Recent work has proposed that language models perform computation by manipulating one-dimensional representations of concepts ("features") in activation space. In contrast, we explore whether some language model representations may be inherently multi-dimensional. We begin by developing a rigorous definition of irreducible multi-dimensional features based on whether they can be decomposed into either independent or non-co-occurring lower-dimensional features. Motivated by these definitions, we design a scalable method that uses sparse autoencoders to automatically find multi-dimensional features in GPT-2 and Mistral 7B. These auto-discovered features include strikingly interpretable examples, e.g. $\textit{circular}$ features representing days of the week and months of the year. We identify tasks where these exact circles are used to solve computational problems involving modular arithmetic in days of the week and months of the year. Next, we provide evidence that these circular features are indeed the fundamental unit of computation in these tasks with intervention experiments on Mistral 7B and Llama 3 8B, and we examine the continuity of the days of the week feature in Mistral 7B. Overall, our work argues that understanding multi-dimensional features is necessary to mechanistically decompose some model behaviors.

Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons''), KANs have learnable activation functions on edges ("weights''). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability, on small-scale AI + Science tasks. For accuracy, smaller KANs can achieve comparable or better accuracy than larger MLPs in function fitting tasks. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful ``collaborators'' helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs. Despite the slow training of KANs, their improved accuracy and interpretability show the potential to improve today's deep learning models which rely heavily on MLPs. More research is necessary to make KANs' training more efficient.

Sparse autoencoders (SAEs) are a recent technique for decomposing neural network activations into human-interpretable features. However, in order for SAEs to identify all features represented in frontier models, it will be necessary to scale them up to very high width, posing a computational challenge. In this work, we introduce Switch Sparse Autoencoders, a novel SAE architecture aimed at reducing the compute cost of training SAEs. Inspired by sparse mixture of experts models, Switch SAEs route activation vectors between smaller "expert" SAEs, enabling SAEs to efficiently scale to many more features. We present experiments comparing Switch SAEs with other SAE architectures, and find that Switch SAEs deliver a substantial Pareto improvement in the reconstruction vs. sparsity frontier for a given fixed training compute budget. We also study the geometry of features across experts, analyze features duplicated across experts, and verify that Switch SAE features are as interpretable as features found by other SAE architectures.

Sparse autoencoders (SAEs) aim to decompose language model representations into a sparse set of linear latent vectors. Recent works have improved SAEs using language model gradients, but these techniques require many expensive backward passes during training and still cause a significant increase in cross entropy loss when SAE reconstructions are inserted into the model. In this work, we improve on these limitations by taking a fundamentally different approach: we use low-rank adaptation (LoRA) to finetune the language model itself around a previously trained SAE. We analyze our method across SAE sparsity, SAE width, language model size, LoRA rank, and model layer on the Gemma Scope family of SAEs. In these settings, our method reduces the cross entropy loss gap by 30% - 55% when SAEs are inserted during the forward pass. We also find that compared to end-to-end (e2e) SAEs, our approach achieves the same downstream cross entropy loss 3$\times$ to 20$\times$ faster on Gemma-2-2B and 2$\times$ to 10$\times$ faster on Llama-3.2-1B. We further show that our technique improves downstream metrics and can adapt multiple SAEs at once. Our results demonstrate that improving model interpretability is not limited to post-hoc SAE training; Pareto improvements can also be achieved by directly optimizing the model itself.

Sparse autoencoders (SAEs) are a popular method for interpreting concepts represented in large language model (LLM) activations. However, there is a lack of evidence regarding the validity of their interpretations due to the lack of a ground truth for the concepts used by an LLM, and a growing number of works have presented problems with current SAEs. One alternative source of evidence would be demonstrating that SAEs improve performance on downstream tasks beyond existing baselines. We test this by applying SAEs to the real-world task of LLM activation probing in four regimes: data scarcity, class imbalance, label noise, and covariate shift. Due to the difficulty of detecting concepts in these challenging settings, we hypothesize that SAEs’ basis of interpretable, concept-level latents should provide a useful inductive bias. However, although SAEs occasionally perform better than baselines on individual datasets, we are unable to design ensemble methods combining SAEs with baselines that consistently outperform ensemble methods solely using baselines. Additionally, although SAEs initially appear promising for identifying spurious correlations, detecting poor dataset quality, and training multi-token probes, we are able to achieve similar results with simple non-SAE baselines as well. Though we cannot discount SAEs’ utility on other tasks, our findings highlight the shortcomings of current SAEs and the need to rigorously evaluate interpretability methods on downstream tasks with strong baselines.

