2D Structural Entropy Minimization

• Two-dimensional structural entropy minimization is a graph-partitioning method that uses a height-2 encoding tree to reduce the coding cost by adaptively grouping nodes into coherent modules.
• The approach integrates graph compression with unsupervised clustering, eliminating the need to pre-specify the number of clusters while ensuring strong internal cohesion and weak external connectivity.
• Optimization strategies such as greedy merging, node-shifting, and hierarchical batching enable scalable and privacy-preserving applications across domains like event detection, image segmentation, and summarization.

Searching arXiv for recent and foundational papers on two-dimensional structural entropy minimization and related usage. arXiv search query: "structural entropy minimization encoding tree two-dimensional graph partitioning" Two-dimensional structural entropy minimization is, in its most common contemporary usage, a graph-partitioning principle in which one seeks a height-2 encoding tree that minimizes the information needed to describe graph structure under a two-level abstraction: root, modules, and individual vertices. In this formulation, the intermediate nodes of the encoding tree are the sought communities, regions, events, sentence groups, or sub-dialogues, and minimization acts as an unsupervised criterion for discovering them without pre-specifying their number (Cao et al., 2023, Zeng et al., 2023, Zhang et al., 31 Jul 2025, Peng et al., 7 Aug 2025). The same phrase also appears in mathematically distinct literatures on entropy for embedded surfaces and on entropy from recurring structural patterns in two-dimensional liquids; those usages concern different entropy functionals and should not be conflated with the encoding-tree formalism (Ketover et al., 2015, Javerzat et al., 28 May 2026).

1. Formal framework

In the encoding-tree formalism, an encoding tree $\mathcal{T}$ of a graph $G=(\mathcal{V},\mathcal{E})$ is a hierarchical partition: the root $\lambda$ corresponds to all vertices, each internal node $\alpha$ corresponds to a subset $T_\alpha\subseteq \mathcal{V}$, and leaves correspond to singleton graph nodes. A standard definition used across papers is

$$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$

where $g_\alpha$ is the cut weight of $T_\alpha$, $vol(\alpha)$ is the weighted volume of $T_\alpha$, and $G=(\mathcal{V},\mathcal{E})$0 is the parent of $G=(\mathcal{V},\mathcal{E})$1. The $G=(\mathcal{V},\mathcal{E})$2-dimensional structural entropy is then

$G=(\mathcal{V},\mathcal{E})$3

For two-dimensional structural entropy, $G=(\mathcal{V},\mathcal{E})$4: the root is the full graph, the height-1 nodes are modules, and the leaves are individual vertices (Cao et al., 2023, Peng et al., 7 Aug 2025).

A closely related partition-level presentation is used in image segmentation. For a partition $G=(\mathcal{V},\mathcal{E})$5 of an undirected weighted graph $G=(\mathcal{V},\mathcal{E})$6, the structural entropy is written as

$G=(\mathcal{V},\mathcal{E})$7

This is explicitly identified as “the two-dimensional SE, i.e., SE of a graph by a partition,” so the “two-dimensional” qualifier there refers to the two-level coding structure rather than to Euclidean image coordinates (Zeng et al., 2023).

Across these formulations, the operative meaning is stable. Two-dimensional structural entropy measures the coding cost of first locating a module and then a node within that module. Minimization therefore favors partitions with strong internal coherence and weak external connectivity. This suggests why the method repeatedly appears in unsupervised clustering settings: it aligns graph compression, community structure, and adaptive partition discovery (Cao et al., 2023, Yang et al., 2022).

2. Optimization principles and algorithmic forms

The basic optimization pattern is greedy agglomeration on the height-1 nodes of a 2D encoding tree. In the event-detection formulation, one initializes each message as a singleton cluster, computes $G=(\mathcal{V},\mathcal{E})$8 for every pair of current height-1 nodes, chooses the pair with the most negative $G=(\mathcal{V},\mathcal{E})$9, merges it, and stops when no pair gives $\lambda$0. The resulting partition is the final event decomposition, and the number of clusters is determined implicitly by the stopping condition rather than supplied externally (Cao et al., 2023).

