Towards Platonic Representation for Table Reasoning: A Foundation for Permutation-Invariant Retrieval

Willy Carlos Tchuitcheu; Tan Lu; Ann Dooms

cs.AIApr 13, 2026

Towards Platonic Representation for Table Reasoning: A Foundation for Permutation-Invariant Retrieval

Authors: Willy Carlos Tchuitcheu, Tan Lu, Ann Dooms

Organizations: Department of Mathematics and Data Science, Vrije Universiteit Brussel (VUB), Brussels, Belgium · Data Science Lab, Royal Library of Belgium (KBR), Brussels, Belgium

Abstract

Historical approaches to Table Representation Learning (TRL) have largely adopted the sequential paradigms of Natural Language Processing (NLP). We argue that this linearization of tables discards their essential geometric and relational structure, creating representations that are brittle to layout permutations. This paper introduces the Platonic Representation Hypothesis (PRH) for tables, positing that a semantically robust latent space for table reasoning must be intrinsically Permutation Invariant (PI). To ground this hypothesis, we first conduct a retrospective analysis of table-reasoning tasks, highlighting the pervasive serialization bias that compromises structural integrity. We then propose a formal framework to diagnose this bias, introducing two principled metrics based on Centered Kernel Alignment (CKA): (i) PI, which measures embedding drift under complete structural derangement, and (ii) rho, a Spearman-based metric that tracks the convergence of latent structures toward a canonical form as structural information is incrementally restored. Our empirical analysis quantifies an expected flaw in modern Large Language Models (LLMs): even minor layout permutations induce significant, disproportionate semantic shifts in their table embeddings. This exposes a fundamental vulnerability in RAG systems, in which table retrieval becomes fragile to layout-dependent noise rather than to semantic content. In response, we present a novel, structure-aware TRL encoder architecture that explicitly enforces the cognitive principle of cell header alignment. This model demonstrates superior geometric stability and moves towards the PI ideal. Our work provides both a foundational critique of linearized table encoders and the theoretical scaffolding for semantically stable, permutation invariant retrieval, charting a new direction for table reasoning in information systems.

Figures & tables

Figure 1. Platonic view of permutation invariance in table embeddings and prospective long-term impact paradigm shift in IR. The left part illustrates the theoretical concept of "Platonic table embedding," investigating whether an original input table and its structurally transposed counterpart (with rows and columns swapped) can be projected by embedding models into a consistent, semantically invariant latent space. The t-SNE visualizations and the question mark between them highlight the core research question regarding the alignment of these embedding spaces across different table representations. The right panel depicts prospective downstream applications that could be built upon such robust embeddings. The top section shows an "Online Table Question Answer" system that uses cell embeddings and nearest-neighbor search to directly answer user queries, while the bottom section outlines a "table-centric RAG application" that leverages table vectors for RAG tasks.

Club	Played	Won	Drawn	Lost	Points for	Points against	Tries for	Tries against	Try bonus	Losing bonus	Points
Bridgend Athletic RFC	22	16	0	6	523	303	68	31	10	4	78
Builth Wells RFC	22	17	0	5	473	305	57	29	7	2	77
Kidwelly RFC	22	14	1	7	532	386	63	45	5	3	66
Loughor RFC	22	13	1	8	532	388	69	43	9	1	64
Ammanford RFC	22	13	0	9	447	394	58	51	6	4	62
Waunarlwydd RFC	22	12	2	8	504	439	57	55	6	3	61

Figure 2. Single-table test example of Rugby Club Performance Statistics ( from WikiSQL dataset, Table ID 560). Identical content values (e.g., “22”) can denote different semantics depending on headers.

Panel of Figure 3 — Figure 3. LLM-only table embeddings across LLMs ( OpenAI , nomic-embed-text , Gemma2 , DeepSeek-R1 , and llama3 ). Rows correspond to models. Columns show: Left t - SNE projections of LLM-based cell embeddings for the baseline table (Figure 2 ); a point indexed by $(\mathrm{row\_id},\mathrm{col\_id})$ denotes the projected embedding of a cell. Center pairwise cosine similarity within the resulting cell-embedding set. Right heatmaps $H[a,b]$ of similarity between the original table and its $169$ permutations $T_{a}^{b}$ .

Figure 4. Distribution of heatmap entries $H[a,b]$ and the mean $\bar{x}$ comparing LLM and TRL-derived cell embeddings under two settings: (i) intra-model , which examines embeddings of permuted tables generated by the same model that was trained on the original table. (ii) Cross-model , where embeddings of permuted tables are extracted from different models trained on each other’s samples.

Panel of Figure 6 — Figure 6. Analysis of the Platonic Representation Hypothesis (PRH) under progressive row and column shuffling. Each curve illustrates the similarity score $H[a,b]$ as the number of fixed rows ( $a$ ) or columns ( $b$ ) increases, ranging from full derangement ( $T_{0}^{0}$ ) to complete identity ( $T_{n}^{m}$ ). Solid lines represent column permutations ( $H[12,b]$ ), while dashed lines represent row permutations ( $H[a,12]$ ). The three panels compare: (left) baseline LLM embeddings, (center) TRL-refined embeddings (intra-model), and (right) TRL-refined embeddings (cross-model). The legend indicates the model size and embedding dimension. The TRL curves are flatter and more stable, demonstrating stronger PI, whereas the LLM-only curves exhibit a steeper decline, particularly under column permutations.

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Department of Computer Science Arizona State University Tempe, Arizona, USA · ScaleAI San Francisco, CA, USA

Retrieval Robustness Information Retrieval Table Retrieval Invariant Representation Learning Dense Retrieval

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Transformer-based table retrieval systems flatten structured tables into token sequences, making retrieval sensitive to the choice of serialization even when table semantics remain unchanged. We show that semantically equivalent serializations, such as

\texttt{csv}

\texttt{tsv}

\texttt{html}

\texttt{markdown}

, and

\texttt{ddl}

, can produce substantially different embeddings and retrieval results across multiple benchmarks and retriever families. To address this instability, we treat serialization embedding as noisy views of a shared semantic signal and use its centroid as a canonical target representation. We show that centroid averaging suppresses format-specific variation and can recover the semantic content common to different serializations when format-induced shifts differ across tables. Empirically, centroid representations outrank individual formats in aggregate pairwise comparisons across

\texttt{MPNet}

\texttt{BGE-M3}

\texttt{ReasonIR}

, and

\texttt{SPLADE}

. We further introduce a lightweight residual bottleneck adapter on top of a frozen encoder that maps single-serialization embeddings towards centroid targets while preserving variance and enforcing covariance regularization. The adapter improves robustness for several dense retrievers, though gains are model-dependent and weaker for sparse lexical retrieval. These results identify serialization sensitivity as a major source of retrieval variance and show the promise of post hoc geometric correction for serialization-invariant table retrieval.

Abstract

Table Reasoning

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