Core Distribution Alignment (CoDA)

• Core Distribution Alignment (CoDA) is a suite of frameworks that identifies and aligns key structural distributions across complex systems and datasets.
• It leverages cooperative training and explicit divergence bounds, geometric, density-based, or spectral techniques to ensure stable and interpretable alignment.
• CoDA has broad applications in unsupervised distribution matching, astrophysical core analysis, training-free dataset distillation, and frequency-aware neural network adaptation.

Core Distribution Alignment (CoDA) refers to families of methodological frameworks across disciplines that quantify, enforce, or exploit the alignment of salient core distributions among a set of complex systems or datasets. The term "Core Distribution Alignment" encompasses (1) cooperative unsupervised distribution matching via explicit variational Jensen–Shannon divergence bounds in machine learning, (2) geometric quantification of spatial alignment in astrophysical core populations, (3) intrinsic core distribution discovery and alignment for training-free dataset distillation with diffusion models, and (4) spectral alignment of frequency-domain components in neural network compression and adaptation. Across these domains, CoDA methods formalize the process of identifying structural cores and then aligning them through explicit mathematical, algorithmic, or statistical means.

1. Cooperative Distribution Alignment via JSD Upper Bound

CoDA was introduced as a unified and generalized framework for unsupervised distribution alignment (UDA), where the objective is to map multiple unknown source distributions $\{p_i\}$ to a common latent distribution via invertible transformations $\{f_i\}$, such that the push-forward distributions $q_i = p_i \circ f_i^{-1}$ align in the latent space (Cho et al., 2022). This problem underpins multi-modal generative modeling, domain adaptation, and fairness.

Mathematically, alignment is measured by the generalized Jensen–Shannon divergence (GJSD),

$\GJSD_w(q_1,\dots,q_K) = \sum_{i=1}^K w_i\,\KL\left(q_i~\|~q_{\rm mix}\right),$

where $q_{\rm mix} = \sum_{i=1}^K w_i q_i$ is the latent mixture and $w_i$ are positive mixing weights.

Instead of adversarial min–max approaches, CoDA leverages a variational upper bound on GJSD by introducing a shared density model $Q(z)$. This provides a tractable, sample-based “Alignment Upper Bound” (AUB) objective: $$\min_{\{f_i\},Q} \sum_{i=1}^K w_i\,\mathbb{E}_{x\sim p_i} \left[ -\log|J_{f_i}(x)| - \log Q(f_i(x)) \right],$$ where $J_{f_i}(x)$ is the Jacobian determinant of $f_i$. Both the transformations $\{f_i\}$ and the density model $Q$ are trained in a min–min (cooperative) fashion, leading to greater stability and interpretability compared to adversarial (min–max) setups.

Notably, the global optimum with infinite model capacity guarantees $q_i \equiv Q$ for all $i$, achieving perfect distributional alignment, which is evidenced by $\GJSD_w = 0$. The empirical AUB score serves as a consistent, model-agnostic evaluation metric.

2. Alignment Parameters for Astronomical Core Distributions

In astrophysics, CoDA specifically denotes a quantitative method for assessing the alignment of dense molecular cores in star-forming regions based on their spatial distribution (Chen et al., 3 Dec 2024). Letting ${\bf x}_i$ denote core positions, the pairwise Euclidean separations $S_{ij} = \|{\bf x}_i - {\bf x}_j\|$ are normalized by the system’s minor-axis length $\sigma_m$ (computed via PCA or beam size) to give dimensionless separations $S'_{ij}$.

The unweighted alignment parameter $A_{L,uw}$ is defined as

$$A_{L,uw} = \frac{1}{N(N-1)} \sum_{i\neq j} S'_{ij},$$

where $N$ is the number of cores. For weighted analyses, where core properties like flux or mass enter, the weighted parameter is

$$A_{L,w} = \frac{\sum_{i\neq j} w_i w_j S'_{ij}}{\sum_{i\neq j} w_i w_j},$$

with $w_i$ as core-specific weights.

A threshold $\gamma_t \approx 3.3$ optimally separates "aligned" (linearly extended) versus "clustered" (compact) morphologies based on visual inspection benchmarks. Systematic studies with ALMA 1.3 mm continuum images reveal that alignment, as quantified by $A_L$, shows little to no robust correlation with bulk clump properties (mass, density, morphology), suggesting the dominance of chaotic fragmentation mechanisms rather than deterministic, global energy reservoirs. The method has broad applicability for statistically comparing observed and simulated star-forming environments.

3. Distribution Alignment in Training-Free Dataset Distillation

In the context of dataset distillation, CoDA describes a two-stage process to facilitate synthetic dataset creation using off-the-shelf text-to-image diffusion models without the need for target-specific generative model training (Zhou et al., 3 Dec 2025).

First, the "intrinsic core distribution" of a dataset is discovered via density-based clustering (UMAP dimensionality reduction followed by HDBSCAN) in latent space (using the VAE encoder of the diffusion model). For each class, cluster representatives are refined with post-processing to match the desired images-per-class (IPC).

