Parameter Identifiability

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Overview

Parameter identifiability asks whether different parameter values or models can produce the same observations, and which aspects of a system can therefore be recovered. Researchers analyze symmetries and equivalence classes, use rank and Fisher-information conditions, and design informative experiments with sufficient excitation, diverse trajectories, or interventions. These ideas apply to dynamical systems, inverse problems, latent-variable models, and neural networks, where tools such as structural analysis, Bayesian inference, and constrained estimation help distinguish unique recovery from uncertainty or partial identification. A central tension is that accurate predictions do not guarantee uniquely or precisely identified parameters: structural ambiguities may persist regardless of data volume, while finite, noisy data can make even identifiable parameters difficult to estimate.

Latest papers

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103 papers13 papers in the last four weeks, up 160% on the four weeks before. 0.1% of all new papers.

Feb 27, 2026

Provable Subspace Identification of Nonlinear Multi-view CCA

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≥3N \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.

Zhiwei Han, Stefan Matthes, Hao Shen

Chair of Data Processing, Technical University of Munich, Germany · fortiss GmbH, Munich, Germany

Representation IdentifiabilityParameter IdentifiabilityMulti-View LearningStatistical Learning TheoryCanonical Correlation Analysis
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Feb 20, 2026

PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Real physical systems are dissipative -- a pendulum slows, a circuit loses charge to heat -- and forecasting their dynamics from partial observations is a central challenge in scientific machine learning. We address the position-only (q-only) problem: given only generalized positions~qtq_t at discrete times (momenta~ptp_t latent), learn a structured model that (a)produces stable long-horizon forecasts and (b)recovers physically meaningful parameters when sufficient structure is provided. The port-Hamiltonian framework makes the conservative-dissipative split explicit via x˙=(J−R)∇H(x)\dot{x}=(J-R)\nabla H(x), guaranteeing dH/dt≤0dH/dt\le 0 when R⪰0R\succeq 0. We introduce PHAST (Port-Hamiltonian Architecture for Structured Temporal dynamics), which decomposes the Hamiltonian into potentialV(q)V(q), massM(q)M(q), and damping~D(q)D(q) across three knowledge regimes (KNOWN, PARTIAL, UNKNOWN), uses efficient low-rank PSD/SPD parameterizations, and advances dynamics with Strang splitting. Across thirteen q-only benchmarks spanning mechanical, electrical, molecular, thermal, gravitational, and ecological systems, PHAST achieves the best long-horizon forecasting among competitive baselines and enables physically meaningful parameter recovery when the regime provides sufficient anchors. We show that identification is fundamentally ill-posed without such anchors (gauge freedom), motivating a two-axis evaluation that separates forecasting stability from identifiability.

Shubham Bhardwaj, Chandrajit Bajaj

Department of Computer Science, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

Time Series ForecastingPort-Hamiltonian SystemsHamiltonian Neural NetworksNonlinear System IdentificationSystem IdentificationParameter Identifiability
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Feb 10, 2026

Stability and Concentration in Nonlinear Inverse Problems with Block-Structured Parameters: Lipschitz Geometry, Identifiability, and an Application to Gaussian Splatting

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.

Joe-Mei Feng, Hsin-Hsiung Kao

Computer Science and Information Engineering, Tamkang University, New Taipei City, Taiwan (R.O.C.) · Department of Information Management, Central Police University, Taoyuan, Taiwan

Inverse RenderingParameter IdentifiabilityGaussian SplattingInverse Problems
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Jan 30, 2026

Complete Identification of Deep ReLU Networks through Łukasiewicz Logic

Two deep ReLU networks can have entirely different architectures and parameters, yet realize the same function. We provide a complete characterization of this nonuniqueness. This is effected by building a symbolic calculus for deep ReLU networks, equivalence and simplification of networks becoming derivation of formulae, in close parallel to Shannon's analysis of switching circuits through Boolean logic. Inspired by Shannon, who turned circuit synthesis into the manipulation of Boolean formulae by the axioms of Boolean algebra, we turn ReLU network identification into the derivation of Łukasiewicz formulae by the axioms of many-valued (MV) logic. Two non-degenerate ReLU networks realize the same function on the unit cube if and only if one is obtained from the other by finitely many applications of the MV axioms for integer weights and biases, the divisible MV axioms for rational ones, and the Riesz MV axioms for real ones. The MV logic axioms characterize all symmetries of ReLU networks, the single-layer ones, which for tanh networks are the only kind, and the deep ones, spanning three or more layers. Our framework consists of three steps, an extraction algorithm turning a network into a substitution graph, whose represented formula has the network's input-output map as its truth function, a completeness theorem, by which functionally equivalent formulae are interderivable, and a construction algorithm returning from graphs to networks. The substitution graph is layered, carrying at each node a formula in the variables of the layer feeding it, encodes the network uniquely, and induces a new normal form for MV logic, compositional rather than flat as in the literature, hence retaining the algebraic structure of the network, with three local operations--node rewrite, layer collapse, layer expansion--realizing every derivation.

