Cross-Component Analysis

• Cross-component analysis is a set of techniques that quantify, characterize, and exploit interdependencies among multiple data sources and system components.
• It employs methods like shared subspace factorization, correlation mapping, and cross-domain adaptation to extract joint and individual latent structures.
• Applications span biomedical data integration, software diagnostics, and multi-domain adaptation, enhancing model selection and system analysis.

Cross-component analysis refers to a broad class of techniques designed to quantify, characterize, and exploit structures that span multiple coupled data sources, system modules, or latent components. These methods articulate the statistical or functional interdependence among subsystems—whether between domain-partitioned data (as in multiview or multisubject biomedical arrays), algorithmic modules (as in cross-domain adaptation), or system components (e.g., software analysis tools, visualization frameworks, or neural network subparts). This survey presents foundational paradigms, methodological innovations, and exemplary applications from contemporary research on arXiv covering domains such as signal processing, machine learning, cognitive modeling, software analytics, and computational biomedicine.

1. Formalization and Theoretical Foundations

Central to cross-component analysis is the formal description of inter-component relations. In its canonical multiview statistical instantiation, one considers data matrices $X^{(v)} \in \mathbb{R}^{n \times p_v}$, $v=1,\ldots,V$ ("views" or "blocks"), and seeks to identify latent structures—common, individual, or correlated components—that capture both within-view variance and cross-view linkage. The structure of these relationships may involve:

• Shared subspace factorization (e.g., linked/joint factor models): Simultaneous decomposition of multiple $X^{(v)}$ emphasizing shared (joint) and view-specific (individual) factors, with constraints imposing agreement, decorrelation, or competition among components (Xiao et al., 17 Jun 2024, Zhou et al., 2015).
• Correlation mapping and rank determination: Using eigen/decomposition of a multi-block "coherence" matrix to establish the dimension and assignment of cross-dataset correlated subcomponents, formalized by counting eigenvalues $\lambda>1$ above unity and examining block-sparsity of associated eigenvectors (Hasija et al., 2019).
• Cross-domain adaptation: Linear projections (e.g., Domain Regularized Component Analysis) maximizing within-domain scatter while minimizing domain discrepancy, formulated as generalized Rayleigh quotients solvable via eigenproblems (Wang et al., 2021).

The guiding objective is not merely variance explanation, but robust characterization of which components are structurally or statistically coupled across components, potentially down to the specific configuration of inter-block linkages.

2. Methodologies and Algorithmic Frameworks

A diverse set of algorithmic tools arise across application areas:

• Joint Linked Component Analysis: Simultaneous estimation of view-specific loadings and rank of joint subspace, using penalized least squares with group penalties for automatic component selection. SVD representations of cross-covariance clarify the clean separation of joint/individual variance, and an explicit group-lasso term enables rank-adaptive procedures. Refitting debiases loadings post-selection (Xiao et al., 17 Jun 2024).
• Bootstrap-based Model Selection: After forming a normalized coherence matrix (whitened in block-diagonal structure), the ordering and blockwise sparsity of eigenvalues and eigenvectors reveal both the number and participation map of correlated components. Bootstrapping forms canonical null distributions for hypothesis testing (Hasija et al., 2019).
• Structured Sparsity and Co-clustering: Cross-product penalized objectives extend Sparse PCA to enforce structured sparsity not only on loadings but also on observation scores, using cross-product matrices encoding sample or feature group structure ("XCAN") (Camacho et al., 2019).
• Wavelet-based Multifractal Joint Analysis: Characterizes long-range cross-component correlations through joint scaling exponents and multifractal spectra, leveraging cross-wavelet transforms and Legendre transforms to capture the geometric structure of co-embedded singularities (Jiang et al., 2016).
• Component-aware Diagnostic Frameworks: In model evaluation (e.g., Transformer-based NLP), explicit partition of the evaluation workflow into per-component and cross-component diagnostic metrics (e.g., type-token ratio, representation silhoutte, entropy of tag distributions, and cross-metric deltas) enables granular analysis of system behavior (Younes, 30 Nov 2025).

3. Model Selection, Interpretation, and Structural Inference

Model selection in cross-component analysis is intrinsically more complex than in single-block or monolithic approaches due to the combinatorial space of possible alignment or correlation structures. State-of-the-art approaches include:

Method	Model Order Criterion	Assignment Recovery
Bootstrap Coherence EVD (Hasija et al., 2019)	p-value over null eigenspectrum	Block pattern of eigenvecs
Group Penalty in LCA (Xiao et al., 17 Jun 2024)	Penalty-driven rank selection	Nonzero pattern in $D_i$
Bayesian Tensor Factorization (Zhou et al., 2015)	Marginal likelihood, sparsity priors	Factor loading structure

Comprehensive interpretation in biomedical applications (e.g., EEG/fMRI block-coupled arrays) requires not only identification of joint and individual features, but their embedding in domain-specific constraints such as independence, nonnegativity, or temporal smoothness (Zhou et al., 2015).

