efficiency bottleneck of obtaining semantic guidance at scale\. This calls for a new paradigm that acquires LLM\-grounded semantic representations without repeated invocation, while still aligning the resulting space with the dataset\-specific clustering structure\.

Guided by this observation, this paper proposes GRACE \(GRoundingAttributes forClustering viaExternal semantics\), a framework that makes LLM world knowledge usable for mixed\-data clustering without embedding the LLM in iterative metric learning or cluster refinement\. The key idea is to shift semantic acquisition from the instance or pair level to the attribute\-value level\. Rather than repeatedly querying the LLM over samples, neighborhoods, or intermediate cluster states, GRACE grounds each possible attribute value once: categorical symbols are expanded into knowledge\-grounded descriptions, and numerical domains are partitioned into domain\-informed intervals described in the same semantic form\. This value\-level design decouples LLM knowledge acquisition from clustering\-time computation and turns heterogeneous values into comparable semantic units for a unified metric space\. External semantics alone, however, do not determine the cluster structure of a specific dataset\. A semantic affinity may be conceptually plausible yet irrelevant to the empirical grouping induced by the observed records\. GRACE therefore complements LLM\-grounded semantics with dataset\-internal statistical evidence\. The semantic space provides external conceptual knowledge, and the statistical view calibrates it by identifying neighborhood relations supported by the original features\. Relations confirmed by both views are selectively enhanced, yielding an affinity structure that preserves world\-knowledge\-aware similarity and remains aligned with dataset\-specific clustering structure\. Finally, eigengap\-guided graph sparsification converts the refined affinity matrix into a spectrally well\-conditioned graph for clustering\. The main contributions of this paper are summarized below:

- •Bridging LLMs and mixed data clustering\.This work is the first to bring external semantic knowledge into effective complementarity with dataset\-internal statistical evidence for mixed\-data clustering, constructing a unified world\-knowledge\-aware metric space that captures both conceptual affinities and distributional structures\.
- •Aligning semantic knowledge with cluster structure\.GRACE resolves the misalignment between semantic proximity and cluster membership through dual\-view neighborhood consistency, providing a principled mechanism for coupling external semantics with dataset\-specific statistical evidence\.
- •Decoupling LLM grounding from metric learning\.GRACE reformulates LLM\-assisted mixed\-data metric learning as a one\-shot value\-level grounding process followed by LLM\-free metric adaptation\. Since the LLM is invoked only over possible attribute values rather than instances, the semantic acquisition cost scales linearly with value cardinality, providing a low\-token\-overhead and scalable route to world\-knowledge\-aware clustering\.

## IIRelated Work

### II\-ADistance Measures for Mixed Data

Categorical values carry no inherent distance, and the earliest measures fill that void by counting matches, treating every mismatch as equally far\. Such matching yields a workable metric, yet it flattens the unequal relatedness that real concepts exhibit\. To recover that relatedness, a long line of work mines it from the data, advancing through co\-occurrence context\[[21](https://arxiv.org/html/2608.07881#bib.bib63)\], entropy weighting, cross\-attribute couplings\[[50](https://arxiv.org/html/2608.07881#bib.bib11),[51](https://arxiv.org/html/2608.07881#bib.bib12)\], learnable intra\-attribute weights, higher\-order value graphs\[[42](https://arxiv.org/html/2608.07881#bib.bib1)\], and dense embeddings of the same couplings\[[22](https://arxiv.org/html/2608.07881#bib.bib14),[23](https://arxiv.org/html/2608.07881#bib.bib15)\]\. Each step reads the statistics more finely than the last, yet every one inherits the same blind spot, namely, any affinity that leaves no trace in the observed sample cannot be recovered\.

Mixed data, where numerical and categorical attributes coexist, raise a second problem, that of placing continuous measurements and discrete symbols on one scale\. Early schemes settle it by hand, weighting a Euclidean term against a matching one \(k\-prototypes\[[19](https://arxiv.org/html/2608.07881#bib.bib16)\], Gower\[[11](https://arxiv.org/html/2608.07881#bib.bib17)\]\), and later ones learn a shared space that absorbs both types \(unified distance learning\[[43](https://arxiv.org/html/2608.07881#bib.bib2)\], quaternion graphs\[[5](https://arxiv.org/html/2608.07881#bib.bib3)\]\)\. The joint metric grows more principled with each design, yet the cure for the scale problem leaves the earlier blind spot untouched, since meaning is still drawn entirely from within the dataset\. Two values that never co\-occur are therefore judged far apart even when domain knowledge would call them close, a verdict no statistical refinement can overturn\.

![Refer to caption](https://arxiv.org/html/2608.07881v1/x1.png)Figure 1:Overview of the GRACE framework\. Given raw tabular data with categorical and numerical attributes, GRACE generates natural\-language descriptions for each attribute value through LLM\-based prompting\. The generated descriptions are fed into a transformer encoderfθf\_\{\\theta\}to obtain semantic representations𝐞i∈ℰ\\mathbf\{e\}\_\{i\}\\in\\mathcal\{E\}\. To improve the reliability of semantic affinity, a statistical encoder produces representations𝐡i∈ℋ\\mathbf\{h\}\_\{i\}\\in\\mathcal\{H\}from the raw attribute values for validating semantic neighbors\. To jointly exploit both spaces, a neighborhood consistency mechanism computesΓi​j=NNi​jsem⋅NNi​jstat\\Gamma\_\{ij\}=\\mathrm\{NN\}\_\{ij\}^\{\\mathrm\{sem\}\}\\cdot\\mathrm\{NN\}\_\{ij\}^\{\\mathrm\{stat\}\}, refining the semantic affinity matrix\. Based on the refined affinity, eigengap\-graph sparsification determines the optimal sparsity for spectral clustering\.
### II\-BLLM\-Enhanced Clustering

LLMs encode world knowledge that statistical distances alone cannot recover, and recent clustering research draws on it\. These efforts, however, remain confined to inherently textual input\. ClusterLLM\[[47](https://arxiv.org/html/2608.07881#bib.bib7)\]elicits triplet judgments of relative proximity, few\-shot prompting\[[38](https://arxiv.org/html/2608.07881#bib.bib8)\]converts limited supervision into pseudo\-labels or pairwise constraints, loop\-based refinement\[[1](https://arxiv.org/html/2608.07881#bib.bib18)\]corrects assignments from model feedback, and contrastive formulations such as SCCL\[[41](https://arxiv.org/html/2608.07881#bib.bib19)\]and TAC\[[25](https://arxiv.org/html/2608.07881#bib.bib20)\]incorporate the language signal into the clustering objective\. Tabular data, by contrast, offers no such linguistic surface, as its categorical entries are discrete symbols and its numerical entries scalar quantities\. A knowledge\-grounded distance over such heterogeneous attributes therefore remains an open question\.

Applying language models to tabular data is itself well studied, though existing work targets prediction or generation instead of a clustering metric\. One line serializes each record into a sentence, as in TabLLM\[[15](https://arxiv.org/html/2608.07881#bib.bib65)\]for classification and GReaT\[[3](https://arxiv.org/html/2608.07881#bib.bib66)\]for generation, and another recasts column semantics as features for a downstream learner, as in CAAFE\[[16](https://arxiv.org/html/2608.07881#bib.bib67)\]and FeatLLM\[[14](https://arxiv.org/html/2608.07881#bib.bib68)\]\. Both lines serve supervised objectives and therefore provide no unsupervised distance over mixed attributes\. The effort closest to such a distance is BREVE\[[40](https://arxiv.org/html/2608.07881#bib.bib69)\], which grounds categorical values in model knowledge to derive a similarity measure\. Its fusion is nonetheless static, embedding raw value identities into a shared space under a fixed, non\-trainable weight, and cannot separate genuine from spurious semantic affinities\. It is moreover restricted to categorical attributes, excluding numerical ones\.

TABLE I:Frequently used symbols and notations\.SymbolDescription𝒟\\mathcal\{D\},nnDataset and the number of samplesxix\_\{i\},𝒜\\mathcal\{A\},mmSample, attribute set, and attribute count𝒜c\\mathcal\{A\}^\{c\},𝒜n\\mathcal\{A\}^\{n\}Categorical and numerical attribute subsetsKK,𝒞\\mathcal\{C\}Number of clusters and partitiont​\(j,o\)t\(j,o\)LLM\-generated description for valueooon attributeAjA\_\{j\}fθf\_\{\\theta\},ded\_\{e\}Sentence encoder and embedding dimensionality𝐞​\(j,o\)\\mathbf\{e\}\(j,o\)Attribute\-level embedding for valueooonAjA\_\{j\}𝐳i\\mathbf\{z\}\_\{i\},𝐡i\\mathbf\{h\}\_\{i\}Semantic and statistical representations of samplexix\_\{i\}𝐒sem\\mathbf\{S\}^\{\\mathrm\{sem\}\},𝐒stat\\mathbf\{S\}^\{\\mathrm\{stat\}\}Semantic and statistical similarity matricesλ\\lambdaNatural neighbor search radius𝐍𝐍sem\\mathbf\{NN\}^\{\\mathrm\{sem\}\},𝐍𝐍stat\\mathbf\{NN\}^\{\\mathrm\{stat\}\}Natural neighbor graphs for two views𝚪\\boldsymbol\{\\Gamma\},τc\\tau\_\{c\}Consistency matrix and adaptive thresholdρi​l\\rho\_\{il\},αi​l\\alpha\_\{il\}Refined affinity and enhancement coefficientθ∗\\theta^\{\*\},𝐀θ∗\\mathbf\{A\}\_\{\\theta^\{\*\}\}Optimal sparsification threshold and adjacency matrixREQ\\mathrm\{REQ\}Relative Eigengap Quality

## IIIProposed Method

GRACE follows a four\-stage pipeline consisting of problem formulation, knowledge\-grounded attribute description, dual\-view neighborhood consistency, and model training with spectral clustering, as illustrated in Fig\.[1](https://arxiv.org/html/2608.07881#S2.F1)\. For clarity, the frequently used symbols are summarized in Table[I](https://arxiv.org/html/2608.07881#S2.T1)\. The complexity of both GRACE and its scalable variant GRACE\-A is analyzed in the complexity analysis subsection\.

### III\-AProblem Formulation

Given a dataset𝒟=\{x1,x2,…,xn\}\\mathcal\{D\}=\\\{x\_\{1\},x\_\{2\},\\ldots,x\_\{n\}\\\}withnnsamples, each samplexi=\[xi1,xi2,…,xim\]⊤x\_\{i\}=\[x\_\{i\}^\{1\},x\_\{i\}^\{2\},\\ldots,x\_\{i\}^\{m\}\]^\{\\top\}is described bymmattributes𝒜=\{A1,A2,…,Am\}\\mathcal\{A\}=\\\{A\_\{1\},A\_\{2\},\\ldots,A\_\{m\}\\\}, wherexijx\_\{i\}^\{j\}denotes the value of samplexix\_\{i\}on attributeAjA\_\{j\}\. These attributes are partitioned into a categorical subset𝒜c\\mathcal\{A\}^\{c\}and a numerical subset𝒜n\\mathcal\{A\}^\{n\}, satisfying𝒜=𝒜c∪𝒜n\\mathcal\{A\}=\\mathcal\{A\}^\{c\}\\cup\\mathcal\{A\}^\{n\}with𝒜c∩𝒜n=∅\\mathcal\{A\}^\{c\}\\cap\\mathcal\{A\}^\{n\}=\\emptyset\. Specifically, a categorical attributeAj∈𝒜cA\_\{j\}\\in\\mathcal\{A\}^\{c\}takes values from a finite unordered set𝒪jc=\{o1c,o2c,…,oVjc\}\\mathcal\{O\}^\{c\}\_\{j\}=\\\{o^\{c\}\_\{1\},o^\{c\}\_\{2\},\\ldots,o^\{c\}\_\{V\_\{j\}\}\\\}of cardinalityVjV\_\{j\}, where values lack intrinsic ordering, and a numerical attributeAj∈𝒜nA\_\{j\}\\in\\mathcal\{A\}^\{n\}takes values from a continuous interval𝒪jn=\[ℓj,uj\]⊆ℝ\\mathcal\{O\}^\{n\}\_\{j\}=\[\\ell\_\{j\},u\_\{j\}\]\\subseteq\\mathbb\{R\}, with observed values\{o1n,o2n,…,oNjn\}\\\{o^\{n\}\_\{1\},o^\{n\}\_\{2\},\\ldots,o^\{n\}\_\{N\_\{j\}\}\\\}\.