Yani Zhang, Helmut Bölcskei

Chair for Mathematical Information Science, ETH Zürich

Parameter IdentifiabilityReLU Neural NetworksFeedforward Neural Networks
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Jan 30, 2026

Uncertainty-aware Causal Decision Making via Effect Bound Decomposition

Causal inference from observational data can provide strong evidence for finding the best action in a decision-making scenario without having to perform expensive randomized trials. The causal effect of an action is often not pointwise identifiable even with infinite data due to unobserved confounding factors. Furthermore, having only finitely many samples adds another layer of uncertainty to causal effect estimation. Several existing methods can be used to obtain upper and lower bounds to the causal effect, ranging from symbolic methods to the more recent neural network-based approaches, which implicitly incorporate both sources of uncertainty. However, these methods do not inform whether collecting more samples may or may not help identify the best action from observational data, leaving experts in the dark about their data collection strategies. We address this problem with a novel framework that can distinguish the range of causal effect values that might be eliminated by collecting more samples from the range of values that, with high probability, cannot be eliminated with more observational samples. We show that this partitioning can be obtained by solving max-min and min-max optimization problems. We leverage neural causal models to approximately recover this decomposition in practice. We demonstrate via experiments on synthetic and real-world datasets that our algorithm can determine when collecting more samples will not help determine the best action. Our framework can help practitioners decide when to resort to non-observational studies or seek to measure some of the unmeasured confounders for optimal decision-making.

Md Musfiqur Rahman, Ziwei Jiang, Hilaf Hasson +1

Electrical and Computer Engineering, Purdue University · Computer Science, Johns Hopkins University · Cohesity

Causal Effect EstimationParameter IdentifiabilityPartial IdentificationCausal IdentifiabilityDecision-Making under UncertaintyCausal Inference with Unobserved Confounding
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Dec 25, 2025

Nonnegative matrix factorizations and related compositional models: Equivalence, identifiability, and an application on the grain-size analysis of sediments

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.

Qianqian Qi, Peter G. M. van der Heijden, Maarten A. Prins

Hangzhou Dianzi University, China · Utrecht University, the Netherlands · University of Southampton, UK +1

Latent Variable ModelsParameter IdentifiabilityNon-Negative Matrix Factorization
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Dec 6, 2025

Variational Physics-Informed Ansatz for Reconstructing Hidden Interaction Networks from Steady States

Inferring interaction structure from steady-state observations is a central inverse problem when transient trajectories are unavailable. Here we formulate this problem as simultaneous compatibility of a single interaction operator with equilibrium constraints generated by heterogeneous perturbations. We introduce a variational physics-informed ansatz that represents the unknown operator as a trainable object and minimizes the resulting steady-state residuals across experiments. In the affine-interaction setting, the stacked equilibrium equations yield explicit finite-sample identifiability conditions: unique recovery is controlled by the rank of the compatibility matrix after elimination of experiment-wise gauge freedom. Synthetic benchmarks on pairwise, directed, weighted, empirical-topology, and selected higher-order systems illustrate this identifiability picture and show how additional heterogeneous steady states improve structural discrimination under the stated assumptions. The results clarify a concrete steady-state reconstruction regime in which equilibrium observations alone can determine hidden interaction operators when the governing dynamics are known and node-level equilibria are fully observed.

Kaiming Luo

School of Information Science and Technology, Fudan University, Shanghai, 200438, China

Parameter Identifiability
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Nov 24, 2025

The Alexander-Hirschowitz theorem for neurovarieties

We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds di≥2ni−1d_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.

A. Massarenti, M. Mella

Polynomial Neural NetworksParameter IdentifiabilityNeural Network Approximation TheoryNeural Network Expressivity
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Oct 6, 2025

Provable Affine Identifiability of Nonlinear CCA under Latent Distributional Priors

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.

