arXiv subjects

Martin Wattenberg

Publications and source records attributed to Martin Wattenberg.

At least 19 recordsLinked to original sources

arXiv2026

Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

cs.LG↗

arXiv2026

Intentmaking and Sensemaking: Human Interaction with AI-Guided Mathematical Discovery

Artificial intelligence offers powerful new tools for scientific discovery, but the interaction paradigms required to effectively harness these systems remain underexplored. In this paper, we present findings from a formative user study with 11 expert mathematicians who used AlphaEvolve, an evolutionary coding agent, to tackle advanced problems in their fields of expertise. We identify and characterize a distinct workflow we term intentmaking, the iterative process of discovering, defining, and refining one's experimental goals through active system interaction. We frame this as a natural extension to sensemaking, the cognitive process of building an understanding of complex or novel data. We suggest that users enter a cycle of intentmaking (defining and updating their experiment) and sensemaking (interpreting the results) which repeats many times during the course of an investigation. Our documentation of these themes suggests an approach to designing AI tools for scientific discovery that goes beyond the existing question/answer model of many current systems, treating them as collaborative instruments rather than opaque black-box assistants.

cs.AI↗

arXiv2026

AI co-mathematician: Accelerating mathematicians with agentic AI

We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research. The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows. In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references. Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated.

cs.AI↗

arXiv2026

Large Language Models Generate Harmful Responses Using a Distinct Mechanism, Shared Across Harm Types

Large language models remain vulnerable to jailbreaks that elicit harmful responses, yet the mechanism behind harmful response generation is poorly understood. Here, we investigate how this capability is organized within model parameters. We identify and prune parameters that specifically support harmful compliance, providing a direct mechanistic analysis at the parameter level. We find that this capability depends on a sparse set of critical parameters: pruning these parameters substantially reduces harmful compliance while causing only limited degradation in benign capabilities, suggesting that key components of harmful generation are separable from those of general utility. Parameters identified from one harm category also reduce harmful responses in others, indicating components shared across harm types. This separability appears primarily in aligned models, suggesting that alignment training internally reshapes the harmful response mechanism even when behavioral safeguards remain brittle. We further show that harmful response generation is dissociable from the ability to recognize and reason about harmfulness. Finally, we extend our analysis to emergent misalignment and identify a sparse set of parameters contributing to it, with substantial sharing across fine-tuning domains. Together, these results reveal a consistent parameter-level organization underlying unsafe behaviors and point toward more principled interventions for improving model safety.

cs.CL↗

arXiv2026

Think Before You Lie: How Reasoning Leads to Honesty

While existing evaluations of large language models (LLMs) measure deception rates, the underlying conditions that give rise to deceptive behavior are poorly understood. We investigate this question using a novel dataset of realistic moral trade-offs where honesty incurs variable costs. Contrary to humans, who tend to become less honest given time to deliberate (Capraro, 2017; Capraro et al., 2019), we find that reasoning consistently increases honesty across scales and for several LLM families. This effect is not only a function of the reasoning content, as reasoning traces are often poor predictors of final behaviors. Rather, we show that the underlying geometry of the representational space itself contributes to the effect. Namely, we observe that deceptive regions within this space are metastable: deceptive answers are more easily destabilized by input paraphrasing, output resampling, and activation noise than honest ones. We interpret the effect of reasoning in this vein: generating deliberative tokens as part of moral reasoning entails the traversal of a biased representational space, ultimately nudging the model toward its more stable, honest defaults.

cs.AI↗

arXiv2026

Decomposing Query-Key Feature Interactions Using Contrastive Covariances

Despite the central role of attention heads in Transformers, we lack tools to understand why a model attends to a particular token. To address this, we study the query-key (QK) space -- the bilinear joint embedding space between queries and keys. We present a contrastive covariance method to decompose the QK space into low-rank, human-interpretable components. It is when features in keys and queries align in these low-rank subspaces that high attention scores are produced. We first study our method both analytically and empirically in a simplified setting. We then apply our method to large language models to identify human-interpretable QK subspaces for categorical semantic features and binding features. Finally, we demonstrate how attention scores can be attributed to our identified features.

