Extended Mahalanobis Norm

• Extended Mahalanobis Norm is a family of covariance-adaptive metrics that generalizes classical distance measures to settings with singular, infinite-dimensional, or structured covariances.
• It utilizes techniques like spectral truncation, Tikhonov regularization, and RKHS kernel mappings to handle functional data and operator-level extensions effectively.
• Applications include enhanced classification, anomaly detection, and manifold learning, supported by consistent statistical inference and computational advances.

The extended Mahalanobis norm encompasses a family of covariance-adaptive norms and distances that generalize the classical Mahalanobis distance well beyond its origin in finite-dimensional Gaussian analysis. These extensions are motivated by practical, mathematical, and statistical challenges—such as singular or infinite-dimensional covariance, functional data, the incorporation of kernel methods, block- and operator-structured geometries, or non-centered coordinate systems. The unifying theme is a norm (or metric) of the general form $$\|x - m\|_{M^{-1}} = [(x{-}m)^T M^{-1} (x{-}m)]^{1/2}$$, interpreted in settings where $M$ is estimated, regularized, structured, or infinite-dimensional. This article delineates formal definitions, structural generalizations, computational approaches, and core properties of extended Mahalanobis norms as established in recent research.

1. Formal Definitions and Canonical Extensions

1.1 Finite-Dimensional Classical and Extended Norms

The classical Mahalanobis norm for $x \in \mathbb{R}^N$ with mean $m \in \mathbb{R}^N$ and positive-definite covariance $\Sigma$ is given by: $$\|x - m\|_{\Sigma} = \sqrt{(x-m)^T \Sigma^{-1} (x-m)}$$ This norm induces an ellipsoidal geometry respecting the scale and correlation structure of $\Sigma$.

Extended Mahalanobis Norm for Non-mean Origin (Spurek et al., 2013): Given an arbitrary origin $m$ (not necessarily $m_Y$), the extended norm re-optimizes the covariance for the fixed $m$ by minimizing the Gaussian cross-entropy of the data relative to $(m, \Sigma)$. The solution is: $$\Sigma(m) = \Sigma_Y\left(\Sigma_Y - \frac{(m - m_Y)(m - m_Y)^T}{1 + \|m - m_Y\|_{\Sigma_Y}^2}\right)^{-1}\Sigma_Y$$ Defining: $$\|x - m\|_{\mathrm{ext}} = \sqrt{(x - m)^T \Sigma(m)^{-1} (x - m)}$$ yields the optimal Mahalanobis-type norm for any origin.

1.2 Infinite-Dimensional and Functional Extensions

Functional Mahalanobis Semi-distance (Joseph et al., 2013): Let $H = L^2(T)$ and let $\Gamma_\chi$ be the covariance operator. Since $\Gamma_\chi^{-1}$ is unbounded, one regularizes by truncating to the first $K$ eigenmodes: $$\Gamma_K^{-\frac12}(\zeta) = \sum_{k=1}^K \lambda_k^{-1/2} \langle \zeta, \psi_k \rangle \psi_k$$ The functional Mahalanobis semi-distance between $f, g \in H$ is: $d_{FM}^K(f, g) = \|\Gamma_K^{-1/2}(f - g)\|$

RKHS-based Mahalanobis Norm (Berrendero et al., 2018): Given a continuous covariance kernel $K(s, t)$, the RKHS norm (extended Mahalanobis) is: $$\|f\|_K^2 = \sum_{i=1}^{\infty} \frac{\langle f, e_i \rangle^2}{\lambda_i}$$ For arbitrary $x \in L^2$, a regularized projection $x_\lambda$ is defined as: $$x_\lambda = (K + \lambda I)^{-1}K x = \sum_{j=1}^\infty \frac{\lambda_j}{\lambda_j + \lambda}\langle x, e_j \rangle e_j$$ The corresponding metric is $d_M(x, g; \lambda) = \|x_\lambda - g_\lambda\|_K$.

Variance (Cameron–Martin) Norm (Zozoulenko et al., 16 Jul 2024): For a probability measure $\mu$ on Banach space $V$ with covariance form $q$, the variance norm is: $$\|x\|_{\mu\text{-cov}} = \sup_{f \in V^*: q(f, f) \le 1} f(x)$$ In Hilbert space, the Cameron–Martin space $H_\mu$ has: $$\|h\|^2_{\mu\text{-cov}} = \sum_{n:\lambda_n>0} \frac{\langle h, e_n \rangle^2}{\lambda_n}$$ with Tikhonov regularization for non-injective covariance.

