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Goutham Rajendran, Simon Buchholz, Bryon Aragam, Bernhard Schölkopf, Pradeep Ravikumar

To build intelligent machine learning systems, there are two broad approaches. One approach is to build inherently interpretable models, as endeavored by the growing field of causal representation learning. The other approach is to build highly-performant foundation models and then invest efforts into understanding how they work. In this work, we relate these two approaches and study how to learn human-interpretable concepts from data. Weaving together ideas from both fields, we formally define a notion of concepts and show that they can be provably recovered from diverse data. Experiments on synthetic data and large language models show the utility of our unified approach.

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Yuhao Wang, Ming Gao, Wai Ming Tai, Bryon Aragam, Arnab Bhattacharyya

We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.

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Zhao Lyu, Wai Ming Tai, Mladen Kolar, Bryon Aragam

Despite numerous years of research into the merits and trade-offs of various model selection criteria, obtaining robust results that elucidate the behavior of cross-validation remains a challenging endeavor. In this paper, we highlight the inherent limitations of cross-validation when employed to discern the structure of a Gaussian graphical model. We provide finite-sample bounds on the probability that the Lasso estimator for the neighborhood of a node within a Gaussian graphical model, optimized using a prediction oracle, misidentifies the neighborhood. Our results pertain to both undirected and directed acyclic graphs, encompassing general, sparse covariance structures. To support our theoretical findings, we conduct an empirical investigation of this inconsistency by contrasting our outcomes with other commonly used information criteria through an extensive simulation study. Given that many algorithms designed to learn the structure of graphical models require hyperparameter selection, the precise calibration of this hyperparameter is paramount for accurately estimating the inherent structure. Consequently, our observations shed light on this widely recognized practical challenge.

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Yibo Jiang, Bryon Aragam, Victor Veitch

Machine learning tools often rely on embedding text as vectors of real numbers. In this paper, we study how the semantic structure of language is encoded in the algebraic structure of such embeddings. Specifically, we look at a notion of ``semantic independence'' capturing the idea that, e.g., ``eggplant'' and ``tomato'' are independent given ``vegetable''. Although such examples are intuitive, it is difficult to formalize such a notion of semantic independence. The key observation here is that any sensible formalization should obey a set of so-called independence axioms, and thus any algebraic encoding of this structure should also obey these axioms. This leads us naturally to use partial orthogonality as the relevant algebraic structure. We develop theory and methods that allow us to demonstrate that partial orthogonality does indeed capture semantic independence. Complementary to this, we also introduce the concept of independence preserving embeddings where embeddings preserve the conditional independence structures of a distribution, and we prove the existence of such embeddings and approximations to them.

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Francesco Montagna, Atalanti A. Mastakouri, Elias Eulig, Nicoletta Noceti, Lorenzo Rosasco, Dominik Janzing, Bryon Aragam, Francesco Locatello

When domain knowledge is limited and experimentation is restricted by ethical, financial, or time constraints, practitioners turn to observational causal discovery methods to recover the causal structure, exploiting the statistical properties of their data. Because causal discovery without further assumptions is an ill-posed problem, each algorithm comes with its own set of usually untestable assumptions, some of which are hard to meet in real datasets. Motivated by these considerations, this paper extensively benchmarks the empirical performance of recent causal discovery methods on observational i.i.d. data generated under different background conditions, allowing for violations of the critical assumptions required by each selected approach. Our experimental findings show that score matching-based methods demonstrate surprising performance in the false positive and false negative rate of the inferred graph in these challenging scenarios, and we provide theoretical insights into their performance. This work is also the first effort to benchmark the stability of causal discovery algorithms with respect to the values of their hyperparameters. Finally, we hope this paper will set a new standard for the evaluation of causal discovery methods and can serve as an accessible entry point for practitioners interested in the field, highlighting the empirical implications of different algorithm choices.

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Chang Deng, Kevin Bello, Bryon Aragam, Pradeep Ravikumar

Recently, a new class of non-convex optimization problems motivated by the statistical problem of learning an acyclic directed graphical model from data has attracted significant interest. While existing work uses standard first-order optimization schemes to solve this problem, proving the global optimality of such approaches has proven elusive. The difficulty lies in the fact that unlike other non-convex problems in the literature, this problem is not "benign", and possesses multiple spurious solutions that standard approaches can easily get trapped in. In this paper, we prove that a simple path-following optimization scheme globally converges to the global minimum of the population loss in the bivariate setting.

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Tianyu Chen, Kevin Bello, Bryon Aragam, Pradeep Ravikumar

Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the true underlying directed acyclic graph (DAG) structure is often unknown, and determining it from observational or interventional data remains a challenging task. However, in many situations, the end goal is to identify changes (shifts) in causal mechanisms between related SCMs rather than recovering the entire underlying DAG structure. Examples include analyzing gene regulatory network structure changes between healthy and cancerous individuals or understanding variations in biological pathways under different cellular contexts. This paper focuses on identifying $\textit{functional}$ mechanism shifts in two or more related SCMs over the same set of variables -- $\textit{without estimating the entire DAG structure of each SCM}$. Prior work under this setting assumed linear models with Gaussian noises; instead, in this work we assume that each SCM belongs to the more general class of nonlinear additive noise models (ANMs). A key contribution of this work is to show that the Jacobian of the score function for the $\textit{mixture distribution}$ allows for identification of shifts in general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences, if any, for the shifted variables. Experiments on synthetic and real-world data are provided to showcase the applicability of this approach.

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Yibo Jiang, Bryon Aragam

We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in a measurement model, i.e. causal graphical models where dependence between observed variables is insignificant compared to dependence between latent representations, without making parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts -- imaginary subsets and isolated edges -- that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. Experiments confirm that the latent graph can be recovered from data using our theoretical results. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness.

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Simon Buchholz, Goutham Rajendran, Elan Rosenfeld, Bryon Aragam, Bernhard Schölkopf, Pradeep Ravikumar

We study the problem of learning causal representations from unknown, latent interventions in a general setting, where the latent distribution is Gaussian but the mixing function is completely general. We prove strong identifiability results given unknown single-node interventions, i.e., without having access to the intervention targets. This generalizes prior works which have focused on weaker classes, such as linear maps or paired counterfactual data. This is also the first instance of causal identifiability from non-paired interventions for deep neural network embeddings. Our proof relies on carefully uncovering the high-dimensional geometric structure present in the data distribution after a non-linear density transformation, which we capture by analyzing quadratic forms of precision matrices of the latent distributions. Finally, we propose a contrastive algorithm to identify the latent variables in practice and evaluate its performance on various tasks.

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Alex Markham, Mingyu Liu, Bryon Aragam, Liam Solus

Factor analysis (FA) is a statistical tool for studying how observed variables with some mutual dependences can be expressed as functions of mutually independent unobserved factors, and it is widely applied throughout the psychological, biological, and physical sciences. We revisit this classic method from the comparatively new perspective given by advancements in causal discovery and deep learning, introducing a framework for Neuro-Causal Factor Analysis (NCFA). Our approach is fully nonparametric: it identifies factors via latent causal discovery methods and then uses a variational autoencoder (VAE) that is constrained to abide by the Markov factorization of the distribution with respect to the learned graph. We evaluate NCFA on real and synthetic data sets, finding that it performs comparably to standard VAEs on data reconstruction tasks but with the advantages of sparser architecture, lower model complexity, and causal interpretability. Unlike traditional FA methods, our proposed NCFA method allows learning and reasoning about the latent factors underlying observed data from a justifiably causal perspective, even when the relations between factors and measurements are highly nonlinear.

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