cs.LG↗

arXiv2025

Priors in Time: Missing Inductive Biases for Language Model Interpretability

Recovering meaningful concepts from language model activations is a central aim of interpretability. While existing feature extraction methods aim to identify concepts that are independent directions, it is unclear if this assumption can capture the rich temporal structure of language. Specifically, via a Bayesian lens, we demonstrate that Sparse Autoencoders (SAEs) impose priors that assume independence of concepts across time, implying stationarity. Meanwhile, language model representations exhibit rich temporal dynamics, including systematic growth in conceptual dimensionality, context-dependent correlations, and pronounced non-stationarity, in direct conflict with the priors of SAEs. Taking inspiration from computational neuroscience, we introduce a new interpretability objective -- Temporal Feature Analysis -- which possesses a temporal inductive bias to decompose representations at a given time into two parts: a predictable component, which can be inferred from the context, and a residual component, which captures novel information unexplained by the context. Temporal Feature Analyzers correctly parse garden path sentences, identify event boundaries, and more broadly delineate abstract, slow-moving information from novel, fast-moving information, while existing SAEs show significant pitfalls in all the above tasks. Overall, our results underscore the need for inductive biases that match the data in designing robust interpretability tools.

cs.LG↗

arXiv2025

Into the Rabbit Hull: From Task-Relevant Concepts in DINO to Minkowski Geometry

DINOv2 is routinely deployed to recognize objects, scenes, and actions; yet the nature of what it perceives remains unknown. As a working baseline, we adopt the Linear Representation Hypothesis (LRH) and operationalize it using SAEs, producing a 32,000-unit dictionary that serves as the interpretability backbone of our study, which unfolds in three parts. In the first part, we analyze how different downstream tasks recruit concepts from our learned dictionary, revealing functional specialization: classification exploits "Elsewhere" concepts that fire everywhere except on target objects, implementing learned negations; segmentation relies on boundary detectors forming coherent subspaces; depth estimation draws on three distinct monocular depth cues matching visual neuroscience principles. Following these functional results, we analyze the geometry and statistics of the concepts learned by the SAE. We found that representations are partly dense rather than strictly sparse. The dictionary evolves toward greater coherence and departs from maximally orthogonal ideals (Grassmannian frames). Within an image, tokens occupy a low dimensional, locally connected set persisting after removing position. These signs suggest representations are organized beyond linear sparsity alone. Synthesizing these observations, we propose a refined view: tokens are formed by combining convex mixtures of archetypes (e.g., a rabbit among animals, brown among colors, fluffy among textures). This structure is grounded in Gardenfors' conceptual spaces and in the model's mechanism as multi-head attention produces sums of convex mixtures, defining regions bounded by archetypes. We introduce the Minkowski Representation Hypothesis (MRH) and examine its empirical signatures and implications for interpreting vision-transformer representations.

cs.CV↗

arXiv2025

Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls

Language models are increasingly capable, yet still fail at a seemingly simple task of multi-digit multiplication. In this work, we study why, by reverse-engineering a model that successfully learns multiplication via \emph{implicit chain-of-thought}, and report three findings: (1) Evidence of long-range structure: Logit attributions and linear probes indicate that the model encodes the necessary long-range dependencies for multi-digit multiplication. (2) Mechanism: the model encodes long-range dependencies using attention to construct a directed acyclic graph to ``cache'' and ``retrieve'' pairwise partial products. (3) Geometry: the model implements partial products in attention heads by forming Minkowski sums between pairs of digits, and digits are represented using a Fourier basis, both of which are intuitive and efficient representations that the standard fine-tuning model lacks. With these insights, we revisit the learning dynamics of standard fine-tuning and find that the model converges to a local optimum that lacks the required long-range dependencies. We further validate this understanding by introducing an auxiliary loss that predicts the ``running sum'' via a linear regression probe, which provides an inductive bias that enables the model to successfully learn multi-digit multiplication. In summary, by reverse-engineering the mechanisms of an implicit chain-of-thought model we uncover a pitfall for learning long-range dependencies in Transformers and provide an example of how the correct inductive bias can address this issue.

