Clustering via Diffusion (CLUDI) Overview

• CLUDI is a framework for unsupervised clustering using diffusion processes that extract and refine intrinsic data structure via similarity graphs and embeddings.
• It employs both classical random-walk diffusion and modern generative diffusion models to capture multiscale and directional data relationships.
• The methodology has been validated across applications like hyperspectral imaging and single-cell analysis, enhancing clustering accuracy and robustness.

Searching arXiv for the most relevant CLUDI papers and closely related diffusion-clustering work. First search: specific CLUDI framework paper. Searching arXiv for "Clustering via Self-Supervised Diffusion". Clustering via Diffusion (CLUDI) denotes a family of unsupervised methods in which cluster structure is extracted from a diffusion process defined on data. In its classical form, CLUDI constructs a similarity graph, forms a Markov operator $P=D^{-1}W$, and uses diffusion distances, diffusion kernels, or diffusion embeddings to recover the intrinsic geometry of the data manifold; in more recent work, the term also covers denoising diffusion models that generate, complete, or regularize cluster-friendly latent representations. The name is additionally used by the 2025 framework "Clustering via Self-Supervised Diffusion" (Uziel et al., 6 Jul 2025), but the broader literature spans diffusion maps, density-mode clustering, subspace affinity diffusion, multiscale diffusion, directed-graph kernels, hyperspectral image clustering, and latent diffusion completion (Damelin et al., 2014).

1. Scope, usage, and historical development

The broad CLUDI paradigm predates the specific 2025 method of the same name. Early representatives include FARDiff, which combined diffusion maps with Fuzzy Adaptive Resonance Theory for high-dimensional clustering (Damelin et al., 2014), and diffusion fingerprints, which embedded subsets of nodes through personalized PageRank-style random walks on directed graphs (Dubuisson et al., 2014). Diffusion then became a recurrent device for repairing weak affinities or revealing connectivity structure: DSSC applied a diffusion process to sparse subspace affinity graphs (Li et al., 2016), DLSS used diffusion distance to estimate modes and propagate labels in hyperspectral images (Murphy et al., 2017), MELD and M-LUND formalized a multiscale family of clusterings parameterized by diffusion time (Murphy et al., 2021), and P-RWDKC generalized diffusion-kernel clustering to directed graphs through a parametrized random walk operator (Sevi et al., 2022).

Later work diversified both the data domains and the meaning of “diffusion.” In hyperspectral imaging, DSIRC and DS$^2$DL used diffusion geometry together with spatial reconstruction, purity estimation, superpixels, and learned latent features (Cui et al., 2022, Buranasiri et al., 14 Apr 2026). In incomplete multi-view clustering, diffusion models were used to complete missing latent views before contrastive clustering (Fang, 2023). Fine-grained image clustering inverted conditional diffusion models to recover textual conditions that were then clustered (Yang et al., 2024). The specific CLUDI framework of 2025 introduced a teacher-student system in which stochastic diffusion-based sampling produces diverse cluster assignments from frozen Vision Transformer features (Uziel et al., 6 Jul 2025). This dual usage creates a persistent terminological ambiguity: in one lineage, diffusion means Markov propagation on a graph; in the other, it means score-based or DDPM-style denoising dynamics.

2. Mathematical foundations of diffusion geometry

In the graph-diffusion formulation, one starts from data points $X=\{x_i\}_{i=1}^N$, constructs a weighted graph with affinity matrix $W$, defines the degree matrix $D$, and row-normalizes to obtain a Markov transition matrix

$P = D^{-1}W.$

FARDiff uses a Gaussian kernel

$$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$

with $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$, and interprets $P^t$ as a $t$-step random walk (Damelin et al., 2014). The associated diffusion distance compares endpoint distributions after $^2$0 steps: $^2$1 Spectrally, if $^2$2 are eigenpairs of $^2$3, diffusion maps use

$^2$4

so Euclidean distance in diffusion coordinates approximates diffusion distance (Damelin et al., 2014).

Several later CLUDI variants use the stationary-distribution-weighted form

$^2$5

which makes explicit that diffusion time $^2$6 acts as a scale parameter: small $^2$7 preserves local geometry, while larger $^2$8 suppresses high-frequency structure and emphasizes coarse connectivity (Murphy et al., 2017, Buranasiri et al., 14 Apr 2026).

For directed graphs, the natural random walk $^2$9 is generally non-reversible and may fail to be irreducible. P-RWDKC addresses this by defining a parametrized random walk

$X=\{x_i\}_{i=1}^N$0

where $X=\{x_i\}_{i=1}^N$1 is a positive vertex measure and $X=\{x_i\}_{i=1}^N$2. The resulting operator is self-adjoint in a weighted space and supports a diffusion kernel

$X=\{x_i\}_{i=1}^N$3

so diffusion distance again becomes a Mahalanobis distance induced by a kernel, but now without discarding edge directionality (Sevi et al., 2022).

3. Core algorithmic families

The CLUDI literature does not implement a single clustering rule. Instead, diffusion supplies a geometry or a propagator, and the downstream clustering mechanism varies substantially.

