We consider the linear causal representation learning setting where we observe a linear mixing of $d$ unknown latent factors, which follow a linear structural causal model. Recent work has shown that it is possible to recover the latent factors as well as the underlying structural causal model over them, up to permutation and scaling, provided that we have at least $d$ environments, each of which corresponds to perfect interventions on a single latent node (factor). After this powerful result, a key open problem faced by the community has been to relax these conditions: allow for coarser than perfect single-node interventions, and allow for fewer than $d$ of them, since the number of latent factors $d$ could be very large. In this work, we consider precisely such a setting, where we allow a smaller than $d$ number of environments, and also allow for very coarse interventions that can very coarsely \textit{change the entire causal graph over the latent factors}. On the flip side, we relax what we wish to extract to simply the \textit{list of nodes that have shifted between one or more environments}. We provide a surprising identifiability result that it is indeed possible, under some very mild standard assumptions, to identify the set of shifted nodes. Our identifiability proof moreover is a constructive one: we explicitly provide necessary and sufficient conditions for a node to be a shifted node, and show that we can check these conditions given observed data. Our algorithm lends itself very naturally to the sample setting where instead of just interventional distributions, we are provided datasets of samples from each of these distributions. We corroborate our results on both synthetic experiments as well as an interesting psychometric dataset. The code can be found at https://github.com/TianyuCodings/iLCS.

翻译：我们考虑线性因果表示学习场景，其中观测到$d$个未知潜在因子的线性混合，这些潜在因子遵循线性结构因果模型。近期研究表明，若至少拥有$d$个实验环境（每个环境对应单个潜在节点/因子的完美干预），则可在置换和尺度变换的意义下恢复潜在因子及其底层结构因果模型。在这一重要成果之后，领域内面临的关键开放问题是如何放宽这些条件：允许比完美单节点干预更粗略的干预方式，并允许少于$d$个干预环境，因为潜在因子数量$d$可能非常庞大。本研究针对此类场景展开，允许环境数量少于$d$，同时允许非常粗略的干预——这些干预能够以极粗略的方式\textit{改变潜在因子间的完整因果图}。另一方面，我们将提取目标放宽为仅识别\textit{在一个或多个环境间发生偏移的节点集合}。我们提出了一个令人惊讶的可识别性结论：在某些非常温和的标准假设下，确实能够识别偏移节点集合。我们的可识别性证明具有构造性：明确给出了节点成为偏移节点的必要充分条件，并证明可通过观测数据验证这些条件。所提算法能自然适用于样本场景——当获得来自各干预分布的样本数据集而非仅干预分布时。我们通过合成实验和有趣的心理测量数据集验证了结果。代码可见https://github.com/TianyuCodings/iLCS。