Clustering such heterogeneous data requires measuring pairwise similarity across both attribute types\. The goal is to partition thennsamples intoKKdisjoint groups𝒞=\{C1,C2,…,CK\}\\mathcal\{C\}=\\\{C\_\{1\},C\_\{2\},\\ldots,C\_\{K\}\\\}such that similar samples fall into the same group\. To achieve the partition, GRACE adopts spectral clustering, which operates on a similarity graph𝒢=\(𝒟,𝐒\)\\mathcal\{G\}=\(\\mathcal\{D\},\\mathbf\{S\}\)in which the edge weightSi​jS\_\{ij\}quantifies the affinity betweenxix\_\{i\}andxjx\_\{j\}\. Then, an indicator matrix𝐘∈ℝn×K\\mathbf\{Y\}\\in\\mathbb\{R\}^\{n\\times K\}encodes the group assignment and is obtained by minimizing the normalized graph cut:

min𝐘⁡Tr⁡\(𝐘⊤​𝐋𝐘\)s\.t\.𝐘⊤​𝐃𝐘=𝐈,\\min\_\{\\mathbf\{Y\}\}\\operatorname\{Tr\}\(\\mathbf\{Y\}^\{\\top\}\\mathbf\{L\}\\mathbf\{Y\}\)\\quad\\text\{s\.t\.\}\\quad\\mathbf\{Y\}^\{\\top\}\\mathbf\{D\}\\mathbf\{Y\}=\\mathbf\{I\},\(1\)where𝐋=𝐈−𝐃−1/2​𝐒𝐃−1/2\\mathbf\{L\}=\\mathbf\{I\}\-\\mathbf\{D\}^\{\-1/2\}\\mathbf\{S\}\\mathbf\{D\}^\{\-1/2\}is the normalized Laplacian and𝐃\\mathbf\{D\}denotes the diagonal degree matrix with entries𝐃i​i=∑jSi​j\\mathbf\{D\}\_\{ii\}=\\sum\_\{j\}S\_\{ij\}\. The standard spectral relaxation solves[Eq\. \(1\)](https://arxiv.org/html/2608.07881#S3.E1)by extracting theKKsmallest eigenvectors of𝐋\\mathbf\{L\}and applyingkk\-means to the rows of the resulting𝐘\\mathbf\{Y\}\. Since𝐋\\mathbf\{L\}is constructed entirely from𝐒\\mathbf\{S\}, the quality of the final partition depends on how𝐒\\mathbf\{S\}captures the true grouping structure\. For categorical attributes whose values lack inherent metric structure, defining meaningful entries in𝐒\\mathbf\{S\}is the core challenge\. GRACE addresses this by grounding each attribute value in LLM world knowledge to derive semantically grounded𝐒\\mathbf\{S\}\.

![Refer to caption](https://arxiv.org/html/2608.07881v1/x2.png)Figure 2:The4​P4Pprompt applied to two representative attributes from the Adult dataset: the categorical valueExec\-managerialofoccupationand the numerical interval\[30,45\]\[30,45\]ofage\. For both attribute types, the prompt elicits four complementary semantic perspectives, producing knowledge\-grounded descriptions in a unified textual format\.
### III\-BLLM\-Grounded Attribute Description

To unify heterogeneous attribute types into a common representation, GRACE converts every attribute value into a natural\-language description generated by an LLM\. The description quality depends on prompt design\. A naive prompt such as “Describe the value has 4 legs” produces nearly identical embeddings for semantically distinct values, because it fails to elicit functional associations \(e\.g\., four\-legged animals are typically mammals, six\-legged ones are insects\) or contrasts with alternatives\.

Motivated by this, GRACE adopts a four\-perspective knowledge elicitation template, denoted4​P4P\([Fig\.2](https://arxiv.org/html/2608.07881#S3.F2)\)\. Within a single query, the template instructs the LLM to produce a structured description for each categorical valueo∈𝒪jco\\in\\mathcal\{O\}^\{c\}\_\{j\}:

t​\(j,o\)=LLM4​P​\(Aj,o\),t\(j,o\)=\\mathrm\{LLM\}\_\{4P\}\(A\_\{j\},\\,o\),\(2\)whereLLM4​P\\mathrm\{LLM\}\_\{4P\}denotes the language model queried under the four\-perspective template,AjA\_\{j\}is the attribute name, andoois the value being described\. The generated descriptiont​\(j,o\)t\(j,o\)consists of four complementary components, denotedtcoret\_\{\\textsc\{core\}\},tindt\_\{\\textsc\{ind\}\},tpatt\_\{\\textsc\{pat\}\}, andtdist\_\{\\textsc\{dis\}\}, corresponding to the definitional, functional, contextual, and discriminative perspectives respectively\. The first two capture the internal semantics of a value, i\.e\., what it means and what it implies, and the latter two encode external relationships, i\.e\., where it co\-occurs and how it contrasts with alternatives\. The four components are concatenated into a single multi\-faceted textual passage that provides rich semantic grounding for downstream encoding\.

The4​P4Ptemplate additionally includes a domain\-adaptive role assignment and quality constraints that require concrete, jargon\-free language with domain\-specific examples\.

The four\-perspective template applies directly to categorical attributes, where the set of possible values is finite, and each value receives its own description\. For numerical attributes, the set of observed values is typically continuous, making per\-value descriptions impractical\. GRACE delegates discretization to the LLM, providing the attribute name, observed range, and domain context\. The LLM determines both the number of intervalsQjQ\_\{j\}and their boundaries from domain knowledge, e\.g\., clinical reference ranges for medical indicators\. Formally, the discretization defines a mapping:

ψj:𝒪jn→ℐj,ℐj=\{I1,I2,…,IQj\},\\psi\_\{j\}:\\mathcal\{O\}^\{n\}\_\{j\}\\;\\to\\;\\mathcal\{I\}\_\{j\},\\quad\\mathcal\{I\}\_\{j\}=\\\{I\_\{1\},I\_\{2\},\\ldots,I\_\{Q\_\{j\}\}\\\},\(3\)whereψj\\psi\_\{j\}assigns each observed value to one ofQjQ\_\{j\}intervals whose union covers the entire observed range\[ℓj,uj\]\[\\ell\_\{j\},u\_\{j\}\]\. As a concrete example, bilirubin is partitioned into three intervals, i\.e\., normal \(below 1\.2 mg/dL\), elevated \(1\.2 to 3\.0 mg/dL\), and severe \(above 3\.0 mg/dL\), corresponding to the standard diagnostic thresholds in hepatology\. Each intervalIkI\_\{k\}is then described through the same four\-perspective query as in[Eq\. \(2\)](https://arxiv.org/html/2608.07881#S3.E2), namelyt​\(j,Ik\)=LLM4​P​\(Aj,Ik\)t\(j,I\_\{k\}\)=\\mathrm\{LLM\}\_\{4P\}\(A\_\{j\},I\_\{k\}\), giving categorical values and numerical intervals one common description format\. All downstream modules therefore operate on a single textual input space, regardless of the original attribute type\.

Given these textual descriptions, GRACE encodes eacht​\(j,o\)t\(j,o\)into aded\_\{e\}\-dimensional dense embedding𝐞​\(j,o\)∈ℝde\\mathbf\{e\}\(j,o\)\\in\\mathbb\{R\}^\{d\_\{e\}\}via a transformer encoderfθf\_\{\\theta\}, which can be expressed as:

𝐞​\(j,o\)=fθ​\(t​\(j,o\)\),\\mathbf\{e\}\(j,o\)=f\_\{\\theta\}\\bigl\(t\(j,o\)\\bigr\),\(4\)To obtain a sample\-level representation𝐳i∈ℝm⋅de\\mathbf\{z\}\_\{i\}\\in\\mathbb\{R\}^\{m\\cdot d\_\{e\}\}from themmattribute\-level embeddings, GRACE concatenates them into a single vector:

𝐳i=𝐞​\(1,oi1\)⊕𝐞​\(2,oi2\)⊕⋯⊕𝐞​\(m,oim\),\\mathbf\{z\}\_\{i\}=\\mathbf\{e\}\(1,o\_\{i\}^\{1\}\)\\oplus\\mathbf\{e\}\(2,o\_\{i\}^\{2\}\)\\oplus\\cdots\\oplus\\mathbf\{e\}\(m,o\_\{i\}^\{m\}\),\(5\)withoijo\_\{i\}^\{j\}denoting the \(possibly discretized\) value of samplexix\_\{i\}on attributeAjA\_\{j\}\. The pairwise*semantic similarity*between two samples is then computed as:

Si​lsem=𝐳i⊤​𝐳l‖𝐳i‖​‖𝐳l‖\.S\_\{il\}^\{\\mathrm\{sem\}\}=\\frac\{\\mathbf\{z\}\_\{i\}^\{\\top\}\\mathbf\{z\}\_\{l\}\}\{\\\|\\mathbf\{z\}\_\{i\}\\\|\\,\\\|\\mathbf\{z\}\_\{l\}\\\|\}\.\(6\)

### III\-CDual\-View Neighborhood Consistency

The semantic similarity matrix𝐒sem\\mathbf\{S\}^\{\\mathrm\{sem\}\}quantifies pairwise conceptual proximity among samples, but in an unsupervised setting, no external labels are available to assess whether specific estimates are accurate\. A principled remedy is to cross\-check against the raw tabular features, which encode sample relationships through observed co\-occurrence patterns\.

GRACE implements this cross\-check through a statistical view that operates on the original feature space\. Because the statistical view serves a purely confirmatory role, an expressive encoding such as entity embeddings\[[13](https://arxiv.org/html/2608.07881#bib.bib57)\]or learned tabular representations\[[10](https://arxiv.org/html/2608.07881#bib.bib58)\]would introduce trainable parameters that require labeled supervision or auxiliary objectives to tune, conflicting with the unsupervised setting\. GRACE therefore adopts a parameter\-free encoding\. For each categorical attributeAj∈𝒜cA\_\{j\}\\in\\mathcal\{A\}^\{c\}, every valueo∈𝒪jco\\in\\mathcal\{O\}^\{c\}\_\{j\}is mapped to a standard basis vector𝐛o∈ℝVj\\mathbf\{b\}\_\{o\}\\in\\mathbb\{R\}^\{V\_\{j\}\}, and for each numerical attributeAj∈𝒜nA\_\{j\}\\in\\mathcal\{A\}^\{n\}, observed values are normalized to\[0,1\]\[0,1\]via min\-max scaling\. Concatenating these attribute\-level representations yields a feature vector𝐡i∈ℝdh\\mathbf\{h\}\_\{i\}\\in\\mathbb\{R\}^\{d\_\{h\}\}with dimensionalitydh=∑Aj∈𝒜cVj\+\|𝒜n\|d\_\{h\}=\\sum\_\{A\_\{j\}\\in\\mathcal\{A\}^\{c\}\}V\_\{j\}\+\|\\mathcal\{A\}^\{n\}\|\. The resulting pairwise statistical similarity between samplesxix\_\{i\}andxlx\_\{l\}, denoted bySi​lstatS^\{\\mathrm\{stat\}\}\_\{il\}, is then defined as:

Si​lstat=𝐡i⊤​𝐡l‖𝐡i‖​‖𝐡l‖\.S\_\{il\}^\{\\mathrm\{stat\}\}=\\frac\{\\mathbf\{h\}\_\{i\}^\{\\top\}\\mathbf\{h\}\_\{l\}\}\{\\\|\\mathbf\{h\}\_\{i\}\\\|\\,\\\|\\mathbf\{h\}\_\{l\}\\\|\}\.\(7\)
Given𝐒sem\\mathbf\{S\}^\{\\mathrm\{sem\}\}and𝐒stat\\mathbf\{S\}^\{\\mathrm\{stat\}\}, the next step is to identify neighbors under each view\. Comparing raw values across views is problematic because the two matrices operate on different scales\. Comparing*local neighborhood structures*is more principled, since clustering depends on relative proximity, not absolute magnitude\. To this end, GRACE extracts neighborhood structures using Natural Neighbor \(NaN\) graphs\[[52](https://arxiv.org/html/2608.07881#bib.bib29)\], which impose a mutual recognition constraint\. Specifically, a samplexlx\_\{l\}qualifies as a natural neighbor ofxix\_\{i\}only ifxix\_\{i\}also appears among the nearest neighbors ofxlx\_\{l\}:

NNi​l​\(𝐒\)=𝕀​\[l∈𝒩λ​\(i;𝐒\)\]⋅𝕀​\[i∈𝒩λ​\(l;𝐒\)\],\\mathrm\{NN\}\_\{il\}\(\\mathbf\{S\}\)=\\mathbb\{I\}\\bigl\[l\\in\\mathcal\{N\}\_\{\\lambda\}\(i;\\mathbf\{S\}\)\\bigr\]\\cdot\\mathbb\{I\}\\bigl\[i\\in\\mathcal\{N\}\_\{\\lambda\}\(l;\\mathbf\{S\}\)\\bigr\],\(8\)in which𝐒\\mathbf\{S\}is a generic similarity matrix standing for either view, and𝒩λ​\(i;𝐒\)\\mathcal\{N\}\_\{\\lambda\}\(i;\\mathbf\{S\}\)denotes theλ\\lambda\-nearest neighbors ofxix\_\{i\}under𝐒\\mathbf\{S\}\. The mutual constraint is well\-suited to the verification task, because one\-directional affinities, i\.e\., cases wherexix\_\{i\}considersxlx\_\{l\}a neighbor but not vice versa, often arise in regions of varying density and tend to be unreliable indicators of cluster membership\. By requiring reciprocation, the NaN graph retains only the most structurally stable relationships in each view, and the neighborhood sizeλ\\lambdais determined automatically by iterating fromλ=1\\lambda=1until every sample possesses at least one natural neighbor\.

