AI 中文总结
该研究针对二元域上受恶意破坏的集合观测问题，提出无承诺尖锐持续子集BSG编译器，结合算法性PFR与确定性提升，实现了鲁棒多项式Freiman-Ruzsa的高效推断。
AI 中文摘要
我们研究在二元域$\boldsymbol{\text{F}}_2^n$上，经敌手恶意破坏的精确集合观测下的结构推断问题。存在一个非空隐藏集合$A$满足$|A+A|\boldsymbol{\text{≤}}K|A|$，算法仅能获取$B$的确定性成员关系及精确均匀采样访问权限，其中对称差$|A\triangle B|\boldsymbol{\text{≤}}\boldsymbol{\text{η}}|A|$。由于损坏会破坏$B$的倍增性质，尖锐存在性BSG与干净输入的算法性PFR无法直接在该模型中组合。我们的主要成果是一个无承诺的尖锐持续子集BSG编译器：给定对$S$和$\boldsymbol{\text{α}}$的采样与查询访问权限，它返回$\boldsymbol{\text{FAIL}}$或一个描述符，该描述符定义了$S$的一个固定子集$Y\boldsymbol{\text{⊆}}S$，满足$|Y|\boldsymbol{\text{≥}}c\boldsymbol{\text{√}}\boldsymbol{\text{α}}|S|$且$|Y+Y|\boldsymbol{\text{≤}}C\boldsymbol{\text{α}}^{-4}|Y|$。在无能量承诺时，所有非失败输出均有效，仅存在规定的可靠性概率；高能量则保证以高概率成功。该描述符在自适应查询下提供持续成员关系，有限时间桥接则给出条件精确乘积采样。因此，尖锐保留质量、无承诺有效性、持续性和精确有限采样构成一个可组合接口。结合无尺寸敏感的算法性PFR与确定性提升，该编译器针对$\boldsymbol{\text{η}}\boldsymbol{\text{=}}O(K^{-1/2})$生成一个随机FPT形式算法，输出$V$满足$|V|\boldsymbol{\text{≤}}|A|$且$\boldsymbol{\text{ℕ}}_V(A)\boldsymbol{\text{≤}}K^{O(1)}$。对于所有给定$\boldsymbol{\text{η}}\boldsymbol{\text{<}}1$，迭代残差算法输出一个公共列表，可服务所有兼容的隐藏集合，且具有多项式覆盖预算；样本数为多项式，成员查询与运行时间复杂度为XP。最后，每个非空兼容性类非构造性地存在一个公共子空间，而精确的双子空间构造则要求公共覆盖成本为$\boldsymbol{\text{Θ}}((1-\boldsymbol{\text{η}})^{-1/2})$。
英文摘要
We study when statistically learnable latent structure can also be recovered efficiently, and how membership queries change the answer. An unknown support $A\subseteq\mathbb F_2^n$ has small additive doubling and is observed through a fixed set $B$ satisfying $|A\triangle B|\leη|A|$. We seek one linear subspace $V$ such that every compatible support $A$ is covered by few $V$-cosets and satisfies $|V|\le|A|$. For every $η<1$, polynomially many uniform samples suffice statistically, with cost polynomial in the doubling constant and proportional to $(1-η)^{-1}$; this radius dependence is sharp. Under a specified hardness assumption for learning parities with noise (search-LPN), however, no polynomial-time sample-only learner achieves even constant covering cost, including when the latent support is unique. At fixed structural parameters and the same constant covering budget, adding exact membership queries to $B$ permits polynomial-time recovery. The general query learner constructs a short structural list and uses fresh samples to select one common output through a majority-coverage rule. Persistent structured cores make this candidate construction possible. At doubling one, a complementary distinction appears at $η=1/3$: coarse recovery remains polynomial time, while exact recovery requires exponentially many accesses in the worst case when latent cardinality is unknown.
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