Zhiwei Han, Stefan Matthes, Hao Shen

fortiss GmbH, Munich, Germany · Technical university of Munich

Representation LearningParameter IdentifiabilityStatistical Learning TheoryCanonical Correlation Analysis
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Jun 22, 2025

Identifiable Convex-Concave Regression via Sub-gradient Regularised Least Squares

We propose a novel nonparametric regression method that models complex input-output relationships as the sum of convex and concave components. The method-Identifiable Convex-Concave Nonparametric Least Squares (ICCNLS)-decomposes the target function into additive shape-constrained components, each represented via sub-gradient-constrained affine functions. To address the affine ambiguity inherent in convex-concave decompositions, we introduce global statistical orthogonality constraints, ensuring that residuals are uncorrelated with both intercept and input variables. This enforces decomposition identifiability and improves interpretability. We further incorporate L1, L2 and elastic net regularisation on sub-gradients to enhance generalisation and promote structural sparsity. The proposed method is evaluated on synthetic and real-world datasets, including healthcare pricing data, and demonstrates improved predictive accuracy and model simplicity compared to conventional CNLS and difference-of-convex (DC) regression approaches. Our results show that statistical identifiability, when paired with convex-concave structure and sub-gradient regularisation, yields interpretable models suited for forecasting, benchmarking, and policy evaluation.

William Chung

Department of Decision Analytics and Operations, City University of Hong Kong, Hong Kong

Parameter IdentifiabilityNonparametric Regression
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Oct 29, 2024

Identifiability Analysis of Linear ODE Systems with Hidden Confounders

The identifiability analysis of linear Ordinary Differential Equation (ODE) systems is a necessary prerequisite for making reliable causal inferences about these systems. While identifiability has been well studied in scenarios where the system is fully observable, the conditions for identifiability remain unexplored when latent variables interact with the system. This paper aims to address this gap by presenting a systematic analysis of identifiability in linear ODE systems incorporating hidden confounders. Specifically, we investigate two cases of such systems. In the first case, latent confounders exhibit no causal relationships, yet their evolution adheres to specific functional forms, such as polynomial functions of time tt. Subsequently, we extend this analysis to encompass scenarios where hidden confounders exhibit causal dependencies, with the causal structure of latent variables described by a Directed Acyclic Graph (DAG). The second case represents a more intricate variation of the first case, prompting a more comprehensive identifiability analysis. Accordingly, we conduct detailed identifiability analyses of the second system under various observation conditions, including both continuous and discrete observations from single or multiple trajectories. To validate our theoretical results, we perform a series of simulations, which support and substantiate our findings.

Yuanyuan Wang, Biwei Huang, Wei Huang +2

The University of Melbourne · University of California, San Diego

Parameter IdentifiabilityCausal IdentifiabilityCausal Inference
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Jan 27, 2023

Epsilon-Identifiability of Causal Quantities

Identifying the effects of causes and causes of effects is vital in virtually every scientific field. Often, however, the needed probabilities may not be fully identifiable from the available data sources. This paper shows how approximate identifiability is still possible for several probabilities of causation. We term this epsilon-identifiability and demonstrate its usefulness in cases where the behavior of certain subpopulations can be restricted within sufficiently narrow bounds. In particular, we show how unidentifiable causal effects and counterfactual probabilities can be epsilon-identified when such allowances are made. Often, these allowances are easily measured and reasonably assumed. Finally, epsilon-identifiability is applied to the unit selection problem.

Ang Li, Scott Mueller, Xin Shu +1

Cognitive Systems Laboratory, Department of Computer Science, University of California, Los Angeles, Los Angeles, California, USA.

Causal IdentifiabilityCausal InferenceCausal Effect EstimationParameter Identifiability
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Oct 15, 2022

Symbolic Recovery of Differential Equations: The Identifiability Problem

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.

Philipp Scholl, Aras Bacho, Holger Boche +1

Ludwig-Maximilians-Universität München Munich Germany · Technische Universität München Munich Center for Quantum Science and Technology (MCQST) Munich Quantum Valley (MQV) Munich Germany · Ludwig-Maximilians-Universität München University of Tromsø DLR-German Aerospace Center Munich Center for Machine Learning (MCML) Munich Germany

Parameter IdentifiabilityAutonomous Scientific DiscoverySymbolic Regression
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