Instantiating[Eq\. \(8\)](https://arxiv.org/html/2608.07881#S3.E8)with the semantic matrix𝐒sem=\[Si​lsem\]\\mathbf\{S\}^\{\\mathrm\{sem\}\}=\[S^\{\\mathrm\{sem\}\}\_\{il\}\]from[Eq\. \(6\)](https://arxiv.org/html/2608.07881#S3.E6)and the statistical matrix𝐒stat=\[Si​lstat\]\\mathbf\{S\}^\{\\mathrm\{stat\}\}=\[S^\{\\mathrm\{stat\}\}\_\{il\}\]from[Eq\. \(7\)](https://arxiv.org/html/2608.07881#S3.E7)produces the two view\-specific neighborhood matrices:

NNi​lsem=NNi​l​\(𝐒sem\),NNi​lstat=NNi​l​\(𝐒stat\)\.\\mathrm\{NN\}^\{\\mathrm\{sem\}\}\_\{il\}=\\mathrm\{NN\}\_\{il\}\(\\mathbf\{S\}^\{\\mathrm\{sem\}\}\),\\qquad\\mathrm\{NN\}^\{\\mathrm\{stat\}\}\_\{il\}=\\mathrm\{NN\}\_\{il\}\(\\mathbf\{S\}^\{\\mathrm\{stat\}\}\)\.\(9\)Their agreement defines the*neighborhood consistency score*of a pair\(xi,xl\)\(x\_\{i\},x\_\{l\}\):

Γi​l=NNi​lsem⋅NNi​lstat\.\\Gamma\_\{il\}=\\mathrm\{NN\}\_\{il\}^\{\\mathrm\{sem\}\}\\cdot\\mathrm\{NN\}\_\{il\}^\{\\mathrm\{stat\}\}\.\(10\)A value ofΓi​l=1\\Gamma\_\{il\}=1thus marks a pair that the semantic and statistical views both recognize as neighbors\. Because the two views are constructed from independent information sources, a pair confirmed by both carries stronger evidence, consistent with the co\-training principle\[[2](https://arxiv.org/html/2608.07881#bib.bib59)\]that agreement between conditionally independent views yields reduced false\-positive rates\.

The reduced false\-positive rate noted above is what makes a confirmed pair worth rewarding\. A proposition in the[supplementary material](https://github.com/develop-yang/GRACE-GRACE-A)formalizes the argument, proving that a dual\-view\-confirmed pair \(Γi​l=1\\Gamma\_\{il\}=1\) carries a strictly higher posterior probability of being a true within\-cluster pair than a pair supported by the semantic view alone\. GRACE therefore rewards dual\-view confirmation by scaling the semantic affinity:

ρi​l=Si​lsem⋅αi​l,\\rho\_\{il\}=S\_\{il\}^\{\\mathrm\{sem\}\}\\cdot\\alpha\_\{il\},\(11\)whose entries form the refined affinity matrix𝐖\\mathbf\{W\}withWi​l=ρi​lW\_\{il\}=\\rho\_\{il\}, through the coefficient:

αi​l=1\+Γi​l,Γi​l∈\{0,1\}\.\\alpha\_\{il\}=1\+\\Gamma\_\{il\},\\qquad\\Gamma\_\{il\}\\in\\\{0,1\\\}\.\(12\)The multiplicative and proportional form is deliberate\. An unconfirmed pair \(Γi​l=0\\Gamma\_\{il\}=0\) keeps its original semantic value, which leaves the knowledge\-grounded signal undistorted, whereas a confirmed pair is boosted in proportion to its existing semantic strength, letting statistical agreement reinforce the semantic evidence instead of overriding it\.

### III\-DModel Training and Spectral Clustering

The semantic affinitySi​lsemS^\{\\mathrm\{sem\}\}\_\{il\}in[Eq\. \(6\)](https://arxiv.org/html/2608.07881#S3.E6)is the cosine similarity between the sample vectors𝐳i\\mathbf\{z\}\_\{i\}and𝐳l\\mathbf\{z\}\_\{l\}, and[Eq\. \(5\)](https://arxiv.org/html/2608.07881#S3.E5)builds each𝐳i\\mathbf\{z\}\_\{i\}from the per\-attribute embeddings𝐞​\(j,o\)=fθ​\(t​\(j,o\)\)\\mathbf\{e\}\(j,o\)=f\_\{\\theta\}\(t\(j,o\)\)of[Eq\. \(4\)](https://arxiv.org/html/2608.07881#S3.E4)\. The encoderfθf\_\{\\theta\}therefore governs the entire metric, and adapting it to the grounded descriptions is what reshapesSsemS^\{\\mathrm\{sem\}\}\. Such adaptation calls for a contrastive objective, one that keeps descriptions of the same value close and drives those of different values apart under the cosine geometry of[Eq\. \(6\)](https://arxiv.org/html/2608.07881#S3.E6)\.

Concretely, GRACE fine\-tunesfθf\_\{\\theta\}with an InfoNCE loss\[[29](https://arxiv.org/html/2608.07881#bib.bib60)\]\. The objective is well suited to attribute descriptions since it preserves the semantic identity of each grounded value and separates alternative values within the same attribute context\. For an anchor descriptiont​\(j,o\)t\(j,o\), the positive examplet\+t^\{\+\}is obtained by encoding the same text with independently sampled dropout masks, following\[[9](https://arxiv.org/html/2608.07881#bib.bib26)\], and the remaining descriptions in the mini\-batchℬ\\mathcal\{B\}serve as negatives\. The per\-anchor loss is:

ℒ​\(t\)=−log⁡exp⁡\(sim​\(fθ​\(t\),fθ​\(t\+\)\)/τe\)∑t′∈ℬexp⁡\(sim​\(fθ​\(t\),fθ​\(t′\)\)/τe\),\\mathcal\{L\}\(t\)=\-\\log\\frac\{\\exp\\bigl\(\\mathrm\{sim\}\(f\_\{\\theta\}\(t\),\\,f\_\{\\theta\}\(t^\{\+\}\)\)/\\tau\_\{e\}\\bigr\)\}\{\\sum\_\{t^\{\\prime\}\\in\\mathcal\{B\}\}\\exp\\bigl\(\\mathrm\{sim\}\(f\_\{\\theta\}\(t\),\\,f\_\{\\theta\}\(t^\{\\prime\}\)\)/\\tau\_\{e\}\\bigr\)\},\(13\)in whichsim​\(⋅,⋅\)\\mathrm\{sim\}\(\\cdot,\\cdot\)denotes cosine similarity andτe\\tau\_\{e\}is a temperature hyperparameter\. Descriptions from different values of the same attribute naturally act as hard negatives, because they share domain context yet refer to distinct categorical states\. The full encoder objective averages over all anchors:

ℒenc=1\|ℬ\|​∑t∈ℬℒ​\(t\)\.\\mathcal\{L\}\_\{\\mathrm\{enc\}\}=\\frac\{1\}\{\|\\mathcal\{B\}\|\}\\sum\_\{t\\in\\mathcal\{B\}\}\\mathcal\{L\}\(t\)\.\(14\)The encoder is optimized with AdamW\[[26](https://arxiv.org/html/2608.07881#bib.bib70)\]forEEepochs using a learning rate ofη\\etaand a batch size\|ℬ\|\|\\mathcal\{B\}\|\.

After training, the refined affinity matrix𝐖\\mathbf\{W\}from[Eq\. \(11\)](https://arxiv.org/html/2608.07881#S3.E11)retains pairwise affinities for all sample pairs, including weak relations with limited contribution to the underlying cluster structure\. Applying spectral clustering directly to such a fully connected affinity graph may propagate low\-confidence edges into the spectral embedding, weakening cluster separation\. GRACE therefore converts𝐖\\mathbf\{W\}into a sparse binary graph by thresholding at a valueθ\\theta, producing the adjacency matrix\(𝐀θ\)i​l=𝕀​\[ρi​l\>θ\]\(\\mathbf\{A\}\_\{\\theta\}\)\_\{il\}=\\mathbb\{I\}\[\\rho\_\{il\}\>\\theta\]\. The thresholdθ\\thetacontrols the trade\-off between edge preservation and graph sparsity\. A smallθ\\thetatends to retain unreliable weak affinities, whereas a largeθ\\thetamay remove structurally necessary edges and fragment the graph into isolated components\. To avoid threshold\-induced fragmentation, GRACE first identifies the candidate setΘ∗\\Theta^\{\*\}of thresholds that achieve minimal graph fragmentation\. Following the spectral\-theoretic analysis of\[[8](https://arxiv.org/html/2608.07881#bib.bib31)\], GRACE then selects fromΘ∗\\Theta^\{\*\}the valueθ∗\\theta^\{\*\}that maximizes the Relative Eigengap Quality \(REQ\) of the normalized Laplacian𝐋θ\\mathbf\{L\}\_\{\\theta\}of𝐀θ\\mathbf\{A\}\_\{\\theta\}:

θ∗=arg⁡maxθ∈Θ∗⁡λK\+1​\(𝐋θ\)−λ¯K​\(𝐋θ\)λ¯K​\(𝐋θ\)\+ϵ,\\theta^\{\*\}=\\arg\\max\_\{\\theta\\in\\Theta^\{\*\}\}\\;\\frac\{\\lambda\_\{K\+1\}\(\\mathbf\{L\}\_\{\\theta\}\)\-\\bar\{\\lambda\}\_\{K\}\(\\mathbf\{L\}\_\{\\theta\}\)\}\{\\bar\{\\lambda\}\_\{K\}\(\\mathbf\{L\}\_\{\\theta\}\)\+\\epsilon\},\(15\)whereλK\+1\\lambda\_\{K\+1\}is the\(K\+1\)\(K\\\!\+\\\!1\)\-th smallest eigenvalue of𝐋θ\\mathbf\{L\}\_\{\\theta\},λ¯K\\bar\{\\lambda\}\_\{K\}denotes the mean of the firstKKeigenvalues, andϵ=10−6\\epsilon=10^\{\-6\}\. A larger REQ indicates a wider spectral gap, corresponding to better\-separated clusters\. TheKKsmallest eigenvectors of𝐋θ∗\\mathbf\{L\}\_\{\\theta^\{\*\}\}are then extracted and row\-normalized, andkk\-means is applied to the rows to produce𝒞=\{C1,…,CK\}\\mathcal\{C\}=\\\{C\_\{1\},\\ldots,C\_\{K\}\\\}, following the normalized spectral clustering algorithm\[[28](https://arxiv.org/html/2608.07881#bib.bib62)\]\.

Input:Dataset𝒟\\mathcal\{D\}with attributes𝒜\\mathcal\{A\}, cluster countKK, threshold candidatesΘ\\Theta

Output:Cluster assignments

𝒞=\{C1,…,CK\}\\mathcal\{C\}=\\\{C\_\{1\},\\ldots,C\_\{K\}\\\}
⊳\\trianglerightKnowledge\-Grounded Attribute Description

1foreach*Aj∈𝒜A\_\{j\}\\in\\mathcal\{A\}*do

2if*Aj∈𝒜cA\_\{j\}\\in\\mathcal\{A\}^\{c\}*then

3Generate description

t​\(j,o\)t\(j,o\)for each

o∈𝒪jco\\in\\mathcal\{O\}^\{c\}\_\{j\}via LLM;

4

5else

6Discretize into

ℐj\\mathcal\{I\}\_\{j\}and generate descriptions via LLM;

7

8

Fine\-tune encoder

fθf\_\{\\theta\}with InfoNCE loss ;

⊳\\triangleright[Eq\. \(14\)](https://arxiv.org/html/2608.07881#S3.E14)

𝐳i←⨁j=1mfθ​\(t​\(j,xij\)\)\\mathbf\{z\}\_\{i\}\\leftarrow\\bigoplus\_\{j=1\}^\{m\}f\_\{\\theta\}\(t\(j,x\_\{i\}^\{j\}\)\)for each

xix\_\{i\};

⊳\\triangleright[Eq\. \(5\)](https://arxiv.org/html/2608.07881#S3.E5)

Compute semantic similarity

𝐒sem\\mathbf\{S\}^\{\\mathrm\{sem\}\};

⊳\\triangleright[Eq\. \(6\)](https://arxiv.org/html/2608.07881#S3.E6)

⊳\\trianglerightDual\-View Neighborhood Consistency

Encode statistical features

\{𝐡i\}\\\{\\mathbf\{h\}\_\{i\}\\\}and compute

𝐒stat\\mathbf\{S\}^\{\\mathrm\{stat\}\};

⊳\\triangleright[Eq\. \(7\)](https://arxiv.org/html/2608.07881#S3.E7)

Build natural neighbor graphs

𝐍𝐍sem,𝐍𝐍stat\\mathbf\{NN\}^\{\\mathrm\{sem\}\},\\mathbf\{NN\}^\{\\mathrm\{stat\}\};

⊳\\triangleright[Eq\. \(8\)](https://arxiv.org/html/2608.07881#S3.E8)

𝚪←𝐍𝐍sem⊙𝐍𝐍stat\\boldsymbol\{\\Gamma\}\\leftarrow\\mathbf\{NN\}^\{\\mathrm\{sem\}\}\\odot\\mathbf\{NN\}^\{\\mathrm\{stat\}\};

⊳\\triangleright[Eq\. \(10\)](https://arxiv.org/html/2608.07881#S3.E10)

Refine affinity

ρi​l\\rho\_\{il\}via hierarchical enhancement ;

⊳\\triangleright[Eq\. \(11\)](https://arxiv.org/html/2608.07881#S3.E11)

⊳\\trianglerightGraph Sparsification and Spectral Clustering

9foreach*θ∈Θ\\theta\\in\\Theta*do

Construct

𝐀θ\\mathbf\{A\}\_\{\\theta\}and evaluate

REQ​\(𝐋θ\)\\mathrm\{REQ\}\(\\mathbf\{L\}\_\{\\theta\}\);

⊳\\triangleright[Eq\. \(15\)](https://arxiv.org/html/2608.07881#S3.E15)

10

11Select

θ∗\\theta^\{\*\}maximizing REQ among thresholds with minimal fragmentation;

return*Normalized spectral clustering on𝐀θ∗\\mathbf\{A\}\_\{\\theta^\{\*\}\}*

Algorithm 1GRACE: GRounding Attributes for Clustering via External Semantics
### III\-EComplexity Analysis

The complete GRACE procedure is summarized in[Algorithm1](https://arxiv.org/html/2608.07881#alg1)\. Because the exact formulation forms a dense affinity over every sample pair and then performs spectral clustering on it, the eigendecomposition of then×nn\\times nLaplacian drives the cost toO​\(n3\)O\(n^\{3\}\), which becomes prohibitive at the scale of large tables\. To retain the same affinity design at a far lower cost, we further develop an approximation variant, denoted GRACE\-A, where “A” stands for Approximation\. GRACE\-A replaces the dense pairwise graph with a sample\-to\-anchor bipartite graph overp≪np\\ll nanchors\.

