潜在字典学习后的真实物理支持推断：碰撞奇点与极小极大分辨率

中文摘要 AI 辅助

稀疏支持的不确定性通常在将字典视为已知的情况下进行量化，当从潜在稀疏混合中学习字典时，这种假设可能会产生过度自信、依赖标签的结论。在相干原子的近碰撞处，测试信号可能会识别出活跃的物理组，即使训练数据无法区分其中的物理射线。我们开发了在潜在字典学习后对活跃物理射线（单位原子模符号）的推断方法。在固定维度的高斯训练-测试实验中，我们保留与稳健训练矩区域兼容的所有字典，在它们之上对测试表示进行剖析，并将幸存的配置投影到置换不变的支持空间上。由此产生的置信对应关系可以报告跨页的不确定性、具有子模糊性的组分辨率或精细支持分辨率。我们刻画了其统计成本和决策理论收益。残差块方向首先以立方阶影响潜在训练密度，产生阶为\(s^6\)的信息，其中\(s\)是块内碰撞尺度。该对应关系提供了高概率的过训练条件测试覆盖，其分辨率分别由父可检测性、测试时支持分离和学习字典方向控制。在已解决的固定壳区域中，其投影豪斯多夫直径以极小极大最优速率\(s \wedge (\sqrt{N}s^2)^{-1}\)收缩，至多相差常数。一个受限任务定理进一步确定了系数不对称何时允许测试复制补充训练信息以及校准不确定性何时仍然不可约。因此，该框架产生了真实的、分辨率自适应的支持陈述，并指导了训练与测试测量的分配。

英文摘要

Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identify the active physical group even though the training data cannot distinguish the physical rays within it. We develop inference for active physical rays, unit atoms modulo sign, after latent dictionary learning. In a fixed-dimensional Gaussian train-test experiment, we retain all dictionaries compatible with a robust training-moment region, profile the test representation over them, and project surviving configurations onto a permutation-invariant support space. The resulting confidence correspondence can report cross-sheet inconclusiveness, group resolution with child ambiguity, or fine-support resolution. We characterize both its statistical cost and decision-theoretic benefit. Residual block orientation first affects the latent training density at cubic order, yielding information of order $s^6$, where $s$ is the within-block collision scale. The correspondence provides high-probability-over-training conditional test coverage, with resolution governed separately by parent detectability, test-time support separation, and learned-dictionary orientation. In the resolved fixed-shell regime, its projective Hausdorff diameter contracts at the minimax-optimal rate $s \wedge (\sqrt{N}s^2)^{-1}$, up to constants. A restricted-task theorem further determines when coefficient asymmetry allows test replication to supplement training information and when calibration uncertainty remains irreducible. The framework thus yields honest, resolution-adaptive support statements and guides the allocation of training versus test measurements.

发表机构

• Institute of Data Science and Information Computing, National Chung Hsing University（国立中兴大学数据科学与信息计算研究所）

机构由 AI 辅助整理，请以论文原文为准。