Multi-Gaussian Expansion

• Multi-Gaussian Expansion is a method that represents complex functions and distributions as weighted sums of Gaussians, combining analytic tractability and high flexibility.
• It employs semi-linear inversion and non-linear optimization to accurately model astrophysical profiles, gravitational potentials, and quantum chemistry kernels.
• Its analytic properties enable efficient PSF convolution and integral evaluation, making MGE a powerful tool in galaxy dynamics and probabilistic modeling.

The multi-Gaussian expansion (MGE) is a method of representing complex functions, fields, or distributions as weighted sums of Gaussian components, exploiting the analytic and computational advantages of the Gaussian basis. MGEs are widely adopted in astrophysics for modeling galaxy surface brightness profiles and gravitational potentials, in quantum chemistry for kernel approximations, and more generally in probabilistic modeling and expansion of probability densities. The approach is characterized by high flexibility, analytic tractability for convolution and integration, and the capacity to approximate smooth target functions or distributions to arbitrary accuracy with a moderate number of components.

1. Mathematical Formalism and Core Concepts

The archetypal MGE represents a two-dimensional surface brightness profile $\Sigma(x', y')$ or image $I(x,y)$ as a sum of $N$ elliptical Gaussian functions: $$\Sigma(x',y') = \sum_{j=1}^N \frac{L_j}{2\pi\,\sigma_j^2\,q'_j} \exp\left(-\frac{1}{2\sigma_j^2}\left[x'^2 + \frac{y'^2}{q_j'^2}\right]\right)$$ where:

• $L_j$ is the luminosity or flux normalization,
• $\sigma_j$ is the width (dispersion) along the major axis,
• $q_j'$ is the projected axis ratio.

For three-dimensional deprojection (oblate axisymmetry), each 2D Gaussian maps to a unique 3D Gaussian: $$\nu(R,z) = \sum_{j=1}^N \frac{L_j}{(2\pi)^{3/2}\,\sigma_j^3\,q_j} \exp\left(-\frac{1}{2\sigma_j^2}\left[R^2 + \frac{z^2}{q_j^2}\right]\right)$$ with the intrinsic axial ratio $q_j$ determined from observed $q_j'$ and the inclination $i$ via

$q_j^2 = \frac{q_j'^2 - \cos^2 i}{\sin^2 i}\,.$

Analogous constructions exist for univariate and multivariate probability density expansions and kernel representations.

In probabilistic settings, the multi-Gaussian (MG) expansion generalizes the normal law as a (possibly infinite) signed sum or mixture: $$p_X^{(\text{MG})}(x) = \frac{1}{C_0(M)\,\sigma\sqrt{2\pi}} \sum_{m=1}^M w_m \exp\left(-\frac{(x-\mu)^2}{2\sigma_m^2}\right)$$ where the weights $w_m$ and widths $\sigma_m = \sigma/\sqrt{m}$ are analytically determined by the expansion parameter $M$ and its combinatorial factors (Korotkova, 2020).

2. Fitting Methodologies and Computational Strategies

Optimizing an MGE involves selecting the widths, centers, axial ratios, and amplitudes of the Gaussian components to best reproduce observed data or target functions. Key fitting techniques include:

• Semi-linear inversion: For fixed Gaussian centers, widths, and shapes, the amplitudes are determined by solving a linear least-squares system, optionally with non-negativity constraints and quadratic regularization (He et al., 24 Mar 2024).
• Non-linear optimization: Parameters such as centers, axis ratios, and position angles are optimized iteratively, often grouped into "sets" to limit the non-linear subspace (He et al., 24 Mar 2024).
• Pre-rendered basis for images: In Bayesian fitting pipelines, the pre-computation of PSF-convolved, pixel-integrated Gaussians enables model evaluation as a matrix-vector multiplication, accelerating parameter inference by orders of magnitude (Miller et al., 2021).

For approximation of radial kernels such as the Yukawa kernel $K_Y(r) = e^{-\omega r}/r$, a least-squares fit is performed in a functional space: $$K_Y(r) \approx \sum_{p=1}^M c_p \frac{e^{-\omega_p r^2}}{r}$$ where the coefficients $c_p$ and exponents $\omega_p$ are optimized by analytic elimination and a subsequent non-linear minimization (Sarcinella et al., 2023).

