In label-noise learning, the noise transition matrix reveals how an instance transitions from its clean label to its noisy label. Accurately estimating an instance's noise transition matrix is crucial for estimating its clean label.

Causal reasoning and discovery, two fundamental tasks of causal analysis, often face challenges in applications due to the complexity, noisiness, and high-dimensionality of real-world data. Despite recent progress in identifying latent causal structures using causal representation learning (CRL), what makes learned representations useful for causal downstream tasks and how to evaluate them are still not well understood. In this paper, we reinterpret CRL using a measurement model framework, where the learned representations are viewed as proxy measurements of the latent causal variables. Our approach clarifies the conditions under which learned representations support downstream causal reasoning and provides a principled basis for quantitatively assessing the quality of representations using a new Test-based Measurement EXclusivity (T-MEX) score. We validate T-MEX across diverse causal inference scenarios, including numerical simulations and real-world ecological video analysis, demonstrating that the proposed framework and corresponding score effectively assess the identification of learned representations and their usefulness for causal downstream tasks.

In label-noise learning, the noise transition matrix reveals how an instance transitions from its clean label to its noisy label. Accurately estimating an instance's noise transition matrix is crucial for estimating its clean label.

Causal representation learning seeks to uncover latent, high-level causal representations from low-level observed data. It is particularly good at predictions under unseen distribution shifts, because these shifts can generally be interpreted as consequences of interventions. Hence leveraging {seen} distribution shifts becomes a natural strategy to help identifying causal representations, which in turn benefits predictions where distributions are previously {unseen}. Determining the types (or conditions) of such distribution shifts that do contribute to the identifiability of causal representations is critical. This work establishes a {sufficient} and {necessary} condition characterizing the types of distribution shifts for identifiability in the context of latent additive noise models. Furthermore, we present partial identifiability results when only a portion of distribution shifts meets the condition. In addition, we extend our findings to latent post-nonlinear causal models. We translate our findings into a practical algorithm, allowing for the acquisition of reliable latent causal representations. Our algorithm, guided by our underlying theory, has demonstrated outstanding performance across a diverse range of synthetic and real-world datasets. The empirical observations align closely with the theoretical findings, affirming the robustness and effectiveness of our approach.

Causal representation learning aims to unveil latent high-level causal representations from observed low-level data. One of its primary tasks is to provide reliable assurance of identifying these latent causal models, known as identifiability. A recent breakthrough explores identifiability by leveraging the change of causal influences among latent causal variables across multiple environments (Liu et al., 2022). However, this progress rests on the assumption that the causal relationships among latent causal variables adhere strictly to linear Gaussian models. In this paper, we extend the scope of latent causal models to involve nonlinear causal relationships, represented by polynomial models, and general noise distributions conforming to the exponential family. Additionally, we investigate the necessity of imposing changes on all causal parameters and present partial identifiability results when part of them remains unchanged. Further, we propose a novel empirical estimation method, grounded in our theoretical finding, that enables learning consistent latent causal representations. Our experimental results, obtained from both synthetic and real-world data, validate our theoretical contributions concerning identifiability and consistency. Causal representation learning, aiming to discover high-level latent causal variables and causal structures among them from unstructured observed data, provides a prospective way to compensate for drawbacks in traditional machine learning paradigms, e.g., the most fundamental limitations that data, driving and promoting the machine learning methods, needs to be independent and identically distributed (i.i.d.) (Sch olkopf, 2015). From the perspective of causal representations, the changes in data distribution, arising from various real-world data collection pipelines (Karahan et al., 2016; Frumkin, 2016; Pearl et al., 2016; Chandrasekaran et al., 2021), can be attributed to the changes of causal influences among causal variables (Sch olkopf et al., 2021). These changes are observable across a multitude of fields. For instance, these could appear in the analysis of imaging data of cells, where the contexts involve batches of cells exposed to various small-molecule compounds.

The task of causal representation learning aims to uncover latent higher-level causal representations that affect lower-level observations. Identifying true latent causal representations from observed data, while allowing instantaneous causal relations among latent variables, remains a challenge, however. To this end, we start from the analysis of three intrinsic properties in identifying latent space from observations: transitivity, permutation indeterminacy, and scaling indeterminacy. We find that transitivity acts as a key role in impeding the identifiability of latent causal representations. To address the unidentifiable issue due to transitivity, we introduce a novel identifiability condition where the underlying latent causal model satisfies a linear-Gaussian model, in which the causal coefficients and the distribution of Gaussian noise are modulated by an additional observed variable. Under some mild assumptions, we can show that the latent causal representations can be identified up to trivial permutation and scaling. Furthermore, based on this theoretical result, we propose a novel method, termed Structural caUsAl Variational autoEncoder, which directly learns latent causal representations and causal relationships among them, together with the mapping from the latent causal variables to the observed ones. We show that the proposed method learns the true parameters asymptotically. Experimental results on synthetic and real data demonstrate the identifiability and consistency results and the efficacy of the proposed method in learning latent causal representations.