cs.LG↗

arXiv2025

Chronotome: Real-Time Topic Modeling for Streaming Embedding Spaces

Many real-world datasets -- from an artist's body of work to a person's social media history -- exhibit meaningful semantic changes over time that are difficult to capture with existing dimensionality reduction methods. To address this gap, we introduce a visualization technique that combines force-based projection and streaming clustering methods to build a spatial-temporal map of embeddings. Applying this technique, we create Chronotome, a tool for interactively exploring evolving themes in time-based data -- in real time. We demonstrate the utility of our approach through use cases on text and image data, showing how it offers a new lens for understanding the aesthetics and semantics of temporal datasets.

cs.HC↗

arXiv2025

Beyond the Rosetta Stone: Unification Forces in Generalization Dynamics

Large language models (LLMs) struggle with cross-lingual knowledge transfer: they sometimes hallucinate when asked in one language about facts expressed in a different language during training. This work introduces a controlled setting to study the causes and training dynamics of this phenomenon by training small Transformer models from scratch on synthetic multilingual datasets. Depending on (1) the correlation between facts and the language they were learned in (informativeness), and (2) the ease of language identification (extractability), models either develop unified representations across languages or separate representations; only when representations are unified do facts transfer across languages. Based on these insights, we propose a unifying perspective which explains a range of prior observations concerning cross-lingual transfer in multilingual LLMs. Our work shows controlled settings can shed light on pre-training dynamics and suggests methods to encourage representational unification as part of training that would improve LLMs' cross-lingual transfer.

cs.CL↗

arXiv2025

Story Ribbons: Reimagining Storyline Visualizations with Large Language Models

Analyzing literature involves tracking interactions between characters, locations, and themes. Visualization has the potential to facilitate the mapping and analysis of these complex relationships, but capturing structured information from unstructured story data remains a challenge. As large language models (LLMs) continue to advance, we see an opportunity to use their text processing and analysis capabilities to augment and reimagine existing storyline visualization techniques. Toward this goal, we introduce an LLM-driven data parsing pipeline that automatically extracts relevant narrative information from novels and scripts. We then apply this pipeline to create Story Ribbons, an interactive visualization system that helps novice and expert literary analysts explore detailed character and theme trajectories at multiple narrative levels. Through pipeline evaluations and user studies with Story Ribbons on 36 literary works, we demonstrate the potential of LLMs to streamline narrative visualization creation and reveal new insights about familiar stories. We also describe current limitations of AI-based systems, and interaction motifs designed to address these issues.

cs.HC↗

arXiv2025

What Does it Mean for a Neural Network to Learn a "World Model"?

We propose a set of precise criteria for saying a neural net learns and uses a "world model." The goal is to give an operational meaning to terms that are often used informally, in order to provide a common language for experimental investigation. We focus specifically on the idea of representing a latent "state space" of the world, leaving modeling the effect of actions to future work. Our definition is based on ideas from the linear probing literature, and formalizes the notion of a computation that factors through a representation of the data generation process. An essential addition to the definition is a set of conditions to check that such a "world model" is not a trivial consequence of the neural net's data or task.

cs.AI↗

arXiv2025

Does visualization help AI understand data?

Charts and graphs help people analyze data, but can they also be useful to AI systems? To investigate this question, we perform a series of experiments with two commercial vision-language models: GPT 4.1 and Claude 3.5. Across three representative analysis tasks, the two systems describe synthetic datasets more precisely and accurately when raw data is accompanied by a scatterplot, especially as datasets grow in complexity. Comparison with two baselines -- providing a blank chart and a chart with mismatched data -- shows that the improved performance is due to the content of the charts. Our results are initial evidence that AI systems, like humans, can benefit from visualization.

cs.AI↗

arXiv2025

Can Interpretation Predict Behavior on Unseen Data?

