PaperlayerView exact revision on arXiv

Source-linked AI summary

Unpaired Canonical Correlation Analysis

Nir Ben-Ari, Ronen Talmon, Uri Shaham

arXiv:2610.09530v1stat.MLcs.LG

TL;DR

CCA's reliance on paired samples is limiting when correspondences are expensive or impossible to obtain. UCCA uses a QAP-based theoretical connection to learn correlation components from strictly unpaired data, and is reported to outperform unpaired alignment baselines across real-world datasets. The paper's current framework is limited to the bi-view setting.

  • Problem

    CCA depends on paired corresponding samples, although paired data can be prohibitively expensive or impossible to obtain while unpaired data is abundant.

    Source: §1 · p. 1

  • Method

    UCCA derives a practical way to learn linear projections for the true pairing from strictly unpaired training data, using theoretical results connecting the task to QAP.

    Source: §1 · p. 2

  • Results

    UCCA significantly outperforms existing unpaired alignment baselines in recovering true underlying correlation components across multiple real-world datasets.

    Source: §7 · p. 10

  • Takeaways & Limitations

    The work demonstrates that latent correlations can be systematically recovered without explicit correspondence in domains where paired samples are prohibitively expensive or impossible to acquire.

    Source: §7 · p. 10

    The current framework focuses on the bi-view setting, leaving extension to multi-view scenarios for future research.

    Source: §7 · p. 10

Abstract

from arXiv · show

Canonical Correlation Analysis (CCA) is a fundamental method for multiview shared space learning. However, its strict reliance on paired data poses a significant limitation, as such data is often difficult to obtain or entirely unavailable. In this paper, we present Unpaired CCA (UCCA), a novel method that learns linear projections to maximize the correlation of the true underlying pairing without access to any paired samples during training. We first establish theoretical results connecting the Quadratic Assignment Problem (QAP) to CCA. Leveraging these theoretical insights, we derive a practical method to maximize correlation exclusively from unpaired data. To the best of our knowledge, UCCA is the first approach to learn maximally correlated projections in a strictly unpaired setting. We validate UCCA on real-world multi-modal datasets, demonstrating that it significantly outperforms recent unpaired alignment baselines in recovering the underlying true correlation. This work fills a critical gap between traditional statistical multiview learning and the growing field of unpaired data learning.

1 Introduction

CCA learns maximally correlated components across views but relies on paired samples, which can be prohibitively expensive or impossible to obtain. UCCA addresses this gap with a QAP-based theoretical framework and a practical method for learning from unpaired data, reporting performance near paired CCA on the Handwritten benchmark.

1 Introduction

  • Paired multimodal samples can be prohibitively expensive or impossible to obtain, while unpaired data is abundantly accessible.CCA remains constrained by its reliance on strictly paired corresponding samples.Source: §1 · p. 1
  • UCCA is presented as an extension of CCA that operates entirely without paired training data, addressing a strict correspondence requirement.The paper describes it as the first such extension to demonstrate or theoretically establish recovery of true correlations in this setting.Source: §1 · p. 2
  • A QAP formulation connects unpaired CCA to a proxy pairing and provides a route to practical approximate solvers.The paper also reports a direct positive relationship between QAP value and total correlation on a 10-sample synthetic dataset.Source: §1 · p. 2, Figure 1 · p. 2
  • On the Handwritten benchmark, UCCA significantly outperforms unpaired alignment baselines and approaches the paired CCA upper bound.The figure caption describes training as fully unpaired.Source: Figure 1 · p. 2

2 Related Work

Prior CCA extensions and many unpaired-learning approaches still depend on exact correspondences or other alignment signals. Fully unpaired methods exist in applied domains, but the cited approaches do not explicitly extract maximally correlated shared representations.

CCA Extensions

  • Sparse CCA reduces the need for large paired datasets but still requires exact correspondences; UCCA instead uses exclusively unpaired data.The cited sparse methods address the case where sample count is smaller than dimensionality.Source: §2 · p. 2

Incorporating Unpaired Data

  • Semi- and weakly supervised approaches, as well as shared-class-label methods, depend on paired examples or complementary alignment signals.These approaches use unpaired data alongside initial paired training, a few exact pairs, or shared labels.Source: §2 · p. 2

Fully Unpaired Methods

  • Fully unpaired applied methods are largely task-specific, while domain-confusion and adaptation methods align marginal distributions rather than extracting maximally correlated components.The paper identifies translation, segmentation, and image restoration as applied examples.Source: §2 · pp. 2–3
  • SCA seeks shared components without pairs, but its framework does not account for correlation; UCCA is reported to capture highly correlated components better in practice.The paper identifies SCA as the most closely related work.Source: §2 · p. 3

3 Background

The background defines CCA as maximizing correlation between views and introduces the notation and QAP formulation used in the paper. CCA returns linear projections for a tunable number of components, while QAP seeks an optimal one-to-one assignment.

