Identifiability

Prediction models trained under the source distribution do not generalize well to a different target distribution. A valid inference about an unseen data distribution must be anchored by the invariance of certain causal mechanisms that generate the source and target data, however, these structural invariances are non-identifiable from the source data alone. Under mild causal assumptions about the data, we show that the optimal prediction in the target is in fact partially identifiable by the source distribution. The result rests on a simple observation: In any domain, the optimal prediction can be factorized into what we call a pair of abduction and deduction maps, where the abduction map makes inference about some unobserved variables (possibly confounders) from the observed variables and the deduction map predicts the label using both the observed and inferred quantities. Access to large source data pins down the optimal prediction, thus constrains the valid abduction-deduction ensembles that produce it -- a non-identifiability that we call the abduction-deduction entanglement. To leverage this, we parameterize the constrained family using what we call a representation transplant, that is a specific linear transformation in the representation space that manipulates the abduction content of the representation while retaining the deduction component. Invariance of the causal mechanism generating the label implies existence of an invariant deduction map between source and target. Thus, we can search the space of plausible target distributions via a parametric transplant. We use this scheme in a learner-adversary game that, under an idealistic optimization, provably terminates with the learner having the minimax-optimal target prediction. Evaluations verify the theory, showing that the method is competitive in DG benchmarks.

We establish four structural results for feature learning in wide two-layer neural networks under the Maximal Update Parametrization ($μ$P). First, we prove global existence and uniqueness of the mean-field limit of noisy gradient descent under $μ$P, identifying the maximal admissible weight $w^*$ on the moment sequence of the initialization as the reciprocal parameter-moment-growth boundary, and hence the largest weighted moment class propagated by the flow. The finite-particle approximation has uniform-in-time squared-Wasserstein rate $O(N^{-1})$. Second, we characterize identifiability of the mean-field limit: two admissible parameter measures induce the same network function in $L^2$ exactly when their active components agree modulo the finite-rank realization symmetry of the architecture. The orbit depth $D^*_{\mathrm{orb}}$ is separated from the moment-variety depth $D^*_{\mathrm{var}}$. Third, under the Barron-Hermite target condition the active support of the long-time limit measure admits a sparse-dictionary decomposition: it is supported on at most $S^*$ atoms modulo finite-rank realization symmetry, with $S^*$ bounded by an explicit coefficient-threshold number. Fourth, we derive the total feature-learning-error decomposition into statistical, optimization, propagation-of-chaos, and sparse-residual components, with a target-dependent Hermite/Barron tail replacing any initialization-only residual. The four results are tied together by an architectural identity: the triple $(w^*, D^*_{\mathrm{orb}}, S^*)$ -- the maximal admissible weight, the orbit identifiability depth, and the sparse-dictionary depth at which the target is realizable -- is the natural learning cell of the architecture-data pair $(σ, ρ)$. The proofs are self-contained except for standard results from $μ$P and mean-field Langevin theory.

Language models trained on observed sequences are often described as learning the conditional distribution of the next token given previous tokens. This description is only conditionally correct. A model trained on realized token trajectories does not observe full conditional laws; it receives sampled continuations. Moreover, real language generation is conditioned not only on previous words but also on non-textual circumstances: facts, events, intentions, goals, beliefs, social context, and task-specific constraints. This paper distinguishes three objects that are often conflated: the full conditional language process conditioned on latent circumstances, the marginal text-only process obtained by integrating those circumstances out, and the model-induced distribution learned from finite observed corpora. The paper argues that interpreting model training as estimating the marginal text-only law requires strong assumptions of stationarity, representativeness, and ergodicity, assumptions that are standard in statistical estimation but problematic when applied to heterogeneous language corpora. Even if these assumptions hold, the marginal text-only law is useful only when the observed prefix is an approximately sufficient statistic for the latent circumstances relevant to continuation. In information-theoretic terms, usefulness requires that the residual conditional mutual information between the next token and the omitted circumstances, given the observed text, be small. The paper then extends this argument to heterogeneous training corpora. Finally, the paper interprets Retrieval Augmented Generation (RAG) and tool use as conditional sufficiency devices.

Conversation logs from AI platforms are increasingly used to measure occupational exposure to artificial intelligence, but the users observed in these logs are not the workforce. We show that platform-derived exposure scores combine task-level AI applicability with the occupational composition of the platform's user base. Holding the empirical design fixed, changing only the platform input changes the post-ChatGPT employment coefficient by a factor of 1.9, and consumer and enterprise channels within the same vendor disagree in sign. We formalize the resulting non-classical measurement error, decompose it into between- and within-occupation selection, and construct workforce-reweighted partial-identification bounds. Reweighting to Bureau of Labor Statistics employment shares attenuates estimates by 42 to 93 percent. The bias captures augmentation among observed users more directly than substitution in the workforce.