GRACE\-A rests on an anchor\-based approximation of the affinity structure\. A set ofppanchors is first selected by farthest\-point sampling, which spreads them across the embedding space and keeps them representative of the latent clusters\. Each sample is then linked only to these anchors through a sample\-to\-anchor affinity matrix𝐙∈ℝn×p\\mathbf\{Z\}\\in\\mathbb\{R\}^\{n\\times p\}, replacing the fulln×nn\\times ngraph\. A truncated factorization of𝐙\\mathbf\{Z\}recovers the leading spectral components of the implied global graph in the Nyström sense, after whichkk\-means on the recovered embedding yields the partition\. The construction thereby involves each sample only a constant number of times onceppis fixed, which lowers the order from cubic to linear innn\.

The following theorems establish the time and space complexity of GRACE and GRACE\-A\. Throughout,nn,mm,KK, andppcount the samples, attributes, clusters, and anchors,ded\_\{e\}denotes the embedding dimension, andVVthe number of distinct attribute values, following[TableI](https://arxiv.org/html/2608.07881#S2.T1)\.

###### Theorem 1\.

Standard GRACE runs inO​\(n3\+n2​m​de\)O\(n^\{3\}\+n^\{2\}md\_\{e\}\)time\.

###### Proof\.

The procedure divides into value grounding, affinity construction, and spectral clustering\. Grounding theVVdistinct attribute values, namely description generation, encoder fine\-tuning, and value\-level embedding, scales withVVinstead ofnnand costsO​\(V​de\)O\(Vd\_\{e\}\)\. Assembling thennsample vectors and evaluating the dense semantic affinity over alln2n^\{2\}pairs then takesO​\(n2​m​de\)O\(n^\{2\}md\_\{e\}\), since every entry is an inner product of two\(m​de\)\(md\_\{e\}\)\-dimensional vectors\. The dual\-view refinement manipulatesn×nn\\times nmatrices atO​\(n2\)O\(n^\{2\}\), a term the spectral stage subsumes\. Eigendecomposing the normalized Laplacian𝐋θ\\mathbf\{L\}\_\{\\theta\}across a fixed budget of candidate thresholds requiresO​\(n3\)O\(n^\{3\}\), whereas the ensuingkk\-means adds onlyO​\(t​n​K2\)O\(tnK^\{2\}\)\. Givenn≫Kn\\gg K, the eigendecomposition and the affinity construction govern the total, which givesO​\(n3\+n2​m​de\)O\(n^\{3\}\+n^\{2\}md\_\{e\}\)\. ∎

###### Theorem 2\.

Standard GRACE requiresO​\(n2\)O\(n^\{2\}\)space\.

###### Proof\.

Storage is dictated by the largest object the pipeline keeps in memory\. The dense affinity matrix occupiesO​\(n2\)O\(n^\{2\}\)and dominates every other term, since the input table𝒟∈ℝn×m\\mathcal\{D\}\\in\\mathbb\{R\}^\{n\\times m\}, the value embeddings, and the spectral embedding𝐘∈ℝn×K\\mathbf\{Y\}\\in\\mathbb\{R\}^\{n\\times K\}need onlyO​\(n​m\)O\(nm\),O​\(n​m​de\)O\(nmd\_\{e\}\), andO​\(n​K\)O\(nK\)respectively, each belowO​\(n2\)O\(n^\{2\}\)whenevernnexceedsm​demd\_\{e\}\. The space complexity is thereforeO​\(n2\)O\(n^\{2\}\)\. ∎

GRACE\-A lowers both bounds to linear order, as the next two theorems show\.

###### Theorem 3\.

GRACE\-A runs inO​\(n​p​\(m​de\+p\)\)O\\bigl\(np\(md\_\{e\}\+p\)\\bigr\)time, linear innnfor fixedpp\.

###### Proof\.

The anchor approximation eliminates the cubic and quadratic terms of[Theorem1](https://arxiv.org/html/2608.07881#Thmtheorem1)\. Farthest\-point selection of theppanchors and construction of the sample\-to\-anchor affinity𝐙∈ℝn×p\\mathbf\{Z\}\\in\\mathbb\{R\}^\{n\\times p\}, which connects each sample to the anchors alone, together costO​\(n​p​m​de\)O\(npmd\_\{e\}\)in place of theO​\(n2​m​de\)O\(n^\{2\}md\_\{e\}\)dense affinity\. A truncated factorization replaces the full eigendecomposition atO​\(n​p2\)O\(np^\{2\}\), and the closingkk\-means contributesO​\(t​n​K2\)O\(tnK^\{2\}\), both staying within the same order for fixedpp\. Collecting the terms yieldsO​\(n​p​\(m​de\+p\)\)O\\bigl\(np\(md\_\{e\}\+p\)\\bigr\), which grows linearly innnbecauseppstays bounded independently ofnn\. ∎

###### Theorem 4\.

GRACE\-A requiresO​\(n​\(m​de\+p\)\)O\\bigl\(n\(md\_\{e\}\+p\)\\bigr\)space\.

###### Proof\.

The same anchoring contracts the footprint\. The sample\-to\-anchor affinity𝐙\\mathbf\{Z\}holdsO​\(n​p\)O\(np\)instead of theO​\(n2\)O\(n^\{2\}\)dense matrix of[Theorem2](https://arxiv.org/html/2608.07881#Thmtheorem2), and the value embeddings together with the input table account for the remainingO​\(n​m​de\)O\(nmd\_\{e\}\)\. Because no retained structure scales withn2n^\{2\}, the dominant term isO​\(n​\(m​de\+p\)\)O\\bigl\(n\(md\_\{e\}\+p\)\\bigr\), again linear innn\. ∎

Taken together, the four bounds capture the trade\-off between the exact model and its scalable variant\. Standard GRACE computes the knowledge\-grounded affinity exactly, yet its cubic time and quadratic space confine it to datasets of moderate size\. GRACE\-A lifts both restrictions at once, reducing the time toO​\(n​p​\(m​de\+p\)\)O\\bigl\(np\(md\_\{e\}\+p\)\\bigr\)and the space toO​\(n​\(m​de\+p\)\)O\\bigl\(n\(md\_\{e\}\+p\)\\bigr\), each linear innnfor a fixed anchor budget\. The anchor design thereby carries the semantic metric of GRACE to corpora whose scale puts a densen×nn\\times naffinity out of reach, where storing it alone would exhaust commodity memory\. As the experiments demonstrate the accuracy traded for the linear cost remains marginal, leaving GRACE\-A with both the world\-knowledge benefit of GRACE and the scalability to cluster tables far beyond the reach of dense spectral methods\.

