NeurIPS 2026 Papers

We study label-free, topology-conditioned zero-shot cross-graph link prediction: at test time the observed target topology is the input graph, but the model receives no target labels, no positive/negative context links, no target fine-tuning, no text attributes, and no post-hoc alignment. This setting is zero-shot with respect to target supervision and adaptation, but it is not topology-free inference. We identify a critical geometric barrier for hyperbolic transfer: radial non-identifiability. Hyperbolic graph learning is motivated by the intuition that radius encodes hierarchy or popularity and angle encodes similarity, yet standard HGNN objectives do not make radii semantically calibrated across disjoint graphs. The same structural role can therefore occupy incompatible radial scales on different target graphs. We propose Invariant Hyperbolic Unfolding (IHU), which restores the intended radial hierarchy channel through selective invariance: radii are fixed by graph-internal structural percentiles while angular representations remain learnable. Its core module, Invariant Structural Anchoring (ISA), rank-canonicalizes structural scores such as coreness into shared hyperbolic radii, producing a comparable radial skeleton without target supervision. In a controlled comparison among matched frozen learned encoders, IHU improves average HR@50 by 2.7 percentage points over the strongest learned hyperbolic baseline and by 7.6 percentage points over the mean of the learned hyperbolic baseline family. The gain amplifies under missing-edge and low-degree regimes, while hierarchy-gap analysis reveals a boundary condition for fixed radial shells: 50% edge sparsification causes 52.5% less degradation, and low-degree nodes show a 4.2× larger advantage. We further position IHU against recent universal link prediction and graph foundation methods through an access-aware taxonomy, showing that the closest recent LP methods use labeled target context links, whereas IHU isolates the geometric effect of fixed versus learned radii in frozen structural hyperbolic encoders.

Language models exhibit strong robustness to paraphrasing, suggesting that semantic information may be encoded through stable internal representations, yet the structure and origin of such invariance remain unclear. We propose a local geometric framework in which semantically equivalent inputs occupy structured regions in latent space, with paraphrastic variation along nuisance directions and semantic identity preserved in invariant subspaces. Building on this view, we make three contributions: (1) a geometric characterization of invariant latent features, (2) a contrastive subspace discovery method that separates semantic-changing from semantic-preserving variation, and (3) an application of invariant representations to zero-shot model attribution. Across models and layers, empirical results support these contributions. Invariant structure emerges in specific depth regions, semantic displacement lies largely outside the nuisance subspace, and representation-level interventions indicate a causal role of invariant components in model outputs. Invariant representations also capture model-specific geometric patterns, enabling accurate attribution. These findings suggest that semantic invariance can be viewed as a local geometric property of latent representations, offering a principled perspective on how language models organize meaning.

Two structural priors—global invariance under a group action and low body-order—are widely used in equivariant architectures on product Lie groups such as $\mathrm{SO}(3)^n$. Whether their statistical gains are fundamental, changing the exponent of the minimax rate, or merely constant-factor, changing only the leading coefficient, has remained open. We show that these gains are statistically fundamental: both priors reduce the effective dimension governing the minimax rate, not merely the leading constant. Specifically, we establish matching upper and lower minimax rates for four nested function classes on $\mathrm{SO}(3)^n$, one for each combination of the two priors. The matching lower bound relies on a pure-$S$ lift construction and a combinatorial packing across the $\binom{n}{K}$ top-order subsets. The resulting effective-dimension formula decomposes additively: invariance contributes a fixed reduction of $3$, while body-order truncation at order $K$ contributes a reduction of $3(n-K)$ that grows with system size. An adaptive sample-splitting procedure selects the body-order from data, achieving the best in-family risk up to an $O(\sqrt{\log N/N})$ remainder with no $N$-dependent tuning. Five experiments using spectral projection estimators corroborate the theory's predicted structural and operational consequences, supporting the four-class hierarchy on synthetic targets and yielding $2$–$17\times$ sample-efficiency gains on the Maier–Saupe benchmark from liquid-crystal theory.