High-fidelity street scene reconstruction is pivotal for end-to-end autonomous driving simulation, where novel-view synthesis (NVS) and time-varying information modeling are two fundamental capabilities to facilitate closed-loop training. However, existing 3DGS methods and their 4D extensions fail to simultaneously achieve both. To bridge this gap, we establish an information-geometric diagnostic framework, revealing that this limitation stems from a credit assignment dilemma between spatial and temporal parameters. Specifically, the deterministic coupling between viewpoint and time in single-source observation creates a low-rank structure that induces massive null-space ambiguity between static view-dependent and dynamic time-varying components. Temporal information overshadows spatial cues, causing the estimation variance of spatial parameters to diverge. To address this issue, we propose Orthogonal Projected Gradient (OPG), a hierarchical training method designed to restore spatial identifiability. OPG prioritizes the integrity of spatial representations by securing them in an initial stage, then restricts temporal updates to the spatial null space, enabling proactive credit assignment. While OPG isolates temporal updates algebraically, Temporal Regularization Strategy is proposed to further refine the temporal solution space by imposing a smoothness constraint based on the physical prior of consistent appearance evolution, ensuring that the reconstructed scene remains physically consistent in closed-loop simulation. Extensive experiments demonstrate that our method not only maintains stable NVS capabilities but also demonstrates superior performance in traditional observation-reproducing metrics, which indirectly reflect the capability of modeling temporal dynamics.

Causal representation learning (CRL) seeks to uncover meaningful latent variables and their corresponding causal structure from high-dimensional observational data. Although its significance, CRL identifiability remains a crucial property, as it ensures the recovery of the mechanisms behind the data generation process, and hence the interpretability and robustness of the representation. Proving identifiability in CRL is intrinsically difficult, and we address in this work an even more challenging setting: multimodality. We consider multimodal observed data with a latent partially shared structure. Each modality is generated, through non linear mixing functions, from a specific subset of causal latent variables. Under flexible assumptions and without imposing any parametric distribution on the latent variables, we establish component-wise identifiability guarantees for the causal latent representation. Our identifiability results, furthermore, apply to the undercomplete scenario where we have, for each modality, more observed than latent variables. To instantiate our theoretical analysis, we introduce a Wasserstein-based module to recover the partially shared latent structure. Due to its differentiability, the latter can be easily integrated into all types of architecture, only requiring minimal changes. Extensive experiments on synthetic and realistic datasets validate the superiority of our approach over SOTA methods.

This paper develops a feedback-based method to preserve the topology privacy of consensus protocols in network systems. The key idea is to intentionally violate topology identifiability conditions, thereby preventing unique or accurate recovery of the true topology from available observations, while preserving the intended consensus behavior. This problem is challenging because the feedback magnitude directly reflects the privacy level of edges, while it is strongly coupled with the consensus convergence and constrained by local communications at each node. To begin with, we derive the feedback conditions of both partial and full observation cases, where the topology unsolvability from observation data is characterized in the former, and the solution space that enforces topology inaccuracy from data is constructed in the latter. Then, we propose a novel distributed topology modification design under limited privacy budgets, and establish the performance guarantees through a controllable tradeoff between the consensus deviation and the topology privacy. Finally, we develop a low-complexity heuristic algorithm to achieve optimal privacy preservation on existing edges. Comparative simulations validate the effectiveness and outperformance of the proposed preservation design.

Causal discovery, the problem of inferring the direction of causality, is generally ill-posed. We use the language of structural causal models (SCM) to show that assuming that the causal relations are acyclic and invariant across multiple environments (e.g., the way minimum wage affects employment rate is stable across different geographical regions), \textit{only} two auxiliary environments are sufficient to infer the causal graph for arbitrary nonlinear mechanisms. Moreover, we demonstrate that this implies identifiability of the SCM functional mechanisms: as a corollary, we show that \textit{two} auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.

Selection bias is pervasive in observational studies. For example, large scale biobanks data can exhibit ``healthy volunteer bias'' when respondents are healthier and of higher socio-economic status than the population they are meant to represent. Recovering causal effects from such sub-population is an important problem in causal inference, as estimating average treatment effects (ATE) from selected populations can result in a severely biased estimate of the ATE from the whole population. In this paper, we investigate the identifiability of the ATE under selection bias. We provide necessary and sufficient conditions for ATE identifiability, leveraging weak assumptions on probability classes to characterize propensity score and selection probability. Compared to previous works, our results extend existing graphical identifiability criteria and offer a more comprehensive understanding of causal effect identification with strictly weaker conditions in the presence of selection bias.

Given a generalist model, learning a task-relevant specialist representation is fundamental for downstream applications. Identifiability, the asymptotic guarantee of recovering the ground-truth representation, is critical because it sets the ultimate limit of any model, even with infinite data and computation. We study this problem in a completely nonparametric setting, without relying on interventions, parametric forms, or structural constraints. We first prove that the structure between time steps and tasks is identifiable in a fully unsupervised manner, even when sequences lack strict temporal dependence and may exhibit disconnections, and task assignments can follow arbitrarily complex and interleaving structures. We then prove that, within each time step, the task-relevant latent representation can be disentangled from the irrelevant part under a simple sparsity regularization, without any additional information or parametric constraints. Together, these results establish a hierarchical foundation: task structure is identifiable across time steps, and task-relevant latent representations are identifiable within each step. To our knowledge, each result provides a first general nonparametric identifiability guarantee, and together they mark a step toward provably moving from generalist to specialist models.