Interpretability research often predicts model responses to targeted mechanistic interventions. But can we predict responses to unseen input data? We propose and demonstrate this alternate objective by using model internals to predict their out-of-distribution (OOD) behavior. We train hundreds of Transformers on simple synthetic tasks, where perfect in-distribution accuracy is compatible with multiple OOD generalization rules. We successfully use attention patterns -- observed only on in-distribution data -- to predict which rule each model follows on OOD data. Our experiments decouple the mechanistic faithfulness of our interpretation from its predictive value; ablations reveal such internal patterns can suppress rather than support the rule they predict, showing observational analysis can forecast behavior even when causal analysis fails to support a simple cause-effect link. Our findings are a proof-of-concept for a new interpretability objective: understanding model internals to predict behavior and assess reliability under distribution shift.

cs.LG↗

arXiv2025

When Bad Data Leads to Good Models

In large language model (LLM) pretraining, data quality is believed to determine model quality. In this paper, we re-examine the notion of "quality" from the perspective of pre- and post-training co-design. Specifically, we explore the possibility that pre-training on more toxic data can lead to better control in post-training, ultimately decreasing a model's output toxicity. First, we use a toy experiment to study how data composition affects the geometry of features in the representation space. Next, through controlled experiments with Olmo-1B models trained on varying ratios of clean and toxic data, we find that the concept of toxicity enjoys a less entangled linear representation as the proportion of toxic data increases. Furthermore, we show that although toxic data increases the generational toxicity of the base model, it also makes the toxicity easier to remove. Evaluations on Toxigen and Real Toxicity Prompts demonstrate that models trained on toxic data achieve a better trade-off between reducing generational toxicity and preserving general capabilities when detoxifying techniques such as inference-time intervention (ITI) are applied. Our findings suggest that, with post-training taken into account, bad data may lead to good models.

cs.LG↗

arXiv2025

The Geometry of Self-Verification in a Task-Specific Reasoning Model

How do reasoning models verify their own answers? We study this question by training a model using DeepSeek R1's recipe on the CountDown task. We leverage the fact that preference tuning leads to mode collapse, yielding a model that always produces highly structured chain-of-thought sequences. With this setup, we do top-down and bottom-up analyses to reverse-engineer how the model verifies its outputs. Top-down, we find Gated Linear Unit (GLU) weights encoding verification-related tokens, such as ``success'' or ``incorrect''. Bottom-up, we find that ``previous-token heads'' are mainly responsible for self-verification in our setup. Our analyses meet in the middle: drawing inspiration from inter-layer communication channels, we use the identified GLU weights to localize as few as three attention heads that can disable self-verification, pointing to a necessary component of a potentially larger verification circuit. Finally, we verify that similar verification components exist in our base model and a general reasoning DeepSeek-R1 model.

cs.AI↗

arXiv2025

Shared Global and Local Geometry of Language Model Embeddings

Researchers have recently suggested that models share common representations. In our work, we find numerous geometric similarities across the token embeddings of large language models. First, we find ``global'' similarities: token embeddings often share similar relative orientations. Next, we characterize local geometry in two ways: (1) by using Locally Linear Embeddings, and (2) by defining a simple measure for the intrinsic dimension of each embedding. Both characterizations allow us to find local similarities across token embeddings. Additionally, our intrinsic dimension demonstrates that embeddings lie on a lower dimensional manifold, and that tokens with lower intrinsic dimensions often have semantically coherent clusters, while those with higher intrinsic dimensions do not. Based on our findings, we introduce EMB2EMB, a simple application to linearly transform steering vectors from one language model to another, despite the two models having different dimensions.

cs.CL↗