Wyner IMVC: Clustering with Common Randomness

• The paper establishes common randomness as a latent cluster variable, enabling direct clustering from observed views without prior imputation.
• It leverages conditional independence and neural variational encoders to handle arbitrary missing-view patterns efficiently.
• Empirical results demonstrate superior accuracy and graceful degradation compared to state-of-the-art methods under varying missing rates.

Searching arXiv for WyIMVC and closely related incomplete multi-view clustering work to ground the article in current literature. Wyner Incomplete MultiView Clustering (WyIMVC) is an incomplete multiview clustering method that extends common information-based multiview learning to the setting in which some views are missing for some samples. Its central formulation treats the cluster assignment itself as the latent “common randomness” shared across views, so that clustering is inferred directly from the available views and missing-value inference is induced from the same latent variable rather than performed as a separate preprocessing stage (Odeh et al., 14 Jul 2025). In the recent incomplete multi-view clustering (IMVC) literature, this places WyIMVC within the broader effort to handle arbitrary missing-view patterns without sacrificing cross-view semantic structure, while distinguishing it from completion-first, alignment-first, graph-recovery, and tensor-recovery paradigms (Wen et al., 2022).

1. Problem setting and placement within IMVC

Incomplete multiview clustering studies unlabeled data described by multiple views

$X^V = (X_1,\dots,X_V),$

where each $X_v$ is a modality or feature representation of the same underlying object, but not every sample is observed in every view (Odeh et al., 14 Jul 2025). For a given sample, if the available views are $x_A$ and the missing views are $x_{\bar A}$, with $A \subseteq [V]$ and $\bar A = [V]\setminus A$, the task is to cluster all samples under arbitrary missing-view patterns (Odeh et al., 14 Jul 2025).

The general IMVC literature frames this as a departure from conventional multi-view clustering, which assumes fully observed views and therefore fails when views are absent (Wen et al., 2022). The 2022 survey organizes IMVC methods into matrix factorization-based, kernel learning-based, graph learning-based, and deep learning-based families, and emphasizes two broad strategies: recovering missing views or structures and then clustering, or avoiding explicit recovery and clustering directly from partially aligned observations (Wen et al., 2022). WyIMVC appears after that survey and is not mentioned there; a plausible placement, given its use of VAEs, neural decoders, and a learned categorical latent variable, is within the deep learning-based end of that taxonomy, but with an explicitly information-theoretic core rather than a purely representation-learning one (Wen et al., 2022, Odeh et al., 14 Jul 2025).

The method is motivated by application settings in which unpaired multiview data with missing values arise in generative learning, cross-modal retrieval, and wireless device identification problems (Odeh et al., 14 Jul 2025). Relative to earlier IMVC workflows, WyIMVC is distinguished by its refusal to treat missing-view completion as a prerequisite: the solver directly estimates the shared cluster variable from observed views, and only then reconstructs missing latent content if desired (Odeh et al., 14 Jul 2025).

2. Wyner common information as the organizing principle

The conceptual basis of WyIMVC is Wyner common information. In the classical Wyner perspective, multiple observations become conditionally independent given a latent common source:

$X_i \rightarrow Z \rightarrow X_j,\qquad i\neq j.$

WyIMVC identifies this latent variable $Z$ with the cluster assignment distribution, so that $Z$ is simultaneously the carrier of shared semantic structure and the clustering target itself (Odeh et al., 14 Jul 2025).

The corresponding relaxed common-information objective is

$$\underset{P_{z|x^V}\in \Omega_{z|X^V}}{\min}\; I(Z;X^V)$$

subject to

$X_v$0

Here $X_v$1 measures how much information the common variable carries about the multiview observation, while the constraints enforce approximate conditional independence between complementary subsets of views (Odeh et al., 14 Jul 2025). When $X_v$2, the exact Wyner-style conditional independence is recovered (Odeh et al., 14 Jul 2025).

This formulation matters because it recasts clustering as common-randomness estimation rather than as late-stage partitioning of a fused embedding. In methods built around imputation or alignment, the cluster variable is typically downstream of recovered data or aligned features. In WyIMVC, by contrast, the cluster distribution is primary: shared information is assumed to align with cluster targets, and estimating that shared information is itself the clustering procedure (Odeh et al., 14 Jul 2025). This suggests a different failure model from completion-based IMVC methods: degradation arises not chiefly from inaccurate view synthesis, but from inaccurate estimation of the common latent source from the available evidence.

