We study an active cluster recovery problem where, given a set of $n$ points and an oracle answering queries like ``are these two points in the same cluster?'', the task is to recover exactly all clusters using as few queries as possible. We begin by introducing a simple but general notion of margin between clusters that captures, as special cases, the margins used in previous works, the classic SVM margin, and standard notions of stability for center-based clusterings. Under our margin assumptions we design algorithms that, in a variety of settings, recover all clusters exactly using only $O(\log n)$ queries. For $\mathbb{R}^m$, we give an algorithm that recovers \emph{arbitrary} convex clusters, in polynomial time, and with a number of queries that is lower than the best existing algorithm by $\Theta(m^m)$ factors. For general pseudometric spaces, where clusters might not be convex or might not have any notion of shape, we give an algorithm that achieves the $O(\log n)$ query bound, and is provably near-optimal as a function of the packing number of the space. Finally, for clusterings realized by binary concept classes, we give a combinatorial characterization of recoverability with $O(\log n)$ queries, and we show that, for many concept classes in $\mathbb{R}^m$, this characterization is equivalent to our margin condition. Our results show a deep connection between cluster margins and active cluster recoverability.

The Stochastic Block Model (SBM) is a widely used random grap h model for networks with communities. Despite the recent burst of inte rest in community detection under the SBM from statistical and computational points of view, there are still gaps in understanding the fundamental limits of re covery. In this paper, we consider the SBM in its full generality, where there is no r estriction on the number and sizes of communities or how they grow with the numb er of nodes, as well as on the connectivity probabilities inside or across c ommunities. For such stochastic block models, we provide guarantees for exact re covery via a semidef-inite program as well as upper and lower bounds on SBM paramet ers for exact recoverability. Our results exploit the tradeoffs among th e various parameters of heterogenous SBM and provide recovery guarantees for man y new interesting SBM configurations.

Is it possible to adapt them in a similar way for use as general-purpose decoders?

Deep learning architectures are highly diverse. To prove their universal approximation properties, existing works typically rely on model-specific proofs. Generally, they construct a dedicated mathematical formulation for each architecture (e.g., fully connected networks, CNNs, or Transformers) and then prove their universal approximability. However, this approach suffers from two major limitations: first, every newly proposed architecture often requires a completely new proof from scratch; second, these proofs are largely isolated from one another, lacking a common analytical foundation. This not only incurs significant redundancy but also hinders unified theoretical understanding across different network families. To address these issues, this paper proposes a general and modular framework for proving universal approximation. We define a basic building block (comprising one or multiple layers) that possesses the universal approximation property as a Universal Approximation Module (UAM). Under this condition, we show that any deep network composed of such modules inherently retains the universal approximation property. Moreover, the overall approximation process can be interpreted as a progressive refinement across modules. This perspective not only unifies the analysis of diverse architectures but also enables a step-by-step understanding of how expressive power evolves through the network.

Reasoning LLMs are trained to verbalize their reasoning process, yielding strong gains on complex tasks. This transparency also opens a promising direction: multiple reasoners can directly collaborate on each other's thinking within a shared trajectory, yielding better inference efficiency and exploration. A key prerequisite, however, is the ability to assess the usefulness and build on another model's partial thinking--we call this off-trajectory reasoning. Our paper investigates a critical question: can standard solo-reasoning training pipelines deliver desired off-trajectory behaviors? We propose twin tests that capture the two extremes of the off-trajectory spectrum, namely Recoverability, which tests whether LLMs can backtrack from "distractions" induced by misleading reasoning traces, and Guid-ability, which tests their ability to build upon correct reasoning from stronger collaborators. Our study evaluates 15 open-weight LLMs (1.5B-32B) and reveals a counterintuitive finding--"stronger" LLMs on benchmarks are often more fragile under distraction. Moreover, all models tested fail to effectively leverage guiding steps from collaborators on problems beyond their inherent capabilities with solve rates remaining under 9.2%. Finally, we conduct control studies to isolate the effects of three factors in post-training on these behaviors: the choice of distillation teacher, the use of RL, and data selection strategy. Our results provide actionable insights for training natively strong reasoning collaborators; e.g., we find that suboptimal recoverability behaviors of teacher models are transferred to distilled students even if the distillation trajectories are correct. Taken together, this work lays the groundwork for evaluating multi-model collaborations in shared reasoning trajectories and highlights the limitations of off-the-shelf reasoning LLMs. Figure 1: Comparison of solo (left) vs. collaborative reasoning (right) setting. LLMs of different sizes and functionalities collaborate on a shared trajectory.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper deals with the problem of identifiability in the presence of various generated mechanisms for missing data, using a framework introduced by Mohan and others. Some existing results are improved upon, and some nice examples are given (such as identifiability of causal distributions even when conditional distributions are unavailable). Quality Technical quality seems high, the results are very involved but appear to be correct. Results like Theorem 3 are clear and interpretable.

