Understanding Task Vectors in In-Context Learning: Emergence, Functionality, and Limitations

Task vectors offer a compelling mechanism for accelerating inference in in-context learning (ICL) by distilling task-specific information into a single, reusable representation. Despite their empirical success, the underlying principles governing their emergence and functionality remain unclear. This work proposes the Linear Combination Conjecture, positing that task vectors act as single in-context demonstrations formed through linear combinations of the original ones. We provide both theoretical and empirical support for this conjecture. First, we show that task vectors naturally emerge in linear transformers trained on triplet-formatted prompts through loss landscape analysis. Next, we predict the failure of task vectors on representing high-rank mappings and confirm this on practical LLMs. Our findings are further validated through saliency analyses and parameter visualization, suggesting an enhancement of task vectors by injecting multiple ones into few-shot prompts. Together, our results advance the understanding of task vectors and shed light on the mechanisms underlying ICL in transformer-based models.

On the Convergence of Gradient Descent on Learning Transformers with Residual Connections

Transformer models have emerged as fundamental tools across various scientific and engineering disciplines, owing to their outstanding performance in diverse applications. Despite this empirical success, the theoretical foundations of Transformers remain relatively underdeveloped, particularly in understanding their training dynamics. Existing research predominantly examines isolated components--such as self-attention mechanisms and feedforward networks--without thoroughly investigating the interdependencies between these components, especially when residual connections are present. In this paper, we aim to bridge this gap by analyzing the convergence behavior of a structurally complete yet single-layer Transformer, comprising self-attention, a feedforward network, and residual connections. We demonstrate that, under appropriate initialization, gradient descent exhibits a linear convergence rate, where the convergence speed is determined by the minimum and maximum singular values of the output matrix from the attention layer. Moreover, our analysis reveals that residual connections serve to ameliorate the ill-conditioning of this output matrix, an issue stemming from the low-rank structure imposed by the softmax operation, thereby promoting enhanced optimization stability. We also extend our theoretical findings to a multi-layer Transformer architecture, confirming the linear convergence rate of gradient descent under suitable initialization. Empirical results corroborate our theoretical insights, illustrating the beneficial role of residual connections in promoting convergence stability.

Analyzing Fine-Grained Alignment and Enhancing Vision Understanding in Multimodal Language Models

Achieving better alignment between vision embeddings and Large Language Models (LLMs) is crucial for enhancing the abilities of Multimodal LLMs (MLLMs), particularly for recent models that rely on powerful pretrained vision encoders and LLMs. A common approach to connect the pretrained vision encoder and LLM is through a projector applied after the vision encoder. However, the projector is often trained to enable the LLM to generate captions, and hence the mechanism by which LLMs understand each vision token remains unclear. In this work, we first investigate the role of the projector in compressing vision embeddings and aligning them with word embeddings. We show that the projector significantly compresses visual information, removing redundant details while preserving essential elements necessary for the LLM to understand visual content. We then examine patch-level alignment -- the alignment between each vision patch and its corresponding semantic words -- and propose a *multi-semantic alignment hypothesis*. Our analysis indicates that the projector trained by caption loss improves patch-level alignment but only to a limited extent, resulting in weak and coarse alignment. To address this issue, we propose *patch-aligned training* to efficiently enhance patch-level alignment. Our experiments show that patch-aligned training (1) achieves stronger compression capability and improved patch-level alignment, enabling the MLLM to generate higher-quality captions, (2) improves the MLLM's performance by 16% on referring expression grounding tasks, 4% on question-answering tasks, and 3% on modern instruction-following benchmarks when using the same supervised fine-tuning (SFT) setting. The proposed method can be easily extended to other multimodal models.

