Multidimensional Density Vector

• Multidimensional density vector is a mathematical construct that generalizes traditional density concepts using vector-valued functions, tensor expansions, and operator measures.
• It employs methods like multiwavelet expansions, low-rank tensor factorizations, and kernel-based estimators to address high-dimensional statistical challenges.
• Practical applications include quantum information, signal processing, and biomedical imaging, where these techniques enhance nonparametric inference and data reconstruction.

A multidimensional density vector is a mathematical and statistical construct that generalizes the concept of density—either in the sense of a probability density, a measure-valued function, or an operator—in settings where multiple components, variables, or attributes are considered simultaneously. Theoretical and applied research has developed a diverse array of methodologies for representing, estimating, and analyzing multidimensional density vectors, motivated by challenges in nonparametric inference, quantum information, statistical physics, signal processing, and high-dimensional data analysis.

1. Mathematical Representation and Theoretical Context

A multidimensional density vector may denote several related, but technically distinct, objects:

• Vector-valued density functions: For example, a probability density function $f: \mathbb{R}^d \to \mathbb{R}$, or more generally, $F: \Omega \to X$ for a Banach space $X$. When $F$ is Pettis or Bochner integrable with respect to a positive measure $\mu$, it defines a vector measure via $v_F(A) = \int_A F d\mu$ on $\delta$-rings, with properties such as semivariation and connections to $L^1$-spaces of vector measures (Avalos-Ramos, 2019).
• Density expansions in functional bases: In nonparametric estimation, densities are expanded in vector-valued bases. For instance, multiwavelet bases provide a representation

$$p(x) = \sum_k \underline{\alpha}_{j_0, k}^T \underline{\phi}_{j_0, k}(x) + \sum_{j = j_0}^J \sum_k \underline{\beta}_{j, k}^T \underline{\psi}_{j, k}(x),$$

where each $$\underline{\alpha}_{j_0, k}, \underline{\beta}_{j,k}$$ is a vector of coefficients and the basis functions $\underline{\phi}, \underline{\psi}$ are themselves vector-valued, preserving symmetry and orthogonality in the density representation (Locke et al., 2012).

• Spectral density vectors: In time series, the spectral density matrix $f(\omega)$ of a $d$-dimensional stationary process can be diagonalized, and its nonzero eigenvalues (smooth functions of frequency) and eigenvectors form the density vector at each frequency. This structure is crucial for factorization and for deriving the Wold representation for best linear prediction purposes (Szabados, 2023).
• Complex vector-valued measures and multifractal densities: In multifractal analysis, a vector-valued measure $\mu = (\mu_1, ..., \mu_k)$ is analyzed via local densities given by scaling limits involving products of the vector's components, such as

$$\overline{d}^{q,t}(x, \mu) = \limsup_{r \to 0} \frac{[\mu(B(x,r))]^q}{[v(B(x,r))]^t},$$

where $[\mu(B(x, r))]^q = \prod_i [\mu_i(B(x, r))]^{q_i}$ (Farhat et al., 2021).

2. Methodologies for Construction and Estimation

A diverse range of methods exists for constructing and estimating multidimensional density vectors, each tailored to the challenges of dimensionality, smoothness, boundary effects, and computational complexity:

• Tensor-product and low-rank tensor factorizations: Densities of multivariate variables $X = (X_1, ..., X_N)$ may be approximated via their characteristic functions, truncated to a finite tensor of Fourier coefficients

$$\Phi_{X}(k_1, ..., k_N) = \mathbb{E}[e^{j 2\pi (k_1 X_1 + ... + k_N X_N)}],$$

and further decomposed by canonical polyadic (CP) or Tucker factorizations to yield computationally parsimonious, low-rank representations (Amiridi et al., 2020, Vandermeulen, 2020). This enables estimation and inference in high dimensions by mitigating the curse of dimensionality, as the number of parameters scales linearly (or polynomially) rather than exponentially in $d$.

