gm_to_dirac_even_closed_form

Gaussian-to-Dirac approximation, quadrature-free (even dimension).

Overview

gm_to_dirac_even_closed_form<T> approximates a standard normal distribution by a Dirac mixture with L components in N dimensions, exactly as gm_to_dirac_short_standard_normal_deviation does, but with every integral over the kernel-width variable b evaluated in closed form.

The quadrature path issues L × N adaptive Gauss-Kronrod integrations per gradient evaluation. This path issues none.

Preconditions

• N must be EVEN and >= 2. Odd N fails loudly (assert in Debug, approximate() returns false in Release); there is no silent fallback.
• The target must be the standard normal N(0, I), or an isotropic N(0, sigma² I) through the approximate_isotropic() overloads. General diagonal covariance with unequal sigma_k is NOT supported: the scaling reduction that makes the isotropic case work does not exist there.
• bMax may be ANY value > 0. Unlike the quadrature path, whose D3 uses a large-bMax logarithmic approximation, D3 here is the exact Ei form, so no "for large bMax" assumption is made anywhere.

Differences from the quadrature path

Two behavioural differences are worth knowing when comparing results against gm_to_dirac_short_standard_normal_deviation:

• The reported distance is the TRUE modified Cramér-von-Mises distance: it includes the sample-independent D1 term, which the quadrature path omits.
• D3 is exact rather than logarithmically approximated.

Interface

Inherits:

• gm_to_dirac_approx_standard_normal_distribution_i<T>

Template parameter:

• T ∈ {float, double}; computation is always performed in double and cast at the interface boundary.

Provides overloads of:

• approximate(...)
• modified_van_mises_distance_sq(...)
• modified_van_mises_distance_sq_derivative(...)
• approximate_isotropic(...)
• modified_van_mises_distance_sq_isotropic(...)

Parameters

Common parameters:

• L -> number of Dirac components
• N -> dimension, must be even
• bMax -> integration bound, any value > 0
• x -> initial guess and output locations (L × N)
• wX -> weights of the Dirac mixture (optional)

If wX is nullptr:

• Uniform weights are assumed

Supported Input Formats

Three overload families are available:

• Raw pointer interface (T*)
• GSL vector interface (gsl_vector / gsl_vector_float)
• GSL matrix interface (gsl_matrix / gsl_matrix_float)

Memory layout:

• x represents L × N Dirac locations, row major

Example (Raw Pointer)

gm_to_dirac_even_closed_form<double> approx;
 
bool ok = approx.approximate(
    L,
    N,        // must be even
    100.0,    // bMax, any value > 0
    x,        // initial guess / output (L × N)
    wX,       // weights (optional)
    &result,
    options
);

Example (true distance)

gm_to_dirac_even_closed_form<double> approx;
 
double distance = 0.0;
approx.modified_van_mises_distance_sq(&distance, L, N, 100.0, x, wX);
// distance includes D1; the quadrature path's value would be short by
// lcd_a_n(N, 100.0)

Example (isotropic sigma² I)

gm_to_dirac_even_closed_form<double> approx;
 
const double sigma = 2.0;
const double bMax  = 100.0;
 
// solves the standard-normal problem at bMax / sigma and scales by sigma
bool ok = approx.approximate_isotropic(
    L, N, sigma, bMax, x, wX, &result, options);

Closed forms

The distance decomposes as D = D1 - 2·D2 + D3, all expressed in units of pi^(N/2) — the same normalisation the quadrature path uses, so the two are directly comparable. The building blocks live in lcd_even_closed_form.h:

• lcd_a_n() -> D1, valid for all N (odd and even)
• lcd_delta_bkk() -> the per-sample D2 term
• lcd_delta_bkk1() -> the per-sample D2 gradient term
• lcd_c_repulsion() -> the exact per-pair D3 term
• lcd_ei() -> Ei(x)

Notes

• Single-threaded implementation
• Uses analytical gradients with no quadrature
• The objective is assembled from reduced, O(1) terms and additionally carries the constant K by default (gm_to_dirac_even_closed_form::includeD1InObjective), so that f is the true distance and ApproximateOptions::ftolRel stays meaningful
• Interface-compatible with gm_to_dirac_short_standard_normal_deviation