Gaussian Combined Distance (GCD)

• Gaussian Combined Distance (GCD) is a metric that encapsulates the shape of Gaussian mixtures by leveraging inter-sample distances and moment generating functions.
• It enables complete recovery of a mixture’s configuration using weighted power sums and provides a solid foundation for geometric and statistical invariance.
• In object detection, GCD offers a scale-invariant similarity measure to optimize bounding box regression and label assignment, especially for tiny objects.

The Gaussian Combined Distance (GCD) is a formal metric originating from probability theory and computer vision, designed to encapsulate geometric relationships in both distributional and object detection settings. GCD traces to two foundational threads: its characterization of the shape of Gaussian mixtures via the distribution of inter-sample distances (Boutin et al., 2021), and its role as a similarity metric for Gaussian-distributed bounding boxes in object detection (Guan et al., 31 Oct 2025). GCD encodes both positional and scale relationships, offering scale invariance and optimized gradient structure for detection algorithms.

1. Characteristic Functionality of GCD in Gaussian Mixtures

For a random variable $x$ distributed by a Gaussian mixture

$\rho(x) = \sum_{i=1}^k \pi_i \rho_i(x)$

with $\rho_i(x)$ a standard normal in $\mathbb{R}^d$ (identity covariance), let $x_1, x_2$ be independent samples drawn from $\rho(x)$, and define the squared Euclidean distance $\Delta = \|x_1-x_2\|^2$. The probability density of $\Delta$, denoted $r(\Delta)$, uniquely determines the mixture’s shape up to rigid motion (translation, rotation) in generic cases. Specifically, $r(\Delta)$ encodes the multiset of pairwise squared distances between the means $\mu_i$, with weights $\pi_i$. Reconstruction is enabled via the moment generating function (MGF) for $\Delta$: $$M(t) = \mathbb{E}[e^{t\Delta}] = \sum_{i,j=1}^k \pi_i \pi_j \frac{1}{(1-4t)^{d/2}} \exp\left( \frac{t\|\mu_i-\mu_j\|^2}{1-4t} \right).$$ Expanding $M(t)$ yields coefficients as weighted power sums of the squared distances, from which the configuration of means (up to labeling and rigid motion) is reconstructible when the means are in generic position and $k \geq d+2$ (or $k=1,2,3$ exceptional cases). A plausible implication is that the "law" of squared distance between random samples, termed the Gaussian Combined Distance, is a complete invariant for the mixture’s shape (Boutin et al., 2021).

2. Generalization via Symmetric Bilinear Forms

The distance measure in GCD may be generalized through symmetric non-degenerate bilinear forms, replacing Euclidean distance. For such forms $\langle\,,\,\rangle$, with

$$\rho(x) = \sum_{i=1}^k \pi_i \frac{1}{(2\pi)^{d/2}} \exp\left(-\frac{1}{2} \langle x-\mu_i, x-\mu_i \rangle\right),$$

and $\Delta = \langle x_1-x_2, x_1-x_2\rangle$, the distribution $r(\Delta)$ again characterizes the mixture shape up to the group of symmetries preserving the form (the orthogonal group $O(V)$ for quadratic forms). This extends the invariance principle underlying GCD beyond the Euclidean setting.

3. Definition and Analytical Properties for Object Detection

In object detection, GCD is formulated as a similarity metric for bounding boxes modeled as axis-aligned 2D Gaussians, with mean $\mu = [x, y]$ and covariance $$\Sigma = \mathrm{diag}(\frac{w^2}{4}, \frac{h^2}{4})$$ for center $(x, y)$, width $w$, and height $h$. For predictive box $\mathcal{N}_p$ and target $\mathcal{N}_t$, the GCD is defined as: $$\begin{aligned} \mathbf{D}_{gc}^2\left(\mathcal{N}_p, \mathcal{N}_t\right) =& \; (\mu_p-\mu_t)^{\top}2\Sigma_p^{-1}(\mu_p-\mu_t) + (\mu_t-\mu_p)^{\top}2\Sigma_t^{-1}(\mu_t-\mu_p) \ & + 2\,\|\Sigma_p^{1/2}-\Sigma_t^{1/2}\|_F^2\,(\Sigma_p^{-1/2})^{\top} + 2\,\|\Sigma_t^{1/2}-\Sigma_p^{1/2}\|_F^2\,(\Sigma_t^{-1/2})^{\top} \end{aligned}$$ For axis-aligned bounding boxes, GCD simplifies component-wise, capturing both center displacement (normalized by scale) and scale difference, with symmetry $$D_{gc}^2(\mathcal{N}_p,\mathcal{N}_t) = D_{gc}^2(\mathcal{N}_t,\mathcal{N}_p)$$. The normalized similarity metric is

$$\mathbf{M}_{gcd} = \exp\left(-\sqrt{\mathbf{D}_{gc}^2(\mathcal{N}_p, \mathcal{N}_t)} \right),$$

enabling usage for label assignment and NMS (Guan et al., 31 Oct 2025).