Notations

  • The notation uses X and Y for whitened datasets, permutation matrices for sample assignments, and σ(A) for the descending singular-value vector of A.The paper also defines the orthogonal group and Frobenius sphere used in its formulations.Source: §3 · p. 3

Canonical Correlation Analysis (CCA)

  • CCA returns projection matrices that maximize total correlation between two views, with the number of components k as a tunable hyperparameter.The projections are constrained to have orthonormal columns, and k is much smaller than the feature dimension d.Source: §3 · p. 3

Quadratic Assignment Problem (QAP)

  • QAP seeks a one-to-one assignment that maximizes structural alignment between an interaction matrix and a similarity matrix.The paper uses the Koopmans-Beckmann formulation and recasts it using permutation matrices and total-correlation notation.Source: §3 · p. 3

4 Correlation of Unpaired Data

The paper links unpaired CCA to a proxy pairing, relaxes its optimization to a QAP, and shows that the relaxed pairing coincides with the proxy under stated assumptions. This equivalence underlies its practical approach to recovering correlation without paired samples.

4.1 Unpaired CCA Proxy Pairing

  • MCPO(d) provides a computable proxy pairing without paired samples, unlike naive MCP, which is sensitive to arbitrary input rotations.MCPO(d) evaluates correlation after optimizing over orthogonal transformations; the proxy need not exactly recover the unknown permutation, but must attain the desired correlation.Source: §4.1 · pp. 4–5

4.2 Connection to QAP

  • Relaxing the orthogonal search to a scaled Frobenius sphere makes pairing optimization feasible, and MCPSF reduces to a QAP with efficient approximate solvers.Under a mild assumption, the relaxation does not alter the optimal pairing.Source: §4.2 · p. 5

4.3 Equivalence of MCPO(d) and MCPSF

  • Under weak majorization of singular values, MCPO(d) and MCPSF select the same pairing; the assumption holds when Y is an isometry of X and asymptotically with probability one for random permutations.Figure 3 reports empirical validation of the majorization behavior and theorem on Flickr.Source: Figure 3 · p. 5, §4.3 · p. 6
  • Under realistic assumptions, the computationally attainable MCPSF pairing coincides with MCPO(d), justifying unpaired CCA via linear-kernel QAP.The paper identifies this equivalence as the theoretical cornerstone of its method.Source: §4 · p. 3, §4.3 · p. 6

5 Unpaired CCA

UCCA uses theoretical insights about QAP to derive a practical way to maximize cross-view correlation without paired training data. It matches representative anchors to form pseudo-pairs, then applies standard CCA.

5 Unpaired CCA

  • UCCA derives a practical algorithm for maximizing correlation using strictly unpaired training data.The approach is built on the theoretical foundations established earlier in the paper.Source: §5 · p. 6
  • Because solving the QAP over all samples is computationally prohibitive, UCCA selects representative anchors from each modality.The QAP is NP-hard, and the anchor-based approach is intended to make the method scalable.Source: §5 · p. 6
  • UCCA matches K-Means centroid anchors with a linear-kernel QAP, repeats matching for robustness, and applies standard CCA to the resulting pseudo-pairs.The procedure does not assume existing correspondences between data samples.Source: §5 · p. 7

6 Experimental Results

Across real-world configurations, UCCA recovers hidden cross-view correlation and performs strongly on downstream classification. Experiments also test the theoretical pairing result, runtime trade-offs, and sensitivity to data conditions.

