超网格Johnson-Lindenstrauss的自适应鲁棒性
Adaptive Robustness of Hypergrid Johnson-Lindenstrauss
AI总结：
针对超网格在随机投影下可能出现任意收缩的问题，本文通过设计在线算法并证明计算硬度，揭示其统计-计算差距，并将结果应用于构造鲁棒性保持哈希函数。
摘要：
Johnson和Lindenstrauss（Contemporary Mathematics，1984）证明，当n > m时，从ℝⁿ到ℝᵐ的缩放随机投影A对任何大小至多为m的指数级的集合S都是近似等距映射。然而，若S更大，其点在A下可能被任意收缩。特别是，超网格([-B, B] ∩ ℤ)ⁿ中预计存在一个点被收缩κ_stat = Θ(B)^(-1/α)倍，其中α = m/n。我们提供证据表明，找到这样的点存在统计-计算差距，精确到κ_comp = Θ̃(√α/B)。在算法方面，受Bansal和Spencer（Random Structures & Algorithms，2020）的差异最小化算法启发，我们设计了一种在线算法实现κ_comp。在硬度方面，我们通过多重叠间隙性质（mOGP）提供证据，该性质特别捕捉在线算法；并通过基于归约的下界，在标准最坏情况格假设下证明硬度。作为密码学应用，我们表明在计算困难区域，舍入Johnson-Lindenstrauss嵌入是超网格上欧几里得度量的鲁棒性保持哈希函数（Boyle、Lavigne和Vaikuntanathan，TCC 2019）。这种哈希函数压缩数据同时保持输入间的ℓ₂距离在一定失真因子内，且保证即使知道哈希函数，计算受限的敌手也无法找到违反失真界的点对。
英文摘要：
Johnson and Lindenstrauss (Contemporary Mathematics, 1984) showed that for $n > m$, a scaled random projection $\mathbf{A}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is an approximate isometry on any set $S$ of size at most exponential in $m$. If $S$ is larger, however, its points can contract arbitrarily under $\mathbf{A}$. In particular, the hypergrid $([-B, B] \cap \mathbb{Z})^n$ is expected to contain a point that is contracted by a factor of $κ_{\mathsf{stat}} = Θ(B)^{-1/α}$, where $α= m/n$. We give evidence that finding such a point exhibits a statistical-computational gap precisely up to $κ_{\mathsf{comp}} = \widetildeΘ(\sqrtα/B)$. On the algorithmic side, we design an online algorithm achieving $κ_{\mathsf{comp}}$, inspired by a discrepancy minimization algorithm of Bansal and Spencer (Random Structures & Algorithms, 2020). On the hardness side, we show evidence via a multiple overlap gap property (mOGP), which in particular captures online algorithms; and a reduction-based lower bound, which shows hardness under standard worst-case lattice assumptions. As a cryptographic application, we show that the rounded Johnson-Lindenstrauss embedding is a robust property-preserving hash function (Boyle, Lavigne and Vaikuntanathan, TCC 2019) on the hypergrid for the Euclidean metric in the computationally hard regime. Such hash functions compress data while preserving $\ell_2$ distances between inputs up to some distortion factor, with the guarantee that even knowing the hash function, no computationally bounded adversary can find any pair of points that violates the distortion bound.
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