Title: PCA++: How Uniformity Induces Robustness to Background Noise in Contrastive Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Contrastive Learning

Principal Component Analysis

Uniformity Regularization

Self-Supervised Learning

Representation Learning
Abstract: High-dimensional data often conceal low-dimensional signals beneath structured background noise, limiting standard PCA. Motivated by contrastive learning, we address the problem of recovering shared signal subspaces from positive pairs--paired observations sharing the same signal but differing in background. Our baseline, PCA+, uses alignment-only contrastive learning and succeeds when background variation is mild, but fails under strong noise or high-dimensional regimes. To address this, we introduce PCA++, a hard uniformity-constrained contrastive PCA that enforces identity covariance on projected features. PCA++ has a closed-form solution via a generalized eigenproblem, remains stable in high dimensions, and provably regularizes against background interference. We provide exact high-dimensional asymptotics in both fixed-aspect-ratio and growing-spike regimes, showing uniformity’s role in robust signal recovery. Empirically, PCA++ outperforms standard PCA and alignment-only PCA+ on simulations, corrupted-MNIST, and single-cell transcriptomics, reliably recovering condition-invariant structure. More broadly, we clarify uniformity’s role in contrastive learning—showing that explicit feature dispersion defends against structured noise and enhances robustness.


Peer reviews:
comments: SUMMARY:
This paper proposes PCA++, a PCA algorithm for contrastive learning, which can robustly extract signal from background under a broad class of scenarios. This is achieved by enforcing feature uniformity via a hard constraint. A baseline PCA+ method is first proposed. Theoretical analysis of both PCA+ and PCA++ are provided, as well as a closed form solution. Numerical experiments verify the improved performance in contrastive learning tasks.

STRENGTHS AND WEAKNESSES:
Strength
1. The proposed PCA++ framework addresses the challenge in contrastive learning in a robust manner with closed form.
2. This paper is very technically solid, and has contribution in the theoretical analysis of high-dimensional data analysis.
3. The insight on the relationship between uniformity and contrastive learning is valuable.
4. Numerical experiments cover different aspects of the method and scenarios.

Weakness
1. The assumptions 1-3 are somewhat restrictive, and may be unrealistic in real scenarios, in particular the orthogonality in Assumption 1. The authors should discuss the dependency and sensitivity of the results on these assumptions, either by theoretical analysis or numerical experiments.
2. The authors should discuss the computational cost of the proposed framework, and it scalability to large datasets.

QUESTIONS:
1. Please include discussion on the computational cost and scalability.
2. Please discuss the restrictiveness of the model assumptions, and their implications on real-world data scenarios.

LIMITATIONS:
Please see above.
TITLE: Official Review
CLARITY: 4
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-06-22

comments: SUMMARY:
This paper introduces PCA++, a contrastive Principal Component Analysis method designed to robustly isolate shared signal subspaces from paired observations under strong background noise. The key innovation is the addition of a hard uniformity constraint , which requires the projected features to have identity covariance. Leveraging a generalized eigen decomposition with low-rank truncation for stability, PCA++ admits a closed-form solution and provides rigorous non-asymptotic and high-dimensional guarantees under a linear contrastive factor model.

STRENGTHS AND WEAKNESSES:
Strengths:

1.	This paper introduces a hard uniformity constraint and leverages paired observations to isolate shared signal subspaces, leading to a closed-form generalized eigenproblem solution.
2.	The authors provide a theoretical explanation for how the uniformity constraint enhances robustness in contrastive learning, especially under structured noise.
3.	Numerical experiments on synthetic datasets, corrupted MNIST, and real single-cell RNA-seq data demonstrate the effectiveness of the proposed algorithm. 

Weaknesses:

1.	The assumption which requires orthogonal signal and background subspaces is quite strong and may not hold in many real-world applications.
2.	The method is limited to linear data, whereas many real-world datasets exhibit nonlinear structures.
3.	The strategy for selecting the hyperparameter s is not reasonable. 
4.	The limitations of PCA++ are not clearly discussed in the paper.
5.	Some typos.

QUESTIONS:
See weaknesses.

LIMITATIONS:
1. Some assumptions might be quite strong and not hold for real-world applications.
2. This method doesn't consider data with non-linear structure.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-06-28