Is it possible to adapt them in a similar way for use as general-purpose decoders?

Self-driving laboratories (SDLs) have shown promise to accelerate materials discovery by integrating machine learning with automated experimental platforms. However, errors in the capture of input parameters may corrupt the features used to model system performance, compromising current and future campaigns. This study develops an automated workflow to systematically detect noisy features, determine sample-feature pairings that can be corrected, and finally recover the correct feature values. A systematic study is then performed to examine how dataset size, noise intensity, and feature value distribution affect both the detectability and recoverability of noisy features. In general, high-intensity noise and large training datasets are conducive to the detection and correction of noisy features. Low-intensity noise reduces detection and recovery but can be compensated for by larger clean training data sets. Detection and correction results vary between features with continuous and dispersed feature distributions showing greater recoverability compared to features with discrete or narrow distributions. This systematic study not only demonstrates a model agnostic framework for rational data recovery in the presence of noise, limited data, and differing feature distributions but also provides a tangible benchmark of kNN imputation in materials data sets. Ultimately, it aims to enhance data quality and experimental precision in automated materials discovery.

Recent advances in large language models (LLMs) have revolutionized natural language processing, yet evaluating their intrinsic linguistic understanding remains challenging. Moving beyond specialized evaluation tasks, we propose an information-theoretic framework grounded in mutual information (MI) to achieve this. We formalize the understanding as MI between an input sentence and its latent representation (sentence-level MI), measuring how effectively input information is preserved in latent representation. Given that LLMs learn embeddings for individual tokens, we decompose sentence-level MI into token-level MI between tokens and sentence embeddings, establishing theoretical bounds connecting these measures. Based on this foundation, we theoretically derive a computable lower bound for token-level MI using Fano's inequality, which directly relates to token-level recoverability-the ability to predict original tokens from sentence embedding. We implement this recoverability task to comparatively measure MI across different LLMs, revealing that encoder-only models consistently maintain higher information fidelity than their decoder-only counterparts, with the latter exhibiting a distinctive late-layer "forgetting" pattern where mutual information is first enhanced and then discarded. Moreover, fine-tuning to maximize token-level recoverability consistently improves understanding ability of LLMs on tasks without task-specific supervision, demonstrating that mutual information can serve as a foundation for understanding and improving language model capabilities.

We address the problem of deciding whether a causal or probabilistic query is estimable from data corrupted by missing entries, given a model of missingness process. We extend the results of Mohan et al. [2013] by presenting more general conditions for recovering probabilistic queries of the form P(y|x) and P(y,x) as well as causal queries of the form P(y|do(x)). We show that causal queries may be recoverable even when the factors in their identifying estimands are not recoverable. Specifically, we derive graphical conditions for recovering causal effects of the form P(y|do(x)) when Y and its missingness mechanism are not d-separable. Finally, we apply our results to problems of attrition and characterize the recovery of causal effects from data corrupted by attrition.