Several papers then customize this baseline. HISEvent replaces global cubic-time greedy merging with a hierarchical procedure: it chunks the current cluster list into subsets of size $\lambda$1, induces subgraphs, initializes a local height-2 tree using the already detected clusters as height-1 nodes, runs vanilla 2D SE minimization locally, and doubles $\lambda$2 if a round stagnates. The stated purpose is to retain the 2D SE objective while making it practical on large social message graphs; the paper reports $\lambda$3 for vanilla 2D SE minimization and $\lambda$4 for the hierarchical version (Cao et al., 2023).

SLED uses a different refinement strategy. It first performs a merge stage, greedily selecting the pair $\lambda$5 with the largest positive entropy decrease $\lambda$6, and then performs a refinement stage in which individual nodes are removed from their current module and inserted into another module according to the objective

$\lambda$7

The paper explicitly argues monotonicity: every accepted merge decreases $\lambda$8, and each node reassignment decreases or preserves it, so the procedure converges when no further improvement exists (Zeng et al., 2023).

Dynamic-graph work turns the static optimizer into an incremental one. Incre-2dSE maintains Structural Data, Structural Expressions, and Adjustment objects, then updates the entropy through an incremental formula rather than recomputing from scratch: $\lambda$9 Its stronger node-shifting strategy defines an optimal preference community (OPC) for a node and greedily moves involved nodes toward entropy-reducing communities, thereby incrementally optimizing the partition toward lower 2D structural entropy over time (Yang et al., 2022).

A further variant appears in privacy-preserving event detection. ADP-SEMEvent constructs “optimal subgraphs” by starting from the largest edge in a coarsened super-graph, expanding the subgraph with the node of highest cut weight until size $\alpha$0, then applying vanilla 2D SE minimization inside each subgraph. The novelty there is not a new entropy formula but a decomposition heuristic meant to preserve strongly connected partitions in noisy privatized graphs (Yang et al., 2024).

3. Canonical graph constructions and application targets

The objects being partitioned vary widely, but the graph-theoretic role of 2D SE is stable: vertices encode atomic units, weighted edges encode task-specific affinity, and the height-1 nodes of the optimal 2D tree are the task outputs.

Domain	Graph nodes and edges	Height-1 clusters represent
Social event detection	Messages; edges from shared social attributes and semantic similarity	Detected events
Skin lesion segmentation	Superpixels; edges from spatial proximity and color similarity	Homogeneous image regions
Multi-document summarization	Sentences; fused semantic and discourse relations	Coherent sentence groups
Dialogue ASQ extraction	Utterances; reply-thread-aware semantic links	Semantically independent sub-dialogues

In HISEvent, the graph is built from messages, common attributes such as sender, mentions, hashtags, and named entities, plus semantic edges chosen through incremental 1D SE minimization; 2D SE then partitions the augmented graph into events. The paper states that this stage “eliminates the noise and reveals the essential 2nd-order (cluster-wise) structure underneath the raw graph with no prior knowledge of the number of event clusters,” and reports closed-set examples of Event2012 ARI $\alpha$1, AMI $\alpha$2, and Event2018 ARI $\alpha$3, AMI $\alpha$4 (Cao et al., 2023).

In SLED, a 2D dermoscopic image is transformed into a superpixel graph. Nodes are SLIC superpixels, edges connect superpixels whose spatial distance is smaller than a threshold $\alpha$5, weights are

$\alpha$6

and the graph is sparsified into a $\alpha$7-nearest-neighbor graph. Minimizing 2D SE yields an adaptive partition into homogeneous regions, after which between-class variance maximization and a multi-scale isolation-forest mechanism separate lesion from healthy skin. The paper reports, for example, on ISIC 2016: SLED$\alpha$8 DI $\alpha$9, JA $T_\alpha\subseteq \mathcal{V}$0, and SLED$T_\alpha\subseteq \mathcal{V}$1 DI $T_\alpha\subseteq \mathcal{V}$2, JA $T_\alpha\subseteq \mathcal{V}$3 (Zeng et al., 2023).

