Contrastive CUR for Case-Control Analysis

• Contrastive CUR (CCUR) is a dimension reduction method that selects actual data columns and rows, ensuring interpretable analysis in case-control studies.
• It employs contrastive leverage scoring by comparing foreground and background singular vectors to isolate features and samples unique to the case group.
• CCUR enhances biological interpretability by directly mapping selected genes and samples to condition-specific signals, facilitating actionable insights in biomedical research.

Contrastive CUR (CCUR) is a dimension reduction and data selection methodology designed for interpretable analysis of high-dimensional case-control datasets. Unlike conventional principal component analysis (PCA) or stochastic feature selection methods, which typically yield low-dimensional projections as linear combinations of features, CUR decomposition selects actual columns and rows from the data matrix for approximation. This property provides direct interpretability of both features and samples. CCUR extends this paradigm by employing contrastive leverage scoring to simultaneously isolate features and samples that are specifically enriched in a foreground (case/treatment) group and not in a background (control/reference) group, directly addressing the demands of biomedical case-control studies (Zhang et al., 15 Aug 2025).

1. Motivation and Context

The principal motivation for CCUR stems from the limitations inherent in existing dimension reduction and feature selection methodologies for case-control analysis. In standard PCA or SVD-based approaches, principal components are typically dense linear combinations of all features, rendering biological interpretation challenging. CUR decomposition addresses this by selecting actual columns (features) and rows (samples), ensuring that the resulting approximation $\tilde{X}$ maintains direct correspondence with interpretable biological entities.

In case-control studies, where the objective is to identify features unique to a condition (such as a disease, treatment, or perturbation) and samples uniquely responsive to that condition, traditional CUR fails to provide contrastive information: the features and samples selected may be equally representative of both foreground and background. CCUR directly incorporates the structure of case-control data to isolate those variables and samples which are highly influential in the foreground but uninformative in the background, enabling precise identification of condition-specific signatures (Zhang et al., 15 Aug 2025).

2. Underlying Methodology

Contrastive CUR decomposes and selects features and samples based on contrastive leverage scoring. Let $X \in \mathbb{R}^{n \times p}$ denote the foreground data (cases), and $Y \in \mathbb{R}^{m \times p}$ the background data (controls). The key steps are as follows:

Step 1: Singular Value Decomposition (SVD) and Standard Leverage Scores

For a general data matrix $X$,

$X = U \Sigma V^\top$

The right singular vectors $V$ used to assess the importance of each feature via leverage scores. For feature (column) $d$, the standard leverage score is

$l_d = \sum_{\xi=1}^{k} (v_d^\xi)^2$

where $v_d^\xi$ is the $d$-th entry of the $\xi$-th top singular vector and $k$ is the target dimension.

Step 2: Contrastive Leverage Scoring

For CCUR, leverage scores are computed for both foreground ($X$) and background ($Y$) using the respective top-$k$ singular vectors. A contrastive leverage score for feature $d$ is defined as

$$l_d^{\text{CCUR}} = \frac{l_d^x}{l_d^y + \varepsilon}$$

where $l_d^x$ and $l_d^y$ are leverage scores obtained from $X$ and $Y$, and $\varepsilon > 0$ is a numerical stability constant. Features that exhibit high influence (large $l_d^x$) in the foreground and low influence (small $l_d^y$) in the background thus attain larger scores and are preferentially selected.

Step 3: Joint Feature and Sample Selection

CCUR proceeds in two stages:

• Feature Selection: The top-$c$ columns (features) are selected according to the sorted contrastive leverage scores. This stage pinpoints features uniquely associated with the foreground group.
• Sample Selection: Restricting to the selected $c$ features, CCUR subsequently re-applies CUR decomposition on the foreground data to select rows (samples) maximally representative of the foreground-specific structure.

This two-stage process enables simultaneous selection of features and samples that together encapsulate the unique biological signals of interest (Zhang et al., 15 Aug 2025).