Experimental Design

  • Evaluation uses four real-world datasets in six configurations, with ground-truth pairings withheld by assigning disjoint training subsets to the two views.Baselines include SCA and linear UCA, plus CCA fitted to pseudo-pairs inferred by J-MDS, SCOTv1, and SCOTv2.Source: §6 · p. 7

Total Correlation

  • UCCA consistently recovers underlying correlation, significantly outperforming baselines and often approaching the paired CCA upper bound across six dataset configurations.Figure 5 reports total correlation for the top two learned components after separately orthogonalizing the views.Source: §6 · p. 7, Figure 5 · p. 8

Cross-view Classification

  • UCCA outperforms all baselines on cross-view classification across four datasets and has comparable performance on the fifth.A kNN classifier is trained on one view's embeddings and evaluated on the other; Flickr is omitted because it lacks class labels.Source: §6 · p. 8, Table 1 · p. 8

Validating MCPO(d) = MCPSF

  • Kendall Tau distances between the approximate QAP pairing and estimated MCPO(d) pairing concentrate near zero, empirically supporting Theorem 2 across datasets.The comparison evaluates pairings across real datasets and different anchor initializations.Source: §6 · pp. 8–9, Figure 6 · p. 8

Computational Efficiency

  • UCCA outperforms baselines even with fewer clustering repetitions and shorter runtimes, requiring only several minutes per dataset.Representative anchors and efficient approximate solvers mitigate the computational overhead of the NP-hard QAP.Source: §6 · p. 9, Figure 7 · p. 9

Ablations

  • On Handwritten PC-PA, UCCA is robust to view imbalance and improves in consistency and performance as the amount of available unpaired data increases.Performance is optimal when all data is accessible during training; additional ablations examine K-Means anchors and anchor counts.Source: §6 · p. 9, Figure 8 · p. 9

7 Conclusions and Summary

The conclusion presents UCCA as a method for recovering correlation components from strictly unpaired data, grounded in a proxy-pairing connection to QAP. Evaluations outperform unpaired alignment baselines, while a gap to paired CCA remains and the framework is currently bi-view.

A.1 Unpaired CCA Proxy Pairing

  • Under realistic data assumptions, the computationally attainable MCPSF pairing coincides with the MCPO(d) target pairing.Source: §7 · p. 9

7 Conclusions and Summary

  • Across multiple real-world datasets, UCCA significantly outperforms existing unpaired alignment baselines in recovering true underlying correlation components.Source: §7 · p. 10
  • UCCA still trails paired CCA, and the current framework addresses only the bi-view setting.Source: §7 · p. 10
  • The work connects classical statistical multiview learning with unpaired data integration for settings where paired samples are prohibitively expensive or impossible to acquire.Source: §7 · p. 10

A.1 Unpaired CCA Proxy Pairing

  • Joint optimization over orthogonal transformations and permutations is computationally prohibitive, so the method relaxes to the scaled Frobenius sphere for an attainable proxy pairing.Source: §A.1 · p. 15

A.2 Connection to QAP

  • The relaxed MCPSF objective is exactly the Koopmans-Beckmann QAP with kernels XXT and YYT.Source: §A.2 · pp. 16–17

A.3 Equivalence of MCPO(d) and MCPSF

The section argues that weak majorization of singular-value spectra is a useful condition for identifying the optimal pairing. It supports this condition in both a finite-sample isometry setting and an asymptotic random-permutation setting.

A.3 Equivalence of MCPO(d) and MCPSF

  • The optimal pairing strictly maximizes the nuclear norm of the permuted cross-correlation matrix.The proof also establishes that it maximizes the Frobenius-norm objective.Source: §A.3 · pp. 17, 21
  • Assumption 1 requires the optimal pairing’s singular values to weakly majorize those of any alternative permutation.The assumption is described as a relatively mild constraint on spectral decay.Source: §A.3 · p. 17
  • For finite samples, Assumption 1 holds when one view is an isometry of the other.The stated setting is Y = XQ for orthogonal Q.Source: §A.3 · p. 17
  • As sample size grows, the optimal pairing weakly majorizes a uniformly sampled permutation with probability approaching 1.This is the asymptotic result stated for fixed feature dimension.Source: §A.3 · pp. 17, 20

B Additional Results

The additional results section points readers to the full results in Table 2 and Figure 5.

B Additional Results

  • The full results are visualized in Figure 5.Source: B Additional Results · p. 21
  • Table 2 reports Total Correlation results as mean ± standard deviation.Paired CCA results are included as an upper-bound reference.Source: Table 2 · p. 21
  • Paired CCA results appear in gray at the bottom of Table 2 as an upper-bound reference.Source: Table 2 · p. 21

B.2 Full Cross-Classification Results

Across the additional evaluations, UCCA retains strong cross-view classification and alignment results, while runtime, solver, anchor, and preprocessing choices affect its practical use. The related-work discussion distinguishes this setting from other alignment approaches.