Modern image encoders achieve high generalization by decoupling semantic meaning from resolution, an ability yet to be fully realized in the 3D domain. We investigate the failure of 3D point cloud encoders to achieve similar generalization and find that existing models are highly sensitive to sampling resolution and scale changes, leading to significant performance degradation. This sensitivity is a major bottleneck for real-world deployment in robotics, as it suggests models overfit to specific quantization densities and object scales rather than learning invariant semantic features. To mitigate this dependency, we propose Invaria, a point cloud encoder that achieves scale and density invariance through next-resolution prediction and receptive field calibration. While our objective is not the explicit generation of high-resolution point clouds, we find that this training objective encourages the model to learn robust, structural invariants. The resulting encoder achieves significant performance gains during resolution shifts while maintaining high efficiency through a compact model size and reduced token requirements. Specifically, on ScanNet, Invaria achieves a 56.0\% higher mIoU at 3$\times$ lower resolution and a 20\% improvement when the objects scale is reduced by a factor of 3. These gains are achieved with a 45\% smaller model size and an average reduction of 40\% in input tokens. The code will be released once the paper is published.

When does training language models (LMs) on explanations yield faithful introspection, rather than superficial imitation? Surprisingly, we find that LMs trained to explain the predictions of similar models frequently produce explanations more faithful to $\textit{their own current behaviors}$ than to those of their training targets. This "introspective'' coupling between the model's explanations and behaviors occurs only when the training target explanations remain sufficiently similar to model behaviors over the course of training. This alignment must be preserved throughout the training process: introspection only emerges when explanation supervision is sufficiently behaviorally compatible with the model as it changes. Finally, we show that introspection generalizes to variants of the training problem: when introspection training is run concurrently with training that shifts a model's behaviors, explanations track those behavioral shifts without requiring updated supervision. This holds across a diverse range of tasks, including sycophancy and refusal, and is robust to label noise. These results suggest that introspection training is a viable component of post-training: explanation labels need not be continually refreshed, and faithfulness extends to regions of input space not explicitly supervised.

We introduce a simple post-training method that makes transformer attention sparse without sacrificing performance. Applying a flexible sparsity regularisation under a constrained-loss objective, we show on models up to 7B parameters that it is possible to retain the original pretraining loss while reducing attention connectivity to $\approx 0.4$% of its edges. Unlike sparse-attention methods designed for computational efficiency, our approach leverages sparsity as a structural prior: it preserves capability while exposing a more organized and interpretable connectivity pattern. We find that this local sparsity cascades into global circuit simplification: task-specific circuits involve far fewer components (attention heads and MLPs) with up to 100× fewer edges connecting them. Additionally, using cross-layer transcoders, we show that sparse attention substantially simplifies attention attribution, enabling a unified view of feature-based and circuit-based perspectives. These results demonstrate that transformer attention can be made orders of magnitude sparser, suggesting that much of its computation is redundant and that sparsity may serve as a guiding principle for more structured and interpretable models.

Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori knowledge of the true variance, and often explicitly scale with the ambient dimension, rendering them vacuous for infinite-dimensional or heavily structured operators. Motivated by these challenges, we establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces. We demonstrate that our empirical bounds achieve asymptotic sharpness with the best known oracle rates. Finally, as an independent byproduct, we derive novel empirical concentration guarantees for the intrinsic dimension itself.