3. Practical Algorithms and Computational Details

The computational framework is based on a standard CUR pipeline with enhancements for contrastive leverage scoring. The explicit algorithmic details are as follows:

• Feature Selection (Algorithm 1): Compute SVDs individually on $X$ and $Y$, obtain leverage scores for each, calculate leverage score ratios for all $d$ based on the above formula, sort features by these ratios, and select the top $c$ with the largest ratios.
• Sample Selection (Algorithm 2): Given the foreground restricted to the selected $c$ features, perform column subset selection followed by sample (row) selection via randomized or deterministic CUR selection strategies (e.g., leverage or norm-based sampling).

This approach efficiently implements joint feature and sample selection, maintaining scalability for large $n$ and $p$. Implementation is compatible with both batched SVD routines and randomized CUR methods.

4. Significance in Case-Control Studies

CCUR explicitly operationalizes contrastive analysis in case-control settings, where the central goal is to recover disease/treatment-specific genes or molecular markers and identify the subset of samples displaying such unique biology. By leveraging differences in the foreground versus background leverage scores, CCUR filters out features that are non-specific (high scores in both groups) and pinpoints those that are truly unique (high foreground, low background).

Similarly, sample selection (row selection in CUR) is restricted to those subjects whose data patterns are driven by the identified features, thus supporting the isolation of unique sample-specific responses to the condition under investigation. This dual selection mechanism addresses the shortcomings of traditional approaches, which may capture only global structure or confounded signals (Zhang et al., 15 Aug 2025).

5. Empirical Performance and Biological Applications

Extensive simulation and real-world experimental evaluations demonstrate that CCUR robustly recovers foreground-specific features and samples in contrast to standard CUR, CPCA, or feature selection based solely on loadings. Examples include:

• Mouse Protein Dataset: CCUR identifies genes such as Sod1, Tau, Il1b—genes connected to stress and neuronal changes in Down Syndrome—and preferentially selects mouse samples with Down Syndrome.
• Small Molecules Dataset: CCUR isolates cell lines (rows) manifesting wild-type p53 activation and selects regulatory genes in relevant pathways.
• Pathogen Infection Data: The method recovers unique immune-related genes and the infected sample cohorts.

Quantitatively, CCUR yields higher cumulative recovery of true foreground signals, demonstrating effectiveness in both synthetic and biological data settings. The selected features are enriched for biologically established functions associated with the case group.

6. Implications and Interpretability in Biomedical Research

By selecting actual genes and samples, rather than arbitrary linear combinations, CCUR provides superior interpretability for biomedical research—facilitating direct mapping from statistical output to biological hypothesis. Features that are sparse in the background and highly variant in the case group can be tracked for mechanistic and diagnostic relevance. Sample-level selection reveals heterogeneity, supporting subsequent exploration in single-cell or subject-level studies.

This method enables actionable follow-up, including experimental design, stratified analysis, and personalized medicine, by systematically narrowing findings to those features and samples exhibiting robust case-specific signatures.

A plausible implication is that CCUR may be further extended to multi-condition analysis or time-series studies by calculating leverage score ratios across multiple backgrounds or time windows, thereby generalizing the interpretability and analytical advantages for complex biomedical datasets.

7. Comparison with Other Contrastive and CUR-based Methods

CCUR distinctly departs from principal component, sparse PCA, and generic CUR methods by integrating contrastive leverage scoring with joint feature/sample selection tailored to foreground-background separation. Earlier methods in contrastive PCA or CPCA modify low-dimensional projection orientation but do not directly select sparse, interpretable subsets of features and samples.

Furthermore, the biological use-cases underscore CCUR's ability to recover context-dependent effects rather than global structure, supporting its role as a key tool in modern case-control processing frameworks (Zhang et al., 15 Aug 2025).

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Contrastive CUR (CCUR) formalizes a mathematically principled, interpretable strategy for dimension reduction and joint feature/sample selection in case-control studies, directly advancing the capacity for discovery and mechanistic interpretation in high-dimensional biomedical research.