ICML 2026 Papers

Explainable artificial intelligence (XAI) aims to provide human-interpretable insights into the behavior of deep neural networks (DNNs), typically by estimating a simplified causal structure of the model. In existing work, this causal structure often includes relationships where the presence of a concept is associated with a strong activation of a neuron. For example, attribution methods primarily identify input pixels that contribute most to a prediction, and feature visualization methods reveal inputs that cause high activation of a target neuron – the former implicitly assuming that the relevant information resides in the input, and the latter that neurons encode the presence of concepts. However, a largely overlooked type of causal relationship is that of encoded absences, where the absence of a concept increases neural activation. In this work, we show that such missing but relevant concepts are common and that mainstream XAI methods struggle to reveal them when applied in their standard form. To address this, we propose two simple extensions to attribution and feature visualization techniques that uncover encoded absences. Across experiments, we show how mainstream XAI methods can be used to reveal and explain encoded absences, how ImageNet models exploit them, and that debiasing can be improved when considering them.

Interpretable machine learning is essential in high-stakes domains where decision-making requires accountability, transparency, and trust. While rule-based models offer global and exact interpretability, learning rule sets that simultaneously achieve high predictive performance and low, human-understandable complexity remains challenging. To address this, we introduce TT-Sparse, a flexible neural building block that leverages differentiable truth tables as nodes to learn sparse, effective connections. A key contribution of our approach is a new soft TopK operator with straight-through estimation for learning discrete, cardinality-constrained feature selection in an end-to-end differentiable manner. Crucially, the forward pass remains sparse, enabling each node (and the entire model) to be transformed exactly into compact, globally interpretable DNF/CNF Boolean formulas via Quine--McCluskey minimization. Extensive empirical results across 28 datasets spanning binary, multiclass, and regression tasks show that the learned sparse rules exhibit superior predictive performance with lower complexity compared to existing state-of-the-art methods.

In standard Transformer architectures, semantic importance is often conflated with activation magnitude, obscuring the geometric structure of latent representations. To disentangle these factors, we introduce PRISM, a complex-valued architecture designed to isolate the computational role of phase. By enforcing a strict unit-norm constraint (|z| = 1) and replacing attention with gated harmonic convolutions, the model is encouraged to utilize subtractive interference in the frequency domain to suppress noise, rather than relying on magnitude-based gating. We utilize this constrained regime to study a hybrid architecture—fusing phase-based routing with standard attention—which achieves improved parameter efficiency and representation quality compared to baselines in our evaluated settings. Mechanistically, interventional ablations indicate that the model carries substantial task-relevant information in phase: preserving phase largely maintains performance, whereas disrupting phase causes severe degradation. Together, these results suggest that phase-based spectral interference is a usable computational mechanism for neural sequence modeling at the evaluated scale.

Algorithm extraction aims to synthesize executable programs directly from models trained on algorithmic tasks, enabling *de novo* recovery of executable mechanisms from weights without relying on human-written target programs. However, applying this paradigm to Transformer is complicated by representation entanglement (e.g., superposition), where features encoded in overlapping directions substantially hinder the recovery of symbolic expressions. We propose the Discrete Transformer, an architecture explicitly designed to bridge the gap between continuous representations and discrete symbolic logic. By injecting discreteness through temperature-annealed sampling, our framework effectively leverages hypothesis testing and symbolic regression to extract human-readable programs. Empirically, the Discrete Transformer achieves performance comparable to the RNN-based MIPS baseline on shared discrete tasks, while broadening extraction to tasks with continuous-valued intermediate computations. Finally, we show that architectural inductive biases provide fine-grained control over synthesized programs, establishing the Discrete Transformer as a controllable testbed for algorithm extraction and Transformer interpretability.

Distinguishing Large Language Model (LLM) generated text from human writing is a critical and difficult challenge. While LLMs are trained to write like humans, we hypothesize that this training leaves an indelible mark. LLMs develop a particularly strong aversion to token repetition very early in training. This bias persists as a ``Vestigial Heuristic'' (a developmental artifact) that is activated in LLM-generated text, separating LLM from human writing. To probe this phenomenon, we introduce Telescope Perplexity, a metric that evaluates the token repetition of the model, $P(s_i | s_{1:i})$. Our empirical investigation reveals that the Telescope Perplexity signature emerges early in pre-training, and Telescope Perplexity empirically enables highly effective zero-shot LLM detection. We show state-of-the-art or competitive performance across diverse datasets (including modern evaluation sets we introduce), reference models, and perturbation schemes with greater efficiency than other methods.

