The objective of generative model inversion is to identify a size-$n$ latent vector that produces a generative model output that closely matches a given target. This operation is a core computational primitive in numerous modern applications involving computer vision and NLP. However, the problem is known to be computationally challenging and NP-hard in the worst case. This paper aims to provide a fine-grained view of the landscape of computational hardness for this problem. We establish several new hardness lower bounds for both exact and approximate model inversion. In exact inversion, the goal is to determine whether a target is contained within the range of a given generative model. Under the strong exponential time hypothesis (SETH), we demonstrate that the computational complexity of exact inversion is lower bounded by $\Omega(2^n)$ via a reduction from $k$-SAT; this is a strengthening of known results. For the more practically relevant problem of approximate inversion, the goal is to determine whether a point in the model range is close to a given target with respect to the $\ell_p$-norm. When $p$ is a positive odd integer, under SETH, we provide an $\Omega(2^n)$ complexity lower bound via a reduction from the closest vectors problem (CVP). Finally, when $p$ is even, under the exponential time hypothesis (ETH), we provide a lower bound of $2^{\Omega (n)}$ via a reduction from Half-Clique and Vertex-Cover.

翻译：生成模型逆问题的目标是在识别一个大小为 $n$ 的潜向量，使得生成模型的输出与给定目标高度匹配。该操作是当前计算机视觉和自然语言处理众多现代应用中的核心计算原型。然而，该问题在最坏情况下已知为计算困难且是NP完全的。本文旨在对该问题的计算硬度格局提供细粒度的视角。我们为精确模型逆问题和近似模型逆问题建立了若干新的下界。在精确逆问题中，目标是判定某个目标是否包含在给定生成模型的值域内。在强指数时间假说（SETH）下，我们通过从$k$-SAT问题归约证明，精确逆问题的计算复杂度下界为$\Omega(2^n)$，这一结果强化了已知结论。对于更具实践意义的近似逆问题，目标是判定模型值域内的点是否在$\ell_p$范数意义下接近于给定目标。当$p$为正奇数时，在SETH下，我们通过从最近向量问题（CVP）归约给出$\Omega(2^n)$的复杂度下界。最后，当$p$为偶数时，在指数时间假说（ETH）下，我们通过从半团问题和顶点覆盖问题归约得到$2^{\Omega(n)}$的下界。