Second, a guided diffusion process is applied to align new samples with the discovered core by steering the diffusion model’s output towards each cluster representative. Given latent code $z_t$ at diffusion step $t$ and representative $s_j$, a guidance update is computed: $\Delta ẑ_0 = \gamma(s_j - \hat{z}_0(z_t)),$ with translation to the noise space, and classifier-free guidance is maintained for stability.

This alignment yields synthetic datasets exhibiting high downstream classification accuracy and low FID, sometimes surpassing approaches using target-trained diffusion models. The framework significantly reduces computational cost and domain adaptation effort by eliminating the prerequisite of target-specific generative model training.

4. Frequency-Aware Alignment for Compression and Domain Adaptation

A separate usage of CoDA refers to "frequency composition-based" domain adaptation and compression for neural networks (Kwon et al., 27 May 2025). Here, “alignment” involves matching low-frequency components (LFC) of input distributions between source and target, while treating high-frequency components (HFC) as domain-specific.

During quantization-aware training (QAT), models are trained on low-pass filtered images: $$\min_{f_q, \theta_q} \mathbb{E}_{(x,y)} \left[ \mathcal{L}_{CE}(f_q(\mathcal{F}^{-1}({\rm LPF}(\mathcal{F}(x); r))), y) \right],$$ where $x$ is the input, $\mathcal{F}$ denotes the Fourier transform, and $r$ the frequency cutoff.

At test time, a Frequency-Aware Batch Normalization (FABN) scheme runs inference on full-frequency inputs, updating BN statistics so that the low-frequency batch statistics leverage running-mean estimates learned from the source, while high-frequency batch statistics adapt immediately to the target batch. This alignment allows compressed models to maintain robustness and generalize rapidly under domain shift, outperforming prior solutions in accuracy and efficiency.

5. Comparative Summary of CoDA Methodologies

The following table synthesizes the principal CoDA variants across different domains:

Application Domain	Core Alignment Object	Alignment Metric / Mechanism
Unsupervised Distribution Alignment (Cho et al., 2022)	Latent distributions ($q_i$)	Upper bound on JSD (AUB), min–min flow + density fit
Astronomical Core Alignment (Chen et al., 3 Dec 2024)	Spatial positions (${\bf x}_i$)	Mean normalized pair separations ($A_{L,uw}, A_{L,w}$)
Dataset Distillation (Zhou et al., 3 Dec 2025)	Latent core samples ($S_r$)	Clustering + guided diffusion toward core
Neural Network Compression & DA (Kwon et al., 27 May 2025)	Frequency bands (LFC/HFC)	Spectral filtering + BN statistic separation

Each CoDA framework operationalizes "core" discovery and alignment with rigor appropriate to the structure of the domain—probabilistic, geometric, density-driven, or spectral.

6. Theoretical Guarantees and Empirical Observations

In cooperative distribution alignment (Cho et al., 2022), upper-bounding the GJSD provides not only a tractable minimization landscape but also theoretical guarantees: global minima correspond to perfect alignment where $\GJSD_w = 0$. Empirically, CoDA achieves lower AUB and improved FID over adversarial and other flow-based baselines on both image and tabular benchmarks.

In astronomical applications (Chen et al., 3 Dec 2024), systematic use of $A_L$ enables reproducible morphometric classification and reveals that fragmentation geometry and core distribution often follow chaotic dynamics, as evidenced by the lack of strong correlations with bulk clump properties.

For dataset distillation (Zhou et al., 3 Dec 2025), CoDA establishes new state-of-the-art results with substantial computational efficiency gains: for ImageNet-1K at 50 IPC, CoDA achieves 60.4% top-1 accuracy compared to 55.2% (D⁴M) and 58.6% (Minimax), requiring no target-dataset generative modeling.

In compressed neural network adaptation (Kwon et al., 27 May 2025), the combined LFC QAT and FABN test-time adaptation yield significant increases over state-of-the-art baselines (e.g., +7.96%p on CIFAR10-C, +5.37%p on ImageNet-C), confirming the efficacy of frequency-domain core alignment.

7. Broader Significance and Future Directions

CoDA frameworks provide systematic, mathematically grounded tools for aligning salient “core” structures across domains and tasks. In machine learning, CoDA’s cooperative min–min objectives enable stable distributional matching without adversarial pitfalls and serve as an interpretable loss metric. In astrophysical studies, alignment parameters facilitate robust classification of spatial configurations, informing theories of chaotic fragmentation. For dataset distillation, core-centric alignment bridges the gap between large-scale generative priors and narrow-domain data, enabling zero-shot transfers. In neural network compression, frequency-domain alignment decouples invariant and environment-specific features for extreme efficiency and adaptation.

A plausible implication is that the principle underlying CoDA—explicitly identifying and aligning succinct core structures—may generalize to other areas where robust, efficient, and interpretable alignment is essential. Future research may further unify these approaches under broader theoretical and algorithmic frameworks, as well as discover new metrics for progressively more structured and high-dimensional alignment tasks.