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness and verify the properties. Our results position Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

Muon and related norm-constrained matrix optimizers have become central to large-scale learning problems. They are formulated as a linear maximization oracle (LMO) over an ambient matrix-norm ball in unconstrained Euclidean space. However, these do not generalize cleanly to manifold-valued parameters such as low-rank factorizations, orthogonality constraints, or symmetric positive definite (SPD) matrices. Naively restricting the Muon LMO to the tangent space (i) breaks quotient symmetries and (ii) couples the tangent-space constraint with an ambient norm bound, thereby obstructing closed-form solutions on various manifolds of interest. We resolve both issues with a single observation: every Riemannian metric canonically lifts a unitarily invariant Euclidean norm to an intrinsic norm on each tangent space, and the resulting intrinsic norm constrained LMO is symmetry preserving. Building on this, we introduce intrinsic Muon (iMuon), a unified framework that yields closed-form updates on the fixed-rank, SPD, Stiefel, and Grassmann manifolds for any unitarily invariant norm, including the spectral, Frobenius, and nuclear norms. We establish convergence guarantees for both deterministic and stochastic iMuon with rate constants that depend only on the manifold dimension. Notably, on the fixed-rank manifold this constant depends only on the rank, making the rate independent of factor conditioning and removing the runtime factor-rescaling required by prior work. Experiments on LoRA finetuning of LLMs, image classification, and subspace learning illustrate the efficacy of the proposed approach.

Mutual information (MI) is fundamental to representation-learning analysis in deep networks, but in high dimensions MI estimators depend on hyperparameters --- bin widths, kernel bandwidths, neighbourhood radii, auxiliary networks --- whose bias often dominates the signal. We show that continuous piecewise-linear (CPWL) networks, including ReLU networks, admit an intrinsic information-theoretic analysis that avoids this pitfall: their inherent partition of the input space into convex linear regions assigns to every input a discrete region label $\Pi$ alongside the continuous representation $T$. We decompose $I(Y; T)$ exactly into geometric terms, where the \emph{routing information} $I(Y; \Pi)$ admits a hyperparameter-free plug-in estimator that approximates a population lower bound on every other MI term in the decomposition. When the partition saturates, a functional-equivalence quotient at relative Frobenius tolerance $\varepsilon \in [0, 2]$ collapses regions implementing the same linear operator and restores informativeness. Empirically, the routing estimator correlates strongly with three established MI baselines {\it without any hyperparameter}, and the functional quotient at moderate $\varepsilon$ recovers informative estimates in the regime where the raw partition is bounded by $\log_2 N$ rather than $H(Y)$. We provide a parameter-free framework for analysing the internal information geometry of CPWL networks, grounded in their inherent piecewise-linear structure rather than in external discretisation.

Off-policy evaluation (OPE) estimates the expected return of a target policy from previously collected data without additional environment interaction. While OPE methods for flat Markovian policies are well studied, little work has addressed evaluating hierarchical policies in which a high-level policy selects temporally extended options that generate primitive actions until a termination condition is met. The primary existing approach applies per-decision importance sampling at the option level, but requires option-annotated trajectories and exhibits variance that grows with the horizon and action dimensionality. For non-hierarchical policies, fitted Q-evaluation (FQE) typically achieves lower error than importance sampling by leveraging the classic Bellman equation for policy evaluation; however, a naive extension of FQE to hierarchical policies results in a biased policy value estimate. In this paper, we introduce Intra-Option Fitted Q-Evaluation (IO-FQE), which estimates a value function in the augmented state-option space; this approach enables OPE of hierarchical policies even when we lack annotations of what option was ran in the data. We develop two continuous-action instantiations of IO-FQE and show on hierarchical continuous control tasks (AntMaze navigation and OGBench Puzzle manipulation) that IO-FQE eliminates the bias incurred by a naive application of FQE to hierarchical policies and substantially lowers mean squared-error compared to option-level importance sampling.

Image-free zero-shot classifier expansion aims to extend recognition to unseen classes by establishing an association mapping between class semantics and visual classifier weights, without accessing original images. However, most existing methods rely on an association-based optimization mechanism, which struggles to meet the demands for fine-grained classification of visually and semantically similar classes. Since co-occurring attributes dominate class semantics, the model gravitates towards co-occurring attributes rather than discriminative attributes. This leads to geometrically entangled classifier weights, resulting in ambiguous decision boundaries. To address this problem, this paper proposes a novel approach that reframes weight generation through the lens of causal intervention. The method first employs comparative reasoning to disentangle discriminative and co-occurring attributes. Subsequently, it synthesizes intervened semantics to impose invariance and sensitivity constraints. This mechanism compels the model to suppress co-occurring attributes while enhancing sensitivity to discriminative attributes. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches, yielding clearer decision boundaries in fine-grained settings.

