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1. Identifiability of deep generative models without auxiliary information

   Kivva, Bohdan; Rajendran, Goutham; Ravikumar, Pradeep; Aragam, Bryon (December 2023, Advances in Neural Information Processing Systems (NeurIPS))

   We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational auto-encoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i.e. unsupervised) models with universal approximation capabilities, the side information u is not necessary: We prove identifiability of the entire generative model where we do not observe u and only observe the data x. The models we consider match auto-encoder architectures used in practice that leverage mixture priors in the latent space and ReLU/leaky-ReLU activations in the encoder, such as VaDE and MFC-VAE. Our main result is an identifiability hierarchy that significantly generalizes previous work and exposes how different assumptions lead to different “strengths” of identifiability, and includes certain “vanilla” VAEs with isotropic Gaussian priors as a special case. For example, our weakest result establishes (unsupervised) identifiability up to an affine transformation, and thus partially resolves an open problem regarding model identifiability raised in prior work. These theoretical results are augmented with experiments on both simulated and real data. 

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2. On Finite-Sample Identifiability of Contrastive Learning-Based Nonlinear Independent Component Analysis

   Lyu, Qi; Fu, Xiao (July 2022, Proceedings of the 39th International Conference on Machine Learning)

   Nonlinear independent component analysis (nICA) aims at recovering statistically independent latent components that are mixed by unknown nonlinear functions. Central to nICA is the identifiability of the latent components, which had been elusive until very recently. Specifically, Hyvärinen et al. have shown that the nonlinearly mixed latent components are identifiable (up to often inconsequential ambiguities) under a generalized contrastive learning (GCL) formulation, given that the latent components are independent conditioned on a certain auxiliary variable. The GCL-based identifiability of nICA is elegant, and establishes interesting connections between nICA and popular unsupervised/self-supervised learning paradigms in representation learning, causal learning, and factor disentanglement. However, existing identifiability analyses of nICA all build upon an unlimited sample assumption and the use of ideal universal function learners—which creates a non-negligible gap between theory and practice. Closing the gap is a nontrivial challenge, as there is a lack of established “textbook” routine for finite sample analysis of such unsupervised problems. This work puts forth a finite-sample identifiability analysis of GCL-based nICA. Our analytical framework judiciously combines the properties of the GCL loss function, statistical generalization analysis, and numerical differentiation. Our framework also takes the learning function’s approximation error into consideration, and reveals an intuitive trade-off between the complexity and expressiveness of the employed function learner. Numerical experiments are used to validate the theorems. 

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3. Nonlinear Dependent Component Analysis: Identifiability and Algorithm

   https://doi.org/10.23919/Eusipco47968.2020.9287354  

   Lyu, Qi; Fu, Xiao (January 2021, EUSIPCO 2020)

   null (Ed.)

   This work studies the model identification problem of a class of post-nonlinear mixture models in the presence of dependent latent components. Particularly, our interest lies in latent components that are nonnegative and sum-to-one. This problem is motivated by applications such as hyperspectral unmixing under nonlinear distortion effects. Many prior works tackled nonlinear mixture analysis using statistical independence among the latent components, which is not applicable in our case. A recent work by Yang et al. put forth a solution for this problem leveraging functional equations. However, the identifiability conditions derived there are somewhat restrictive. The associated implementation also has difficulties-the function approximator used in their work may not be able to represent general nonlinear distortions and the formulated constrained neural network optimization problem may be challenging to handle. In this work, we advance both the theoretical and practical aspects of the problem of interest. On the theory side, we offer a new identifiability condition that circumvents a series of stringent assumptions in Yang et al.'s work. On the algorithm side, we propose an easy-to-implement unconstrained neural network-based algorithm-without sacrificing function approximation capabilities. Numerical experiments are employed to support our design. 

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4. RevUp: Revise and Update Information Bottleneck for Event Representation

   https://doi.org/10.18653/v1/2023.eacl-main.56  

   Rezaee, Mehdi; Ferraro, Francis (May 2023, 17th Conference of the European Chapter of the Association for Computational Linguistics (EACL))

   The existence of external (“side”) semantic knowledge has been shown to result in more expressive computational event models. To enable the use of side information that may be noisy or missing, we propose a semi-supervised information bottleneck-based discrete latent variable model. We reparameterize the model’s discrete variables with auxiliary continuous latent variables and a light-weight hierarchical structure. Our model is learned to minimize the mutual information between the observed data and optional side knowledge that is not already captured by the new, auxiliary variables. We theoretically show that our approach generalizes past approaches, and perform an empirical case study of our approach on event modeling. We corroborate our theoretical results with strong empirical experiments, showing that the proposed method outperforms previous proposed approaches on multiple datasets. 

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5. Learning Linear Causal Representations from Interventions under General Nonlinear Mixing

   Buchholz, Simon; Rajendran, Goutham; Rosenfeld, Elan; Aragam, Bryon; Schölkopf, Bernhard; Ravikumar, Pradeep (December 2023, Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track)

   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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