3. Incomplete-view formulation and posterior inference

For sample $X_v$3, WyIMVC denotes the observed subset by $X_v$4 and infers a categorical latent variable from the available views alone:

$X_v$5

The paper emphasizes that this latent variable is the cluster distribution itself, not merely an auxiliary shared representation (Odeh et al., 14 Jul 2025).

For complete multiview data, the relaxed Wyner objective admits a difference-of-convex update of the form

$X_v$6

where $X_v$7 is a normalization constant and the mixture weights satisfy $X_v$8 and $X_v$9 (Odeh et al., 14 Jul 2025).

WyIMVC extends this update to incomplete data by replacing missing-view likelihoods with the prior:

$x_A$0

The explicit interpretation given is that the first term corresponds to the observed/missing bipartite structure and the second term accounts for bipartitions within the available set (Odeh et al., 14 Jul 2025).

A key consequence is

$x_A$1

so the cluster posterior depends only on the observed views. The paper states the resulting Markov relation as

$x_A$2

meaning that missing views are not required for estimating the common randomness (Odeh et al., 14 Jul 2025). This is one of the clearest points of contrast with completion-first IMVC pipelines: missing views do not need to be synthesized before clustering can proceed.

For scalability, the implementation uses a conditional-independence simplification,

$x_A$3

under which the growth in complexity drops from exponential in $x_A$4 to linear in $x_A$5 (Odeh et al., 14 Jul 2025). The paper also defines a generalized class probability $x_A$6 for the incomplete-view setting and a simplified variant $x_A$7 under this conditional-independence assumption (Odeh et al., 14 Jul 2025).

4. Neural implementation and DC-based optimization

WyIMVC couples the common-information posterior with a neural implementation consisting of per-view variational encoders, sampling of the categorical common variable, latent reconstruction, and contrastive label matching (Odeh et al., 14 Jul 2025). Each view is encoded as

$x_A$8

with Gaussian noise $x_A$9 (Odeh et al., 14 Jul 2025). The per-view objective is an ELBO-type loss

$x_{\bar A}$0

where the reconstruction and KL terms are computed only over observed instances through the missingness indicators $x_{\bar A}$1 (Odeh et al., 14 Jul 2025).

Because $x_{\bar A}$2 is categorical, the method uses Gumbel-softmax rather than Gaussian reparameterization:

$x_{\bar A}$3

with $x_{\bar A}$4 (Odeh et al., 14 Jul 2025). In WyIMVC, the class probabilities $x_{\bar A}$5 are replaced by the probabilities produced by the incomplete-view common-information model (Odeh et al., 14 Jul 2025).

The same common variable then drives missing-value inference in latent space through view-specific decoders,

$x_{\bar A}$6

with reconstruction penalty

$x_{\bar A}$7

The paper’s stated interpretation is that common-randomness estimation has two effects at once: clustering and missing-value inference (Odeh et al., 14 Jul 2025).

To align predictions across views, WyIMVC adds a contrastive loss over view pairs with complete overlap. Positive examples are diagonal cosine similarities and negatives are off-diagonal similarities, yielding a pairwise contrastive term $x_{\bar A}$8 (Odeh et al., 14 Jul 2025). The full implementation loss is

$x_{\bar A}$9

Optimization is organized around the difference-of-convex structure of the common-information objective. The solver uses the difference-of-convex algorithm (DCA), with a closed-form update for the common-randomness posterior and a convergence guarantee independent of initialization (Odeh et al., 14 Jul 2025). In practical terms, the optimization alternates over view encoding, posterior computation from available views, Gumbel-softmax sampling of $A \subseteq [V]$0, latent reconstruction from $A \subseteq [V]$1, and contrastive alignment of cluster predictions (Odeh et al., 14 Jul 2025).

5. Empirical results and reported advantages

The paper evaluates WyIMVC on three incomplete multiview benchmarks: Handwritten with 2,000 samples, 6 views, and 10 clusters; Fashion with 10,000 samples, 3 views, and 10 clusters; and MSRC-v1 with 210 samples, 5 views, and 7 clusters (Odeh et al., 14 Jul 2025). Incompleteness is simulated by randomly masking subsets of views at missing rates from 0.1 to 0.7 (Odeh et al., 14 Jul 2025).

The reported baselines are DCP-CV, described as contrastive incomplete multiview representation learning, and DSIMVC, described as deep safe incomplete multiview clustering (Odeh et al., 14 Jul 2025). Across all three datasets, WyIMVC is reported to achieve the highest clustering accuracy over the full range of missing rates, to degrade more gracefully as missingness increases, and to show especially strong gains on datasets with more views and higher-dimensional view spaces (Odeh et al., 14 Jul 2025). Dataset-specific descriptions state that the method consistently outperforms both baselines on Handwritten, maintains an advantage on Fashion even with fewer views but high-dimensional features, and yields the best performance on MSRC-v1 across missingness settings (Odeh et al., 14 Jul 2025).