TABLE II:Statistics of the twenty\-two datasets\.dcd\_\{c\},dnd\_\{n\}, andnndenote the numbers of categorical attributes, numerical attributes, and samples, respectively\.VVis the total number of distinct values across all categorical attributes\.KKequals the true number of clusters\.No\.DatasetAbbr\.\(dc,dn\)\(d\_\{c\},\\,d\_\{n\}\)nnVVKK1LensesLE\(4,0\)\(4,\\,0\)24932Caesarian SectionCS\(3,2\)\(3,\\,2\)80823ZooZO\(16,0\)\(16,\\,0\)1013674Autism AdolescentAA\(18,2\)\(18,\\,2\)1047625LymphographyLY\(18,0\)\(18,\\,0\)1486046Teaching AssistantTA\(4,1\)\(4,\\,1\)1515537AmphibiansAM\(12,3\)\(12,\\,3\)1897128Soybean \(Large\)SO\(35,0\)\(35,\\,0\)266100159SPECT HeartSH\(22,0\)\(22,\\,0\)26744210Breast CancerBC\(9,0\)\(9,\\,0\)28651211Heart DiseaseHD\(8,5\)\(8,\\,5\)30323512Primary TumorPT\(17,0\)\(17,\\,0\)339372213DermatologyDE\(33,1\)\(33,\\,1\)366130614Chronic Kidney DiseaseCK\(13,11\)\(13,\\,11\)40037215Congressional VotingCV\(16,0\)\(16,\\,0\)43532216Statlog AustralianSA\(8,6\)\(8,\\,6\)69037217Car EvaluationCA\(6,0\)\(6,\\,0\)1,72821418Auction VerificationAV\(7,1\)\(7,\\,1\)2,04354219Obesity LevelsOL\(8,8\)\(8,\\,8\)2,11123720SpliceSP\(60,0\)\(60,\\,0\)3,190240321MushroomMU\(21,0\)\(21,\\,0\)8,124115222AdultAD\(8,6\)\(8,\\,6\)48,842992TABLE III:Clustering results on 11 categorical datasets and two reconstructed datasets \(in dark red\) for probing LLM label leakage, reported as mean±\\pmstandard deviation\.Bestandsecond\-bestresults are highlighted \(GRACE\-A included\)\.A​R¯\\overline\{AR\}is the average rank over all 13 datasets; GRACE and GRACE\-A are each ranked against the competitors only, not against each other\.†indicates statistical significance \(p<0\.05p<0\.05\) against GRACE via the Wilcoxon signed\-rank test\.MetricDatasetKMOADCGUDMM\-SMCDCCOForestAMPHMSigDTHARRDiSCOCLGRACEGRACE\-A\[DMKD’98\]\[TNNLS’23\]\[PR’23\]\[ICDCS’24\]\[ECAI’24\]\[CAIS’25\]\[Inf\.Sci\.’25\]\[ESWA’25\]\[AAAI’26\]\[SIGMOD’26\]\[Ours\]\[Approx\.\]ARILE0\.043±\.060\.061±\.100\.130±\.09\-0\.076±\.000\.119±\.11\-0\.013±\.00\-0\.043±\.000\.125±\.160\.065±\.180\.137±\.100\.163±\.170\.213±\.10ZO0\.584±\.100\.494±\.000\.602±\.040\.602±\.000\.639±\.130\.379±\.000\.696±\.000\.778±\.130\.644±\.250\.721±\.120\.827±\.000\.660±\.00LY0\.011±\.020\.112±\.040\.117±\.03\-0\.010±\.000\.118±\.030\.129±\.000\.015±\.000\.152±\.050\.080±\.040\.159±\.050\.215±\.010\.161±\.00SO0\.227±\.040\.358±\.020\.353±\.030\.340±\.000\.397±\.040\.157±\.000\.120±\.000\.355±\.030\.293±\.150\.409±\.020\.454±\.000\.456±\.00SH\-0\.073±\.01\-0\.007±\.02\-0\.023±\.00\-0\.039±\.00\-0\.048±\.030\.063±\.000\.094±\.000\.023±\.000\.312±\.00\-0\.088±\.000\.294±\.000\.143±\.00BC0\.000±\.000\.003±\.010\.053±\.00\-0\.008±\.000\.013±\.050\.013±\.000\.058±\.000\.000±\.000\.036±\.08\-0\.003±\.000\.156±\.000\.147±\.00PT0\.068±\.020\.079±\.010\.072±\.010\.076±\.000\.111±\.010\.084±\.000\.078±\.000\.103±\.010\.085±\.020\.101±\.010\.115±\.000\.110±\.00CV0\.422±\.020\.495±\.170\.550±\.000\.537±\.000\.566±\.000\.008±\.000\.459±\.000\.550±\.000\.563±\.000\.590±\.050\.628±\.000\.573±\.00CA0\.021±\.030\.030±\.030\.050±\.050\.001±\.000\.044±\.050\.194±\.00\-0\.041±\.00\-0\.002±\.080\.032±\.080\.048±\.040\.131±\.000\.087±\.00SP0\.025±\.000\.078±\.000\.075±\.000\.071±\.000\.013±\.01–0\.386±\.000\.107±\.080\.000±\.000\.078±\.010\.396±\.000\.438±\.00MU0\.373±\.050\.344±\.200\.273±\.000\.066±\.010\.111±\.10–0\.266±\.140\.384±\.270\.261±\.230\.435±\.260\.606±\.010\.581±\.00BC0\.020±\.000\.002±\.000\.054±\.000\.003±\.000\.006±\.000\.013±\.000\.011±\.00\-0\.004±\.000\.047±\.000\.028±\.000\.065±\.000\.057±\.00LY0\.029±\.000\.051±\.000\.045±\.000\.025±\.000\.029±\.000\.038±\.000\.052±\.000\.020±\.000\.050±\.000\.025±\.000\.059±\.000\.054±\.00A​R¯\\overline\{AR\}8\.23†6\.54†5\.31†9\.00†5\.93†6\.73†6\.54†5\.77†5\.54†4\.61†1\.151\.54NMILE0\.165±\.060\.232±\.110\.210±\.100\.200±\.000\.287±\.150\.129±\.000\.027±\.000\.183±\.160\.226±\.150\.287±\.100\.341±\.130\.339±\.11ZO0\.679±\.050\.702±\.000\.745±\.020\.803±\.000\.793±\.050\.601±\.000\.786±\.000\.869±\.040\.736±\.200\.810±\.040\.854±\.000\.819±\.00LY0\.117±\.030\.156±\.020\.212±\.020\.108±\.000\.161±\.030\.330±\.000\.141±\.000\.174±\.050\.173±\.050\.196±\.050\.253±\.000\.218±\.00SO0\.518±\.030\.638±\.010\.644±\.020\.634±\.000\.697±\.020\.562±\.000\.538±\.000\.642±\.020\.569±\.280\.704±\.020\.756±\.000\.731±\.00SH0\.059±\.000\.011±\.030\.068±\.000\.062±\.000\.069±\.020\.061±\.000\.119±\.000\.115±\.000\.185±\.000\.094±\.000\.144±\.000\.135±\.00BC0\.002±\.000\.005±\.000\.041±\.000\.001±\.000\.009±\.020\.030±\.000\.053±\.000\.000±\.000\.033±\.040\.001±\.000\.079±\.000\.076±\.00PT0\.250±\.010\.263±\.020\.304±\.010\.322±\.000\.360±\.010\.308±\.000\.304±\.000\.351±\.010\.345±\.020\.354±\.010\.370±\.000\.364±\.00CV0\.438±\.000\.436±\.150\.444±\.000\.475±\.000\.491±\.000\.006±\.000\.526±\.000\.479±\.000\.446±\.000\.503±\.050\.538±\.000\.483±\.00CA0\.041±\.020\.049±\.030\.054±\.070\.003±\.000\.111±\.060\.124±\.000\.034±\.000\.065±\.060\.084±\.070\.104±\.040\.188±\.000\.133±\.00SP0\.050±\.000\.091±\.000\.076±\.000\.082±\.000\.015±\.01–0\.329±\.000\.115±\.090\.000±\.000\.128±\.030\.404±\.000\.399±\.00MU0\.329±\.020\.361±\.170\.303±\.000\.052±\.010\.098±\.10–0\.332±\.110\.323±\.230\.272±\.190\.377±\.230\.554±\.010\.553±\.00BC0\.110±\.000\.062±\.000\.138±\.000\.099±\.000\.111±\.000\.090±\.000\.136±\.000\.080±\.000\.129±\.000\.144±\.000\.149±\.000\.155±\.00LY0\.179±\.000\.234±\.000\.227±\.000\.191±\.000\.206±\.000\.172±\.000\.209±\.000\.185±\.000\.213±\.000\.186±\.000\.226±\.000\.227±\.00A​R¯\\overline\{AR\}9\.00†7\.15†5\.54†7\.69†5\.23†7\.73†5\.92†6\.00†5\.85†4\.00†1\.381\.58ACCLE0\.517±\.070\.529±\.060\.545±\.090\.417±\.000\.554±\.090\.458±\.000\.417±\.000\.567±\.100\.500±\.120\.558±\.100\.575±\.110\.600±\.11ZO0\.655±\.090\.545±\.000\.526±\.010\.673±\.000\.702±\.090\.554±\.000\.802±\.000\.788±\.110\.717±\.150\.746±\.100\.822±\.000\.756±\.00LY0\.410±\.030\.433±\.040\.443±\.030\.446±\.000\.508±\.040\.561±\.000\.493±\.000\.487±\.040\.470±\.070\.513±\.040\.546±\.000\.509±\.00SO0\.424±\.060\.483±\.020\.452±\.020\.564±\.000\.602±\.040\.436±\.000\.323±\.000\.510±\.030\.473±\.160\.595±\.040\.658±\.000\.616±\.00SH0\.430±\.020\.766±\.080\.547±\.000\.502±\.000\.544±\.020\.629±\.000\.397±\.000\.592±\.000\.843±\.000\.559±\.000\.824±\.000\.793±\.00BC0\.516±\.020\.544±\.010\.575±\.000\.531±\.000\.531±\.070\.567±\.000\.523±\.000\.708±\.000\.609±\.080\.509±\.000\.711±\.000\.704±\.00PT0\.191±\.020\.280±\.020\.287±\.010\.265±\.000\.306±\.020\.286±\.000\.256±\.000\.287±\.020\.279±\.020\.299±\.020\.293±\.000\.287±\.00CV0\.762±\.010\.846±\.080\.871±\.000\.867±\.000\.877±\.000\.618±\.000\.657±\.000\.871±\.000\.876±\.000\.884±\.020\.897±\.000\.879±\.00CA0\.262±\.030\.393±\.050\.375±\.040\.270±\.000\.383±\.050\.514±\.000\.554±\.000\.600±\.080\.544±\.120\.372±\.040\.487±\.000\.398±\.00SP0\.377±\.000\.513±\.000\.510±\.000\.556±\.000\.392±\.02–0\.687±\.000\.554±\.050\.519±\.000\.474±\.010\.748±\.000\.776±\.00MU0\.678±\.020\.782±\.080\.678±\.000\.628±\.010\.650±\.07–0\.551±\.040\.759±\.170\.732±\.110\.806±\.120\.889±\.000\.813±\.00BC0\.206±\.000\.191±\.000\.219±\.000\.188±\.000\.217±\.000\.206±\.000\.213±\.000\.191±\.000\.227±\.000\.217±\.000\.246±\.000\.238±\.00LY0\.247±\.000\.251±\.000\.254±\.000\.243±\.000\.257±\.000\.257±\.000\.267±\.000\.250±\.000\.277±\.000\.250±\.000\.270±\.000\.270±\.00A​R¯\\overline\{AR\}9\.31†6\.73†6\.50†8\.16†5\.30†6\.36†7\.08†4\.69†4\.54†5\.08†1\.692\.15

TABLE IV:Clustering results on 11 mixed datasets\. All notations are consistent with those defined in[TableIII](https://arxiv.org/html/2608.07881#S3.T3)\.MetricDatasetKPRADCGUDMM\-SAMPHMHARRGRACEGRACE\-A\[PAKDD’97\]\[TNNLS’23\]\[PR’23\]\[CAIS’25\]\[ESWA’25\]\[Ours\]\[Approx\.\]ARICS0\.003±\.00\-0\.003±\.010\.010±\.000\.050±\.000\.044±\.040\.094±\.000\.078±\.00AA0\.478±\.020\.029±\.020\.509±\.000\.040±\.000\.131±\.230\.528±\.000\.588±\.00TA0\.004±\.010\.012±\.010\.016±\.000\.016±\.00\-0\.008±\.010\.065±\.000\.052±\.00AM\-0\.005±\.000\.008±\.000\.067±\.000\.010±\.000\.001±\.010\.109±\.000\.090±\.00HD0\.019±\.000\.229±\.000\.372±\.000\.242±\.000\.337±\.000\.394±\.000\.390±\.00DE0\.101±\.010\.351±\.080\.705±\.000\.679±\.000\.805±\.080\.921±\.000\.914±\.00CK0\.112±\.000\.664±\.000\.510±\.000\.567±\.000\.626±\.000\.895±\.000\.815±\.00SA0\.004±\.000\.077±\.010\.347±\.00\-0\.009±\.000\.417±\.010\.387±\.000\.363±\.00AV\-0\.100±\.000\.004±\.010\.003±\.00\-0\.047±\.00\-0\.009±\.050\.053±\.00\-0\.002±\.00OL0\.214±\.000\.079±\.010\.231±\.000\.163±\.000\.090±\.020\.255±\.000\.264±\.00AD\-0\.012±\.000\.084±\.000\.086±\.01–0\.100±\.000\.093±\.030\.156±\.01A​R¯\\overline\{AR\}5\.09†4\.36†2\.82†4\.00†3\.36†1\.181\.27NMICS0\.018±\.000\.006±\.010\.017±\.000\.077±\.000\.053±\.030\.108±\.000\.098±\.00AA0\.347±\.010\.019±\.010\.432±\.000\.058±\.000\.122±\.220\.460±\.000\.492±\.00TA0\.014±\.000\.021±\.010\.046±\.000\.084±\.000\.004±\.010\.074±\.000\.065±\.00AM0\.016±\.000\.004±\.000\.079±\.000\.005±\.000\.002±\.000\.113±\.000\.100±\.00HD0\.014±\.000\.174±\.000\.294±\.010\.189±\.000\.262±\.000\.306±\.000\.308±\.00DE0\.218±\.020\.403±\.080\.749±\.000\.740±\.000\.822±\.060\.913±\.000\.904±\.00CK0\.246±\.000\.639±\.000\.528±\.000\.568±\.000\.611±\.000\.848±\.000\.767±\.00SA0\.016±\.000\.059±\.000\.288±\.000\.040±\.000\.323±\.010\.300±\.000\.278±\.00AV0\.003±\.000\.004±\.000\.003±\.000\.009±\.000\.020±\.030\.006±\.000\.000±\.00OL0\.281±\.000\.135±\.020\.338±\.000\.249±\.000\.146±\.020\.346±\.000\.354±\.00AD0\.001±\.000\.077±\.000\.057±\.01–0\.082±\.000\.064±\.050\.125±\.01A​R¯\\overline\{AR\}4\.73†4\.45†3\.36†3\.50†3\.18†1\.551\.73ACCCS0\.562±\.000\.540±\.030\.575±\.000\.625±\.000\.606±\.050\.662±\.000\.650±\.00AA0\.784±\.010\.592±\.040\.828±\.000\.612±\.000\.631±\.140\.867±\.000\.885±\.00TA0\.383±\.010\.426±\.020\.404±\.000\.384±\.000\.352±\.020\.470±\.000\.459±\.00AM0\.561±\.000\.566±\.000\.635±\.000\.571±\.000\.554±\.010\.669±\.000\.654±\.00HD0\.576±\.000\.741±\.000\.806±\.000\.748±\.000\.791±\.000\.815±\.000\.813±\.00DE0\.363±\.020\.494±\.080\.746±\.000\.768±\.000\.832±\.070\.953±\.000\.947±\.00CK0\.669±\.000\.908±\.000\.858±\.000\.877±\.000\.896±\.000\.973±\.000\.952±\.00SA0\.562±\.000\.642±\.010\.792±\.000\.520±\.000\.823±\.000\.812±\.000\.801±\.00AV0\.660±\.000\.552±\.030\.527±\.000\.814±\.000\.582±\.050\.801±\.000\.562±\.00OL0\.324±\.000\.274±\.010\.429±\.000\.390±\.000\.283±\.020\.437±\.000\.423±\.00AD0\.613±\.010\.654±\.000\.690±\.01–0\.660±\.000\.739±\.040\.701±\.00A​R¯\\overline\{AR\}4\.82†4\.45†3\.18†3\.50†3\.64†1\.181\.45TABLE V:ARI of GRACE and GRACE\-A versus LLM\-based clustering methods on 22 datasets\. All notations are consistent with those defined in[TableIII](https://arxiv.org/html/2608.07881#S3.T3)\.DatasetTabLLMClusterLLMFewShotGenericDesc\.BREVEGRACEGRACE\-ALE0\.0260\.000\-0\.0430\.0940\.2170\.1630\.213CS0\.003\-0\.0140\.0640\.064–0\.0940\.078ZO0\.6310\.0000\.8050\.4470\.7740\.8270\.660AA\-0\.0040\.0740\.0240\.466–0\.5280\.588LY0\.0720\.0000\.0610\.0200\.2070\.2150\.161TA0\.0450\.0060\.039\-0\.011–0\.0650\.052AM0\.0220\.1100\.0220\.129–0\.1090\.090SO0\.1730\.0000\.1390\.1500\.4280\.4540\.456SH\-0\.0160\.000\-0\.013\-0\.010\-0\.0220\.2940\.143BC0\.1170\.0080\.008\-0\.0020\.1690\.1560\.147HD0\.174\-0\.0040\.0020\.306–0\.3940\.390PT0\.0560\.0000\.0580\.0360\.1030\.1150\.110DE0\.159\-0\.0050\.1040\.314–0\.9210\.914CK0\.3960\.6720\.1010\.639–0\.8950\.815CV0\.0060\.0000\.6060\.0610\.5780\.6280\.573SA0\.069\-0\.0040\.0000\.193–0\.3870\.363CA0\.0090\.0110\.1690\.0070\.0850\.1310\.087AV0\.0140\.286\-0\.0020\.003–0\.053\-0\.002OL0\.2220\.1310\.1420\.130–0\.2550\.264SP0\.0300\.0000\.0060\.0330\.3530\.3960\.438MU0\.0810\.0000\.6480\.0930\.5930\.6060\.581AD0\.1040\.0260\.0320\.094–0\.0930\.156A​R¯\\overline\{AR\}3\.61†4\.48†3\.66†3\.57†2\.55†1\.411\.73

## IVExperiments

### IV\-AExperimental Settings

Four experimental studies and one case studyare conducted to evaluate GRACE from complementary perspectives\.1\) Clustering effectiveness\([§IV\-B](https://arxiv.org/html/2608.07881#S4.SS2)\) compares GRACE with eleven representative counterparts on twenty UCI datasets and reports Wilcoxon signed\-rank tests to assess statistical reliability\.2\) Component analysis\([§IV\-C](https://arxiv.org/html/2608.07881#S4.SS3)\) isolates the contribution of semantic representation and dual\-view neighborhood consistency\.3\) Generality and robustness\([§IV\-D](https://arxiv.org/html/2608.07881#S4.SS4),[§IV\-E](https://arxiv.org/html/2608.07881#S4.SS5)\) examines whether the learned representation remains effective across clustering algorithms and whether the results are stable across different LLM backends\.4\) Scalability evaluation\([§IV\-F](https://arxiv.org/html/2608.07881#S4.SS6)\) studies runtime and clustering quality under increasing sample size, attribute cardinality, and cluster count\.5\) Case study\([§IV\-G](https://arxiv.org/html/2608.07881#S4.SS7)\) analyzes the learned semantic metric on the CK dataset to illustrate how GRACE improves cluster separation\. All methods run on an NVIDIA RTX 4090, except the scalability study, which is timed on a single commodity RTX 3060 \(6GB\) to provide an independent efficiency measurement\.

