Query Identifiability

Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

A reasoning model asked whether a causal effect is recoverable from observational data can fail in two ways: it refuses an identifiable query or answers a nonidentifiable one. The latter is more consequential, as no observational data can validate the claimed formula. Measuring this failure requires queries that are provably non-identifiable, which prior evaluations lack, and grading that accepts correct formulas in any equivalent form, which string matching cannot provide. We build CERTID, a formal identification pipeline that addresses both limitations. CERTID uses the sound and complete causal identification algorithm ID to certify whether an effect is identifiable from a given graph and query, and verifies returned formulas against structural causal models whose interventional distributions are known exactly. CERTID further develops theoretical results to mitigate structural leakage, repair non-identifiable queries, and establish grading guarantees. We evaluate three frontier reasoning models (Gemini Flash, Gemini Pro, and GPT5.5) on 1,200 certified instances spanning 4 to 50 vertices. Accuracy proves a poor proxy for soundness: on identical instances, the false-claim rate on non-identifiable queries varies by seventeen-fold across models. We also find that models decide identifiability with 97-100% accuracy on graphs generated after the strongest model's training snapshot. Instances, the certification procedure, the verifier, and per-instance records are available at https://anonymous.4open.science/r/certid-D718.

Linear probes that decode the truth from a language model's activations have been proposed as deception monitors, and a truth probe scoring below chance on a model trained to deceive is naturally read as evidence that the model hid or stopped representing the truth. We show that a fitting-label ambiguity can produce the same readout. In a controlled game, a model should report a secret bit to an ally and its complement to a rival. On compliant (ally) contexts the true bit and the answer the task prescribes are identical labels, so a probe fitted there cannot tell which of the two it measures. We call this complete agreement perfect aliasing. The labels are complements on rival contexts, so one ally-fitted probe scored against each has rival AUROCs that sum to one. Mixed fitting, on ally and rival contexts together, makes the two labels differ; randomized output codebooks also decouple the prescribed answer from the output letter. For a reward-trained Gemma-2-9B policy that answers falsely on every evaluated rival trial, the ally-fitted probe scores $0.006 \pm 0.005$ AUROC at the final layer (mean $\pm$ sample SD over three RL training seeds), while mixed-fit probes score 1.000 on the same held-out activations. Mixed fitting also uses more examples and labelled rival data, so this shows the true bit remains linearly recoverable, not that separating the labels alone explains the gain. In instructed Llama-3.1-8B, a probe refitted on one prompt variant and one frozen from a reference variant both have held-out ally accuracy 1.000 but rival truth AUROCs of 0.080 and 0.986 on the same trials. Our main experiments use one single-token game that states the bit in the prompt, so the recovered direction may read a retained copy of it; where the model must infer the bit, no tested arm deceives reliably. We study what a probe measures, not whether the model uses that information.

Sparse pursuit after dictionary learning can yield a precise atom support even when its physical interpretation is not justified by the calibration data, especially for highly coherent dictionaries where alternative calibration-compatible dictionaries may assign different physical meanings to the same selected support. We develop resolution-aware physical-support inference that jointly accounts for uncertainty in the learned dictionary and in the representation of a deployment signal. Our cross-dictionary confidence correspondence retains calibration-compatible dictionaries and deployment-compatible sparse representations, then projects the surviving explanations onto physical-support space. For local coherent-atom classes with separation scale s, once the deployment data resolve the coherent-block explanation and its atom support, the minimax physical resolution from N calibration signals satisfies $δ_{\mathrm{opt}}(N,s)\asymp\min\{s,\frac{1}{\sqrt{N}s^2}\}$, with relative resolution governed by the orientation-information scale $Ns^6$. Deployment replication improves physical localization only when orientation changes cannot be absorbed by adjusting the active coefficients. For computation, we introduce active endpoint bracketing (AEB), an adaptive finite-bank procedure that evaluates only candidates that can still affect the physical report and otherwise safely coarsens or abstains. Finite-bank experiments, including a four-region synthetic application, show that a point-valued plug-in selector can be physically overprecise, whereas AEB avoids unsupported refinement with fewer candidate evaluations.