B.2 Full Cross-Classification Results

  • In the alternate kNN classification direction, UCCA is best except on COCO, where it is second-best; linear-kernel classification preserves the trends.The reported findings complement the primary cross-view classification results.Source: §B.2 · p. 22, Table 3 · p. 22

B.3 Validating MCPO(d) = MCPSF

  • Across evaluated domains, Kendall Tau distances are concentrated at zero, supporting the practical equality of MCPO(d) and MCPSF.This extends the validation beyond the PC-PA results.Source: Figure 9 · p. 22, §B.3 · p. 23

B.4 Running Times

  • Even with 1 or 20 clustering repetitions, UCCA outperforms unsupervised baselines, often in under a minute; 500 repetitions rarely take more than a few minutes.More repetitions reduce run-to-run variance and move alignment quality toward the paired CCA upper bound.Source: Figure 10 · p. 23, §B.4 · p. 24

B.5 Direct Optimization of MCPO(d)

  • The QAP formulation provides a more robust optimization landscape than directly optimizing the MCPO(d) objective.The alternating optimization methods were each run 50 times with different initializations.Source: §B.5 · p. 24, Table 4 · p. 24

B.6 Effect of QAP Solver

  • 2-Opt and FAQ achieve almost identical alignment quality when FAQ converges, but FAQ failed in two instances, motivating 2-Opt as the default.The comparison reports Total Correlation across multiple initializations.Source: §B.6 · p. 24, Figure 11 · p. 25

B.7 Comparison with Non-linear UCA

  • UCCA significantly outperforms nonlinear UCA across datasets on Total Correlation and cross-view classification.This comparison tests whether nonlinear mappings improve the baseline’s unpaired structural alignment.Source: §B.7 · p. 25, Table 5 · p. 25, Table 6 · p. 25

B.8 Anchors Choice

  • K-Means anchors outperform random sampling, paired K-Means approaches the paired CCA bound on five of six datasets, and GMM anchors give comparable results.Increasing anchor count improves and then stabilizes performance, with 20 chosen to balance accuracy and computation.Source: §B.8 · p. 26, Table 7 · p. 26, Table 8 · p. 26

B.9 Effect of PCA dimension; B.10 Visualizations

  • Performance remains stable across PCA whitening dimensions unless dimension is reduced excessively, while UCCA’s global manifold alignment yields class clusters that support near-oracle classification.The embedding comparison notes slightly lower Total Correlation than paired CCA because UCCA lacks pairwise supervision.Source: §B.9 · p. 27, Figure 13 · p. 27, §B.10 · p. 28

D Complexity Analysis

The appendix gives a complexity analysis and empirical checks of the relationship between QAP optimization and CCA, as well as the weak-majorization assumption.

D Complexity Analysis

  • When k and d are much smaller than n and iteration and clustering counts are treated as constants, UCCA is linear in sample count n.The stated overall complexity is O(nd^2 + ctnkd + cIk^3 + ckd^2 + d^3).Source: D Complexity Analysis · p. 29

E.1 QAP Alignment with CCA

  • Across all 10! pairings of a 10-sample synthetic dataset, higher QAP values correspond to higher Total Correlation, and the QAP maximum nearly matches the maximum-correlation pairing.The exhaustive evaluation used a linear kernel for the QAP objective.Source: §E.1 · p. 29

E.2 Assumption 1 Validation

  • Across the evaluated real-world domains, singular-value cumulative sums are lower as random shuffles increase, corroborating weak majorization.The extended validation includes PC-KL, PC-PA, KL-PA, SNARE, Flickr, and COCO.Source: §E.2 · p. 30, Figure 16 · p. 30

E.3 Theorem 2 Validation

Across the evaluated datasets, the Nuclear Norm and Squared Frobenius Norm objectives decay together as permutations move away from the true alignment, supporting their shared local maximum at the ground-truth permutation.

E.3 Theorem 2 Validation

  • Both objectives strictly decay as random shuffles increase across PC-KL, PC-PA, KL-PA, SNARE, Flickr, and COCO, indicating a shared local maximum at the ground-truth permutation.The comparison extends the Flickr validation and corroborates Theorem 2's equivalence of the Nuclear Norm and Squared Frobenius Norm objectives.Source: §E.3 · p. 30, Figure 17 · p. 31

F Implementation Details

F.1 Preprocessing and Dimensionality Reduction

  • PCA independently reduces each view to 10 whitened components by default, producing latent spaces for structural matching.Source: §F.1 · p. 32