The deployment of Artificial Intelligence in high-risk domains, such as finance and healthcare, necessitates models that are both fair and transparent. While regulatory frameworks, including the EU's AI Act, mandate bias mitigation, they are deliberately vague about the definition of bias. In line with existing research, we argue that true fairness requires addressing bias at the intersections of protected groups. We propose a unified framework that leverages Mixed-Integer Optimization (MIO) to train intersectionally fair and intrinsically interpretable classifiers. We prove the equivalence of two measures of intersectional fairness (MSD and SPSF) in detecting the most unfair subgroup and empirically demonstrate that our MIO-based algorithm improves performance in finding bias. We train high-performing, interpretable classifiers that bound intersectional bias below an acceptable threshold, offering a robust solution for regulated industries and beyond.

While Neural Combinatorial Optimization (NCO) achieves strong performance, its black-box nature remains a key roadblock. Standard interpretability tools, such as Concept Bottleneck Models (CBMs), are ill-equipped for NCO, whose decisions are dynamic, state-dependent, and lack proper concept definition. To close this gap, we introduce Evolving Programmatic Bottlenecks (EPB), the first framework that distills black-box NCO models into human-readable program portfolios. EPB extends CBMs for sequential decision problems: it employs an LLM to autonomously evolve a bank of programs, where each program's per-step action distribution serves as the bottleneck. We realize this through an iterative framework: Block I fixes program bank capacity and introduces a hybrid textual-numerical gradient descent scheme to backpropagate gradients across the student router model and the LLM in-context learning; Block II dynamically adapts bank capacity via fault-targeted boosting and redundancy pruning. Extensive experiments demonstrate EPB's effectiveness and broad applicability, where the interpreted program portfolios largely match original performance. EPB also reveals that NCO behavior shifts across optimization stages and can be approximated as a composition of classic heuristic variants. Our work advances interpretable NCO and establishes EPB as a promising tool for interpreting complex sequential decision-making models.

Protein language models (pLMs) encode information about protein sequences which enable downstream tasks such as structure prediction, but their internal representations are not well understood. Sparse autoencoders (SAEs) provide a promising tool to disentangle latent pLM representations into interpretable features, but existing annotation pipelines largely rely on protein-level annotations derived from database labels and LLM annotations of top activating sequences. Such annotations can overlook the localized residue-level and geometric patterns encoded by sparse features. We introduce an automated and scalable method for interpreting SAE features in ESM-2 by using geometrically inspired features of the protein $\text{C}_{\alpha}$ backbone. Across ESM-2 8M layers, geometric annotations explain a large fraction of SAE features, expanding coverage beyond database and sequence-based methods. In particular, geometry can distinguish SAE features sharing the same database annotation, revealing substructure within known biological labels. A significant portion of SAE features activate on unannotated metagenomic protein sequences enabling us to use our SAE annotations to better understand these sequences. In addition, ablation experiments at the level of contact predictions hint toward SAE features controlling protein geometry. This provides a robust method of annotating proteins activated within SAE neurons at a residue level, providing a bridge between mechanistic interpretability and structural biology.