TABLE VI:Ablation study over Semantic Representation \(SR\) and Dual\-View Neighborhood Consistency \(DVNC\)\.MetricComponentsMarkDatasetsA​R¯\\overline\{AR\}SRDVNCLECSZOAALYTAAMSOSHBCHDPTDECKCVSACAAVOLSPARII0\.0640\.0940\.8070\.3660\.1920\.0190\.0680\.4320\.0680\.1730\.3700\.0990\.7780\.7550\.5850\.3540\.0000\.0490\.2900\.0002\.65✓\\checkmarkII0\.0640\.1110\.6560\.5290\.2380\.0650\.0860\.4560\.0680\.1510\.4200\.0990\.9210\.7680\.5990\.3870\.1310\.0530\.2410\.3791\.85✓\\checkmark✓\\checkmarkFull0\.1300\.0940\.8270\.5280\.2170\.0650\.1080\.4540\.2940\.1560\.3940\.1160\.9210\.8950\.6280\.3870\.1310\.0530\.2550\.3961\.50NMII0\.1370\.0780\.8030\.2640\.2430\.0260\.0660\.7250\.1190\.0890\.2800\.3530\.8510\.7130\.5050\.2800\.0030\.0340\.3540\.0012\.65✓\\checkmarkII0\.1370\.1310\.7940\.4370\.2750\.0740\.0880\.7480\.1190\.0760\.3290\.3540\.9130\.7250\.5200\.3000\.1880\.0060\.3640\.4011\.88✓\\checkmark✓\\checkmarkFull0\.3370\.1080\.8540\.4600\.2550\.0740\.1120\.7560\.1440\.0790\.3060\.3710\.9130\.8480\.5380\.3000\.1880\.0060\.3460\.4041\.48ACCI0\.5830\.6620\.8010\.8060\.5130\.4300\.6350\.6330\.6330\.7220\.7950\.2830\.8130\.9350\.8830\.8060\.2660\.7020\.5150\.5192\.60✓\\checkmarkII0\.5830\.6750\.6730\.8670\.5400\.4700\.6510\.6800\.6330\.7080\.8250\.2820\.9530\.9390\.8870\.8120\.4870\.8010\.4270\.7371\.93✓\\checkmark✓\\checkmarkFull0\.5830\.6620\.8220\.8670\.5470\.4700\.6680\.6580\.8240\.7110\.8150\.2940\.9530\.9730\.8970\.8120\.4870\.8010\.4370\.7481\.48

TABLE VII:Mechanism ablation on ten mixed\-type datasets \(ACC\)\. The two components indicate whether semantic grounding \(✓\\checkmark\) replaces the default distance for categorical \(Cat\., default Hamming\) and numerical \(Num\., default Euclidean\) attributes\. Row I \(neither\) reduces to KPR and Full \(both\) is the complete GRACE\.A​R¯\\overline\{AR\}is the average rank; the best result per dataset is inbold\.MetricComponentsMarkDatasetsA​R¯\\overline\{AR\}Cat\.Num\.CSAATAAMHDDECKSAAVOLACCI0\.5620\.7840\.3830\.5610\.5760\.3630\.6690\.5620\.6600\.3243\.60✓\\checkmarkII0\.5990\.6630\.3480\.6460\.7540\.3770\.9150\.8010\.5950\.3832\.90✓\\checkmarkIII0\.6650\.8530\.4580\.6260\.8270\.6230\.8650\.7940\.5090\.3942\.30✓\\checkmark✓\\checkmarkFull0\.6620\.8670\.4700\.6700\.8150\.9530\.9730\.8120\.8010\.4371\.20Eleven representative clustering counterparts, five LLM\-based baselines, and four LLM backendsare selected for comparison\. Partition\-based methods includekk\-Modes \(KMO\)\[[19](https://arxiv.org/html/2608.07881#bib.bib16)\]andkk\-Prototypes \(KPR\)\[[18](https://arxiv.org/html/2608.07881#bib.bib32)\]\. Distance metric learning methods consist of DiSC\[[49](https://arxiv.org/html/2608.07881#bib.bib33)\], ADC\[[42](https://arxiv.org/html/2608.07881#bib.bib1)\], HARR\[[43](https://arxiv.org/html/2608.07881#bib.bib2)\], MCDC\[[4](https://arxiv.org/html/2608.07881#bib.bib34)\], and GUDMM\-S\[[27](https://arxiv.org/html/2608.07881#bib.bib39)\]\. Tree\-based approaches include COForest\[[48](https://arxiv.org/html/2608.07881#bib.bib35)\]and SigDT\[[17](https://arxiv.org/html/2608.07881#bib.bib36)\]\. OCL\[[44](https://arxiv.org/html/2608.07881#bib.bib37)\]adopts ordinalization, and AMPHM\[[46](https://arxiv.org/html/2608.07881#bib.bib38)\]employs multi\-resolution condensation\. Among these counterparts, KPR, ADC, HARR, AMPHM, and GUDMM\-S natively support mixed data\. The five LLM\-based baselines adapt representative LLM\-for\-clustering paradigms to our mixed\-attribute setting\. Following the*row\-serialization*paradigm of TabLLM\[[15](https://arxiv.org/html/2608.07881#bib.bib65)\], we serialize each record into a natural\-language string over its attribute names and values, encode it withfθf\_\{\\theta\}, and cluster the resulting embeddings\. Adapting the*triplet\-guided representation*idea of ClusterLLM\[[47](https://arxiv.org/html/2608.07881#bib.bib7)\], we elicit LLM triplet judgments over records to shape the clustering representation\. Following the*few\-shot pairwise\-constraint*paradigm of\[[38](https://arxiv.org/html/2608.07881#bib.bib8)\], we derive LLM\-generated must\-link/cannot\-link constraints and run constrained clustering\.*GenericDescribe*prompts the LLM for a single free\-form description per attribute value and clusters the description embeddings, serving as a description\-only variant\. Finally, BREVE\[[40](https://arxiv.org/html/2608.07881#bib.bib69)\], preliminary study on*purely categorical*data, generates per\-value descriptions and linearly fuses a semantic view with a one\-hot identity view; GRACE extends it to mixed data via LLM\-based numerical discretization, dual\-view neighborhood consistency, and spectral clustering\. Four LLM backends, namely GPT\-5\.1, Claude Opus 4\.5, DeepSeek V3\.2, and Gemini 3 Pro, are considered and evaluated in[§IV\-E](https://arxiv.org/html/2608.07881#S4.SS5)\. For sentence encoding,fθf\_\{\\theta\}is instantiated as all\-mpnet\-base\-v2\.

Twenty\-two public datasetsfrom the UCI Machine Learning Repository111[https://archive\.ics\.uci\.edu/](https://archive.ics.uci.edu/)\[[7](https://arxiv.org/html/2608.07881#bib.bib40)\]are used for evaluation\. Eleven contain purely categorical attributes, and the remaining eleven mix categorical and numerical attributes\.[TableII](https://arxiv.org/html/2608.07881#S3.T2)reports the statistics of each dataset\.

Three validity indicesare adopted to evaluate clustering quality, namely the Adjusted Rand Index \(ARI\)\[[20](https://arxiv.org/html/2608.07881#bib.bib52)\], Normalized Mutual Information \(NMI\)\[[36](https://arxiv.org/html/2608.07881#bib.bib53)\], and Clustering Accuracy \(ACC\)\[[39](https://arxiv.org/html/2608.07881#bib.bib54)\]\. Higher values indicate better performance for all three metrics\. All experiments are repeated 10 times with different random seeds\.

### IV\-BClustering Performance Evaluation

[TableIII](https://arxiv.org/html/2608.07881#S3.T3)and[TableIV](https://arxiv.org/html/2608.07881#S3.T4)report the categorical and mixed benchmarks, where GRACE attains the bestA​R¯\\overline\{AR\}on every metric, with a categorical ARI rank of1\.151\.15, i\.e\., a first\-place finish on nearly every dataset\. The size of the lead points to a qualitative gain, since the LLM\-grounded semantics recover relations that co\-occurrence statistics cannot supply, e\.g\., the ordering of clinical severities or the kinship among biological traits\. No competing method stays on top of both benchmarks, because the strongest categorical specialists do not carry over to mixed inputs, whereas the mixed\-capable GUDMM\-S slips to mid\-pack once symbolic attributes dominate, leaving GRACE and GRACE\-A as the only methods placed first and second throughout\. GRACE\-A trails GRACE by a fraction of a rank and still outranks every competitor, with its rare sizeable drops confined to small datasets such as ZO and SH, where the anchor set draws on too few points to preserve the fine structure\.

A potential concern is that prompting the LLM exposes the dataset identity and lets the model retrieve the published labels online, shaping the value descriptions toward them\. To rule out such leakage, we relabel two datasets \(shown in dark red\) with a composite label that a LLM builds from attribute names and value distributions, with no access to the original\. The two attributes that define it are held out from every method, and an NMI audit keeps only labels near\-independent of the published one \(NMI≤0\.13\\leq 0\.13here\), which places the new label beyond the reach of memory or a web search\. GRACE still ranks first in ARI on both, which a method exploiting retrieved labels could not achieve, i\.e\., the advantage does not stem from label leakage\. Every method scores low on these intrinsically hard targets, and the evidence therefore rests on the relative order, where GRACE leads even the baselines fed the ground\-truth label as a feature\.

Against methods built specifically for LLM\-driven clustering \([TableV](https://arxiv.org/html/2608.07881#S3.T5)\), GRACE and GRACE\-A again rank first and second, and the strongest prior design BREVE trails at2\.552\.55\. Since those baselines also query an LLM yet land well behind, the gain traces to the4​P4Pdescription and the cross\-view verification, not to LLM access itself\.

TABLE VIII:ARI performance of representations generated by One\-Hot \(OH\) encoding, HARR, and GRACE underkk\-means, hierarchical clustering with average linkage \(HC\-avg\), and spectral clustering algorithms\. “↑\\uparrow” indicates that GRACE outperforms both baselines\.Datasetkk\-MeansHC\-avgSpectral ClusteringOHHARRGRACEOHHARRGRACEOHHARRGRACELE0\.0850\.2270\.166\-0\.043\-0\.0430\.074↑\\uparrow0\.1820\.1660\.002CS0\.0780\.0940\.078\-0\.0060\.0630\.112↑\\uparrow0\.0640\.0940\.078ZO0\.7020\.5010\.6740\.8890\.9600\.7990\.5230\.6570\.650AA0\.5660\.0170\.0900\.3250\.2990\.561↑\\uparrow0\.5920\.4360\.595↑\\uparrowLY0\.1970\.1940\.199↑\\uparrow0\.1110\.1440\.154↑\\uparrow0\.2340\.2280\.238↑\\uparrowTA0\.037\-0\.0080\.0100\.0370\.0170\.063↑\\uparrow\-0\.0020\.0110\.065↑\\uparrowAM0\.0530\.0990\.0920\.059\-0\.003\-0\.0010\.0620\.0660\.067↑\\uparrowSO0\.4140\.4230\.476↑\\uparrow0\.3430\.3030\.522↑\\uparrow0\.3440\.4360\.443↑\\uparrowSH\-0\.0070\.0020\.123↑\\uparrow\-0\.103\-0\.1070\.031↑\\uparrow\-0\.061\-0\.0630\.118↑\\uparrowBC\-0\.003\-0\.002\-0\.0100\.2360\.0160\.1450\.1240\.1850\.149HD0\.3830\.4120\.4020\.2700\.2840\.1980\.3860\.4200\.412PT0\.1000\.1010\.115↑\\uparrow0\.0670\.0580\.129↑\\uparrow0\.1020\.0970\.107↑\\uparrowDE0\.7980\.8820\.940↑\\uparrow0\.5140\.6160\.836↑\\uparrow0\.7010\.7090\.940↑\\uparrowCK0\.7060\.8090\.837↑\\uparrow0\.0700\.7020\.880↑\\uparrow0\.5320\.4770\.745↑\\uparrowCV0\.5780\.5330\.606↑\\uparrow0\.0040\.0040\.599↑\\uparrow0\.5640\.5850\.571MA0\.4250\.4420\.4260\.0000\.3140\.319↑\\uparrow0\.4260\.4190\.432↑\\uparrowCA0\.0580\.0130\.209↑\\uparrow0\.0130\.0130\.147↑\\uparrow0\.0490\.0680\.115↑\\uparrowAV\-0\.0020\.030\-0\.0070\.0900\.0540\.0330\.0850\.1010\.053OL0\.2770\.2860\.2640\.0010\.0010\.270↑\\uparrow0\.2560\.2860\.260SP0\.6120\.6420\.683↑\\uparrow\-0\.0170\.0000\.277↑\\uparrow0\.3110\.4360\.405MU0\.2070\.1660\.215↑\\uparrow0\.5990\.1720\.5760\.6020\.1460\.596AD0\.1910\.0330\.170\-0\.005\-0\.0060\.158↑\\uparrow0\.159\-0\.0060\.140A​R¯\\overline\{AR\}2\.302\.001\.702\.182\.411\.412\.412\.001\.59

### IV\-CAblation Study

To isolate each component,[TableVI](https://arxiv.org/html/2608.07881#S4.T6)compares the statistical view alone \(I\), that view plus semantic representation \(SR, giving II\), and the full model that adds dual\-view neighborhood consistency \(DVNC, giving Full\)\. SR is the primary driver, as theA​R¯\\overline\{AR\}falls from roughly2\.62\.6to1\.91\.9on all three metrics\. DVNC then contributes a further drop to near1\.51\.5by cross\-checking the semantic affinities against statistical evidence\. The benefit is clearest on SH and ZO\. On SH, SR alone adds nothing, yet the full model lifts ARI from0\.0680\.068to0\.2940\.294once cross\-view confirmation removes unreliable semantic neighbors\. On ZO, SR even lowers the score, whereas the complete model climbs back above the baseline because the statistical view down\-weights the misleading semantic affinities\. DVNC therefore serves a dual purpose, reinforcing the correct semantic signals and repairing the harmful ones\.