3. Applications in Astrophysical and Physical Modeling

Galaxy Morphology and Dynamics

• Surface brightness modeling: MGE provides a non-parametric yet analytic model for galaxy light profiles, offering superior flexibility over single-component profiles (e.g., Sérsic) (Miller et al., 2021, He et al., 24 Mar 2024).
• Galaxy dynamics and mass modeling: The Jeans Anisotropic MGE (JAM) approach leverages the analytic potential of 3D MGEs to solve axisymmetric Jeans equations, allowing dynamical inference of mass profiles, anisotropy, and dark matter fractions. Recovery of total mass within $2.5\,R_e$ is robust (~10% scatter for oblate systems), though decomposition into stellar vs. dark matter remains degenerate (30–50% scatter), with sensitivity to data quality and galaxy shape (Li et al., 2015).
• Gravitational lensing: Embedding MGE models for lens galaxy light within semi-linear source inversion schemes yields unbiased mass parameter recovery and residuals below 5%, outperforming standard multiple-Sérsic models, especially for noise propagation and capturing isophotal complexity (radial twists, boxiness/diskiness) (He et al., 24 Mar 2024).

Kernel and Functional Expansion in DFT

• Yukawa kernel expansion: The MGE enables efficient implementation of non-local functionals in real-space quantum chemistry codes using Gaussian integral technology by expressing $e^{-\omega r}/r$ as a compact sum of screened Coulomb Gaussians. This expansion achieves tunable accuracy ($10^{-4}$ to $10^{-8}$) and is density-independent for the homogeneous electron gas (Sarcinella et al., 2023).

Extended Probabilistic Modeling

• General probability densities: MG expansions provide a rigorous analytic family interpolating between flat-topped (for $M>1$) and cusp-like (for $0Korotkova, 2020).
• Multivariate and log-transformed distributions: The formalism extends naturally to the multivariate case and to log-normal mixtures (LMG), supporting a broader class of probabilistic models.

4. Analytic Properties and Tractability

MGEs are analytically closed under convolution, linear transforms, and multiplication by polynomials, yielding:

• Analytic PSF convolution: Convolution of an MGE with a PSF (itself an MGE) preserves the sum-of-Gaussians form, with resultant widths and axial ratios computed by quadrature (Miller et al., 2021).
• Integral evaluation: Surface integrals, potential calculation, and projection are reduced to analytic or efficiently vectorized forms, facilitating computation in dynamical and lensing applications (Li et al., 2015, He et al., 24 Mar 2024).
• Variance and moment control: In the MG framework, higher-order terms modulate the kurtosis and flattening, continuously deforming the shape between Gaussian and non-Gaussian extremes (Korotkova, 2020).

5. Performance, Accuracy, and Practical Considerations

Empirical studies demonstrate that:

• Photometric modeling: MGEs recover galaxy total fluxes and effective radii with sub-percent bias and can reproduce surface-brightness and color profiles over a broad dynamic range, including strong color gradients and complex morphologies (Miller et al., 2021).
• Dynamical modeling: For axisymmetric galaxies with high-resolution imaging, the JAM-MGE pipeline constrains total enclosed mass within 10%; however, the accuracy is degraded for prolate/triaxial systems and at lower resolution (Li et al., 2015).
• Kernel expansions: Gaussian expansions of kernels converge exponentially with the number of terms, and a moderate number ($M=6$–9) achieves kernel errors $<10^{-6}$, well below methodological errors in DFT applications (Sarcinella et al., 2023).
• Limitations: MGEs are less effective for highly asymmetric profiles (e.g., pronounced spiral arms, tidal disturbances) and in the presence of strong triaxiality or insufficient spatial resolution; the fitting of negative Gaussians can introduce overfitting artifacts unless constrained (He et al., 24 Mar 2024).
• Computational efficiency: Pre-rendered basis sets and semi-linear inversion enable rapid fitting even for large image datasets and are implemented in open-source tools such as imcascade (Miller et al., 2021).

6. Extensions, Variants, and Future Directions

The MGE formalism is actively extended to:

• Complex photometric decomposition: Fitting of radially varying ellipticity/position-angle, inclusion of isophote higher-order terms, and two-component (stars + dark halo/ISM) joint modeling (He et al., 24 Mar 2024).
• Automated large-sample pipelines: Integration of MGE-based mass/light modeling into pipelines for surveys (Euclid, Rubin) is ongoing, aiming at precise structural and population inferences at scale.
• Multiband and multi-modal expansions: Incorporation of color gradients, multiband imaging data, and adaptation to spectroscopic modeling or more general kernel approximations in electronic structure theory (Sarcinella et al., 2023).
• Free-form and non-parametric adaptations: Use of Gauss–Hermite or other expansions to flexibly represent lopsidedness, bars, and dust features—potentially addressing the remaining limitations in reproducing strongly asymmetric systems (He et al., 24 Mar 2024).

A plausible implication is that the analytic, modular, and data-driven nature of the MGE approach will remain central in high-precision photometric, kinematic, and lens modeling, as well as in computational physics applications requiring tractable non-local kernel representations.