In MRGSEM-Sum, sentence nodes are connected by semantic and discourse relations. Because 2D SE “cannot be directly applied to multi-relational graphs,” the relation layers are fused into an integrated weighted graph by elementwise max, and 2D SE minimization is then used to obtain root-level sentence clusters for downstream compression. The paper’s ablation against spectral clustering reports substantial degradation when 2D SE clustering is removed; on Multi-News, MRGSEM-Sum obtains $T_\alpha\subseteq \mathcal{V}$4 versus $T_\alpha\subseteq \mathcal{V}$5 for the variant without 2D SE clustering (Zhang et al., 31 Jul 2025).

In dialogue aspect-based sentiment quadruple extraction, DSEM builds an undirected weighted utterance graph in which edges follow reply relations and thread-reachable utterance pairs, with weights

$T_\alpha\subseteq \mathcal{V}$6

The optimal 2D encoding tree defines sub-dialogues; quadruple matching is then restricted to entities inside the same sub-dialogue. The ablations report that replacing DSEM with K-means degrades EN Micro-F1 from $T_\alpha\subseteq \mathcal{V}$7 to $T_\alpha\subseteq \mathcal{V}$8 and ZH Micro-F1 from $T_\alpha\subseteq \mathcal{V}$9 to $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$0 (Peng et al., 7 Aug 2025).

4. Scalability, privacy, and dynamic maintenance

A recurring theme in the literature is that the 2D SE objective is attractive precisely because it does not require a predefined cluster count, but naïve optimization is expensive. The main engineering innovations therefore concern search-space restriction, incremental maintenance, and robustness to graph noise.

HISEvent’s hierarchical 2D SE minimization is one such response. The paper emphasizes that vanilla 2D SE minimization is cubic in $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$1 and reports that it can take more than 5 days on a dense Event2012 block and more than 10 days in one closed-set ablation, whereas the hierarchical version reduces runtime by $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$2 on $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$3 while preserving performance. It also notes that the split subgraphs can be processed concurrently, making the method parallelizable (Cao et al., 2023).

ADP-SEMEvent addresses a different failure mode: the message graph is already noisy because it has been privatized. Its first stage uses an adaptive differential privacy mechanism to construct a private message graph, with 1D structural entropy used to choose $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$4-NN similarity edges; its second stage uses “a novel 2-dimensional structural entropy minimization algorithm based on optimal subgraphs” to recover events from the noisy graph. The clustering stage is explicitly framed as post-processing of the released private graph, so it “does not compromise differential privacy” (Yang et al., 2024).

Incre-2dSE addresses evolving graphs rather than privatized ones. It proves that, under the naïve adjustment strategy, the Local Difference between the Global Invariant approximation and the exact updated entropy satisfies

$$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$5

and it gives an $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$6 AIUA update procedure for incremental size $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$7. The node-shifting strategy then augments this measurement framework with entropy-guided partition repair, and the experiments report large speedups over offline recomputation, such as $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$8 on SBM-Hawkes versus Infomap and $$\mathcal{H}^{\mathcal{T}}(G) = -\sum_{\alpha\in\mathcal{T},\,\alpha\neq\lambda} \frac{g_\alpha}{vol(\lambda)} \log \frac{vol(\alpha)}{vol(\alpha^-)},$$9 on DBLP versus Infomap (Yang et al., 2022).

MRGSEM-Sum provides a summarization-specific batching analogue. Its 2D SE clustering algorithm groups current clusters into subsets of size $g_\alpha$0, optimizes locally, and doubles $g_\alpha$1 when no updates occur. The paper does not give a formal asymptotic complexity theorem, but the batching logic is explicitly presented as the mechanism that makes sentence-level SE clustering tractable beyond a monolithic full-graph search (Zhang et al., 31 Jul 2025).

5. Distinct non-graph usages of entropy minimization on two-dimensional structures

The phrase is not uniform across the broader arXiv literature. In geometric analysis, “entropy” refers to the Colding–Minicozzi entropy for hypersurfaces or self-shrinkers, defined from the Gaussian area or $g_\alpha$2-functional rather than from an encoding tree. For a smooth embedded surface $g_\alpha$3,

$g_\alpha$4

and Ketover and Zhou prove the sharp genus-zero statement

$g_\alpha$5

for every smooth closed embedded $g_\alpha$6-sphere in $g_\alpha$7. Their method uses a canonical four-parameter translate-and-dilate family, Gaussian min-max theory, and a degree argument ruling out degeneration to a plane (Ketover et al., 2015).