When people share the same documents and observations yet reach different conclusions, the disagreement often shifts into a judgment that the other party is cognitively defective, irrational, or acting in bad faith. This paper argues that such divergence is better described as a form of non-identifiability inherent in inference and learning, rather than as a defect of the other party. We organize the phenomenon into two levels: (i) $θ$-level non-identifiability, where conclusions diverge under the same world model $W$ because inference settings differ; and (ii) $W$-level non-identifiability, where repeated use of an inference setting $θ$ biases data exposure and update rules, causing the learned world model $W$ itself to diverge. We introduce an inference profile $θ= (R, E, S, D)$, consisting of Reference, Exploration, Stabilization, and Horizon, and show how outputs can split even for the same observation $o$ and the same $W$. We further explain why disagreements tend to project onto a small number of bases -- abstract versus concrete, externalizability, and order versus freedom -- as a consequence of general constraints on learning systems: computational, observational, and coordination constraints. Finally, we relate the framework to deep representation learning, including representation hierarchy, latent-state estimation, and regularization-exploration trade-offs, and illustrate the framework through a case study on AI regulation debates.

Learning governing dynamics from data is a common goal across the sciences, yet it is only well-posed when the underlying mechanisms are identifiable. In practice, many data-driven methods implicitly assume identifiability; when this assumption fails, estimated models can yield spurious predictions and invalid mechanistic conclusions. Classical identifiability guarantees for controlled linear time-invariant (LTI) systems provide sufficient conditions -- controllability and persistent excitation -- but leave open whether identifiability holds when these conditions fail, and which parts of the system remain identifiable without full identifiability. We show that the experimental setup, i.e., the realized initial state and control input, dictates a fundamental limit on the information recoverable from the observed trajectory. We develop a geometric characterization of this limit and derive a closed-form description of all systems consistent with the experimental setup. Crucially, we prove that even when the full system is not identifiable, the restricted dynamics on the subspace reachable by the experiment remain uniquely determined.

Recent work on causal abstraction, in particular graphical approaches focusing on causal structure between clusters of variables, aims to summarize a high-dimensional causal structure in terms of a low-dimensional one. Existing methods for learning such summaries from data assume that both the high- and low-dimensional structures are acyclic, which is helpful for causal effect identification and reasoning but excludes many high-dimensional models and thus limits applicability. We show that in the linear non-Gaussian (LiNG) setting, the high-dimensional acyclicity assumption can be relaxed while still allowing recovery of a low-dimensional causal directed acyclic graph (DAG). We further connect identifiability of this low-dimensional DAG to existing results: LiNG models with cycles are observationally identifiable only up to an equivalence class whose members differ by reversals of directed cycles; our low-dimensional DAG, which is invariant across all members of a given equivalence class, thus forms a natural representative of the class. While existing approaches for learning this observational equivalence class over high-dimensional variables have exponential time complexity, our low-dimensional summary is learned in worst-case cubic time and comes with explicit bounds on the sample complexity. We provide open source code and experiments on synthetic data to corroborate our theoretical results.

A valid tool call is not necessarily a valid intervention. Tool-using language agents are guarded by schema validators, policy filters, provenance checks, state predictors, and self-verification, yet such safeguards do not certify that a state-changing action has an identifiable causal effect. In confounded workflows, the action that looks optimal in observational logs can reduce utility when executed. We introduce CIVeX, a causal intervention verifier that maps proposed actions to structural causal queries over a committed action-state graph, checks identifiability, and returns one of four auditable verdicts: EXECUTE, REJECT, EXPERIMENT, or ABSTAIN. Execution requires an assumption-scoped causal certificate carrying graph commitments, an identification argument, a one-sided lower confidence bound (LCB), provenance, and risk limits. On Causal-ToolBench (1,890 instances, 7 seeds), CIVeX yields zero observed false executions across moderate and adversarial confounding. Under adversarial confounding it reaches 84.9% accuracy and 81.1% of oracle utility (+2.23 vs +2.76) and is the only non-oracle method whose constrained utility under a zero-false-execution constraint exceeds the AlwaysAbstain floor. On IHDP and ZOZO Open Bandit (real production logs with uniform-random ground truth), CIVeX matches Oracle correct-execution within 0.1pp and cuts per-execute false-execution by >=50x over naive baselines. A chain-of-thought LLM verifier (Claude Opus, Sonnet) cuts false-execution by an order of magnitude over a terse baseline, yet under adversarial confounding Opus's utility falls to 74% of CIVeX's. Intervention identifiability, not action validity, is the missing primitive for reliable tool use.

