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In this paper, we study the gradient descent dynamics for jointly training both layers of a one- hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse prob- lems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient de- scent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progres- sively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local re- finement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory- level control arguments for the end-to-end dynamics. In addition, we establish novel uniform con- centration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations. 

Modern large language models (LLMs) excel at tasks that require storing and retrieving knowledge, such as factual recall and question answering. Transformers are central to this capability because they can encode information during training and retrieve it at inference. Existing theoretical analyses typically study transformers under idealized assumptions such as infinite data or orthogonal embeddings. In realistic settings, however, models are trained on finite datasets with non-orthogonal (random) embeddings. We address this gap by analyzing a single-layer transformer with random embeddings trained with (empirical) gradient descent on a simple token-retrieval task, where the model must identify an informative token within a length-L sequence and learn a one-to-one mapping from tokens to labels. Our analysis tracks the ``early phase'' of gradient descent and yields explicit formulas for the model's storage capacity -- revealing a multiplicative dependence between sample size N, embedding dimension d, and sequence length L. We validate these scalings numerically and further complement them with a lower bound for the underlying statistical problem, demonstrating that this multiplicative scaling is intrinsic under non-orthogonal embeddings. 

Large Language Models (LLMs) typically represent numbers using multiple tokens, which requires the model to aggregate these tokens to interpret numerical values. This fragmentation makes both training and inference less efficient and adversely affects the model's performance on number-related tasks. Inspired by the observation that pre-trained LLMs internally learn Fourier-like features for number tokens, we propose Fourier Number Embedding (FoNE), a novel method that directly maps numbers into the embedding space with their Fourier features. FoNE encodes each number as a single token with only two embedding dimensions per digit, effectively capturing numerical values without fragmentation. This compact representation accelerates both training and inference. Compared to traditional subword and digit-wise embeddings, FoNE not only reduces computational overhead but also achieves higher accuracy across various numerical tasks including addition, subtraction and multiplication. On 6-digit decimal addition, FoNE requires 64× less data to achieve 99% accuracy than subword and digit-wise embeddings while using 3× and 6× fewer tokens per number, respectively. Furthermore, FoNE is the only method that yields 100% accuracy on over 100,000 test examples for addition, subtraction, and multiplication. 

Diffusion models have become the go-to method for text-to-image generation, producing high-quality images from pure noise. However, the inner workings of diffusion models is still largely a mystery due to their black-box nature and complex, multi-step generation process. Mechanistic interpretability techniques, such as Sparse Autoencoders (SAEs), have been successful in understanding and steering the behavior of large language models at scale. However, the great potential of SAEs has not yet been applied toward gaining insight into the intricate generative process of diffusion models. In this work, we leverage the SAE framework to probe the inner workings of a popular text-to-image diffusion model, and uncover a variety of human-interpretable concepts in its activations. Interestingly, we find that even before the first reverse diffusion step is completed, the final composition of the scene can be predicted surprisingly well by looking at the spatial distribution of activated concepts. Moreover, going beyond correlational analysis, we design intervention techniques aimed at manipulating image composition and style, and demonstrate that (1) in early stages of diffusion image composition can be effectively controlled, (2) in the middle stages image composition is finalized, however stylistic interventions are effective, and (3) in the final stages only minor textural details are subject to change. 

Mental visualization, the ability to construct and manipulate visual representations internally, is a core component of human cognition and plays a vital role in tasks involving reasoning, prediction, and abstraction. Despite the rapid progress of Multimodal Large Language Models (MLLMs), current benchmarks primarily assess passive visual perception, offering limited insight into the more active capability of internally constructing visual patterns to support problem solving. Yet mental visualization is a critical cognitive skill in humans, supporting abilities such as spatial navigation, predicting physical trajectories, and solving complex visual problems through imaginative simulation. To bridge this gap, we introduce Hyperphantasia, a synthetic benchmark designed to evaluate the mental visualization abilities of MLLMs through four carefully constructed puzzles. Each puzzle is procedurally generated and presented at three difficulty levels, enabling controlled analysis of model performance across increasing complexity. Our comprehensive evaluation of state-of-the-art models reveals a substantial gap between the performance of humans and MLLMs. Additionally, we explore the potential of reinforcement learning to improve visual simulation capabilities. Our findings suggest that while some models exhibit partial competence in recognizing visual patterns, robust mental visualization remains an open challenge for current MLLMs. 

An increasingly popular machine learning paradigm is to pretrain a neural network (NN) on many tasks offline, then adapt it to downstream tasks, often by re-training only the last linear layer of the network. This approach yields strong downstream performance in a variety of contexts, demonstrating that multitask pretraining leads to effective feature learning. Although several recent theoretical studies have shown that shallow NNs learn meaningful features when either (i) they are trained on a single task or (ii) they are linear, very little is known about the closer-to-practice case of nonlinear NNs trained on multiple tasks. In this work, we present the first results proving that feature learning occurs during training with a nonlinear model on multiple tasks. Our key insight is that multi-task pretraining induces a pseudo-contrastive loss that favors representations that align points that typically have the same label across tasks. Using this observation, we show that when the tasks are binary classification tasks with labels depending on the projection of the data onto an 𝑟-dimensional subspace within the 𝑑 ≫𝑟-dimensional input space, a simple gradient-based multitask learning algorithm on a two-layer ReLU NN recovers this projection, allowing for generalization to downstream tasks with sample and neuron complexity independent of 𝑑. In contrast, we show that with high probability over the draw of a single task, training on this single task cannot guarantee to learn all 𝑟 ground-truth features.