Operator-level (Unitized Hilbert–Schmidt) Extensions (Goomanee et al., 12 Nov 2025): For $X, M$ (compact, self-adjoint operators) on Hilbert space $\mathcal{H}$, the "extended Mahalanobis norm" is defined via: $$\|X + \mu I\|_{M_\infty^{-1}} = \left[\mathrm{tr}\{(X + \mu I) (M + \rho I)^{-1} (X + \mu I)\}\right]^{1/2}$$ where $\mu, \rho > 0$ ensure positive-definiteness and well-posedness.

2. Structural and Algorithmic Generalizations

2.1 Block and Covariance-structure Adaptation

The clustering-informed Mahalanobis norm (Lahav et al., 2017) leverages the structure of high-dimensional data, exploiting coordinate clusters via k-means. Rows of the data are grouped into $K+1$ clusters; the leading principal directions are projected onto cluster indicator spaces, creating a block-structured covariance estimator: $$\Sigma_C = \widetilde U_K \Lambda_K \widetilde U_K^T$$ Distance computations then employ $\Sigma_C^{-1}$, yielding improved stability and reduced estimation error, especially for $m \gg n$.

2.2 Operator and Kernel Generalizations

In operator-based frameworks (Goomanee et al., 12 Nov 2025), covariance and data may be infinite-dimensional or non-invertible, and the Mahalanobis norm is defined via trace inner products in extended Hilbert–Schmidt algebras, ensuring regularity via additive identity operators. Kernel-based (RKHS) formulations allow the extension of Mahalanobis-type distances to feature spaces for applications such as time-series anomaly detection (Zozoulenko et al., 16 Jul 2024).

2.3 Regularization and Consistency

Regularization is essential for the practical and theoretical tractability of the infinite-dimensional case. Approaches include spectral truncation (Joseph et al., 2013), Tikhonov regularization (Berrendero et al., 2018, Zozoulenko et al., 16 Jul 2024), and additive ridge shifts (Goomanee et al., 12 Nov 2025). Empirical consistency is established for the variance norm and plug-in estimators, with uniform convergence in operator norm as sample size increases.

3. Theoretical Properties

Property	Classical Mahalanobis	Functional/Infinite-Dim	Cluster-informed / Operator extension
Invariance	Orthonormal / Affine	Isometric (RKHS)	Unitary (operator); block structure (k-means/PCA)
Definiteness	Positive	Semi-metric/Metric	Metric (under regularization)
Consistency	Yes	Plug-in (a.s.) (Berrendero et al., 2018, Zozoulenko et al., 16 Jul 2024)	Yes (if regularized spectrum)
Computational Cost	$O(N^3)$	Spectral truncation, $O(M^2)$	Truncated SVD, operator inverse, Nyström approx
Regularization Parameter	Not needed	$K$ (truncation), $\lambda$ (Tikhonov)	$\mu$, $\rho$ (operator shift); learnable

Extended norms retain invariance properties analogous to the classical case: under orthonormal or unitary transformation, the norm is preserved. The RKHS and operator-level extensions ensure that the metric structure is preserved up to isometry or unitary equivalence. Regularized versions are designed to yield finite values even when the covariance operator is non-injective.

4. Computational Implementation

Finite-dimensional extended Mahalanobis norms require, at most, a rank-one update and matrix inversion (e.g., via Sherman–Morrison–Woodbury formula (Spurek et al., 2013)). Functional and operator extensions are practically realized via:

• Basis Expansion/Discretization: Observed functions are projected onto a finite basis (Fourier, B-splines, wavelets), covariance operators are estimated via empirical basis coefficients.
• Spectral Truncation or SVD: Only the leading $K$ components are retained; distances are then computed via $\ell^2$ norms of standardized scores (Joseph et al., 2013).
• RKHS Gram Matrices: Covariance is constructed in feature space by computing (centered) Gram matrices (Zozoulenko et al., 16 Jul 2024).
• Operator Approximation: Eigen-decomposition and spectrum truncation are applied to compact operators; the extended Mahalanobis norm is approximated by a finite sum over leading eigenpairs (Goomanee et al., 12 Nov 2025).