Inferring latent interaction structures from observed dynamics is a fundamental inverse problem in many-body interacting systems. Most neural approaches rely on black-box surrogates over trainable graphs, achieving accuracy at the expense of mechanistic interpretability. Symbolic regression offers explicit dynamical equations and stronger inductive biases, but typically assumes known topology and a fixed function library. We propose $\textbf{COSINE}$ ($\textbf{C}$o-$\textbf{O}$ptimization of $\textbf{S}$ymbolic $\textbf{I}$nteractions and $\textbf{N}$etwork $\textbf{E}$dges), a differentiable framework that jointly discovers interaction graphs and sparse symbolic dynamics. To overcome the limitations of fixed symbolic libraries, COSINE further incorporates an outer-loop large language model that adaptively prunes and expands the hypothesis space using feedback from the inner optimization loop. Experiments on synthetic systems and large-scale real-world epidemic data demonstrate robust structural recovery and compact, mechanism-aligned dynamical expressions. Code: https://anonymous.4open.science/r/COSINE-6D43.

Decision tree optimization is fundamental to interpretable machine learning, yet most tree learning algorithms are restricted to binary splits. This restriction can produce unnecessarily deep trees when the underlying decision logic is naturally multi-valued. Multiway-split trees address this, but finding sparse yet accurate ones is more challenging, as the number of possible partitions at each node grows combinatorially with the number of feature bins. We introduce SPLINTER (SParse Lookahead for Interpretable N-way Trees by Eliminative Ranking), a dynamic programming and branch-and-bound-based framework that combines efficient candidate generation with pruning bounds to find sparse, accurate multiway-split trees. Our empirical results show that our methods produce trees that achieve higher accuracy at a greater decision sparsity (shorter path lengths) than greedy multiway and near-optimal binary-split tree baselines, while remaining practical to train. We further extend the framework to approximate the Rashomon set of near-optimal multiway-split trees, allowing users to inspect multiple sparse and accurate alternatives.

Generalized additive models (GAMs) offer interpretability through independent univariate feature effects but underfit when interactions are present in data. GA$^2$Ms add selected pairwise interactions which improves accuracy, but sacrifices interpretability and limits model auditing. We propose \emph{Conditionally Additive Local Models} (CALMs), a new model class, that balances the interpretability of GAMs with the accuracy of GA$^2$Ms. CALMs allow multiple univariate shape functions per feature, each active in different regions of the input space. These regions are defined independently for each feature as simple logical conditions (thresholds) on the features it interacts with. As a result, effects remain locally additive while varying across subregions to capture interactions. We further propose a principled distillation-based training pipeline that identifies homogeneous regions with limited interactions and fits interpretable shape functions via region-aware backfitting. Experiments on diverse classification and regression tasks show that CALMs consistently outperform GAMs and achieve accuracy broadly comparable to GA$^2$Ms, while preserving the univariate auditability that GA$^2$Ms forfeit. Overall, CALMs offer a favorable trade-off between predictive accuracy and interpretability.

What if you could edit an image simply by drawing on it---circling an object, writing a short label, or sketching an arrow---with minimal or even no typed prompt? We first reveal a surprising capability: when a text-driven image editor is paired with a strong Vision-Language Model (VLM) at inference time, the combined system can directly interpret visual instructions embedded in the image itself, such as on-image text, bounding boxes, and directional arrows. However, this pipeline incurs additional VLM-planning latency, introduces brittle cross-model error propagation, and depends on external APIs or auxiliary model weights. To address these limitations, we introduce Siphon, a framework that internalizes this externally elicited visual-instruction-following competence into a single diffusion-based editor. Rather than relying on an external VLM at inference time, Siphon first uses a VLM planner to synthesize paired visual-instruction supervision, and then transfers this competence into the editor through lightweight LoRA fine-tuning. The resulting model reads, grounds, and executes on-image annotations directly from pixels, while keeping the diffusion sampling cost close to that of the base editor. Across multiple diffusion architectures, Siphon substantially improves spatial controllability, instruction adherence, and marker removal over text-driven baselines, while matching VLM-assisted pipelines without their additional planning latency, API dependence, and cross-model fragility. We do not position rendered long text as a replacement for conventional text prompts; rather, on-image text is mainly intended for short labels and spatially grounded edit intents, while long or complex instructions can still be provided through the standard text channel.