From Compression to Expansion: A Layerwise Analysis of In-Context Learning

In-context learning (ICL) enables large language models (LLMs) to adapt to new tasks without weight updates by learning from demonstration sequences. While ICL shows strong empirical performance, its internal representational mechanisms are not yet well understood. In this work, we conduct a statistical geometric analysis of ICL representations to investigate how task-specific information is captured across layers. Our analysis reveals an intriguing phenomenon, which we term *Layerwise Compression-Expansion*: early layers progressively produce compact and discriminative representations that encode task information from the input demonstrations, while later layers expand these representations to incorporate the query and generate the prediction. This phenomenon is observed consistently across diverse tasks and a range of contemporary LLM architectures. We demonstrate that it has important implications for ICL performance -- improving with model size and the number of demonstrations -- and for robustness in the presence of noisy examples. To further understand the effect of the compact task representation, we propose a bias-variance decomposition and provide a theoretical analysis showing how attention mechanisms contribute to reducing both variance and bias, thereby enhancing performance as the number of demonstrations increases. Our findings reveal an intriguing layerwise dynamic in ICL, highlight how structured representations emerge within LLMs, and showcase that analyzing internal representations can facilitate a deeper understanding of model behavior.

Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling

This work addresses the critical question of why and when diffusion models, despite being designed for generative tasks, can excel at learning high-quality representations in a self-supervised manner. To address this, we develop a mathematical framework based on a low-dimensional data model and posterior estimation, revealing a fundamental trade-off between generation and representation quality near the final stage of image generation. Our analysis explains the unimodal representation dynamics across noise scales, mainly driven by the interplay between data denoising and class specification. Building on these insights, we propose an ensemble method that aggregates features across noise levels, significantly improving both clean performance and robustness under label noise. Extensive experiments on both synthetic and real-world datasets validate our findings.

Optimal Error Analysis of Channel Estimation for IRS-assisted MIMO Systems

As intelligent reflecting surface (IRS) has emerged as a new and promising technology capable of configuring the wireless environment favorably, channel estimation for IRS-assisted multiple-input multiple-output (MIMO) systems has garnered extensive attention in recent years. While various algorithms have been proposed to address this challenge, there is a lack of rigorous theoretical error analysis. This paper aims to address this gap by providing theoretical guarantees in terms of stable recovery of channel matrices for noisy measurements. We begin by establishing the equivalence between IRS-assisted MIMO systems and a compact tensor train (TT)-based tensor-on-tensor (ToT) regression. Building on this equivalence, we then investigate the restricted isometry property (RIP) for complex-valued subgaussian measurements. Our analysis reveals that successful recovery hinges on the relationship between the number of user terminals (in the uplink scenario) or base stations (in the downlink scenario) and the number of time slots during which channel matrices remain invariant. Utilizing the RIP condition, we analyze the theoretical recovery error for the solution to a constrained least-squares optimization problem, including upper error bound and minimax lower bound, demonstrating that the error decreases inversely with the number of time slots and increases proportionally with the number of unknown elements in the channel matrices. In addition, we extend our error analysis to two more specialized IRS-assisted MIMO systems, incorporating low-rank channel matrices or an unknown IRS. Furthermore, we explore a multi-hop IRS scheme and analyze the corresponding recovery errors. Finally, we introduce and implement two nonconvex optimization algorithms--alternating least squares and alternating gradient descent--to validate our conclusions through simulations.

Analyzing and Improving Model Collapse in Rectified Flow Models

Generative models aim to produce synthetic data indistinguishable from real distributions, but iterative training on self-generated data can lead to \emph{model collapse (MC)}, where performance degrades over time. In this work, we provide the first theoretical analysis of MC in Rectified Flow by framing it within the context of Denoising Autoencoders (DAEs). We show that when DAE models are trained on recursively generated synthetic data with small noise variance, they suffer from MC with progressive diminishing generation quality. To address this MC issue, we propose methods that strategically incorporate real data into the training process, even when direct noise-image pairs are unavailable. Our proposed techniques, including Reverse Collapse-Avoiding (RCA) Reflow and Online Collapse-Avoiding Reflow (OCAR), effectively prevent MC while maintaining the efficiency benefits of Rectified Flow. Extensive experiments on standard image datasets demonstrate that our methods not only mitigate MC but also improve sampling efficiency, leading to higher-quality image generation with fewer sampling steps.

