A Symplectic Method to Generate Multivariate Normal Distributions

Abstract: The AMAS group at the Paul Scherrer Institute developed an object oriented library for high performance simulation of high intensity ion beam transport with space charge. Such particle-in-cell (PIC) simulations require a method to generate multivariate particle distributions as starting conditions. In a preceeding publications it has been shown that the generators of symplectic transformations in two dimensions are a subset of the real Dirac matrices (RDMs) and that few symplectic transformations are required to transform a quadratic Hamiltonian into diagonal form. Here we argue that the use of RDMs is well suited for the generation of multivariate normal distributions with arbitrary covariances. A direct and simple argument supporting this claim is that this is the "natural" way how such distributions are formed. The transport of charged particle beams may serve as an example: An uncorrelated gaussian distribution of particles starting at some initial position of the accelerator is subject to linear deformations when passing through various beamline elements. These deformations can be described by symplectic transformations. Hence, if it is possible to derive the symplectic transformations that bring up these covariances, it is also possible to produce arbitrary multivariate normal distributions without Cholesky decomposition. The method allows the use of arbitrary uncoupled distributions. The functional form of the coupled multivariate distributions however depends in the general case on the type of the used random number generator. Only gaussian generators always yield gaussian multivariate distributions.

• The paper introduces a unique symplectic transformation framework using RDMs to generate MVN distributions while preserving covariance properties.
• The method decouples and couples independent Gaussian samples through a finite sequence of symplectic transformations, enabling accurate covariance reconstruction.
• Empirical tests validate the algorithm's accuracy in replicating targeted covariances, with applications in beam physics and Hamiltonian dynamics simulations.

Symplectic Methods for Generating Multivariate Normal Distributions

Introduction

This paper, "A Symplectic Method to Generate Multivariate Normal Distributions" (1205.3601), introduces an alternative computational framework for generating multivariate normal (MVN) distributions with specified covariance structure, leveraging symplectic transformations and the algebra of real Dirac matrices (RDMs). Unlike canonical approaches such as Cholesky decomposition, this method is naturally motivated by the preservation of covariances under linear (particularly symplectic) transformations, exploiting the formal structure encountered in Hamiltonian mechanics and coupled linear optics.

Symplectic Structure and Decoupling

The method arises from the context of coupled linear dynamics, as in charged particle beam optics, where state vectors combine canonical position and momentum variables, and evolution is dictated by quadratic Hamiltonians. The transport (or force) matrices and the covariance (second moments, $\sigma$) matrices can be expressed and manipulated within the symplectic group, with the RDMs serving as a basis for all real $4 \times 4$ matrices and, more specifically, for characterizing symplectic generators ("symplices").

The decoupling transformation, previously developed for linear Hamiltonians, is inverted to generate desired coupled second-moment (covariance) matrices from initial diagonal (uncorrelated) covariance—thus, independent Gaussian variables are coupled via a sequence of symplectic similarity transformations, determined by the RDM parameterization of the target covariance.

Algorithmic Construction

The procedure for constructing an MVN sample with arbitrary covariances is as follows:

1. Diagonalization via RDM Symplectic Decomposition: The desired covariance matrix $\sigma$ is transformed to a diagonal form using a finite, explicitly constructible sequence of at most six symplectic transformations, realized as matrix exponentials of generators (RDMs). The parameters and sequence are computed iteratively using auxiliary quantities derived from the RDM representation of the covariance.
2. Generation and Coupling: Independent Gaussian samples are created, scaled to the variances specified by the diagonalized second-moment matrix, then mapped to the target distribution by applying the inverse of the transformation sequence (the computed symplectic transport matrix).
3. Iterative Extension: The basic algorithm handles variables in canonical pairs (with $4 \times 4$ matrices), with higher dimensionality managed by iterative blockwise application, analogous to Jacobi diagonalization for symmetric matrices.

Unlike the Cholesky or eigen-decomposition, the approach maintains canonical invariance and allows efficient simulation of dynamics or distributions within phase space, with immediate application to particle-in-cell (PIC) codes and high-intensity beam physics.

Numerical Validation and Properties

Empirical tests demonstrate the method accurately reconstructs the prescribed covariance from random samples, as evidenced by explicit numerical results in six-dimensional cases. The produced covariance matrices closely match the specified input after transformation, confirming the fidelity of the algorithm.

A key technical claim is that the structure of the output distribution will mimic the structure of the initial, decoupled variable distribution only in the Gaussian case. For non-Gaussian priors (e.g., uniform distributions), the algorithm still enforces the desired second moments, but the full joint pdf is not Gaussian—rather, the resulting marginals and correlations reflect both the initial distribution and the linear transformation, with central-limit-like "smoothing" in highly coupled cases.

Practical and Theoretical Implications

The methodology presents several practical advantages in computational physics and simulation:

• Direct Physical Interpretability: The symplectic transformation framework naturally aligns with the transport of canonical variables in Hamiltonian dynamics, directly encoding the physical evolution and coupling encountered in beamlines and other systems.
• Algorithmic Generality: The technique accommodates arbitrary covariance structures and is not limited by positive-definiteness constraints beyond those required by the covariance.
• Beyond Gaussians: The transformation allows deterministic imposition of covariance also on non-Gaussian distributions, supporting more general simulation requirements.
• Parameter Identification: Given a linear transport process and observed output distributions, the method is invertible, offering avenues for estimating initial distribution parameters from observed data.

Theoretically, the work highlights the utility of symplectic and Clifford algebra structures for stochastic simulation, bridging linear algebra and group-theoretic perspectives in a way rarely exploited in statistics or data science.

Future Directions

Potential extensions include:

• Generalization to infinite-dimensional systems (field theories)
• Efficient GPU implementations leveraging the structured group operations
• Statistical inference for non-Gaussian and non-stationary processes, using the invertibility and algebraic transparency of the symplectic method

Additionally, the formalism may be beneficial for structure-preserving machine learning architectures or model-based control in dynamical systems involving symplectic in/out mappings.

Conclusion

This paper provides a detailed exposition and implementation pathway for generating multivariate normal distributions via symplectic transformations and RDM algebra, with both rigorous theoretical grounding and empirical validation. The approach is particularly suited for physics-based simulations, enabling direct translation from uncoupled to fully coupled variable sets with prescribed covariances, supporting a broad range of applications in computational accelerator physics and beyond.