A further ablation in[TableVII](https://arxiv.org/html/2608.07881#S4.T7)turns to mixed data, where the baseline \(I\) reduces to k\-prototype with Hamming for categorical and Euclidean for numerical attributes\. Grounding the numerical distance alone \(II\) or the categorical distance alone \(III\) already improves on I, which shows the semantic metric suits either attribute type\. Applying it to both \(Full\) wins by the widest margin, and the average rank drops steadily from3\.603\.60at I to1\.201\.20at Full across the four settings\. The monotonic trend traces the gain to a single cause, i\.e\., the two attribute types come to share one semantic metric instead of the separate distances that k\-prototype fuses by hand, which unifies them in a common metric space\. Finally, three additional ablations on the four prompt perspectives, the dual\-channel design, and the graph\-construction and spectral\-clustering steps appear in the[supplementary material](https://github.com/develop-yang/GRACE-GRACE-A)\.

### IV\-DGeneral\-Purpose Representation

To test whether the GRACE representation generalizes beyond the spectral setting,[TableVIII](https://arxiv.org/html/2608.07881#S4.T8)pairs three encodings, namely one\-hot \(OH\), HARR, and GRACE, withkk\-means, hierarchical clustering under average linkage \(HC\-avg\), and spectral clustering\. The three representations differ in the information they carry, since OH treats every value as equidistant, HARR computes statistical distances from conditional distributions, and GRACE grounds the value relations in external semantic knowledge\. GRACE holds the bestA​R¯\\overline\{AR\}under all three algorithms, whereas OH and HARR trade the runner\-up position from one to the next, which shows that only the added world knowledge delivers a reliable gain\. The margin widens most under HC\-avg, because greedy agglomeration runs no iterative refinement and therefore exposes the quality of the input distances most directly\. GRACE thus serves as a general\-purpose representation that transfers across clustering paradigms, not a component tuned to the spectral case alone\.

### IV\-ERobustness to LLM Selection

Robustness to the LLM backend is examined by replacing the default GPT with Claude, DeepSeek, and Gemini, recording the ARI deviation on every benchmark \([TableIX](https://arxiv.org/html/2608.07881#S4.T9)\)\. For GRACE,\|Δ\|\|\\Delta\|stays within0\.020\.02on the vast majority of datasets, and the Friedman test returnsp=0\.41p\{=\}0\.41, i\.e\., switching the model leaves performance statistically unchanged\. The consistency follows from the design of the4​P4Pprompt, because it queries established factual knowledge that LLMs trained on overlapping corpora encode in much the same way, and the three alternatives therefore produce nearly identical value descriptions\. GRACE\-A shows larger swings on a few datasets, e\.g\., a drop on CA under Gemini, because the anchor approximation magnifies small representation differences\. Those swings favor no particular backend, however, and the test for the approximate variant returns an even higherp=0\.96p\{=\}0\.96\. Therefore, the effectiveness of GRACE rests on the description design and the cross\-view verification, not on any single LLM\.

TABLE IX:LLM robustness evaluation \(ARI\) on 22 benchmarks\. The GPT column reports absolute performance, and eachΔ\\Deltacolumn gives the deviation of an alternative backend from the GPT baseline, where Cla\., DS, and Gem\. denote Claude, DeepSeek, and Gemini\. The Friedmanpprow reports a Friedman test assessing whether the choice of backend significantly affects performance, with n\.s\. denoting a non\-significant result \(p≥0\.05p\\geq 0\.05\)\.DatasetGRACEGRACE\-AGPTΔ\\DeltaCla\.Δ\\DeltaDSΔ\\DeltaGem\.GPTΔ\\DeltaCla\.Δ\\DeltaDSΔ\\DeltaGem\.LE0\.1640\.000\.000\.000\.2010\.000\.000\.00CS0\.094\+0\.02\+0\.05\+0\.050\.0940\.000\.000\.00ZO0\.8270\.000\.000\.000\.617\-0\.010\.00\-0\.01AA0\.5290\.00\+0\.010\.000\.608\+0\.01\+0\.01\+0\.01LY0\.194\+0\.02\-0\.01\+0\.010\.148\+0\.020\.00\+0\.01TA0\.0660\.000\.00\-0\.010\.060\-0\.010\.00\-0\.01AM0\.093\+0\.02\-0\.01\-0\.010\.086\-0\.010\.00\-0\.01SO0\.4560\.000\.000\.000\.444\+0\.01\-0\.01\+0\.02SH0\.2940\.000\.000\.000\.1340\.000\.000\.00BC0\.145\+0\.01\+0\.010\.000\.155\+0\.01\+0\.010\.00HD0\.409\-0\.01\-0\.020\.000\.417\-0\.02\-0\.040\.00PT0\.1140\.000\.000\.000\.116\-0\.010\.000\.00DE0\.895\+0\.030\.00\+0\.010\.887\+0\.020\.00\+0\.01CK0\.8960\.00\+0\.01\-0\.020\.8140\.000\.000\.00CV0\.6280\.000\.000\.000\.5800\.000\.000\.00MA0\.4020\.00\-0\.02\-0\.010\.403\-0\.01\-0\.030\.00CA0\.1310\.00\+0\.04\+0\.010\.092\+0\.01\+0\.04\-0\.08AV0\.0530\.000\.000\.00\-0\.0020\.000\.000\.00OL0\.2550\.00\+0\.02\-0\.010\.250\+0\.03\+0\.03\+0\.02SP0\.3960\.00\-0\.03\-0\.010\.3660\.00\-0\.02\-0\.01Friedmanpp0\.410\.41\(n\.s\.\)0\.960\.96\(n\.s\.\)

### IV\-FScalability Evaluation

Scalability is evaluated on synthetic data generated from the Dermatology dataset by sampling from its per\-class conditional distributions, with the sample\-size experiment varyingnnfrom 10,000 to 100,000 in steps of 10,000 ats=20s\{=\}20attributes andk=5k\{=\}5clusters \([Figure3](https://arxiv.org/html/2608.07881#S4.F3)\)\. As an exact spectral method, GRACE inherits anO​\(n2\)O\(n^\{2\}\)memory cost and gives out beyond 20,000, later than dense baselines such as AMPHM that fail earlier\. GRACE\-A removes the ceiling and reaches 100,000 with near\-linear scaling, a scale no competing method attains without sacrificing clustering quality\. In fact, the approximate variant attains the highest ARI throughout and edges slightly above GRACE at the two overlapping scales, because the anchor sampling acts as an implicit regularizer on the well\-separated synthetic clusters\. The supporting analysis is deferred to the[supplementary material](https://github.com/develop-yang/GRACE-GRACE-A)\. The cardinality and cluster\-count experiments fixn=2,000n\{=\}2\{,\}000ands=20s\{=\}20, sweepingVVfrom 100 to 1,000 \(atk=5k\{=\}5\) andkkfrom 20 to 200, respectively\. Several counterparts show steep time growth or memory failures as either quantity rises, and the full model itself climbs steadily with the value count, whereas GRACE\-A stays nearly flat across both sweeps\. Taken together, the approximate variant offers a practical drop\-in replacement once the quadratic cost of GRACE turns prohibitive\.

![Refer to caption](https://arxiv.org/html/2608.07881v1/x3.png)Figure 3:Scalability evaluation on synthetic data\. The sample\-size experiment variesnnfrom 10,000 to 100,000, reporting execution time and ARI\. The cardinality experiment variesVVfrom 100 to 1,000\. The cluster\-count experiment varieskkfrom 20 to 200\. Missing points indicate out\-of\-memory failures\.
### IV\-GCase Study on CK Dataset

![Refer to caption](https://arxiv.org/html/2608.07881v1/x4.png)Figure 4:Case study on the serum creatinine attribute of the CK dataset, where GRACE is our method and HARR is the strongest statistics\-only baseline\. The top row reports the value\-level affinity among the five severity levels, \(a\) under HARR and \(b\) under GRACE\. The bottom row shows the UMAP of the sample representations, \(c\) under HARR and \(d\) under GRACE, colored by cluster assignment\.Among the categorical attributes considered so far, ordinal ones additionally carry an inherent order over their values, which makes them a clean test of whether a method’s affinity reflects genuine structure \([Figure4](https://arxiv.org/html/2608.07881#S4.F4)\)\. The serum creatinine attribute of the CK \(Chronic Kidney Disease\) dataset offers one such case, with five levels running from normal to critical\. Because GRACE grounds every value in external world knowledge before consulting dataset statistics, it encodes each level into a description representation, and the cosine similarity between two such representations defines the affinity reported in Panel \(b\)\. The grounded affinity therefore tracks clinical meaning, falling off as severity distance grows, e\.g\., severe remains near critical at 0\.76 yet drops to 0\.61 against normal, and the five levels settle into a smooth gradient along the scale\. Reading the recovered structure then uncovers an ordinal arrangement that GRACE was never instructed to produce, i\.e\., the gradient emerges as a by\-product of reasoning over meaning instead of counting co\-occurrences\. HARR, by comparison, derives its affinity from dataset statistics alone, and Panel \(a\) reveals the resulting distortion\. It assigns mild and moderate an identical score of 1\.00 and pushes normal away from every abnormal level, thereby flattening the fine\-grained progression that separates one clinical stage from the next\. The contrast at the value level in turn carries over to the sample level without further intervention\. The faithful gradient of GRACE divides the patients into two compact clusters in Panel \(d\), whereas the collapsed affinity of HARR leaves the same patients interleaved in Panel \(c\)\. Overall, semantic grounding enables GRACE to recover the genuine structure of an ordinal attribute, and the correct linear order follows as a natural product of that grounding and not as a target built into the model\.

## VConcluding Remarks

In modern data engineering, clustering mixed tabular data serves as a fundamental technique for the label\-free organization, cohort discovery, and exploratory analysis of heterogeneous records\. To advance this critical capability, this paper presents GRACE, a scalable framework that practically injects LLM\-derived world knowledge into the clustering pipeline\. To circumvent the prohibitive token and temporal overheads inherent to iterative LLM queries, GRACE shifts semantic acquisition to the attribute\-value level\. This one\-shot grounding extracts reusable semantic anchors, breaking the scalability bottleneck of LLM\-assisted metric learning\. Furthermore, to guarantee the faithfulness of the external knowledge and prevent semantic hallucinations from distorting actual data groupings, GRACE strictly cross\-validates these conceptual affinities against dataset\-internal statistical evidence\. This dual\-view calibration yields a robust metric space, ensuring the resulting clusters achieve superior accuracy alongside conceptual interpretability\. Building upon this architecture, we further introduce GRACE\-A, an approximation\-based variant that reduces the computational complexity to linear time\. GRACE\-A delivers an ultra\-efficient yet precise solution, ensuring that LLM\-enhanced metric learning remains computationally feasible even for massive\-scale datasets\.

Despite the above\-mentioned merits, the current one\-shot LLM grounding mechanism inherently assumes an offline batch setting, requiring the complete attribute value space to be fully observable before metric construction\. Given the increasingly frequent updates in modern databases, extending our framework to efficiently accommodate emerging distributional shifts of mixed data on the fly presents an important avenue for future research\.