F.2 Unsupervised Anchor Selection

  • K-Means independently selects 20 anchor centroids per view, reduced to 12 for SNARE because of its lower data amount.Source: §F.2 · p. 32

F.3 Anchor Matching via QAP

  • UCCA matches anchor sets by maximizing a linear-kernel QAP objective with a 2-Opt solver and 50 random restarts.The selected permutation is the one with the highest objective value.Source: §F.3 · p. 32

F.4 Learning the Projections

  • After estimating the structural permutation, UCCA concatenates pseudo-paired anchors from 500 repeated clustering-and-alignment trials and fits standard paired CCA to learn the final projections.The learned projections are applied to the original high-dimensional samples.Source: §F.4 · p. 32

F.5 Data Split and Unpaired Setting

The evaluation protocol withholds all training correspondences while retaining paired test data. The extended algorithm constructs matched anchor sets from unpaired views and uses them to learn CCA projectors.

F.5 Data Split and Unpaired Setting

  • Figure 18 depicts a protocol that retains a common paired test set but guarantees zero sample correspondence between the two training views.Source: Figure 18 · p. 33, §F.5 · p. 33
  • Typically, 256 paired samples are held out for testing, while the remaining samples are split into disjoint subsets supplying the first and second training views, respectively.Source: §F.5 · p. 33

F.6 Extended Algorithm

  • The extended algorithm repeatedly clusters each unpaired view, approximates QAP anchor matching with 2-opt restarts, concatenates matched anchors, and fits CCA to return projectors.Its inputs are unpaired views, and its specified parameters include the centroid count, clustering repetitions, and restart count.Source: §F.6 · pp. 33–34

F.7 Datasets

This appendix documents the datasets and experimental setup used to evaluate UCCA, including protocols, baseline implementations, computational details, and reproducibility and reporting checklist statements. It uses existing public benchmarks and official baseline implementations, with no new datasets or models reported.

F.7 Datasets

  • UCCA is evaluated on four datasets spanning handwritten views, single-cell measurements, and image-text pairs.Handwritten Digits uses PC, KL, and PA views (872 training and 256 test samples); SNARE uses ATAC and RNA views (395 training and 257 test cells). Flickr uses 2,744 training and 512 test pairs, while COCO uses 872 training and 256 test pairs. The image and text representations are 384-dimensional for Flickr and COCO; COCO also uses object-category labels and a top-10-PCA paired CCA upper bound.Source: §F.7 · pp. 34–35

F.8 Experiment-Specific Details

  • Experiments use kNN with k = 5 for cross-view classification and test robustness across unpaired-data amounts and anchor choices.The classifier is fitted on specified source views and, in a separate evaluation, on the alternate views. Unpaired training data ranges from 50% to 100% of the available set; paired K-Means anchors are nearest training samples to single-view centroids, with their pairs excluded from the standard unpaired split.Source: §F.8 · p. 35

F.9 Baseline Implementations

  • Because direct unpaired CCA baselines are unavailable, related alignment methods are tuned on Handwritten PC-PA test data and their hyperparameters are frozen for other datasets.For J-MDS and SCOT variants, the authors extract pseudo-pairs from point-cloud alignments and fit standard CCA; UCA is self-implemented, with its linear form used in the main text and its nonlinear form in App. B.7.Source: §F.9 · pp. 35–36

F.10 Hardware and Software Infrastructure

  • Experiments use NumPy, scikit-learn, SciPy’s quadratic-assignment solver, and PyTorch on a server with an RTX 2080 Ti GPU and eight Xeon CPU cores.The authors attribute UCCA’s low memory and compute footprint to dimensionality reduction and applying the QAP solver only to a small anchor subset.Source: §F.10 · p. 36

NeurIPS Paper Checklist

  • The checklist states that limitations are discussed, datasets are public, code is released, experimental settings are documented, and results report means, standard deviations, and visualized variance.The code is released at github.com/shaham-lab/UCCA; the checklist points to App. F for implementation details and data splits, and to App. F.10, App. B.4, and App. D for compute resources and complexity.Source: NeurIPS Paper Checklist · pp. 37–40
  • Public datasets and official baseline implementations are cited in Sec. 6 and App. F.Source: NeurIPS Paper Checklist · p. 41
  • The paper uses existing benchmarks and does not release new datasets or models.The checklist marks documentation of new assets as not applicable.Source: NeurIPS Paper Checklist · p. 42
  • The research does not involve crowdsourcing or human subjects.Source: NeurIPS Paper Checklist · p. 42

Related papers

Loading 2610.09530v1…