The earliest and most direct family uses diffusion embeddings followed by a conventional clustering module. FARDiff computes diffusion coordinates and then applies Fuzzy ART, replacing the more common choice of $X=\{x_i\}_{i=1}^N$4-means with an adaptive resonance network governed by choice, vigilance, and learning-rate parameters (Damelin et al., 2014). Diffusion fingerprints take a related but seed-centered view: a personalized PageRank vector becomes the embedding of a subset or object, after which any standard clustering method can operate in fingerprint space (Dubuisson et al., 2014).

A second family uses diffusion distance for mode seeking and density-based propagation rather than explicit embedding-then-$X=\{x_i\}_{i=1}^N$5-means. DLSS estimates a density $X=\{x_i\}_{i=1}^N$6, defines the diffusion-distance-to-higher-density quantity $X=\{x_i\}_{i=1}^N$7, and scores modes by

$X=\{x_i\}_{i=1}^N$8

The top $X=\{x_i\}_{i=1}^N$9 maximizers become cluster modes, and the remaining points inherit the label of their diffusion-distance nearest labeled neighbor of higher density (Murphy et al., 2017). DSIRC retains this structure but augments density with purity, using a harmonic-mean quality score $W$0 and the mode criterion $W$1 on shape-adaptively reconstructed hyperspectral pixels (Cui et al., 2022). DS$W$2DL continues the same pipeline in a learned latent space, with

$W$3

mode selection on superpixel representatives, and diffusion-based label propagation back to all pixels (Buranasiri et al., 14 Apr 2026).

A third family diffuses the affinity itself. DSSC begins with the sparse subspace affinity $W$4 obtained by $W$5-SSC and then iterates

$W$6

This realizes diffusion on a tensor-product graph and converges to a refined affinity that preserves the approximate block structure of SSC while strengthening within-subspace connectivity (Li et al., 2016).

A fourth family is explicitly multiscale. MELD treats clustering as a family $W$7 indexed by diffusion time, and M-LUND runs diffusion-based clustering over a range of $W$8 values before selecting a representative partition by minimizing total variation of information across nontrivial clusterings (Murphy et al., 2021).

Family	Core diffusion object	Representative papers
Embedding then cluster	$W$9, diffusion maps $D$0	FARDiff (Damelin et al., 2014)
Density-mode diffusion	$D$1, $D$2, $D$3	DLSS (Murphy et al., 2017), DSIRC (Cui et al., 2022), DS$D$4DL (Buranasiri et al., 14 Apr 2026)
Affinity diffusion	$D$5	DSSC (Li et al., 2016)
Multiscale diffusion	time-indexed family $D$6	MELD, M-LUND (Murphy et al., 2021)
Directed-graph diffusion kernels	$D$7, $D$8	P-RWDKC (Sevi et al., 2022)
Generative diffusion for clustering	latent DDPMs, conditional denoising	IMVCDC (Fang, 2023), CLUDI (Uziel et al., 6 Jul 2025), DiFiC (Yang et al., 2024)

4. Spatial, hyperspectral, and graph-structured variants

Hyperspectral image analysis has become one of the most technically developed application areas for CLUDI. DLSS builds a $D$9-nearest-neighbor graph in spectral space, computes diffusion distances, estimates density modes, and then applies a two-stage spectral-spatial labeling procedure (Murphy et al., 2017). DSIRC inserts a shape-adaptive reconstruction stage before diffusion and combines density with spectral unmixing purity; on Indian Pines it reported $P = D^{-1}W.$0 and $P = D^{-1}W.$1, compared with DLSS at $P = D^{-1}W.$2 and $P = D^{-1}W.$3 (Cui et al., 2022). DS$P = D^{-1}W.$4DL then replaced raw spectra by a masked-autoencoder latent space, used entropy rate superpixels and representative selection, and kept the diffusion pipeline of S$P = D^{-1}W.$5DL essentially unchanged except for operating in the latent manifold. It reported higher OA, AA, $P = D^{-1}W.$6, purity, and NMI than S$P = D^{-1}W.$7DL on Botswana and KSC, and reduced runtime from $P = D^{-1}W.$8 to $P = D^{-1}W.$9 seconds on KSC and from $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$0 to $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$1 seconds on Botswana (Buranasiri et al., 14 Apr 2026).

Directed and structured graphs required a distinct set of modifications. P-RWDKC addresses the non-reversibility and non-ergodicity of digraph random walks by constructing a parametrized reversible walk, then clustering rows of the diffusion kernel $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$2; it was reported to outperform existing directed-graph baselines on K-NN graphs from real-world datasets and on real-world graphs in most tested cases (Sevi et al., 2022). Outside graph-spectral settings, ClusTEK applies Laplacian-kernel diffusion imputation on a uniform spatial grid and then performs origin-constrained connected-component analysis; its fixed-resolution spatial indexing yields scaling of $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$3, and the reported polymer benchmarks range from $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$4k to $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$5k atoms (Tourani et al., 18 Dec 2025). These variants show that CLUDI is not restricted to one graph construction or one notion of node: points, superpixel representatives, grid cells, and directed-network vertices have all been used.