Personalized alignment aims to adapt large language models to heterogeneous user preferences, yet the precise theoretical conditions for its statistical efficiency have not been formally established. This paper characterizes the conditions under which personalized alignment achieves O(1) online regret and log(1/epsilon) offline sample complexity. We show that these optimal rates depend on a specific user-diversity condition: the population of user-specific heads must span the latent reward directions that can alter the optimal response. We prove that this condition is both necessary and sufficient. When it holds, simple greedy algorithms achieve benchmark efficiency; when it fails, every learner in a natural admissible class incurs at least logarithmic regret. Our results identify user diversity as the fundamental driver of personalized identifiability.

Mechanistic interpretability papers increasingly use causal vocabulary: circuits, mediators, causal abstraction, monosemanticity. Such claims require explicit identification assumptions. A purposive audit of 10 papers across four methodological strands finds no dedicated identification-assumptions section and a recurring pattern: validation metrics such as faithfulness, completeness, monosemanticity, alignment, or ablation effects are reported as causal support without stating the assumptions that make them identifying. A two-human-coder audit on $n=30$ reproduces the direction of the main finding: dedicated identification sections are absent, and validation-metric substitution is common, though exact Dim B/D counts are coding-rule sensitive. The paper proposes a disclosure norm: state whether the claim is causal, name the identification strategy, enumerate assumptions, stress at least one, and explain how conclusions shift if assumptions fail. Validation is not identification.

Test-Time Adaptation (TTA) aims to improve time series forecasting under distribution shifts by using limited observations revealed during inference. However, forecasting TTA must operate in a source-free online setting, where the adaptation signal is short, temporally correlated, and potentially noisy. Existing methods can therefore suffer from weak identifiability, error accumulation, and unstable long-horizon corrections when the revealed prefix is sparse or contaminated. To address these issues, we propose STEPS, a Smooth Temporal Error Propagation Solver for TTA in time-series forecasting. STEPS reformulates forecasting TTA as a Dirichlet Boundary Value Problem on a temporal manifold, where the revealed prefix error serves as the boundary condition for the unknown future error field. Then, STEPS solves a smooth and bounded correction field in prediction space: a Local Solver propagates prefix errors under temporal smoothness, a Global Solver retrieves stable cross-window error memory and Spatiotemporal Manifold Fusion (SMF) integrates both solutions into the final correction. Across six standard benchmarks and four frozen backbones, STEPS achieves an average relative MSE reduction of 26.82% over the zero-shot backbone, exceeding the strongest compared TTA baseline by 12.77%. Additional sparse prefix and contamination tests confirm the robustness of STEPS under limited and noisy prefixes.

Successful deep neural networks discover salient features of data. We show when and why they fail to learn out-of-distribution (OOD)-relevant representations from an in-distribution (ID) training window. This requires decoupling feature learning from data-generating-process (DGP) identifiability. From a single training window, OOD extrapolation is non-identifiable: infinitely many DGPs are $\varepsilon$-observationally equivalent on the training data but diverge arbitrarily outside it, and no in-distribution criterion alone reliably breaks the tie. A structural commitment, the feature map, label map, and model class $(\varphi, ψ, \mathcal{M})$, dictates the assumed DGP and governs OOD generalization while leaving ID performance essentially unchanged. When architecture, pretraining, augmentation, input formats, or domain knowledge implicitly inject the missing commitment, the model succeeds. When it cannot infer OOD-relevant structure from ID evidence, it fails. Changing only the representation can make the same architecture, at the same in-distribution loss, differ by ${\sim}520\times$ out of distribution. When the commitment is correct and identifiable, OOD error vanishes. For example, Fourier coordinates turn periodic extrapolation into interpolation on $\mathbb{S}^1$. The same mechanism predicts outcomes in three natural-science settings (mass-action chemistry; Kepler's-third-law exoplanet prediction, $n=2{,}362$; and cross-species coding-DNA detection) and in a 264-run positional-encoding study across Transformer, Mamba, and S4D. Finally, a controlled study shows: correct features are necessary but not sufficient. The model class must express the target, and the transformed training data must cover the relevant representation space.

Causal queries are often only partially identifiable from observational data, and experiments that could tighten the resulting bounds are typically costly. We study the problem of selecting, prior to observing experimental outcomes, a cost-constrained subset of experiments that maximally tightens bounds on a target query. We formalize this as the max-potency problem, where epistemic potency measures the worst-case reduction in bound width guaranteed by an experiment, and show that this problem is NP-hard via a reduction from 0-1 knapsack. Building on the polynomial-programming framework of Duarte et al. (2023), we give a general procedure for evaluating epistemic potency in discrete settings. To control the super-exponential search space, we introduce two graphical pruning criteria that depend only on the causal graph and the query: a novel path-interception rule that exploits district structure to certify zero potency in linear time, and an identifiability check based on the ID algorithm. On Erdos-Renyi random graphs and 11 bnlearn benchmark networks, the two criteria together prune 50-88% of candidate experiments on average without solving a single polynomial program. For the general subset search, we show that ID-pruned experiments are combinatorially inert, yielding a super-exponential reduction in the number of subsets evaluated. We close with an end-to-end demonstration on observational NHANES data, selecting optimal experiments for estimating the effect of physical activity on diabetes.

