Curvature-Adjusted PCA (CA-PCA)

• CA-PCA is a technique that extends traditional PCA by explicitly modeling second-order curvature effects to improve intrinsic dimension estimation.
• It employs quadratic regression methods like OLS and TLS to capture the bending of data manifolds and correct linear bias.
• CA-PCA outperforms classical PCA in handling curved and nonlinear data, thereby enhancing manifold learning and representation tasks.

Curvature-Adjusted Principal Component Analysis (CA-PCA) is a family of techniques that generalize classical Principal Component Analysis (PCA) to better handle data that reside on or near nonlinear manifolds, explicitly incorporating effects due to curvature. Unlike conventional PCA, which implicitly assumes locally flat (Euclidean) geometry, CA-PCA corrects for local second-order (curvature-driven) deviations from the tangent space, thereby improving the fidelity of dimension estimation, representation learning, and regression on curved manifolds. CA-PCA developments span a range of methodological innovations, including theoretical analysis, regression-based curvature modeling, and validation strategies.

1. Conceptual Foundations and Mathematical Framework

Classical (global) PCA presumes that data are distributed close to a linear subspace of ℝᵖ and measures intrinsic dimension by the rank of the sample covariance and the decay of its eigenvalues. Local PCA extends this to neighborhood-based tangent space approximations. However, both approaches are predicated on local flatness: they ignore or only implicitly account for nonlinearity and curvature in the data manifold.

CA-PCA addresses this limitation by including second-order (curvature) effects in the local approximation. Given a d-dimensional Riemannian manifold M embedded in ℝᵖ, a sufficiently small neighborhood around a point x₀ ∈ M can be locally modeled as a graph:

$x' = (x'_1, ..., x'_d, \; g(x'_1, ..., x'_d)),$

where the first d coordinates span Tₓ₀M (the tangent space), and the function $g: \mathbb{R}^d \to \mathbb{R}^{p-d}$ captures deviation from the linear tangent approximation. In CA-PCA, this deviation is systematically modeled using quadratic (second-order Taylor) expansions:

$$q_\ell(x'_1, ..., x'_j) = (x'_1, ..., x'_j)^T Q_\ell (x'_1, ..., x'_j),$$

where $Q_\ell$ is a symmetric matrix capturing curvature in the $\ell$-th normal direction. The resulting model thus corrects the tangent approximation via an explicit quadratic embedding, rather than assuming $g \equiv 0$ as in traditional PCA or local PCA (Bi et al., 16 Oct 2025, Gilbert et al., 2023).

2. Methodology: Regression-Based Curvature Modeling

CA-PCA fits a quadratic model to local PCA coordinates, representing both tangent (linear) and normal (curvature) directions. The primary procedural workflow is:

• For each point $x_0$, extract its $k$-nearest neighbor neighborhood.
• Center the neighborhood at $x_0$ and apply PCA to obtain principal (tangent) directions as the first d components.
• Represent points in local PCA coordinates and fit quadratic models to the remaining $p-d$ "normal" components using the tangent coordinates as inputs.
• For each normal direction $\ell$, solve for $Q_\ell$ (or equivalently a third-order tensor $\mathcal{Q}$) via regression.

Two representative estimation techniques are employed within this CA-PCA framework:

Estimator	Model	Goodness-of-fit Test
Quadratic Embedding (QE)	OLS regression in one output direction	F-statistic and p-value for quadratic term significance
Total Least Squares (TLS)	TLS regression in orthogonal representation	Relative drop in orthogonal error when increasing modeled dimension

The intrinsic dimension $d$ is declared as the smallest $j$ for which the regression-based curvature model achieves a significant improvement in fit, as indicated by F-statistic jumps (QE) or a sharp drop in TLS error (Bi et al., 16 Oct 2025).

3. Improvements over Classical and Local PCA

Traditional PCA and local PCA estimate intrinsic dimension through eigenvalue decay, implicitly assuming local flatness. This leads to instability or overestimation when neighborhoods are large and curvature is appreciable. CA-PCA’s explicit quadratic model corrects the linear bias by capturing nontrivial second-order dependencies (i.e., the “bending” of the manifold in normal directions).

In practice:

• For neighborhoods where the manifold is nearly flat, CA-PCA and local PCA agree.
• As neighborhood size increases or for highly curved data, CA-PCA remains stable whereas local PCA estimates increase with neighborhood radius.
• In cases of nonlinear or deformed manifolds (e.g., deformed spheres), CA-PCA maintains dimensional consistency, whereas classical methods are misled by curvature-induced variance in tail eigenvalues (Gilbert et al., 2023).

4. Integration with Quadratic Embedding (QE) and Total Least Squares (TLS)

Within the CA-PCA paradigm, QE and TLS serve as concrete implementations for curvature modeling and intrinsic dimension estimation:

• Quadratic Embedding (QE): An OLS regression fits a quadratic model to one of the output (normal) dimensions. Significance is assessed via F-statistic and p-value. A significant jump at $j = d$ indicates the right number of tangent coordinates.
• Total Least Squares (TLS): TLS minimizes orthogonal error and is robust to noise in predictors and responses. The key diagnostic is the maximum relative drop in total TLS error when increasing $j$, with the peak at $d$.

These approaches are algorithmically summarized in the source and provide efficient means to select the correct dimension through regression rather than solely eigenvalue thresholding (Bi et al., 16 Oct 2025).

5. Empirical Performance and Comparative Results

Experimental benchmarks on both synthetic and real datasets demonstrate that CA-PCA and its regression-based variants provide superior performance compared to classical local PCA, especially for curved and nonlinearly embedded data. Key findings include:

• Local PCA estimates drift upward as neighborhood size increases, violating the flatness assumption, while CA-PCA, QE, and TLS estimates are stable.
• In high ambient dimension or highly curved scenarios, CA-PCA, QE, and TLS provide more accurate and less variable estimates.
• QE and TLS frequently outperform state-of-the-art alternatives including TwoNN and DanCo, particularly with limited samples or strong curvature (Bi et al., 16 Oct 2025).

6. Applications and Practical Implications

Curvature-adjusted PCA and its extensions apply broadly to the estimation of intrinsic dimension of data manifolds, manifold learning, and geometric representation. Explicit modeling of curvature enables:

• More robust and accurate estimation of intrinsic dimension in practical manifold learning workflows.
• Correction of bias introduced by curvature in traditional local PCA.
• Improved performance in non-Euclidean embedding tasks and geometric data analysis.
• Direct extension to regression-based frameworks for handling noise and capturing higher-order dependencies.

7. Theoretical and Methodological Significance

CA-PCA provides a principled means for accommodating curvature in intrinsic dimension estimation on general Riemannian manifolds. It establishes a clear mathematical relationship between variance in normal directions and local curvature, and it enables the use of statistical regression tools to augment geometric analysis. Empirical evidence shows superior consistency and interpretability in both synthetic and real-world nonlinear geometric data scenarios. This perspective generalizes naturally to a range of other manifold learning and dimension reduction applications, suggesting substantial potential for future research and adaptation across various disciplines (Gilbert et al., 2023, Bi et al., 16 Oct 2025).