Scalable oversight, the process by which weaker AI systems supervise stronger ones, has been proposed as a key strategy to control future superintelligent systems. However, it is still unclear how scalable oversight itself scales. To address this gap, we propose a framework that quantifies the probability of successful oversight as a function of the capabilities of the overseer and the system being overseen. Specifically, our framework models oversight as a game between capability-mismatched players; the players have oversight-specific Elo scores that are a piecewise-linear function of their general intelligence, with two plateaus corresponding to task incompetence and task saturation. We validate our framework with a modified version of the game Nim and then apply it to four oversight games: Mafia, Debate, Backdoor Code and Wargames. For each game, we find scaling laws that approximate how domain performance depends on general AI system capability. We then build on our findings in a theoretical study of Nested Scalable Oversight (NSO), a process in which trusted models oversee untrusted stronger models, which then become the trusted models in the next step. We identify conditions under which NSO succeeds and derive numerically (and in some cases analytically) the optimal number of oversight levels to maximize the probability of oversight success. We also apply our theory to our four oversight games, where we find that NSO success rates at a general Elo gap of 400 are 13.5\% for Mafia, 51.7\% for Debate, 10.0\% for Backdoor Code, and 9.4\% for Wargames; these rates decline further when overseeing stronger systems.

We investigate the robustness of Large Language Models (LLMs) to structural interventions by deleting and swapping adjacent layers during inference. Surprisingly, models retain 72–95\% of their original top-1 prediction accuracy without any fine-tuning. We find that performance degradation is not uniform across layers: interventions to the early and final layers cause the most degradation, while the model is remarkably robust to dropping middle layers. This pattern of localized sensitivity motivates our hypothesis of four stages of inference, observed across diverse model families and sizes: (1) detokenization, where local context is integrated to lift raw token embeddings into higher-level representations; (2) feature engineering, where task- and entity-specific features are iteratively refined; (3) prediction ensembling, where hidden states are aggregated into plausible next-token predictions; and (4) residual calibration, where irrelevant features are suppressed to finalize the output distribution. Synthesizing behavioral and mechanistic evidence, we provide a hypothesis for interpreting depth-dependent computations in LLMs.

We study the problem of creating strong, yet narrow, AI systems. While recent AI progress has been driven by the training of large general-purpose foundation models, the creation of smaller models specialized for narrow domains could be valuable for both efficiency and safety. In this work, we explore two challenges involved in creating such systems, having to do with basic properties of how neural networks learn and structure their representations. The first challenge regards when it is possible to train narrow models from scratch. Through experiments on a synthetic task, we find that it is sometimes necessary to train networks on a wide distribution of data to learn certain narrow skills within that distribution. This effect arises when skills depend on each other hierarchically, and training on a broad distribution introduces a curriculum which substantially accelerates learning. The second challenge regards how to transfer particular skills from large general models into small specialized models. We find that model skills are often not perfectly localized to a particular set of prunable components. However, we find that methods based on pruning can still outperform distillation. We investigate the use of a regularization objective to align desired skills with prunable components while unlearning unnecessary skills.

Sparse autoencoders (SAEs) are designed to extract interpretable features from language models by enforcing a sparsity constraint. Ideally, training an SAE would yield latents that are both sparse and semantically meaningful. However, many SAE latents activate frequently (i.e., are *dense*), raising concerns that they may be undesirable artifacts of the training procedure. In this work, we systematically investigate the geometry, function, and origin of dense latents and show that they are not only persistent but often reflect meaningful model representations. We first demonstrate that dense latents tend to form antipodal pairs that reconstruct specific directions in the residual stream, and that ablating their subspace suppresses the emergence of new dense features in retrained SAEs---suggesting that high density features are an intrinsic property of the residual space. We then introduce a taxonomy of dense latents, identifying classes tied to position tracking, context binding, entropy regulation, letter-specific output signals, part-of-speech, and principal component reconstruction. Finally, we analyze how these features evolve across layers, revealing a shift from structural features in early layers, to semantic features in mid layers, and final to output-oriented signals in the last layers of the model. Our findings indicate that dense latents serve functional roles in language model computation and should not be dismissed as training noise.