4. Applications Across Domains

Cross-component analysis underpins several advanced analytical regimes:

• Cross-user and cross-domain adaptation: Domain Regularized Component Analysis addresses taste sensation recognition under sEMG by unsupervised projection, with demonstrated statistical improvements in cross-subject classification accuracy (Wang et al., 2021).
• Multi-block biomedical data integration: Joint and linked analysis methods (CCA, JIVE, CIFA) extract biomarkers and harmonize multi-modal recordings (EEG, MRI), supporting robust feature extraction and denoising across blocks (Zhou et al., 2015).
• Multifaceted visualization and system diagnostics: Formalizing data flows and interactions with cross-flow analysis graphs supports scalable, high-fidelity performance diagnosis in large software systems at both component and API granularity (Steven et al., 1 Sep 2024); visual component analysis decomposes layouts into focus/context/overview modules to support information-rich, flexible exploratory visualization (Guchev et al., 2023).
• Functional and time-varying factor models: Temporal cross-volatility matrices, estimated via quadratic variation or Fourier methods, reveal the geometry and real-time factor evolution (e.g., in financial modeling), with error and computational complexity characterized under competing estimation strategies (Liu et al., 2014).
• Neuroscience and cognition: Hierarchical Bayesian cross-component modeling explicitly quantifies cross-task latent parameter correlations, providing principled borrowing of strength and interpretable coherence metrics in cognitive process modeling (Wall et al., 2019).

5. Contemporary Innovations: Unifying PCA–CCA Spectra and Structural Penalties

Recent advances have conceived frameworks interpolating between variance-maximizing (PCA) and correlation-maximizing (CCA) extremes, such as Cooperative Component Analysis (CoCA), which balances within-view and cross-view objectives via tunable penalties, admits data-driven model selection of tuning parameters, and extends to sparse formulations that are provably more effective than classical methods for detecting shared structure in noisy or finite-sample settings (Ding et al., 23 Jul 2024). These approaches blur the boundary between data-driven latent structure discovery and hypothesis-driven model selection, enabling adaptive, interpretable, and validated inference of cross-component relations.

6. Limitations, Challenges, and Best Practices

Several domain-independent limitations and best practices are recognized:

• Identifiability: Recovery of fine-grained correlation structure requires assumptions (e.g., diagonal cross-covariances, eigenvalue simplicity) and may fail or merge groups for repeated or nearly identical cross-component strengths (Hasija et al., 2019).
• Computational Complexity: Cross-block or multi-tensor factorization, especially with bootstrapped model order selection or high-dimensional joint SVDs, is cubic or higher in the system size; blockwise methods and randomized algorithms can mitigate, but not eliminate, these costs (Hasija et al., 2019, Xiao et al., 17 Jun 2024).
• Domain Adaptivity: For practical deployment (e.g., in video coding or software profiling), lightweight or approximate realizations (e.g., multiplier-free LUT filters, O(1) binary instrumentation, or sub-sampled border regression) are implemented to maintain scalability and operational efficiency (Gao et al., 3 Jun 2024, Li et al., 2020, Steven et al., 1 Sep 2024).
• Interpretability: Visualization frameworks should maintain explicit decomposition into base, focus, and context modules, support multiple component correspondences, and embed semantic linking between data and visual subcomponents (Guchev et al., 2023, Younes, 30 Nov 2025).
• Evaluation and Validation: Empirical and simulation benchmarks, including cross-validation, bootstrapped statistical inference, and domain-specific performance metrics, are essential to validate model selection, structural recovery, and predictive or explanatory utility (Ding et al., 23 Jul 2024, Wang et al., 2021).

7. Future Prospects and Extensions

Ongoing directions include integration of Bayesian inference for automatic rank selection under uncertainty, extension to high-order tensor joint analysis for complex applications (e.g., spatio-temporal biomedicine, multimodal machine learning), and generalization of cross-component analysis to neural models and instance-level diagnostics (e.g., LLMs, advanced sequence models) (Zhou et al., 2015, Younes, 30 Nov 2025). Advancements in scalable algorithms for high-dimensional, multi-block, and streaming data scenarios will further shape the centrality of cross-component analysis in computational science.

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References:

• "Unsupervised cross-user adaptation in taste sensation recognition based on surface electromyography with conformal prediction and domain regularized component analysis" (Wang et al., 2021)
• "Determining the Dimension and Structure of the Subspace Correlated Across Multiple Data Sets" (Hasija et al., 2019)
• "Joint Linked Component Analysis for Multiview Data" (Xiao et al., 17 Jun 2024)
• "Cross-product Penalized Component Analysis (XCAN)" (Camacho et al., 2019)
• "Linked Component Analysis from Matrices to High Order Tensors: Applications to Biomedical Data" (Zhou et al., 2015)
• "CoCA: Cooperative Component Analysis" (Ding et al., 23 Jul 2024)
• "Scaler: Efficient and Effective Cross Flow Analysis" (Steven et al., 1 Sep 2024)
• "Multifractal cross wavelet analysis" (Jiang et al., 2016)
• "Identifying relationships between cognitive processes across tasks, contexts, and time" (Wall et al., 2019)
• "Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis" (Liu et al., 2014)
• "Combining Multiple View Components for Exploratory Visualization" (Guchev et al., 2023)
• "DeformAr: Rethinking NER Evaluation through Component Analysis and Visual Analytics" (Younes, 30 Nov 2025)
• "Video Coding with Cross-Component Sample Offset" (Gao et al., 3 Jun 2024)
• "Sub-sampled Cross-component Prediction for Emerging Video Coding Standards" (Li et al., 2020)