5. Generative diffusion and the modern meaning of CLUDI

In recent work, “diffusion” increasingly refers to denoising diffusion probabilistic models rather than Markov diffusion geometry. IMVCDC is an early clustering example in this sense: it trains autoencoders for each view, performs conditional latent-space diffusion completion of missing views, and then applies contrastive clustering. On Multi-Coil20 with missing rate $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$6, the ablation $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$7 reached $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$8 ACC, $$w(x_i,x_j)=\exp\!\left(-\frac{\|x_i-x_j\|^2}{\sigma^2}\right),$$9 NMI, and $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$0 ARI, while the full model $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$1 reached $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$2 ACC, $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$3 NMI, and $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$4 ARI (Fang, 2023).

DiFiC uses a pretrained text-to-image diffusion model differently: it infers a proxy word $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$5 that acts as the textual condition explaining an image under the frozen diffusion model, regularizes the diffusion target with an attention-derived object mask, and applies neighborhood-similarity guidance in the proxy-word space (Yang et al., 2024). On Stanford Cars, the full method reported $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$6 ACC and $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$7 NMI, substantially above its raw Stable Diffusion feature baseline (Yang et al., 2024). DiEC pursues a related but distinct route by searching over diffusion U-Net layer and timestep, fixing a clustering-friendly middle layer and then selecting an optimal timestep before DEC-style KL self-training with graph and entropy regularization (Hu, 24 Dec 2025).

The framework specifically named CLUDI introduced in 2025 combines frozen DINO Vision Transformer features with a conditional diffusion model over assignment embeddings $D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$8, trained in a teacher-student configuration (Uziel et al., 6 Jul 2025). The teacher uses stochastic DDIM-style backward sampling to produce diverse soft cluster assignments, and the student learns to denoise noisy assignment embeddings while matching non-collapsing cluster probabilities. At inference, multiple samples are averaged,

$D_{ii}=d(x_i)=\sum_j w(x_i,x_j)$9

which turns diffusion stochasticity into a cluster-space augmentation mechanism. On ImageNet-50 with ViT-B/16, it reported $P^t$0 NMI, $P^t$1 ACC, and $P^t$2 ARI (Uziel et al., 6 Jul 2025). In this lineage, diffusion is no longer merely a geometry on an externally defined graph; it is the generator of the clustering representation itself.

6. Limitations, misconceptions, and emerging directions

A common misconception is that CLUDI simply means $P^t$3-means after diffusion maps. The literature is broader. FARDiff uses Fuzzy ART rather than $P^t$4-means (Damelin et al., 2014); DLSS, DSIRC, and DS$P^t$5DL perform density-mode discovery and diffusion-based label propagation (Murphy et al., 2017, Cui et al., 2022, Buranasiri et al., 14 Apr 2026); DSSC diffuses affinities before spectral clustering (Li et al., 2016); and the 2025 CLUDI framework performs stochastic assignment generation in latent space (Uziel et al., 6 Jul 2025). Another misconception is that diffusion removes the need for model selection. In practice, performance depends on graph construction, kernel bandwidth, neighbor counts, diffusion time, spatial radius, masking strategy, noise scale, and the assumed number of clusters (Murphy et al., 2017, Sevi et al., 2022, Yang et al., 2024).

The limitations are correspondingly method-specific. In random-walk CLUDI, incorrect affinities or weakly connected graphs can distort diffusion geometry; directed graphs require special constructions precisely because the natural walk is often non-reversible or non-ergodic (Sevi et al., 2022). In hyperspectral variants, DS$P^t$6DL explicitly assumes that ERS superpixels align reasonably with semantic classes; over-segmentation or under-segmentation weakens spatial regularization (Buranasiri et al., 14 Apr 2026). In generative variants, raw diffusion features may be poorly clusterable unless additional objectives are imposed, as DiFiC’s ablations make clear (Yang et al., 2024). In plug-and-play latent diffusion for single-cell data, large $P^t$7 can over-weight the prior and map genuinely novel structure into previously learned modes, while repeated sampling exposes meaningful uncertainty near cluster boundaries (Meier et al., 26 Oct 2025).

Current directions indicate a widening role for diffusion beyond classical partitioning. DS$P^t$8DL proposes fine-tuning its masked autoencoder with unsupervised contrastive learning and extending the framework to semi-supervised and active settings (Buranasiri et al., 14 Apr 2026). MELD suggests that diffusion time can be treated as a coordinate on a family of valid clusterings rather than a nuisance hyperparameter (Murphy et al., 2021). Work on latent group sparsity via heat-flow penalties shows that diffusion on networks can interpolate between lasso and group lasso without explicit pre-clustering, which suggests a broader interpretation of CLUDI as soft group-structure discovery rather than only hard partition recovery (Ghosh et al., 20 Jul 2025). Taken together, these developments indicate that diffusion now functions both as a metric geometry and as a probabilistic prior, and that the boundary between clustering, denoising, completion, and representation learning is becoming increasingly porous.