4. Scale Invariance and Joint Optimization Mechanisms

GCD offers formal scale invariance: if both $\mathcal{N}_p, \mathcal{N}_t$ are scaled by any full-rank affine transformation, $D_{gc}^2$ remains invariant. This property addresses shortcomings of Wasserstein Distance (WD), which is not scale-invariant and may require dataset-dependent normalization constants. Gradient analysis reveals that GCD couples box center and scale parameters, yielding amplified and joint adaptation for boxes with small scale (as in tiny object detection). For center $x_p$, the gradient is

$$\frac{\partial \mathbf{D}_{gc}^2}{\partial x_p} \propto \frac{w_t^2+w_p^2}{w_t^2w_p^2}(x_p-x_t),$$

indicating sensitivity inversely proportional to size, a requisite feature for locating small objects.

5. Theoretical Recovery via Weighted Power Sums

A central lemma in the probabilistic characterization asserts that if one is given $p_n = \sum_{i=1}^k a_i x_i^n$ for $n=0,\ldots,2k-1$ with pairwise distinct $x_i$ and nonzero $a_i$, the configuration $\{(a_i,x_i)\}$ is recoverable (up to order). Such recovery is achieved by constructing and factoring a generalized Vandermonde system. This result underpins both shape reconstruction for mixtures and the completeness of GCD representation in the statistical setting (Boutin et al., 2021).

6. Application in Object Detection and Experimental Outcomes

Empirical evaluation demonstrates GCD’s efficacy when used as both bounding box regression loss and label assignment metric in leading object detectors. On the AI-TOD-v2 dataset for tiny object detection, GCD achieves superior average precision (AP) scores compared to GIoU, DIoU, KLD, WD, and NWD metrics:

Metric	AP	AP$_{50}$	AP$_{75}$	AP$_{vt}$	AP$_{t}$	AP$_{s}$	AP$_{m}$
GIoU	6.8	17.9	4.1	2.6	8.3	7.7	23.4
DIoU	6.9	19.5	3.6	3.8	7.3	8.4	23.4
KLD	7.3	20.0	4.1	3.2	7.4	10.8	23.7
WD	9.1	24.2	4.9	2.2	8.4	14.9	25.4
NWD	8.0	21.0	4.4	2.7	8.3	13.0	25.1
GCD	11.5	31.2	5.7	3.6	9.7	16.0	28.5

Substantial improvement is noted for very tiny and tiny object classes ($2$–$16$ pixels). On VisDrone-2019 and MS-COCO-2017, GCD consistently outperforms WD and maintains competitiveness with IoU-type metrics, demonstrating universality in variable object scale environments.

7. Summary Table: Group of Equivalence for GCD Characterization

Setting	Sufficient information?	Group of equivalence
Equally weighted, identity covariance, generic means	Yes, by distance distribution $r(\Delta)$	Euclidean group $E(d)$
Arbitrary weights, identity covariance, generic means	Yes, via weighted power sums from $r(\Delta)$	$E(d)$
Bilinear symmetric form	Yes, by "quadratic distance" distribution	Orthogonal group of the form

8. Significance and Potential Extensions

GCD unifies several concepts: in statistical geometry, it is a complete invariant for the shape of generic Gaussian mixtures; in detection, it provides a formally justified, empirically validated loss and assignment metric with scale invariance and joint optimization properties. The formalism does not require hyperparameters or dataset-dependent scaling and is directly integrated into existing detection frameworks. A plausible implication is that GCD may extend its utility to detection tasks requiring orientation modeling or nontrivial covariance structures, as suggested by observed benefits in oriented bounding box scenarios.

Further information and implementations are available at the cited repositories. For rigorous computational procedures or extensions, the foundational papers provide detailed theoretical and empirical analyses (Boutin et al., 2021, Guan et al., 31 Oct 2025).