## References

- \[1\]\(2024\)Generalized category discovery with large language models in the loop\.InProceedings of the 62nd Annual Meeting of the Association for Computational Linguistics: Findings,pp\. 8653–8665\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p1.1)\.
- \[2\]A\. Blum and T\. Mitchell\(1998\)Combining labeled and unlabeled data with co\-training\.InProceedings of the 7th annual conference on Computational learning theory,pp\. 92–100\.Cited by:[§III\-C](https://arxiv.org/html/2608.07881#S3.SS3.p4.4)\.
- \[3\]V\. Borisov, K\. Seßler, T\. Leemann, M\. Pawelczyk, and G\. Kasneci\(2023\)Language models are realistic tabular data generators\.InProceedings of the 11th International Conference on Learning Representations,Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p2.1)\.
- \[4\]S\. Cai, Y\. Zhang, X\. Luo,et al\.\(2024\)Robust categorical data clustering guided by multi\-granular competitive learning\.InProceedings of the 44th IEEE International Conference on Distributed Computing Systems,pp\. 288–299\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[5\]J\. Chen, Y\. Ji, R\. Zou, Y\. Zhang,et al\.\(2024\)QGRL: quaternion graph representation learning for heterogeneous feature data clustering\.InProceedings of the 30th ACM Special Interest Group on Knowledge Discovery and Data Mining Conference on Knowledge Discovery and Data Mining,pp\. 297–306\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1),[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p2.1)\.
- \[6\]X\. Du and K\. Tanaka\-Ishii\(2025\)Information\-theoretic generative clustering of documents\.InProceedings of the 39th AAAI Conference on Artificial Intelligence,pp\. 16408–16417\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p3.1)\.
- \[7\]D\. Dua and C\. Graff\(2017\)UCI machine learning repository\.University of California, Irvine, School of Information and Computer Sciences\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p3.1)\.
- \[8\]J\. Fan, Y\. Tu, Z\. Zhang, M\. Zhao, and H\. Zhang\(2022\)A simple approach to automated spectral clustering\.InProceedings of the 35th International Conference on Neural Information Processing Systems,pp\. 9907–9921\.Cited by:[§III\-D](https://arxiv.org/html/2608.07881#S3.SS4.p3.12)\.
- \[9\]T\. Gao, X\. Yao, and D\. Chen\(2021\)SimCSE: simple contrastive learning of sentence embeddings\.InProceedings of the 25th Conference on Empirical Methods in Natural Language Processing,pp\. 6894–6910\.Cited by:[§III\-D](https://arxiv.org/html/2608.07881#S3.SS4.p2.4)\.
- \[10\]Y\. Gorishniy, I\. Rubachev, V\. Khrulkov, and A\. Babenko\(2021\)Revisiting deep learning models for tabular data\.Proceedings of the 35th International Conference on Neural Information Processing Systems34,pp\. 18932–18943\.Cited by:[§III\-C](https://arxiv.org/html/2608.07881#S3.SS3.p2.10)\.
- \[11\]J\. C\. Gower\(1971\)A general coefficient of similarity and some of its properties\.Biometrics27\(4\),pp\. 857–871\.Cited by:[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p2.1)\.
- \[12\]G\. Guérard and S\. Djebali\(2025\)Mixed data clustering survey and challenges\.SN Computer Science6\(8\),pp\. 939\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1)\.
- \[13\]C\. Guo and F\. Berkhahn\(2016\)Entity embeddings of categorical variables\.arXiv:1604\.06737\.Cited by:[§III\-C](https://arxiv.org/html/2608.07881#S3.SS3.p2.10)\.
- \[14\]S\. Han, J\. Yoon, S\. O\. Arik, and T\. Pfister\(2024\)Large language models can automatically engineer features for few\-shot tabular learning\.InProceedings of the 41st International Conference on Machine Learning,pp\. 17454–17479\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p2.1)\.
- \[15\]S\. Hegselmann, A\. Buendia, H\. Lang,et al\.\(2023\)Tabllm: few\-shot classification of tabular data with large language models\.InProceedings of the 26th International Conference on Artificial Intelligence and Statistics,pp\. 5549–5581\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p2.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[16\]N\. Hollmann, S\. Müller, and F\. Hutter\(2023\)Large language models for automated data science: introducing caafe for context\-aware automated feature engineering\.InProceedings of the 37th International Conference on Neural Information Processing Systems,pp\. 44753–44775\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p2.1)\.
- \[17\]L\. Hu, M\. Jiang, X\. Liu, and Z\. He\(2025\)Significance\-based decision tree for interpretable categorical data clustering\.Information Sciences690,pp\. 121588\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[18\]Z\. Huanget al\.\(1997\)Clustering large data sets with mixed numeric and categorical values\.InProceedings of the 1st pacific\-asia conference on knowledge discovery and data mining,pp\. 21–34\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[19\]Z\. Huang\(1998\)Extensions to the k\-means algorithm for clustering large data sets with categorical values\.Data Mining and Knowledge Discovery2\(3\),pp\. 283–304\.Cited by:[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p2.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[20\]L\. Hubert and P\. Arabie\(1985\)Comparing partitions\.Journal of classification2\(1\),pp\. 193–218\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p4.1)\.
- \[21\]D\. Ienco, R\. G\. Pensa, and R\. Meo\(2012\)From context to distance: learning dissimilarity for categorical data clustering\.ACM Transactions on Knowledge Discovery from Data6\(1\),pp\. 1–25\.Cited by:[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1)\.
- \[22\]S\. Jian, L\. Cao, G\. Pang, K\. Lu,et al\.\(2017\)Embedding\-based representation of categorical data by hierarchical value coupling learning\.InProceedings of the 26th International Joint Conference on Artificial Intelligence,pp\. 1937–1943\.Cited by:[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1)\.
- \[23\]S\. Jian, G\. Pang, L\. Cao, K\. Lu,et al\.\(2019\)CURE: flexible categorical data representation by hierarchical coupling learning\.IEEE Transactions on Knowledge and Data Engineering31\(5\),pp\. 853–866\.Cited by:[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1)\.
- \[24\]J\. Kim, K\. Lee, and T\. Park\(2025\)To predict or not to predict? proportionally masked autoencoders for tabular data imputation\.InProceedings of the 39th AAAI Conference on Artificial Intelligence,pp\. 17886–17894\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[25\]Y\. Li, P\. Hu, D\. Peng, J\. Lv,et al\.\(2024\)Image clustering with external guidance\.InProceedings of the 41st International Conference on Machine Learning,pp\. 27890–27902\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p1.1)\.
- \[26\]I\. Loshchilov and F\. Hutter\(2019\)Decoupled weight decay regularization\.InProceedings of the 7th International Conference on Learning Representations,Cited by:[§III\-D](https://arxiv.org/html/2608.07881#S3.SS4.p2.9)\.
- \[27\]E\. Mousavi and M\. Sehhati\(2023\)A generalized multi\-aspect distance metric for mixed\-type data clustering\.Pattern Recognition138,pp\. 109353\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[28\]A\. Ng, M\. Jordan, and Y\. Weiss\(2001\)On spectral clustering: analysis and an algorithm\.Proceedings of the 15th International Conference on Neural Information Processing Systems14\.Cited by:[§III\-D](https://arxiv.org/html/2608.07881#S3.SS4.p3.22)\.
- \[29\]A\. v\. d\. Oord, Y\. Li, and O\. Vinyals\(2018\)Representation learning with contrastive predictive coding\.arXiv:1807\.03748\.Cited by:[§III\-D](https://arxiv.org/html/2608.07881#S3.SS4.p2.4)\.
- \[30\]B\. Pu, X\. Lv, J\. Yang, X\. Dong, Y\. Lin,et al\.\(2025\)Leveraging anatomical consistency for multi\-object detection in ultrasound images via source\-free unsupervised domain adaptation\.InProceedings of the 39th AAAI Conference on Artificial Intelligence,pp\. 6532–6540\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[31\]J\. Shi, M\. Xu, H\. Hua, H\. Zhang,et al\.\(2025\)TabDiff: a mixed\-type diffusion model for tabular data generation\.InProceedings of the 13th International Conference on Learning Representations,Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[32\]Y\. Shi, Y\. Wang, X\. Tian, T\. Zhang, B\. Yao,et al\.\(2025\)SPEHEATAL: a cluster\-enhanced segmentation method for sperm morphology analysis\.InProceedings of the 39th AAAI Conference on Artificial Intelligence,pp\. 729–737\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[33\]S\. Shields, O\. Withington, and E\. Melcer\(2025\)Designer difficulties: visualizing the possibility spaces of dynamic difficulty adjustment systems\.InProceedings of the 21st AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment,pp\. 315–325\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[34\]F\. Spurio, E\. Bahrami, G\. Francesca, and J\. Gall\(2025\)Hierarchical vector quantization for unsupervised action segmentation\.InProceedings of the 39th AAAI Conference on Artificial Intelligence,pp\. 6996–7005\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[35\]A\. Stephan, L\. Miklautz, K\. Sidak, J\. P\. Wahle,et al\.\(2024\)Text\-guided image clustering\.InProceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics,pp\. 2960–2976\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p1.1)\.
- \[36\]A\. Strehl and J\. Ghosh\(2002\)Cluster ensembles—a knowledge reuse framework for combining multiple partitions\.Journal of machine learning research3\(Dec\),pp\. 583–617\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p4.1)\.
- \[37\]J\. Svirsky and O\. Lindenbaum\(2024\)Interpretable deep clustering for tabular data\.InProceedings of the 41st International Conference on Machine Learning,pp\. 47314–47330\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1)\.
- \[38\]V\. Viswanathan, K\. Gashteovski, C\. Lawrence, T\. Wu,et al\.\(2024\)Large language models enable few\-shot clustering\.Transactions of the Association for Computational Linguistics12,pp\. 321–333\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p4.1),[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p1.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[39\]W\. Xu, X\. Liu, and Y\. Gong\(2003\)Document clustering based on non\-negative matrix factorization\.InProceedings of the 26th annual international ACM Special Interest Group on Information Retrieval conference on Research and development in informaion retrieval,pp\. 267–273\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p4.1)\.
- \[40\]Z\. Yang, X\. Liao, Y\. Zhang, and Y\. Cheung\(2026\)Bridging the semantic gap for categorical data clustering via large language models\.arXiv preprint arXiv:2601\.01162\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p2.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[41\]D\. Zhang, F\. Nan, X\. Wei, S\. Li,et al\.\(2021\)Supporting clustering with contrastive learning\.InProceedings of the 10th Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies,pp\. 5419–5430\.Cited by:[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p1.1)\.
- \[42\]Y\. Zhang and Y\. Cheung\(2023\)Graph\-based dissimilarity measurement for cluster analysis of any\-type\-attributed data\.IEEE Transactions on Neural Networks and Learning Systems34\(9\),pp\. 6530–6544\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1),[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[43\]Y\. Zhang, M\. Zhao, Y\. Chen, Y\. Lu,et al\.\(2025\)Learning unified distance metric for heterogeneous attribute data clustering\.Expert Systems with Applications273,pp\. 126738\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1),[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p2.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[44\]Y\. Zhang, M\. Zhao, H\. Jia, M\. Li, Y\. Lu,et al\.\(2025\)Categorical data clustering via value order estimated distance metric learning\.InProceedings of the 52nd ACM Special Interest Group on Management of Data Conference,pp\. 1–24\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[45\]Y\. Zhang, M\. Zhao, Y\. Zhang, and Y\. Cheung\(2025\)Trending applications of large language models: a user perspective survey\.IEEE Transactions on Artificial Intelligence6\(1\),pp\. 1–20\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p3.1)\.
- \[46\]Y\. Zhang, R\. Zou, Y\. Zhang, Y\. Zhang,et al\.\(2025\)Adaptive micro partition and hierarchical merging for accurate mixed data clustering\.Complex & Intelligent Systems11\(1\),pp\. 84\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[47\]Y\. Zhang, Z\. Wang, and J\. Shang\(2023\)ClusterLLM: large language models as a guide for text clustering\.InProceedings of the 27th Conference on Empirical Methods in Natural Language Processing,pp\. 13903–13920\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p4.1),[§II\-B](https://arxiv.org/html/2608.07881#S2.SS2.p1.1),[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[48\]M\. Zhao, S\. Feng, Y\. Zhang, M\. Li,et al\.\(2024\)Learning order forest for qualitative\-attribute data clustering\.InProceedings of the 27th European Conference on Artificial Intelligence,pp\. 1943–1950\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[49\]M\. Zhao, Z\. Huang, Y\. Lu, M\. Li, Y\. Zhang,et al\.\(2025\)Break the tie: learning cluster\-customized category relationships for categorical data clustering\.arXiv:2511\.09049\.Cited by:[§IV\-A](https://arxiv.org/html/2608.07881#S4.SS1.p2.4)\.
- \[50\]C\. Zhu, L\. Cao, Q\. Liu, J\. Yin,et al\.\(2018\)Heterogeneous metric learning of categorical data with hierarchical couplings\.IEEE Transactions on Knowledge and Data Engineering30\(7\),pp\. 1254–1267\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1),[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1)\.
- \[51\]C\. Zhu, L\. Cao, and J\. Yin\(2022\)Unsupervised heterogeneous coupling learning for categorical representation\.IEEE Transactions on Pattern Analysis and Machine Intelligence44\(1\),pp\. 533–549\.Cited by:[§I](https://arxiv.org/html/2608.07881#S1.p2.1),[§II\-A](https://arxiv.org/html/2608.07881#S2.SS1.p1.1)\.
- \[52\]Q\. Zhu, J\. Feng, and J\. Huang\(2016\)Natural neighbor: a self\-adaptive neighborhood method without parameter k\.Pattern recognition letters80,pp\. 30–36\.Cited by:[§III\-C](https://arxiv.org/html/2608.07881#S3.SS3.p3.6)\.