A normal--inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For $σ^2\sim\mathrm{InvGamma}(α,β)$, $μ\midσ^2\sim\mathcal{N}(γ,σ^2/ν)$, and $z\midμ,σ^2\sim\mathcal{N}(μ,σ^2)$, the marginal law of $z$ depends on $(ν,β)$ only through $c=β(1+1/ν)$. Hence the reconstruction-visible parameter space is the three-dimensional quotient $(γ,α,c)$, with a one-dimensional fiber degree of freedom. For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective. Writing $ρ_0=2β_0/ν_0$ and $T=c/\{α[(γ-γ_0)^2+ρ_0]\}$, we show that inverse canonical allocation $1/ν_{\rm can}$ is an explicit strictly increasing function of $T$. For $α>1$, the ratio $u_{\rm epi}/u_{\rm var}=1/ν_{\rm can}$ is therefore determined by the quotient state and prior; for rank-only use under a common calibration, $T$ contains the same coordinatewise ordinal information. The residual prior gauge is characterized rather than eliminated: $(γ_0,ρ_0)$ govern ordinal dependence, while $(ν_0,α_0)$ determine numerical calibration and the analytic ceiling of $1/ν_{\rm can}$.

In-context learning is commonly formalized as inference from examples of a function. Partial orders instead combine transitivity, antisymmetry, and incomparability, so a finite prompt may not determine a queried comparison. We develop a theory of in-context learning on partial orders that separates logical identifiability, prompt teaching cost, structural complexity, and the exact capacity of a formal coordinate-decoder class. A version-space semantics makes background knowledge and open- versus closed-world assumptions explicit. For finite open-world prompts with positive and negative comparisons, we prove an exact completion trichotomy: after taking the reflexive transitive closure of the positive demonstrations, a query is forced true, forced false because every true completion creates a cycle or violates a negative demonstration, or remains genuinely ambiguous. For a known $n$-element universe, we characterize the open-world teaching number as the number of covers plus a blocker-set hitting number, prove that its maximum over all $n$-element posets is $n(n-1)$ and is uniquely attained by the antichain, and identify the blocker term as the exact cost of open-world rather than complete-Hasse semantics. We formalize prompt-dependent $s$-coordinate decoders and use the classical coordinate-order equivalence to obtain an exact representation boundary: dimension at most $s$ is necessary and sufficient, while width at most $s$ is a convenient sufficient condition.

When data sources are integrated through a shared interface, a downstream query may or may not be determined by what the interface exposes: two globally consistent worlds can agree on every shared attribute yet disagree on the query answer. This ambiguity is structural -- a property of the interface design, not the data volume -- and cannot be resolved by collecting more records or training a larger model. We formalize query identifiability for data integration under interface laws (functional dependencies that hold uniformly across all legal worlds rather than within a single instance) and prove three results. (i) A polynomial-time certificate (CheckCert) decides identifiability via attribute closure, and is exact on instances that expose any residual ambiguity (closure-separable). (ii) Non-identifiable queries face an irreducible 1/2 minimax error floor for any estimator using only interface evidence, bounding multi-view pretraining systems from below. (iii) A minimum-augmentation algorithm (Greedy-MinAug) finds the smallest set of interface additions to certify a query, reducing to Set Cover (logarithmic approximation). Experiments on synthetic benchmarks, real integration datasets spanning three domains (scholarly, product, restaurant), and schemas up to 10^3 attributes confirm CheckCert is exact, both algorithms run in single-digit milliseconds, and ML classifiers exhibit the predicted error floor and abrupt capability gains.