Conditional independence (CI) is central to causal inference, feature selection, and graphical modeling, yet it is untestable in many settings without additional assumptions. Existing CI tests often rely on restrictive structural conditions, limiting their validity. Kernel methods using partial covariance operators offer a more principled approach but suffer from limited adaptivity and scalability. In this work, we explore whether representation learning can help address these limitations. Specifically, we focus on representations derived from the singular value decomposition of partial covariance operators and use them to construct a simple test statistic. We also introduce a bi-level contrastive algorithm to learn these representations. Our theory links representation learning error to test performance and establishes asymptotic validity and power guarantees. Experiments on real and synthetic data suggest that this approach offers a principled and statistically grounded path toward scalable CI testing, bridging kernel-based theory with modern representation learning.

In statistical applications it has become increasingly common to encounter data structures that live on non-linear spaces such as manifolds. For data living on such non-linear spaces geodesic regression emerged as a natural extension of linear regression where the response variable lives on a Riemannian manifold. The parameters of geodesic regression capture the relationship of sensitive data, and hence, one should consider the privacy protection practices of said parameters. We consider releasing Differentially Private (DP) parameters of geodesic regression via the K-Norm Gradient (KNG) mechanism for Riemannian manifolds. We derive theoretical bounds for the sensitivity of the parameters showing they are tied to their respective Jacobi fields and hence the curvature of the space. We demonstrate the efficacy of our methodology on the sphere, $S_2\subset\mathbb{R}^3$, the space of symmetric positive definite matrices, and Kendall's planar shape space. Our methodology is general to any Riemannian manifold, and thus it is suitable for data in domains such as medical imaging and computer vision.

Recent work describes what transformers can and cannot compute through connections to boolean circuits, but existing results lack exact characterizations and are sensitive to modeling choices. Padded transformers---to whose input filler symbols such as ``...'' are appended---emerge as a useful gadget for establishing equivalences to circuit classes by providing polynomial space for adaptive parallel computation. However, only a limited set of padded transformer idealizations has been studied, leaving open how robustly these equivalences hold under changes to attention type, model width, and uniformity. We find that, under practical assumptions, padded transformers are surprisingly robust to all of these, and identify numeric precision and model depth as the main factors affecting expressivity. Concretely, we prove that polynomially padded $\text{L-uniform}$ constant-precision transformers are equivalent to $\text{L-uniform AC}^0$, while growing-precision ones achieve $\text{L-uniform TC}^0$ regardless of width. Furthermore, looping enables sequential processing analogous to circuits: $\log^d N$-looped constant-precision transformers reach $\text{FO-uniform AC}^d$, and growing-precision ones reach $\text{FO-uniform TC}^d$. Interestingly, growing width or precision beyond logarithmic does not increase expressivity, and all our results hold for both softmax and average hard attention transformers.

Representing the past in a compressed, efficient, and informative manner is a central problem for systems trained on sequential data. The \textit{HiPPO} framework, originally proposed by Gu \& Dao et al., provides a principled approach to sequential compression by projecting signals onto orthogonal polynomial (OP) bases via structured linear ordinary differential equations. Subsequent works have embedded these dynamics in state space models (SSMs), where HiPPO structure serves as an initialization. Nonlinear successors of these SSM methods such as Mamba are state-of-the-art for many tasks with long-range dependencies, but the mechanisms by which they represent and prioritize history remain largely implicit. In this work, we revisit the HiPPO framework with the goal of making these mechanisms explicit. We show how polynomial representations of history can be extended to support capabilities of modern SSMs such as adaptive memory allocation and associative memory, while retaining direct interpretability in the OP basis. We introduce a unified framework comprising five such extensions, which we collectively refer to as a ``HiPPO zoo.'' Each extension exposes a specific modeling capability through an explicit, interpretable modification of the HiPPO framework. The resulting models adapt their memory online and train in streaming settings with efficient updates. We illustrate the behaviors and modeling advantages of these extensions through a range of synthetic sequence modeling tasks, demonstrating that capabilities typically associated with modern SSMs can be realized through explicit, interpretable polynomial memory structures.