The capabilities of large language models (LLMs) have sparked debate over whether such systems just learn an enormous collection of superficial statistics or a set of more coherent and grounded representations that reflect the real world. We find evidence for the latter by analyzing the learned representations of three spatial datasets (world, US, NYC places) and three temporal datasets (historical figures, artworks, news headlines) in the Llama-2 family of models. We discover that LLMs learn linear representations of space and time across multiple scales. These representations are robust to prompting variations and unified across different entity types (e.g. cities and landmarks). In addition, we identify individual "space neurons" and "time neurons" that reliably encode spatial and temporal coordinates. While further investigation is needed, our results suggest modern LLMs learn rich spatiotemporal representations of the real world and possess basic ingredients of a world model.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N+D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D=1 and to diffusion models when D$\to\infty$. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D$\to\infty$) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64$\times$64 datasets/LSUN Churches 256$\times$256, with median Ds. In class-conditional setting, D=2048 yields current state-of-the-art FID of 1.74 on CIFAR-10 without additional training. Furthermore, we demonstrate that models with smaller $D$ exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

We propose the Quantization Model of neural scaling laws, explaining both the observed power law dropoff of loss with model and data size, and also the sudden emergence of new capabilities with scale. We derive this model from what we call the Quantization Hypothesis, where network knowledge and skills are "quantized" into discrete chunks (quanta). We show that when quanta are learned in order of decreasing use frequency, then a power law in use frequencies explains observed power law scaling of loss. We validate this prediction on toy datasets, then study how scaling curves decompose for large language models. Using language model gradients, we automatically decompose model behavior into a diverse set of skills (quanta). We tentatively find that the frequency at which these quanta are used in the training distribution roughly follows a power law corresponding with the empirical scaling exponent for language models, a prediction of our theory.

Do neural networks, trained on well-understood algorithmic tasks, reliably rediscover known algorithms? Several recent studies, on tasks ranging from group operations to in-context linear regression, have suggested that the answer is yes. Using modular addition as a prototypical problem, we show that algorithm discovery in neural networks is sometimes more complex: small changes to model hyperparameters and initializations can induce discovery of qualitatively different algorithms from a fixed training set, and even learning of multiple different solutions in parallel. In modular addition, we specifically show that models learn a known *Clock* algorithm, a previously undescribed, less intuitive, but comprehensible procedure we term the *Pizza* algorithm, and a variety of even more complex procedures. Our results show that even simple learning problems can admit a surprising diversity of solutions, motivating the development of new tools for mechanistically characterizing the behavior of neural networks across the algorithmic phase space.

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations, whose training dynamics and dependence on training set size can be predicted by our effective theory (in a toy setting). We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a "Goldilocks zone" (including comprehension and grokking) between memorization and confusion. Compared to the comprehension phase, the grokking phase stays closer to the memorization phase, leading to delayed generalization. The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning.

We propose a new "Poisson flow" generative model~(PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the $z=0$ hyperplane in a space augmented with an additional dimension $z$, generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the $z=0$ plane transforms into a distribution on the hemisphere of radius $r$ that becomes uniform in the $r \to\infty$ limit. To learn the bijective transformation, we estimate the normalized field in the augmented space. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the (unaugmented) data manifold when the $z$ reaches zero. Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of $9.68$ and a FID score of $2.35$. It also performs on par with the state-of-the-art SDE approaches while offering $10\times $ to $20 \times$ acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method. The code is available at https://github.com/Newbeeer/poisson_flow .