Learning identifiable representations in deep generative models remains a fundamental challenge, particularly for sequential data with regime-switching dynamics. Existing approaches establish identifiability under restrictive assumptions, such as stationarity or limited emission models, and typically rely on variational autoencoder (VAE) estimators, which introduce approximation gaps that limit the recovery of the latent structure. In this work, we address both the theoretical and practical limitations of this setting. First, we establish identifiability of a broad class of recurrent nonlinear switching dynamical systems under flexible assumptions, significantly extending prior results. Second, we introduce $Ω$SDS, a flow-based estimator that enables exact likelihood optimization using expectation-maximisation. Through empirical validation on both synthetic and real-world data, our results demonstrate that $Ω$SDS achieves improved disentanglement compared to VAE-based estimators and more accurate forecasting of underlying dynamics.

Domain adaptation aims to transfer knowledge from a labeled source domain to an unlabeled target domain with different distributions. In real-world scenarios, the label spaces of the two domains often have an inclusion relationship, where some classes exist only in one domain but not the other. These non-overlapping classes are referred to as private classes. Identifying private class samples and mitigating their adverse effects is critical in the literature. Existing methods rely on the assumption that shifts in private classes are large enough to be considered outliers. However, the variance within a single shared class can be significantly larger than the difference between a private class and another shared class, challenging this assumption. Consequently, private classes substantially increase the difficulty of cross-domain classification. To address these issues, based on local transportation and metric properties of optimal transport (OT), a locality-aware private class identification approach is proposed in the form of a score function on transport mass. The effectiveness of the proposed approach is theoretically proven, highlighting the score function's strong ability to distinguish between shared and private class samples. Building on this, we introduce a reliable OT-based method (ReOT) for domain adaptation under severe label shift. ReOT minimizes classification risk while learning the separated cluster structure between the identified shared classes and private classes, effectively avoiding mismatch between shared-private sample pairs, thus ensuring that important knowledge is reliably transported intra-class to mitigate class-conditional discrepancy. Furthermore, a generalization upper bound of the target risk is provided for extreme label shift scenarios, which can be minimized by ReOT. Extensive experiments on benchmarks validate the effectiveness of ReOT.

Data-driven equation discovery is fundamentally an inverse problem that seeks to infer the governing differential equations of a system directly from time-series measurements. A known issue is the ill-conditioned nature of the inverse problem, which frequently produces multiple mathematical models that fit the data similarly well. One path to address this issue is by incorporating known hypotheses and constraints into the training phase beforehand. While this approach effectively reduces the search space, it still results in multiple candidate models, forcing practitioners to rely on post-hoc manual filtering based on their own domain expertise. A recent approach incorporates structural `skeletons' inspired by characteristic curves (CCs), defining a hypothesis-driven methodology. In this methodology, practitioners define a skeleton, which is associated with a family of ordinary differential equations (ODEs), and then add their hypotheses and priors based on their domain knowledge to refine the obtained model iteratively. An important advantage of this approach is that some skeletons have demonstrable structural identifiability properties, which are useful for checking whether the skeleton is correct or should be discarded. Furthermore, this formalism enables the use of multiple equation discovery paradigms due to its modularity (such as neural networks, symbolic regression, and sparse regression). In this work, we present the Python library PyCC, which condenses these efforts into a flexible tool that allows researchers and engineers to seamlessly define their skeletons and hypotheses to discover ODEs from time-dependent data.

Causal representation learning (CRL) seeks to recover latent variables with identifiability guarantees, typically up to permutation and component-wise reparameterization under appropriate assumptions. However, identifiability does not imply interpretability: latent semantics are typically assigned post hoc by alignment with known ground-truth factors. This limitation is particularly acute in scientific time series, where underlying mechanisms are unknown and discovering interpretable structure is a primary goal. In contrast, scientific observations (such as residue-pair distances, climate indices, or process sensors) are inherently semantic, as they correspond to named physical quantities. This raises a key question: can the interpretability of observations be transferred to the identifiable latent space? We propose MOSAIC (Module discovery via Sparse Additive Identifiable Causal learning), a sparse temporal VAE that integrates temporal CRL identifiability with support recovery over observed variables. MOSAIC identifies latent variables via regime-conditioned temporal variation, and recovers for each latent a sparse set of associated observations through an additive decoder, yielding module-level interpretability. We show that ANOVA main-effect supports are identifiable under general smooth mixing functions, and provide finite-sample recovery guarantees for a tractable sparse-additive variant. Empirically, MOSAIC recovers domain-consistent variable groups across RNA molecular dynamics, solar wind, ENSO climate, the Tennessee Eastman process, and a synthetic tokamak benchmark, enabling interpretable discovery of latent mechanisms in scientific time series.