The abstract summarizes this as outperformance over state-of-the-art solvers on incomplete multiview datasets with varying numbers of views and dimensions (Odeh et al., 14 Jul 2025). The mechanism offered for this robustness is direct estimation of the common randomness rather than explicit imputation or post-hoc clustering (Odeh et al., 14 Jul 2025). A plausible interpretation is that the method benefits when the available views already contain enough shared information to infer the cluster posterior, because under that regime missing views can be marginalized rather than reconstructed first.

6. Relation to alternative IMVC paradigms

The broader IMVC literature contains several competing strategies for handling missing views. WyIMVC is most usefully understood by contrast with those strategies.

Paradigm	Representative method	Characteristic mechanism
Prototype-based imputation	ProImp (Li et al., 2023)	Recovers missing views from missing-view prototypes and observed-view sample–prototype relationships
Diffusion completion	IMVCDC (Fang, 2023)	Reconstructs per-view latent features, imputes missing views with conditional diffusion, then performs contrastive clustering
Partial sample/prototype alignment	CPSPAN (Jin et al., 2023)	Uses pair-observed data as proxy supervision, aligns prototypes, then performs post-hoc structure embedding imputation
Relation-transfer recovery	CRTC (Wang et al., 2021)	Transfers cross-view KNN relations, recovers missing data with a GCN, then fuses views with attention
Anchor/structure alignment	SIMVC-SA (Wen et al., 2023)	Learns view-specific anchors and anchor graphs, aligns structures through a shared latent representation
Tensor low-rank graph recovery	JTIV-LRR (Wang et al., 4 Mar 2025)	Recovers a complete similarity graph tensor via low-tubal-rank and inter-view low-rank constraints
Common-randomness estimation	WyIMVC (Odeh et al., 14 Jul 2025)	Infers a latent cluster variable directly from observed views and reconstructs missing latent content from that variable

The main conceptual dividing line is whether clustering is downstream of recovery or whether recovery is downstream of clustering. ProImp, IMVCDC, CRTC, CPSPAN, SIMVC-SA, and JTIV-LRR all incorporate some form of view reconstruction, graph completion, prototype transfer, or structure alignment before or alongside final clustering (Li et al., 2023, Fang, 2023, Wang et al., 2021, Jin et al., 2023, Wen et al., 2023, Wang et al., 4 Mar 2025). WyIMVC inverts that logic: it estimates the cluster distribution first and uses the resulting common randomness to reconstruct latent view content (Odeh et al., 14 Jul 2025).

A common misconception is therefore to treat WyIMVC as merely another imputation model. The paper explicitly argues otherwise: missing-value inference is enabled by the common randomness and occurs jointly with clustering, rather than serving as a prerequisite for clustering (Odeh et al., 14 Jul 2025). Another misconception is that the method requires complete multiview samples to infer cluster identity. The incomplete-view update is constructed so that the posterior depends only on observed views, formalized by $A \subseteq [V]$2 and the Markov relation $A \subseteq [V]$3 (Odeh et al., 14 Jul 2025).

7. Open questions and research context

The 2022 IMVC survey identifies several field-level open problems: large-scale efficiency, information imbalance among views, mixed data types, noise robustness, missing-view recovery with theory, and the partially aligned plus missing-view setting (Wen et al., 2022). These are not presented as WyIMVC-specific limitations, but they remain relevant background questions for evaluating any IMVC method, including one grounded in common information.

Within that context, WyIMVC contributes a specific answer to two long-standing IMVC design choices. First, it declines the recover-then-cluster paradigm by making the cluster variable itself the sufficient statistic for missing-value inference (Odeh et al., 14 Jul 2025). Second, it replaces purely heuristic fusion rules with a common-information formulation optimized by DCA and equipped, in the paper’s statement, with a convergence guarantee independent of initialization (Odeh et al., 14 Jul 2025). This suggests a research trajectory in which information-theoretic latent-variable models are used not only for multiview representation learning but also for incomplete-data clustering under arbitrary missingness patterns.

Taken as a whole, WyIMVC represents a common-randomness interpretation of incomplete multiview clustering: the latent source shared across available views is identified with the cluster distribution, estimated directly from observed data, sampled differentiably, and reused to reconstruct missing latent view content (Odeh et al., 14 Jul 2025). In the current IMVC landscape, its significance lies less in a new completion heuristic than in a redefinition of what the primary latent object should be.