In the surveying of subpopulations (e.g. by the US Census Bureau) one can improve noisy estimates (e.g.

We analyzed the multi-layer case beyond what is reported in the submitted paper. Equations to get the optimal error in the multi-layer case are in page 10-11 of the SM. The vertical lines show the PCA and the optimal threshold respectively. Our claims of optimality of AMP are indeed limited to the cases investigated numerically. We will make a statement collecting all the assumptions in the final version.

This is also confirmed by our experiments with the MNIST and synthetic dataset.

Estimating distributions from samples is a fundamental statistical task.

We thank the referees for their comments. Following the suggestions of R1 and R2, we will add a conclusion to the paper and modify the broader impact statement. This will clarify our contribution, its practical impact, but also the results described in Section 5. We will, of course, Our "trap-avoidance" result suggests that, at high precision, artificial critical points are not met in practice. On the other hand, at low precision, "genericity results" may partly collapse. Y et the theory says uniqueness is generic.

About Assumption K1 - Y es, the inequality should be strict.

Fair, transparent, and explainable decision-making remains a critical challenge in Olympic and Paralympic combat sports. This paper presents \emph{FST.ai 2.0}, an explainable AI ecosystem designed to support referees, coaches, and athletes in real time during Taekwondo competitions and training. The system integrates {pose-based action recognition} using graph convolutional networks (GCNs), {epistemic uncertainty modeling} through credal sets, and {explainability overlays} for visual decision support. A set of {interactive dashboards} enables human--AI collaboration in referee evaluation, athlete performance analysis, and Para-Taekwondo classification. Beyond automated scoring, FST.ai~2.0 incorporates modules for referee training, fairness monitoring, and policy-level analytics within the World Taekwondo ecosystem. Experimental validation on competition data demonstrates an {85\% reduction in decision review time} and {93\% referee trust} in AI-assisted decisions. The framework thus establishes a transparent and extensible pipeline for trustworthy, data-driven officiating and athlete assessment. By bridging real-time perception, explainable inference, and governance-aware design, FST.ai~2.0 represents a step toward equitable, accountable, and human-aligned AI in sports.

Abstract--Although existing cross-domain continual learning approaches successfully address many streaming tasks having domain shifts, they call for a fully labeled source domain hindering their feasibility in the privacy constrained environments. This paper goes one step ahead with the problem of source-free cross-domain continual learning where the use of source-domain samples are completely prohibited. We propose the idea of rehearsal-free frequency-aware dynamic prompt collaborations (REFEREE) to cope with the absence of labeled source-domain samples in realm of cross-domain continual learning. REFEREE is built upon a synergy between a source-pre-trained model and a large-scale vision-language model, thus overcoming the problem of sub-optimal generalizations when relying only on a source pre-trained model. The domain shift problem between the source domain and the target domain is handled by a frequency-aware prompting technique encouraging low-frequency components while suppressing high-frequency components. This strategy generates frequency-aware augmented samples, robust against noisy pseudo labels. The noisy pseudo-label problem is further addressed with the uncertainty-aware weighting strategy where the mean and covariance matrix are weighted by prediction uncertainties, thus mitigating the adverse effects of the noisy pseudo label. Besides, the issue of catastrophic forgetting (CF) is overcome by kernel linear discriminant analysis (KLDA) where the backbone network is frozen while the classification is performed using the linear discriminant analysis approach guided by the random kernel method. Our rigorous numerical studies confirm the advantage of our approach where it beats prior arts having access to source domain samples with significant margins. HE goal of continual learning (CL) is to deal with lifelong learning environments where a sequence of non-stationary tasks is observed.

We analyzed the multi-layer case beyond what is reported in the submitted paper. Equations to get the optimal error in the multi-layer case are in page 10-11 of the SM. The vertical lines show the PCA and the optimal threshold respectively. Our claims of optimality of AMP are indeed limited to the cases investigated numerically. We will make a statement collecting all the assumptions in the final version.