BACKGROUND: Determining cell identity in volumetric images of tagged neuronal nuclei is an ongoing challenge in contemporary neuroscience. Frequently, cell identity is determined by aligning and matching tags to an "atlas" of labeled neuronal positions and other identifying characteristics. Previous analyses of such C. elegans datasets have been hampered by the limited accuracy of such atlases, especially for neurons present in the ventral nerve cord, and also by time-consuming manual elements of the alignment process. RESULTS: We present a novel automated alignment method for sparse and incomplete point clouds of the sort resulting from typical C. elegans fluorescence microscopy datasets. This method involves a tunable learning parameter and a kernel that enforces biologically realistic deformation. We also present a pipeline for creating alignment atlases from datasets of the recently developed NeuroPAL transgene. In combination, these advances allow us to label neurons in volumetric images with confidence much higher than previous methods. CONCLUSIONS: We release, to the best of our knowledge, the most complete full-body C. elegans 3D positional neuron atlas, incorporating positional variability derived from at least 7 animals per neuron, for the purposes of cell-type identity prediction for myriad applications (e.g., imaging neuronal activity, gene expression, and cell-fate).

We present an improved method for symbolic regression that seeks to fit data to formulas that are Pareto-optimal, in the sense of having the best accuracy for a given complexity. It improves on the previous state-of-the-art by typically being orders of magnitude more robust toward noise and bad data, and also by discovering many formulas that stumped previous methods. We develop a method for discovering generalized symmetries (arbitrary modularity in the computational graph of a formula) from gradient properties of a neural network fit. We use normalizing flows to generalize our symbolic regression method to probability distributions from which we only have samples, and employ statistical hypothesis testing to accelerate robust brute-force search.

We present a novel recurrent neural network (RNN)-based model that combines the remembering ability of unitary evolution RNNs with the ability of gated RNNs to effectively forget redundant or irrelevant information in its memory. We achieve this by extending restricted orthogonal evolution RNNs with a gating mechanism similar to gated recurrent unit RNNs with a reset gate and an update gate. Our model is able to outperform long short-term memory, gated recurrent units, and vanilla unitary or orthogonal RNNs on several long-term-dependency benchmark tasks. We empirically show that both orthogonal and unitary RNNs lack the ability to forget. This ability plays an important role in RNNs. We provide competitive results along with an analysis of our model on many natural sequential tasks, including question answering, speech spectrum prediction, character-level language modeling, and synthetic tasks that involve long-term dependencies such as algorithmic, denoising, and copying tasks.

Using unitary (instead of general) matrices in artificial neural networks (ANNs) is a promising way to solve the gradient explosion/vanishing problem, as well as to enable ANNs to learn long-term correlations in the data. This approach appears particularly promising for Recurrent Neural Networks (RNNs). In this work, we present a new architecture for implementing an Efficient Unitary Neural Network (EUNNs); its main advantages can be summarized as follows. Firstly, the representation capacity of the unitary space in an EUNN is fully tunable, ranging from a subspace of SU(N) to the entire unitary space. Secondly, the computational complexity for training an EUNN is merely $\mathcal{O}(1)$ per parameter. Finally, we test the performance of EUNNs on the standard copying task, the pixel-permuted MNIST digit recognition benchmark as well as the Speech Prediction Test (TIMIT). We find that our architecture significantly outperforms both other state-of-the-art unitary RNNs and the LSTM architecture, in terms of the final performance and/or the wall-clock training speed. EUNNs are thus promising alternatives to RNNs and LSTMs for a wide variety of applications.

Although there is growing interest in measuring integrated information in computational and cognitive systems, current methods for doing so in practice are computationally unfeasible. Existing and novel integration measures are investigated and classified by various desirable properties. A simple taxonomy of Φ-measures is presented where they are each characterized by their choice of factorization method (5 options), choice of probability distributions to compare (3 × 4 options) and choice of measure for comparing probability distributions (7 options). When requiring the Φ-measures to satisfy a minimum of attractive properties, these hundreds of options reduce to a mere handful, some of which turn out to be identical. Useful exact and approximate formulas are derived that can be applied to real-world data from laboratory experiments without posing unreasonable computational demands.