• Multiwavelet and orthogonal expansions: Multiwavelet density estimation uses bases of vector-valued, compactly supported, symmetric, and orthogonal functions. For the multidimensional case, tensor products of univariate bases provide efficient expansions, and the coefficients are projected directly onto these bases from data (Locke et al., 2012).
• Regularized spline-based models: When the data is sampled with heterogeneous sensitivity (e.g., in PET imaging or variable detector response), the density is modeled as $\hat{f}(x; c) = \exp(\sum_m c[m] Y_m(x))$ with tensor-product B-spline basis $Y_m(x)$. The estimation involves maximizing a log-likelihood term regularized by the nuclear norm of the Hessian, $R(c) \approx \sum_m \|H\{\log \hat{f}(c)\}(m)\|_*$ to control curvature and promote spatial adaptivity (Boquet-Pujadas et al., 2 Jun 2025).
• Kernel and tree-based estimators: In settings with complex or high-dimensional geometry, kernel density estimators (KDE) are adapted via boundary correction or through the use of an approximation PDF for relative KDE (Poluektov, 2014). Density estimation trees (DETs) partition the space adaptively and self-optimizing, providing nonparametric estimators that scale more favorably with data size and dimensionality (Anderlini, 2015).
• Voronoi and radial tessellation estimators: The Compactified Voronoi Density Estimator (CVDE) and Radial Voronoi Density Estimator (RVDE) exploit local geometric adaptiveness by partitioning space according to Voronoi cells and employing kernel or radial ansatz for continuity and computational tractability in high-dimensional settings (Polianskii et al., 2022, Marchetti et al., 2022).

3. Statistical, Physical, and Quantum Interpretations

Density vectors play crucial roles in the interpretation and analysis of structural and statistical properties in several domains:

• Quantum information: The entropy vector formalism encodes all marginal entropic information associated with a density matrix (state) in multipartite quantum systems. The entropy vector can characterize genuine multipartite entanglement, $k$-separability, decomposability, and entanglement dimensionality, using witness functions and convex-roof constructions (Huber et al., 2013).
• Stochastic dynamics and large deviations: In multidimensional stochastic systems (e.g., hyperbolic conservation laws), the cost of fluctuations in the density profile—quantified by a large deviation function—depends on the full structure of the density vector and transport coefficients. Explicit formulas for step-like profiles involve integration over the relevant directional components of the vector-valued diffusivity and mobility matrices (Barré et al., 2017).
• Signal processing and nonparametric statistics: Low-rank tensor-based density estimates are used for recovering distributions in sparse regimes, imputing missing data, and dimension reduction. Innovations are uniquely identified and exploited for best linear prediction in stationary time series via spectral factorization and Wold representation constituents (Szabados, 2023, Amiridi et al., 2020).
• Fractal geometry: Mixed multifractal densities for quasi-Ahlfors vector-valued measures allow for the exact computation of multifractal spectra in scenarios with multiple interacting attributes, leveraging the local $d^{q,t}(x, \mu)$ densities and global generalized measures (Farhat et al., 2021).

4. Computational and Regularization Strategies

The estimation of multidimensional density vectors is subject to significant practical and statistical challenges. Modern approaches address these by leveraging structure, regularization, and computational architecture:

• Spatial adaptivity and stability: Nuclear norm regularization of the Hessian of the log-density within a spline framework introduces affine regions in under-sampled areas, stabilizing the estimate and ensuring a form of locally adaptive bandwidth selection. This approach is robust against choices of the regularization parameter $\lambda$, unlike naïve smoothing parameter selection in KDE (Boquet-Pujadas et al., 2 Jun 2025).
• Variance-reduced sketching: Variance-Reduced Sketching (VRS) represents high-dimensional functions as infinite-size tensors and projects onto judiciously chosen subspaces to reduce estimator variance and mitigate the curse of dimensionality. Hilbert–Schmidt SVD and operator-theoretic tools (with projection-based sketching) yield minimax-optimal error rates for nonparametric estimation, as the rates depend on the dimension of the active variable and smoothness, not the full tensor order (Peng et al., 22 Jan 2024).
• Monte Carlo and efficient search: Estimators such as the CVDE or RVDE rely on efficient nearest-neighbor searches, spherical ray integration, or fast Newton–Raphson root-finding, and can be efficiently implemented using parallelizable routines like convolutional filtering or FFTs on regular grids (Polianskii et al., 2022, Marchetti et al., 2022, Boquet-Pujadas et al., 2 Jun 2025).

