Causal representation learning (CRL) aims to identify the underlying latent variables from high-dimensional observations, even when variables are dependent with each other. We study this problem for latent variables that follow a potentially degenerate Gaussian mixture distribution and that are only observed through the transformation via a piecewise affine mixing function. We provide a series of progressively stronger identifiability results for this challenging setting in which the probability density functions are ill-defined because of the potential degeneracy. For identifiability up to permutation and scaling, we leverage a sparsity regularization on the learned representation. Based on our theoretical results, we propose a two-stage method to estimate the latent variables by enforcing sparsity and Gaussianity in the learned representations. Experiments on synthetic and image data highlight our method's effectiveness in recovering the ground-truth latent variables.

However, crucially, we do not make two assumptions used to derive the AVAE objective. Firstly, we do not assume that the decoder is a one-to-one mapping between latent samples and a corresponding generated sample. Interestingly,theDCIresponsibility matrices dooftenresemble theconditioned response matrices, suggesting that relying on correlations instead of a full causal analysis, can yield similar results. Infact, then the learned causal structure estimated using the latent response matrix may be used in tandem to develop a structure-aware disentanglementmetric. Given the complexity of the underlying data manifold, aviable alternativeisbased on riemanian geometry [78]which has previously been investigated for alternativeprobabilistic models likeGaussian Process regression [79].

The design of efficient nonparametric estimators has long been a central problem in statistics, machine learning, and decision making. Classical optimal procedures often rely on strong structural assumptions, which can be misspecified in practice and complicate deployment. This limitation has sparked growing interest in structure-agnostic approaches -- methods that debias black-box nuisance estimates without imposing structural priors. Understanding the fundamental limits of these methods is therefore crucial. This paper provides a systematic investigation of the optimal error rates achievable by structure-agnostic estimators. We first show that, for estimating the average treatment effect (ATE), a central parameter in causal inference, doubly robust learning attains optimal structure-agnostic error rates. We then extend our analysis to a general class of functionals that depend on unknown nuisance functions and establish the structure-agnostic optimality of debiased/double machine learning (DML). We distinguish two regimes -- one where double robustness is attainable and one where it is not -- leading to different optimal rates for first-order debiasing, and show that DML is optimal in both regimes. Finally, we instantiate our general lower bounds by deriving explicit optimal rates that recover existing results and extend to additional estimands of interest. Our results provide theoretical validation for widely used first-order debiasing methods and guidance for practitioners seeking optimal approaches in the absence of structural assumptions. This paper generalizes and subsumes the ATE lower bound established in \citet{jin2024structure} by the same authors.

As graph-structured data grow increasingly large, evaluating their robustness under adversarial attacks becomes computationally expensive and difficult to scale. To address this challenge, we propose to compress graphs into compact representations that preserve both topological structure and robustness profile, enabling efficient and reliable evaluation. We propose Cutter, a dual-agent reinforcement learning framework composed of a Vital Detection Agent (VDA) and a Redundancy Detection Agent (RDA), which collaboratively identify structurally vital and redundant nodes for guided compression. Cutter incorporates three key strategies to enhance learning efficiency and compression quality: trajectory-level reward shaping to transform sparse trajectory returns into dense, policy-equivalent learning signals; prototype-based shaping to guide decisions using behavioral patterns from both high- and low-return trajectories; and cross-agent imitation to enable safer and more transferable exploration. Experiments on multiple real-world graphs demonstrate that Cutter generates compressed graphs that retain essential static topological properties and exhibit robustness degradation trends highly consistent with the original graphs under various attack scenarios, thereby significantly improving evaluation efficiency without compromising assessment fidelity.

We present a JIT PL semantics for ReLU-type networks that compiles models into a guarded CPWL transducer with shared guards. The system adds hyperplanes only when operands are affine on the current cell, maintains global lower/upper envelopes, and uses a budgeted branch-and-bound. We obtain anytime soundness, exactness on fully refined cells, monotone progress, guard-linear complexity (avoiding global $\binom{k}{2}$), dominance pruning, and decidability under finite refinement. The shared carrier supports region extraction, decision complexes, Jacobians, exact/certified Lipschitz, LP/SOCP robustness, and maximal causal influence. A minimal prototype returns certificates or counterexamples with cost proportional to visited subdomains.

We hereby provide a summarizing figure of the training and inference stages used in TIM. Figure 2: Outline of TIM framework (best viewed in color).

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