Information retrieval is a core component of many intelligent systems as it enables conditioning of outputs on new and large-scale datasets. While effective, the standard practice of encoding data into high-dimensional representations for similarity search entails large memory and compute footprints, and also makes it hard to inspect the inner workings of the system. Hierarchical retrieval methods offer an interpretable alternative by organizing data at multiple granular levels, yet do not match the efficiency and performance of flat retrieval approaches. In this paper, we propose Retreever, a tree-based method that makes hierarchical retrieval viable at scale by directly optimizing its structure for retrieval performance while naturally providing transparency through meaningful semantic groupings. Our method offers the flexibility to balance cost and utility by indexing data using representations from any tree level. We show that Retreever delivers strong coarse (intermediate levels) and fine representations (terminal level), while achieving the highest retrieval accuracy at the lowest latency among hierarchical methods. These results demonstrate that this family of techniques is viable in practical applications.

Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretisation. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as a comparison we analyse a natural analogue of SGLD utilising gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.

Recently, sparse autoencoders (SAEs) have emerged as an attractive tool for interpreting and interacting with representations in practical neural networks. While it is common empirical folklore, we also show theoretically that SAEs are highly unstable: different training runs are likely to produce different concept dictionaries and sparse codes. We characterize the model properties that hinder the stability of real-world SAEs, and address each of these problems through minimal changes to the architecture and training procedure. Together, these changes yield two versions of an **i**dentifiable SAE (iSAE), a variant of the standard TopK SAE with lower reconstruction error and improved stability. We explain this improvement theoretically by connecting SAEs with traditional dictionary learning approaches, and show that the dictionaries learned in practice satisfy an approximate restricted isometry condition, rendering the corresponding sparse codes in those models near-identifiable.

Physics-informed neural networks (PINNs) have emerged as a powerful deep learning approach for solving partial differential equations (PDEs) in the physical sciences, yet their behavior remains largely opaque and is typically understood through failure mode analyses rather than explicit interpretability. To address this issue, we introduce PINNfluence, a training data attribution framework for interpreting PINNs based on influence functions. By extending influence functions to composite physics-informed training objectives, we enable fine-grained attribution between predictions, loss components, and training data points. Through benchmark experiments across various PDEs, we demonstrate that influence patterns provide granular diagnostics that distinguish structural characteristics across well-trained and poorly-trained PINNs. PINNfluence thus opens a new avenue for understanding and improving the reliability of PINNs through the lens of their data.

Multilingual safety remains significantly imbalanced, leaving non-high-resource (NHR) languages vulnerable compared to robust high-resource (HR) ones. Moreover, the neural mechanisms driving safety alignment remain unclear despite observed cross-lingual representation transfer.In this paper, we find that LLMs contain a set of cross-lingual shared safety neurons (SS-Neurons), a remarkably small yet critical neuronal subset that jointly regulates safety behavior across languages. We first identify monolingual safety neurons (MS-Neurons) and validate their causal role in safety refusal behavior through targeted activation and suppression. Our cross-lingual analyses then identify SS-Neurons as the subset of MS-Neurons shared between HR and NHR languages, serving as a bridge to transfer safety capabilities from HR to NHR domains. We observe that suppressing these neurons causes concurrent safety drops across NHR languages, whereas reinforcing them improves cross-lingual defensive consistency. Building on these insights, we propose a simple neuron-oriented training strategy that targets SS-Neurons based on language resource distribution and model architecture. Experiments demonstrate that fine-tuning this tiny neuronal subset outperforms state-of-the-art methods, significantly enhancing NHR safety while maintaining the model's general capabilities.