5. Applications and Empirical Demonstrations

The application of multidimensional density vectors and their estimation underpins modern practice across diverse scientific disciplines:

• Particle physics: Multidimensional efficiency profiles in decay kinematics are estimated using relative KDE with an approximation PDF to correct for biases at boundaries and preserve narrow structures in high-variance, high-dimensional phase spaces (Poluektov, 2014). DETs are used for modeling signal and background in multi-variable distributions for optimization and likelihood analyses (Anderlini, 2015).
• Neuroimaging and demography: Wasserstein covariance matrices quantify the dependency structure between multiple functional densities, as in the analysis of intra-regional brain connectivity (using fMRI) or the co-evolution of mortality densities across countries (Petersen et al., 2018).
• PET imaging: Sensitivity-aware density estimation using exponential B-splines regularized by the nuclear norm leads to improved sinogram rebinning and image reconstruction, outperforming traditional histogram or kernel-based methods, particularly in low-sensitivity or undersampled regions (Boquet-Pujadas et al., 2 Jun 2025).
• High-dimensional embeddings and local density queries: Locality-sensitive hashing–based estimators efficiently estimate point densities or local neighbor counts in high-dimensional vector spaces, with precise bounds on sample complexity and space, critical for applications in word embedding and network analysis (Wu et al., 2018).
• Causal inference in multidimensional RDD: Multidimensional density vectors of the assignment variable enter into tests for manipulation in regression discontinuity designs, where the continuity of each marginal density at the boundary is tested jointly to bolster causal identification (Crippa, 16 Feb 2024).

6. Theoretical Guarantees and Performance Bounds

A key concern across methodologies is the statistical efficiency and optimality of density vector estimation:

• Minimax lower bounds: Information-theoretic constructions establish that any algorithm achieving $\epsilon$-accuracy in $L_1$-distance must have sample complexity at least $\Omega((1/\epsilon)^d)$ for shift-invariant multidimensional densities, indicating that under natural regularity conditions the presented methods are rate-optimal up to polynomial factors in $d$ (De et al., 2018).
• Convergence and adaptivity: Estimators such as RVDE and CVDE provably converge to the true density without the need for vanishing bandwidth or kernel width, overcoming limitations of classical schemes in high dimensions (Polianskii et al., 2022, Marchetti et al., 2022).
• Separation rates in testing: For mixture detection in high-dimensional Gaussian settings, optimal detection boundaries for the multidimensional density vector alternative are given by $\epsilon \|\mu\| \gtrsim d^{1/4}/\sqrt{n}$ ($\ell_2$-norm) and $\epsilon \|\mu\|_\infty \gtrsim \sqrt{\log d/n}$ ($\ell_\infty$-norm) (Laurent et al., 2015).

7. Summary and Broader Implications

The notion of a multidimensional density vector serves as both a conceptual framework and a computational target for representing, estimating, and analyzing structure in complex, high-dimensional stochastic systems. Advances in vector-valued expansions, low-rank and tensor factorizations, spatially adaptive regularization, and efficient geometric and sketching-based estimators have enabled the systematic treatment of density vectors well beyond the limitations of classical histogram or kernel approaches. These tools find broad applicability in areas ranging from quantum information, functional data analysis, and statistical physics to biomedical imaging, machine learning, and econometric identification.

The development of multidimensional density vector methodologies continues to shape the modern landscape of high-dimensional statistics, nonparametric inference, and their applications to scientific and engineering problems characterized by large-scale, structured, and interactive data.