A related higher-codimension paper studies smooth, complete, embedded, boundaryless self-shrinking surfaces $g_\alpha$8, $g_\alpha$9, and proves compactness for the low-entropy class $T_\alpha$0. It then uses that compactness to establish the existence of entropy minimizers among non-flat self-shrinking surfaces, again in a geometric sense unrelated to 2D encoding trees (Lee, 2021).

A third, conceptually separate usage appears in the study of two-dimensional supercooled liquids. There entropy is estimated from the decay of recurring structural patterns with patch size through a higher-dimensional Grassberger–Procaccia construction. The central scaling law is

$T_\alpha$1

so entropy is read off from the exponential decay rate of patch recurrence. This is a measurement framework for configurational entropy in 2D amorphous systems, not an optimization over graph partitions (Javerzat et al., 28 May 2026).

These literatures share the language of entropy minimization over two-dimensional objects, but they optimize or estimate fundamentally different quantities. A plausible implication is that “two-dimensional structural entropy minimization” is now a cross-disciplinary label rather than a single canonical formalism.

6. Interpretation, misconceptions, and limitations

A persistent misconception is that “two-dimensional” means planar geometry. In the dominant graph-partitioning literature, it usually means a height-2 encoding tree: root, clusters, leaves. This is explicit in social event detection, dialogue partitioning, and dynamic-graph work, and the image-segmentation paper goes out of its way to clarify that 2D SE there is “the structural entropy of a graph under a partition,” not entropy over pixel coordinates directly (Cao et al., 2023, Peng et al., 7 Aug 2025, Yang et al., 2022, Zeng et al., 2023).

A second common misunderstanding is that 2D SE requires the number of clusters in advance. The opposite is central to several papers. HISEvent, SLED, MRGSEM-Sum, and the DiaASQ partitioner all present adaptive cluster-number selection as a primary advantage over $T_\alpha$2-means, NCut, spectral clustering, or graph-clustering pipelines that require a fixed cluster count (Cao et al., 2023, Zeng et al., 2023, Zhang et al., 31 Jul 2025, Peng et al., 7 Aug 2025).

The major technical limitation is that most practical optimizers are heuristic. HISEvent’s local subgraph processing, ADP-SEMEvent’s “optimal subgraphs,” MRGSEM-Sum’s batched merging, and DSEM’s greedy parallel merges are all approximate procedures rather than exact global solvers for the 2D SE objective. Incre-2dSE is especially explicit on this point: node shifting is a greedy local optimizer, may not converge, and requires an iteration cap (Yang et al., 2024, Zhang et al., 31 Jul 2025, Peng et al., 7 Aug 2025, Yang et al., 2022).

Another limitation is dependence on graph construction. SLED reports that performance degrades when the spatial threshold $T_\alpha$3 is too small, because graph construction underuses color-similarity structure; HISEvent’s entire design separates 1D SE for graph augmentation from 2D SE for graph partitioning, implicitly acknowledging that a poor graph can undermine even a strong partition objective (Zeng et al., 2023, Cao et al., 2023).

Finally, some papers present formulas with noticeable typesetting corruption. The privacy-preserving event-detection paper states that its printed 2D structural entropy formula is malformed but recoverable from surrounding notation; the summarization paper’s fusion and merge equations are likewise described as corrupted; and the dialogue paper’s merge-score sign convention appears inconsistent between formula and prose. This suggests that, for implementation, the surrounding algorithmic definitions and the earlier Li–Pan structural-information formalism remain essential interpretive anchors (Yang et al., 2024, Zhang et al., 31 Jul 2025, Peng et al., 7 Aug 2025).

Taken together, the literature presents two-dimensional structural entropy minimization as a flexible unsupervised partition principle whose main strengths are adaptive cluster discovery, compatibility with weighted relational graphs, and extensibility to privacy-preserving, dynamic, and multi-relational settings. Its main weaknesses are optimizer heuristics, graph-construction sensitivity, and occasional formal ambiguity in presentation.