We propose dgMARK, a decoding-guided watermarking method for discrete diffusion language models (dLLMs). Unlike autoregressive models, dLLMs can generate tokens in arbitrary order. While an ideal conditional predictor would be invariant to this order, practical dLLMs exhibit strong sensitivity to the unmasking order, creating a new channel for watermarking. dgMARK steers the unmasking order toward positions whose high-reward candidate tokens satisfy a simple parity constraint induced by a binary hash, without explicitly reweighting the model’s learned probabilities. The method is plug-and-play with common decoding strategies (e.g., confidence, entropy, and margin-based ordering) and can be strengthened with a one-step lookahead variant. Watermarks are detected via elevated parity-matching statistics, and a sliding-window detector ensures robustness under post-editing operations including insertion, deletion, substitution, and paraphrasing. Project website: https://dgmark-watermarking.github.io

Ensuring scalable input-to-state stability (sISS) is critical for the safety and reliability of large-scale interconnected systems, especially in the presence of communication delays. While learning-based controllers can achieve strong empirical performance, their black-box nature makes it difficult to provide formal and scalable stability guarantees. To address this gap, we propose a framework to synthesize and verify neural vector Lyapunov-Razumikhin certificates for discrete-time delayed interconnected systems. Our contributions are three-fold. First, we establish a sufficient condition for discrete-time sISS via vector Lyapunov-Razumikhin functions, which enables certification for large-scale delayed interconnected systems. Second, we develop a scalable synthesis and verification framework that learns the neural certificates and verifies the certificates on reachability-constrained delay domains with scalability analysis. Third, we validate our approach on mixed-autonomy platoons, drone formations, and microgrids against multiple baselines, showing improved verification efficiency with competitive control performance.

While Contrastive Learning (CL) has revolutionized self-supervised representation learning, its latent representations remain highly entangled and opaque, limiting their interpretability in safety-critical applications. We identify that a fundamental cause of this entanglement is the reliance on deterministic similarity measures, which treat all feature dimensions equally. In compositional scenes, this creates an Optimization Conflict: common background features, such as, "blue sky", are encouraged to align in positive pairs but simultaneously repelled in negative pairs, causing gradient oscillations that hinder precise semantic disentanglement. To address this, we propose **BayesNCL** (Bayesian Gated Non-Negative Contrastive Learning). Unlike standard approaches, BayesNCL introduces a probabilistic gating mechanism that dynamically filters out task-irrelevant, high-frequency common features while selectively retaining discriminative semantics. By formalizing feature selection as a variational inference problem with a sparse Bernoulli prior, our method effectively resolves the optimization conflict. Empirical experimental results on Imagenet-100 demonstrate that BayesNCL achieves a remarkable 142.1\% improvement in semantic consistency compared to state-of-the-art baselines, yielding highly interpretable representations without compromising downstream task performance. Code is available at https://github.com/Cui-Peng-624/BayesNCL.

Unlearnable examples (UEs) aim to compromise model training by injecting imperceptible perturbations to clean samples. However, existing UE schemes exhibit limited robustness against advanced defenses due to their heuristic design or narrowly scoped domain perturbations. To address this, we propose DUNE, a Dual-branch UNlearnable Ensemble perturbation optimization approach. Specifically, DUNE separately optimizes perturbations in the spatial and color domains to establish the mapping between perturbations and shift-induced labels. This design extends the perturbation domain to increase noise intensity for improving robustness and drives the models to learn perturbation-oriented features with degraded generalization, thereby achieving unlearnability. To strengthen DUNE's performance, we further propose an unlearnability-enhancing ensemble strategy that aggregates diverse pre-trained models during the dual-branch optimization. Extensive experiments on benchmark datasets CIFAR-10 and ImageNet verify that DUNE's robustness outperforms 12 SOTA UE schemes under 7 mainstream defenses, yielding a lower average test accuracy of 14.95% to 50.82%.

Data attribution estimates how a trained model would change if a subset of training points were removed, and is a central primitive for tasks such as interpretability, data valuation, and machine unlearning. Despite its widespread use, our theoretical understanding of key data attribution methods -- Influence Functions (IF) and a single Newton Step (NS) -- remains limited: existing guarantees heavily rely on *global* strong convexity and yield bounds with pessimistic dependence on the parameter dimension $d$ and the number of removed samples $k$. We give a new analysis of IF and NS for convex ERM that replaces global assumptions with *local* conditions: it suffices that the loss is strongly convex and sufficiently smooth only in a neighborhood of the first Newton step. As a concrete validation, we analyze logistic regression with Gaussian features and show that our bounds capture the correct scaling up to polylogarithmic factors, yielding matching upper and lower bounds and explaining observed regimes in which NS is markedly more accurate than IF, thereby resolving open questions raised by (Koh et al., 2019).