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper utilizes ideas from contrastive learning literature to propose PCA methods for robust signal recovery under strong background noise. Firstly, the authors introduce PCA with alignment-only contrastive learning (they name it ), analyze the finite-sample performance of the proposed method PCA+, and identify some cases (high background spikes relative to signal spikes or ambient dimension comparable to the number of samples) where PCA+ is not enough for robust signal recovery. To solve this issue, they propose to use a “hard uniformity” constraint, leading to a PCA method that they named PCA++. Then, the authors provide high-dimensional characterizations of the proposed method (PCA++) and a theoretical explanation for the uniformity’s power in contrastive learning. Finally, the paper includes experimental results with simulated data (based on contrastive factor model), corrupted MNIST, and single-cell transcriptomics, indicating PCA++ outperforms standard PCA.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n1. An easily understandable paper\n2. Studies an interesting and relevant problem: robust signal recovery under strong background noise.\n3. Provides theoretical and empirical results regarding the proposed PCA methods. \n4. Gives a high-dimensional analysis of a uniformity-constrained estimator together with some insights regarding the role of uniformity in contrastive learning, which is valuable from my point of view. \n\n### Weaknesses\n- The main problem with this paper is its positioning in the literature. With the current presentation of the paper, it sounds like this is the first paper to study contrastive PCA. However, there exists some work studying contrastive PCA in the literature (see the examples below). Furthermore, these highly relevant papers are not mentioned or cited in this paper. Therefore, evaluating the contributions of this work is challenging without any comparison with the prior contrastive PCA literature. Due to this major issue, I recommend rejection.\n- Even if the presentation of the paper is fixed by adding the relevant citations and so on, the fundamental differentiating point of this work compared to the contrastive PCA literature seems to be \"uniformity constraint\", which is also an existing idea in the contrastive learning literature (e.g., see [41] from the references of the submitted paper). Therefore, I think the originality/contribution of this work is questionable at this point.\n\n**Example papers about \"contrastive PCA\" in the literature:**\n- \"Contrastive principal component analysis\", A Abid, MJ Zhang, VK Bagaria, J Zou, arXiv:1709.06716, 2017.\n- \"Probabilistic contrastive principal component analysis\", D Li, A Jones, B Engelhardt, arXiv:2012.07977, 2020.\n- \"Exploring high-dimensional biological data with sparse contrastive principal component analysis\", P Boileau, NS Hejazi, S Dudoit, Bioinformatics, 2020.\n- \"Sparse discriminant PCA based on contrastive learning and class-specificity distribution\", Q Zhou, Q Gao, Q Wang, M Yang, X Gao, Neural Networks, 2023.\n- \"Contrastive Functional Principal Component Analysis\", E Zhang, D Li, AAAI, 2025.\n\nQUESTIONS:\n1. Are the authors aware of the existence of the mentioned papers about \"contrastive PCA\"?\n\n2. What is the additional contribution of this work compared to the existing contrastive learning and contrastive PCA literature?\n\nLIMITATIONS:\nLimitations of the proposed method (PCA++) are not discussed explicitly in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a new contrastive PCA algorithm, PCA++, that isolates shared signals in positively related datasets under strong background noise. PCA++ enforces hard uniformity constraints in the projected features. And it results in closed-form solutions with stability and non-asymptotic guarantees. The paper compares this approach with alignment only contrastive PCA and reports improved performance on synthetic (simulation, MNIST) and one real-world (sequencing data) experiments. The paper further provides justification and proof for the importance of uniformity in improving performance and makes PCA robust to high background noise.\n\nSTRENGTHS AND WEAKNESSES:\n**S1**: The paper presents a well-motivated problem. It presents the classical and alignment-only contrastive PCA and their limitations with examples, setting up a perfect ground for the proposed method, PCA++. In addition, the paper is easier to follow, with a clear outline of its contribution.  \n \n**S2**: The paper presents a novel contribution to contrastive learning theory by analysing the role of uniformity constraints on projected features in high-dimensional strong noise setting. This result could have an impact on other machine learning areas.  \n \n \n**S3**: The paper precisely shows the limitation of PCA  and the utility of the proposed techniques, with the real-world experiments showing isolating noise and extracting shared space. Specifically, the synthetic experiments showed clear improvement in error trends with aspect ratio and truncation levels.  \n\n**Weakness**\n\n**W1**: While the authors have mentioned the work could be extended to other branches of contrastive learning, the authors did not discuss the limitations of the work, for instance, with regards to the assumption or settings considered (Assumptions 4.2, and 4.3), and how real-world data conforms to these assumptions.  \n \n \n**W2**: The simulation experiment shows the effect of varying aspect ratio d/n, and truncation etc.. However, there seems to be a missing discussion on how these factors affect the performance of PCA++ in real-world experiments (RNA sequencing). \n \n \n**W3**: I think the only real-world experiment presented in the paper ( using sequencing data ) is limiting in judging how important the uniformity constraint is in contrastive PCA. I think it would be important to at least show how well the alignment-only PCA+ does on the data. This limited real-world experiment and the considered baselines make it hard to assess the utility of the method in real-world experiments.\n\nQUESTIONS:\n* In Figure 3, only classical PCA seems to be considered as a baseline for comparison. This limited real-world experiment and the considered baselines make it hard to assess the utility of the method in real-world experiments.  \n\n\n* I encourage authors to consider citing an important early work on contrastive PCA[1] and comparing the method with existing baselines. \n\n[1] Contrastive Principal Component Analysis https://arxiv.org/pdf/1709.06716\n\nLIMITATIONS:\nWhile the authors have mentioned extensions to other existing contrastive learning methods, the limitations of this work are not clearly outlined.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}