Identification conditions describe the computability of a target query or parameter of interest as a function of the type and amount of information available. In causal identification, this information is often expressed in the form of a causal graph, and data are observed or collected for some subset of variables in the graph. Target queries may be for a single effect alone or for a class of effects in a given model. The derivation of an identification algorithm then defines mathematically the process by which the desired causal effect(s) can be uniquely determined, theoretically, in expectation. Identifiability in expectation, or 'theoretical identifiability,' generally assumes asymptotic properties, infinite data, or other mathematically idealized conditions. In this paper, we explore a fundamental distinction between this theoretical, idealized notion of identifiability and a proposed alternative that is computation-bound. The framework we propose - 'computational identifiability' - is to instead define a finite computational search procedure for an empirical estimator. If this process finds an estimator empirically, within a desired error tolerance, then identifiability is satisfied, conditional on the specified assumptions of the search (i.e., a prior distribution over the parameters) and conditional on the search procedure itself. Through several experiments, we demonstrate how this framework allows us to answer fine-grained, practical identification questions, such as identification with small finite samples, with ambiguous graphical criteria, with mixed observational-interventional data, and across counterfactual data and estimands. Code is available at https://github.com/lbynum/metadentify.

Independent Component Analysis (ICA) is a fundamental unsupervised learning technique foruncovering latent structure in data by separating mixed signals into their independent sources. While substantial progress has been made in establishing asymptotic identifiability guarantees for nonlinear ICA, the finite-sample statistical properties of learning algorithms remain poorly understood. This gap poses significant challenges for practitioners who must determine appropriate sample sizes for reliable source recovery. This paper presents a comprehensive finite-sample analysis of nonlinear ICA with neural network encoders, providing the first complete characterization with matching upper and lower bounds. Our theoretical development introduces three key technical contributions. First, we establish a direct relationship between excess risk and identification error that bypasses parameter-space arguments, thereby avoiding the rate degradation that would otherwise yield suboptimal scaling. Second, we prove matching information-theoretic lower bounds that confirm the optimality of our sample complexity results. Third, we extend our analysis to practical SGD optimization, showing that the same sample efficiency can be achieved with finite-iteration gradient descent under standard landscape assumptions. We validate our theoretical predictions through carefully designed simulation experiments. This gap points toward valuable future research on finite-sample behavior of neural network training and highlights the importance of our validated scaling laws for dimension and diversity.

Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. To find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns a distribution over connection weights that is maximally unbiased over unidentifiable components while matching the observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.

When used in automated decision-making systems, machine learning (ML) models are vulnerable to data-manipulation attacks. Some defense mechanisms (e.g., adversarial regularization) directly affect the ML models while others (e.g., anomaly detection) act within the broader system. In this paper we consider a different task for defending the adversary, focusing on the attacker, rather than the attack. We present and demonstrate a framework for identifying characteristics about the attacker from an observed attack. We prove that, without additional knowledge, the attacker is non-identifiable (multiple potential attackers would perform the same observed attack). To address this challenge, we propose a domain-agnostic framework to identify the most probable attacker. This framework aids the defender in two ways. First, knowledge about the attacker can be leveraged for exogenous mitigation (i.e., addressing the vulnerability by altering the decision-making system outside the learning algorithm and/or limiting the attacker's capability). Second, when implementing defense methods that directly affect the learning process (e.g., adversarial regularization), knowledge of the specific attacker improves performance. We present the details of our framework and illustrate its applicability through specific instantiations on a variety of learners.

Coarse data arise when learners observe only partial information about samples; namely, a set containing the sample rather than its exact value. This occurs naturally through measurement rounding, sensor limitations, and lag in economic systems. We study Gaussian mean estimation from coarse data, where each true sample $x$ is drawn from a $d$-dimensional Gaussian distribution with identity covariance, but is revealed only through the set of a partition containing $x$. When the coarse samples, roughly speaking, have ``low'' information, the mean cannot be uniquely recovered from observed samples (i.e., the problem is not identifiable). Recent work by Fotakis, Kalavasis, Kontonis, and Tzamos [FKKT21] established that sample-efficient mean estimation is possible when the unknown mean is identifiable and the partition consists of only convex sets. Moreover, they showed that without convexity, mean estimation becomes NP-hard. However, two fundamental questions remained open: (1) When is the mean identifiable under convex partitions? (2) Is computationally efficient estimation possible under identifiability and convex partitions? This work resolves both questions. [...]