We consider the problem of learning data-driven replicas for stiff systems of ordinary differential equations arising in chemical kinetics that can be evaluated with high computational efficiency. We first focus on training emulators for families of reaction equations under varying reaction rates, using conditional residual networks or long-short term memory architectures. We then apply a recently proposed data-driven framework known as ``inVAErt networks'' to address the ill-posed inverse problem of inferring reaction rates, integration time, and possibly initial conditions from a target set of species concentrations - a problem that has received relatively little attention in the literature. The proposed approach is demonstrated on chemical systems with reversible and irreversible kinetics, spanning 2 to 20 differential equations, 3 to 20 chemical species, and 3 to 25 reaction rate parameters. Relative root mean squared errors produced by the proposed emulators range from $10^{-5}$ for lower-dimensional systems to $10^{-4}$ and $10^{-3}$ for an air pollution model and a hydrogen-air reaction system, respectively. Manifolds of non-identifiable reaction rates recovered by the proposed approach can be analytically verified for simple systems and are consistent with local identifiability analysis in higher dimensions.

Here we introduce Partially Observed Structural Causal Models (POSCMs) as an extension of structural causal models (SCMs) to settings where upstream contexts co-determine both the interaction structure and downstream mechanisms on observed variables. POSCMs thus provide a self-contained causal modeling framework for endogenous graphs, allowing for an intervention hierarchy spanning node- and edge-level contexts and endogenous variable interventions. To define edge interventions, we separate node mechanisms into edge-local transmission channels that can be modified without changing the source node or the rest of the target mechanism. We provide an identifiability theory that clarifies which intervention families would suffice to disentangle structure formation from mechanisms. We then empirically validate these theoretical results in two external simulators: a biophysically detailed virtual human retina and a gene-regulatory analogue. The experiments reproduce non-identifiability under latent context, expose structure-mechanism confounding under latent edges, and recover pathway-level input-output relationships under targeted interventions, consistent with our positive Markov kernel identifiability results. Together, POSCMs provide an intervention-oriented framework for causal systems in which contexts, graph structure, mechanisms, and measurements are jointly generated and only partially observed.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for integrating data-driven learning with physical knowledge. In this work, we propose a coupled PINN approach for the joint reconstruction of indoor temperature and humidity dynamics in greenhouse environments, together with simultaneous identification of key model parameters. The method incorporates a reduced-order physically motivated model into the learning process, enabling consistent estimation under sparse and noisy observations. The artificial intelligence contribution lies in the development of a coupled physics-informed neural learning framework that integrates governing dynamical constraints into neural network training, while the engineering application focuses on greenhouse climate state reconstruction and parameter identification. The proposed framework is evaluated on a controlled synthetic benchmark that mimics diurnal forcing conditions. Compared with a purely data-driven neural network baseline, the coupled PINN achieves improved reconstruction accuracy, reducing temperature and humidity errors while maintaining high coefficients of determination. The improvement is particularly pronounced in the humidity channel, where latent moisture dynamics are more difficult to infer from limited measurements. In addition to accurate state reconstruction, the method successfully recovers the dominant physical parameters governing the system dynamics, demonstrating its ability to learn interpretable representations beyond data interpolation. These results highlight the potential of physics-informed learning for greenhouse climate modeling and, more broadly, for data-scarce environmental systems.

Techniques for concept extraction, such as sparse autoencoders and transcoders, aim to extract high-level symbolic concepts from low-level nonsymbolic representations. When these extracted concepts are used for downstream tasks such as model steering and unlearning, it is essential to understand their guarantees, or lack thereof. In this work, we present a unified theoretical framework for unsupervised concept extraction, in which we frame the task of concept extraction as identifying a generative model. We present a general meta-theorem for identifiability, which reduces the problem of establishing identifiability guarantees to the problem of characterizing the intersection of two sets. As we demonstrate on a range of widely-used approaches, this meta-theorem substantially simplifies the task of proving such guarantees, thus paving the way for the development of new, principled approaches for concept extraction.

This paper studies the identifiability and stability of drifting fields in the framework of Generative Modeling via Drifting. The motivating question is whether a zero-drift equilibrium identifies the target distribution and whether an approximately vanishing drift implies weak distributional convergence. Since the original drifting model employs the Laplace kernel by default, we first analyze why Gaussian score-based arguments fail to apply. This analysis motivates the introduction of companion-elliptic kernel families, which are characterized by a companion potential satisfying an elliptic closure relation. We show that this class naturally contains the Laplace kernel and consists precisely of Gaussian and Matérn kernels with smoothness parameter $ν>0$. Within this class, we establish field identifiability for arbitrary Borel probability measures on $R^d$: if the drifting field between two such measures vanishes identically, then they must coincide. For stability, we demonstrate that convergence of the field alone does not guarantee weak convergence, since mass may escape to infinity while remaining invisible to the field. Although tightness directly removes this obstruction and restores weak stability, we prove that, even without tightness, every $C_0$-vague cluster point lies exactly on the defect ray $\{cp:0\le c\le1\}$. Consequently, a single scalar $C_0$ observable suffices to detect the missing mass and recover weak convergence.

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

We analyze identifiability in co-adaptive human-machine systems. We show that closed-loop encoder estimates do not uniquely identify user adaptation, but instead reflect properties of the joint system. We discuss implications for interpreting behavioral adaptation and propose conditions for identification.