Optimal Allocation of Pauli Measurements for Low-rank Quantum State Tomography

The process of reconstructing quantum states from experimental measurements, accomplished through quantum state tomography (QST), plays a crucial role in verifying and benchmarking quantum devices. A key challenge of QST is to find out how the accuracy of the reconstruction depends on the number of state copies used in the measurements. When multiple measurement settings are used, the total number of state copies is determined by multiplying the number of measurement settings with the number of repeated measurements for each setting. Due to statistical noise intrinsic to quantum measurements, a large number of repeated measurements is often used in practice. However, recent studies have shown that even with single-sample measurements--where only one measurement sample is obtained for each measurement setting--high accuracy QST can still be achieved with a sufficiently large number of different measurement settings. In this paper, we establish a theoretical understanding of the trade-off between the number of measurement settings and the number of repeated measurements per setting in QST. Our focus is primarily on low-rank density matrix recovery using Pauli measurements. We delve into the global landscape underlying the low-rank QST problem and demonstrate that the joint consideration of measurement settings and repeated measurements ensures a bounded recovery error for all second-order critical points, to which optimization algorithms tend to converge. This finding suggests the advantage of minimizing the number of repeated measurements per setting when the total number of state copies is held fixed. Additionally, we prove that the Wirtinger gradient descent algorithm can converge to the region of second-order critical points with a linear convergence rate. We have also performed numerical experiments to support our theoretical findings.

Captions Speak Louder than Images (CASLIE): Generalizing Foundation Models for E-commerce from High-quality Multimodal Instruction Data

Leveraging multimodal data to drive breakthroughs in e-commerce applications through Multimodal Foundation Models (MFMs) is gaining increasing attention from the research community. However, there are significant challenges that hinder the optimal use of multimodal e-commerce data by foundation models: (1) the scarcity of large-scale, high-quality multimodal benchmark datasets; and (2) the lack of effective multimodal information integration methods. To address these challenges, in this paper, we introduce MMECInstruct, the first-ever, large-scale, and high-quality multimodal instruction dataset for e-commerce. We also develop CASLIE, a simple, lightweight, yet effective framework for integrating multimodal information for e-commerce. Leveraging MMECInstruct, we fine-tune a series of e-commerce MFMs within CASLIE, denoted as CASLIE models. Our comprehensive evaluation demonstrates that CASLIE models substantially outperform 5 categories of advanced baseline models in the in-domain evaluation. Moreover, CASLIE models show strong generalizability to out-of-domain settings. MMECInstruct and CASLIE models are publicly accessible through https://ninglab.github.io/CASLIE/.

Robust Low-rank Tensor Train Recovery

Tensor train (TT) decomposition represents an $N$-order tensor using $O(N)$ matrices (i.e., factors) of small dimensions, achieved through products among these factors. Due to its compact representation, TT decomposition has found wide applications, including various tensor recovery problems in signal processing and quantum information. In this paper, we study the problem of reconstructing a TT format tensor from measurements that are contaminated by outliers with arbitrary values. Given the vulnerability of smooth formulations to corruptions, we use an $\ell_1$ loss function to enhance robustness against outliers. We first establish the $\ell_1/\ell_2$-restricted isometry property (RIP) for Gaussian measurement operators, demonstrating that the information in the TT format tensor can be preserved using a number of measurements that grows linearly with $N$. We also prove the sharpness property for the $\ell_1$ loss function optimized over TT format tensors. Building on the $\ell_1/\ell_2$-RIP and sharpness property, we then propose two complementary methods to recover the TT format tensor from the corrupted measurements: the projected subgradient method (PSubGM), which optimizes over the entire tensor, and the factorized Riemannian subgradient method (FRSubGM), which optimizes directly over the factors. Compared to PSubGM, the factorized approach FRSubGM significantly reduces the memory cost at the expense of a slightly slower convergence rate. Nevertheless, we show that both methods, with diminishing step sizes, converge linearly to the ground-truth tensor given an appropriate initialization, which can be obtained by a truncated spectral method.