连续LWE与LWE一样难&在高斯混合模型学习中的应用
Continuous LWE is as Hard as LWE & Applications to Learning Gaussian Mixtures
摘要：
我们展示了经典错误学习（LWE）问题与其连续模拟CLWE（Bruna，Regev，Song和Tang，STOC 2021）之间的直接和概念上简单的简化。这使我们能够将基于LWE的密码学的强大机制应用于CLWE的应用。例如，我们得到的硬度CLWE下的经典最坏情况下的硬度间隙最短向量问题。以前，这是已知的量子最坏情况下的硬度晶格问题。更广泛地说，随着我们在两个问题之间的减少，LWE的任何未来发展也将适用于CLWE及其下游应用。作为一个具体的应用，我们显示了改进的硬度结果的混合高斯密度估计。在这个计算问题中，给定样本访问高斯混合，目标是输出一个估计混合密度函数的函数。在经典LWE问题的指数硬度下（合理且被广泛认为），我们证明了在$\mathsf{poly}(n)$样本下具有大致$\log n$高斯分量的$\mathbb{R}^n$中的高斯混合密度估计在$n$中需要时间准多项式。在LWE的（保守）多项式硬度下，我们显示了对于任何常数$ε> 0$的$n^ε$高斯密度估计的硬度，这改进了Bruna，Regev，Song和Tang（STOC 2021），他们在多项式（量子）硬度假设下显示了至少$\sqrt{n}$高斯的硬度。我们的关键技术工具是从经典LWE减少到具有$k$的LWE-稀疏秘密，其中噪声的乘法增加仅为$O(\sqrt{k})$，独立于环境维度$n$。
英文摘要：
We show direct and conceptually simple reductions between the classical learning with errors (LWE) problem and its continuous analog, CLWE (Bruna, Regev, Song and Tang, STOC 2021). This allows us to bring to bear the powerful machinery of LWE-based cryptography to the applications of CLWE. For example, we obtain the hardness of CLWE under the classical worst-case hardness of the gap shortest vector problem. Previously, this was known only under quantum worst-case hardness of lattice problems. More broadly, with our reductions between the two problems, any future developments to LWE will also apply to CLWE and its downstream applications. As a concrete application, we show an improved hardness result for density estimation for mixtures of Gaussians. In this computational problem, given sample access to a mixture of Gaussians, the goal is to output a function that estimates the density function of the mixture. Under the (plausible and widely believed) exponential hardness of the classical LWE problem, we show that Gaussian mixture density estimation in $\mathbb{R}^n$ with roughly $\log n$ Gaussian components given $\mathsf{poly}(n)$ samples requires time quasi-polynomial in $n$. Under the (conservative) polynomial hardness of LWE, we show hardness of density estimation for $n^ε$ Gaussians for any constant $ε> 0$, which improves on Bruna, Regev, Song and Tang (STOC 2021), who show hardness for at least $\sqrt{n}$ Gaussians under polynomial (quantum) hardness assumptions. Our key technical tool is a reduction from classical LWE to LWE with $k$-sparse secrets where the multiplicative increase in the noise is only $O(\sqrt{k})$, independent of the ambient dimension $n$.
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