Determining identifiability of causal effects from observational data under latent confounding is a central challenge in causal inference. For linear structural causal models, identifiability of causal effects is decidable through symbolic computation. However, standard approaches based on Gröbner bases become computationally infeasible beyond small settings due to their doubly exponential complexity. In this work, we study how to practically use symbolic computation for deciding rational identifiability. In particular, we present an efficient algorithm that provably finds the lowest degree identifying formulas. For a causal effect of interest, if there exists an identification formula of a prespecified maximal degree, our algorithm returns such a formula in quasi-polynomial time.

Given only observational data $X = g(Z)$, where both the latent variables $Z$ and the generating process $g$ are unknown, recovering $Z$ is ill-posed without additional assumptions. Existing methods often assume linearity or rely on auxiliary supervision and functional constraints. However, such assumptions are rarely verifiable in practice, and most theoretical guarantees break down under even mild violations, leaving uncertainty about how to reliably understand the hidden world. To make identifiability actionable in the real-world scenarios, we take a complementary view: in the general settings where full identifiability is unattainable, what can still be recovered with guarantees, and what biases could be universally adopted? We introduce the problem of diverse dictionary learning to formalize this view. Specifically, we show that intersections, complements, and symmetric differences of latent variables linked to arbitrary observations, along with the latent-to-observed dependency structure, are still identifiable up to appropriate indeterminacies even without strong assumptions. These set-theoretic results can be composed using set algebra to construct structured and essential views of the hidden world, such as genus-differentia definitions. When sufficient structural diversity is present, they further imply full identifiability of all latent variables. Notably, all identifiability benefits follow from a simple inductive bias during estimation that can be readily integrated into most models. We validate the theory and demonstrate the benefits of the bias on both synthetic and real-world data.

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as reverse engineering," or parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.

We study identifiability in continuous-time linear stationary stochastic differential equations with a known causal structure. Unlike existing approaches, we relax the assumption of a known diffusion matrix, thereby respecting the model's intrinsic scale invariance. Therefore, rather than recovering drift coefficients themselves, we introduce edge-sign identifiability: for a given causal structure, we ask whether the sign of a given drift entry is uniquely determined across all observational covariance matrices induced by parametrisations compatible with that structure. This leads to a trichotomy of edge-sign identifiability: identifiable, non-identifiable, and partially identifiable. This trichotomy introduces the new notion of partial identifiability to the literature, which we show is a genuine category in our setting. Under a notion of faithfulness, we derive criteria to identify membership of each category for general graphs. Applying our criteria to specific causal structures, both analogous to classical causal settings (e.g., instrumental variables) and novel cyclic settings, we determine their edge-sign identifiability and, in some cases, obtain explicit expressions for the sign of a target edge in terms of the observational covariance matrix.

We investigate the identifiability of nonlinear canonical correlation analysis (CCA) in a multi-view setup, in which each view is generated by applying an unknown nonlinear map to a linear mixture of shared latent variables plus view-private noise. Rather than pursuing exact unmixing, which is known to be ill-posed under general nonlinear mixing, we instead reframe multi-view CCA as a basis-invariant subspace identification problem. Under suitable latent priors and spectral separation conditions, we prove that the pairwise population CCA objective recovers correlated signal subspaces up to view-wise orthogonal ambiguity. For $N \geq 3$ views, their multi-view aggregation provably isolates the jointly correlated subspaces shared across all views while eliminating view-private variation. We further establish finite-sample statistical consistency guarantees by translating the concentration of empirical cross-covariances into explicit subspace error bounds via spectral perturbation theory. Experiments on synthetic and rendered image datasets support our theoretical findings and illustrate the necessity of the assumed conditions.

Causal identification functionals often require integration over conditional densities of continuous variables, such as those arising in nonparametric identification theory of total and mediated causal effects in DAGs with hidden variables. Estimating these densities and evaluating the resulting integrals can be statistically and computationally demanding. A common workaround is to discretize the continuous variable and replace integrals with finite sums. Although convenient, discretization alters the population-level functional and can induce non-negligible approximation bias, even when identification is correct. Under smoothness conditions, we show that the resulting coarsening error is first order in the bin width and arises at the level of the target functional, distinct from statistical estimation error. We propose a simple debiased coarsened functional that evaluates the outcome regression at within-bin conditional means, eliminating the leading coarsening error term and yielding a second-order approximation error. We derive plug-in and one-step estimators for this debiased coarsened functional. Simulations demonstrate substantial bias reduction and near-nominal confidence interval coverage, even under coarse binning. Our results provide a simple framework for controlling the impact of variable discretization on both parameter approximation and statistical estimation.