Regularization parameters ($K$, $\lambda$, $\rho$) balance bias and variance: smaller values capture more structure but risk instability; larger values provide numerical robustness and possibly isotropy.

5. Applications and Empirical Performance

Extended Mahalanobis norms have demonstrable advantages in:

• Classification and Outlier Detection: Functional data classification, using the functional Mahalanobis semi-distance, achieves higher accuracy and robustness against alternative distances (e.g., $L^2$, unstandardized fPC) in both simulated and real data (Joseph et al., 2013, Berrendero et al., 2018).
• Anomaly and Novelty Detection in High-Dimensional and Infinite-Dimensional Settings: The operator and kernelized variants (variance norm, operator-based norm) provide principled, covariance-adaptive anomaly metrics in Banach, Hilbert, and RKHSs (Zozoulenko et al., 16 Jul 2024, Goomanee et al., 12 Nov 2025).
• Manifold Learning, Embedding, and Clustering: Clustering-informed Mahalanobis metrics improve the recovery of latent manifold structures, accurately recovering latent Euclidean geometry in nonlinear generative models and improving Kaplan–Meier risk separation in gene expression studies (Lahav et al., 2017).
• Statistical Inference on SPD Manifolds: In operator-valued settings, the extended Mahalanobis norm enables generalized Procrustes, Bures–Wasserstein, and log-Hilbert–Schmidt distances for robust geometric comparison between infinite-dimensional SPD operators (Goomanee et al., 12 Nov 2025).

Empirical results confirm improvements—often nontrivial—over classical or unstructured approaches, with the variance norm and cluster-informed operator regularization yielding state-of-the-art or near-optimal results on challenging benchmarks, including multivariate time series, functional classification, and biological data stratification.

6. Limitations and Parameter Selection

• Origin Sensitivity: Extended norms that relocate the origin ($m \neq m_Y$) bring extra flexibility at the cost of requiring a meaningful, data-independent reference point (Spurek et al., 2013).
• Parameter Tuning: Regularization parameters must be chosen judiciously. Strategies include cross-validation, maximizing statistical power, or optimizing geometric stability (Berrendero et al., 2018, Goomanee et al., 12 Nov 2025).
• Computational Complexity: High- or infinite-dimensional settings necessitate spectral truncation or approximation, which introduces trade-offs between fidelity and cost; operator inversion and trace computation may become computational bottlenecks if not approximated efficiently.
• Non-metricity and Semi-distances: Some function-space extensions yield semi-distances rather than strict metrics unless additional regularity or injectivity conditions are imposed (Joseph et al., 2013).

7. Future Directions and Theoretical Integration

The extended Mahalanobis norm framework subsumes a broad class of covariance-sensitive geometries, kernel-induced distances, and operator-theoretic metrics. Ongoing developments include:

• Unified Operator and Kernel Frameworks: Integration of variance norm/Cameron–Martin approach with operator-based metrics enables a seamless passage between finite and infinite-dimensional settings (Zozoulenko et al., 16 Jul 2024, Goomanee et al., 12 Nov 2025).
• Learnable Regularization and Adaptive Geometry: Operator-level and block-structured norms allow embedding of learnable hyperparameters (e.g., $\rho$) to adaptively regularize geometry for improved stability and task-specific performance (Goomanee et al., 12 Nov 2025).
• Empirical Consistency and Statistical Inference: Recent work rigorously establishes convergence rates, plug-in consistency, and the sampling distribution of extended Mahalanobis distances under broad conditions, enabling principled statistical inference (Berrendero et al., 2018, Zozoulenko et al., 16 Jul 2024).
• Applications in Functional, Structured, and Structured Data Science: Extended Mahalanobis norms underlie robust classification, clustering, and detection methods in functional data analysis, biomedical analysis, high-dimensional genomics, SPD geometry, and more.

A plausible implication is that as the mathematical, computational, and statistical theory of extended Mahalanobis norms matures, their role as the default metric structure in high- and infinite-dimensional data analysis will continue to expand, often bridging classical multivariate analysis, functional data, RKHS methodology, and operator-theoretic geometry.