We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, we establish deterministic stability inequalities, global Lipschitz bounds for least-squares misfit functionals, and nonasymptotic concentration estimates. These results yield high-probability parameter error bounds that are intrinsic to the forward operator and independent of any specific reconstruction algorithm. As a concrete instantiation, we verify that the Gaussian Splatting rendering operator satisfies the proposed assumptions and derive explicit constants governing its Lipschitz continuity and resolution-dependent observability. This leads to a fundamental stability--resolution tradeoff, showing that estimation error is inherently constrained by the ratio between image resolution and model complexity. Overall, the analysis characterizes operator-level limits for a broad class of high-dimensional nonlinear inverse problems arising in modern imaging and differentiable rendering.

Across fields such as machine learning, social science, and geology, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extent similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five models, latent budget analysis (LBA) and latent class analysis (LCA) from social science, end-member analysis (EMA) from geology, probabilistic latent semantic analysis (PLSA) and nonnegative matrix factorization (NMF) from machine learning. We focus on the identifiability of these models. We prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. Consequently, existing uniqueness theorems for NMF directly apply to LBA, EMA, LCA, PLSA, and vice versa. We also provide a brief review of algorithms for the estimation of these models. We illustrate NMF on a sedimentary grain-size distribution dataset from sedimentary geology, and end the paper with a discussion of closely related model: archetypal analysis.

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds $d_i\geq 2n_i-1$ on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability. The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.

In this work, we establish the sufficient conditions under which nonlinear Canonical Correlation Analysis (CCA) recovers ground-truth latent factors up to an affine transformation. By transporting the analysis from the observation space to the source space, we extend classical statistical results on orthogonal polynomial expansions of bivariate distributions to representation learning, proving affine identifiability under specific distributional priors. We formally demonstrate that whitening is strictly necessary to ensure the boundedness and well-conditioning of the learned mappings. Furthermore, we bridge the gap between theory and practice by proving that ridge-regularized empirical CCA converges to its population counterpart in the finite-sample regime. Finally, our findings provide a rigorous theoretical foundation explaining the empirical success of recent correlation-based non-contrastive learning methods. Experiments on synthetic and rendered image datasets, alongside systematic ablations, validate the predicted recovery behavior and illustrate the failure modes that arise when the assumptions are violated.

We study operator relevance in data-driven partial differential equation (PDE) discovery. Sparse residual methods can select terms that improve residual fit, but residual contribution is not the same as functional necessity. We formalize this distinction through counterfactual operator interventions, where a candidate term is deleted or perturbed and the factual and intervened trajectories, or observables, are compared. The resulting theory gives six reusable results. A residual--counterfactual gap theorem shows that deletion effects are governed by the inverse linearized PDE map, not by residual magnitude alone. A certified decision theorem gives error margins for relevance, irrelevance, and abstention under neural or numerical surrogate error. An aliasing theorem characterizes experiment-dependent non-identifiability through the null space of the operator-evaluation design. A constraint-manifold theorem shows that operators vanishing on invariant constraint classes cannot be identified from trajectories restricted to those classes. A pruning-consistency theorem proves that sparse screening followed by counterfactual deletion recovers the functionally relevant support under a recall and margin condition. An observable-level adjoint theorem extends relevance testing from full-state deviations to scientific quantities of interest. Validation experiments test these mechanisms on synthetic PDEs with known support and on public geophysical fields from atmospheric reanalysis and NOAA OISST. The real-data results are reported as operator-surrogate diagnostics, not as unconditional recovery of physical laws. The framework provides a rigorous diagnostic layer for distinguishing residual usefulness from counterfactual operator relevance within a specified library, experiment class, norm, and tolerance.

Data fusion, the process of combining observational and experimental data, can enable the identification of causal effects that would otherwise remain non-identifiable. Although identification algorithms have been developed for specific scenarios, do-calculus remains the only general-purpose tool for causal data fusion, particularly when variables are present in some data sources but not others. However, approaches based on do-calculus may encounter computational challenges as the number of variables increases and the causal graph grows in complexity. Consequently, there exists a need to reduce the size of such models while preserving the essential features. For this purpose, we propose pruning (removing unnecessary variables) and clustering (combining variables) as preprocessing operations for causal data fusion. We generalize earlier results on a single data source and derive conditions for applying pruning and clustering in the case of multiple data sources. We give sufficient conditions for inferring the identifiability or non-identifiability of a causal effect in a larger graph based on a smaller graph and show how to obtain the corresponding identifying functional for identifiable causal effects. Examples from epidemiology and social science demonstrate the use of the results.

Causal discovery is a difficult problem that typically relies on strong assumptions on the data-generating model, such as non-Gaussianity. In practice, many modern applications provide multiple related views of the same system, which has rarely been considered for causal discovery. Here, we leverage this multi-view structure to achieve causal discovery with weak assumptions. We propose a multi-view linear Structural Equation Model (SEM) that extends the well-known framework of non-Gaussian disturbances by alternatively leveraging correlation over views. We prove the identifiability of the model for acyclic SEMs. Subsequently, we propose several multi-view causal discovery algorithms, inspired by single-view algorithms (DirectLiNGAM, PairwiseLiNGAM, and ICA-LiNGAM). The new methods are validated through simulations and applications on neuroimaging data, where they enable the estimation of causal graphs between brain regions.

Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.