Large language models (LLMs) can be adapted to new tasks using parameter-efficient fine-tuning (PEFT) methods that modify only a small number of trainable parameters, often through low-rank updates. In this work, we adopt a quantum-information-inspired perspective to understand their effectiveness. From this perspective, low-rank parameterizations naturally correspond to low-dimensional Matrix Product States (MPS) representations, which enable entanglement-based characterizations of parameter structure. Thereby, we term and measure "Artificial Entanglement", defined as the entanglement entropy of the parameters in artificial neural networks (in particular the LLMs). We first study the representative low-rank adaptation (LoRA) PEFT method, alongside full fine-tuning (FFT), using LLaMA models at the 1B and 8B scales trained on the Tulu3 and OpenThoughts3 datasets, and uncover: (i) Internal artificial entanglement in the updates of query and value projection matrices in LoRA follows a volume law with a central suppression (termed as the "Entanglement Valley"), which is sensitive to hyper-parameters and is distinct from that in FFT; (ii) External artificial entanglement in attention matrices, corresponding to token-token correlations in representation space, follows an area law with logarithmic corrections and remains robust to LoRA hyper-parameters and training steps. Drawing a parallel to the No-Hair Theorem in black hole physics, we propose that although LoRA and FFT induce distinct internal entanglement signatures, such differences do not manifest in the attention outputs, suggesting a "no-hair" property that results in the effectiveness of low rank updates. We further provide theoretical support based on random matrix theory, and extend our analysis to an MPS Adaptation PEFT method, which exhibits qualitatively similar behaviors.

Private data analysis suffers a costly curse of dimensionality.

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the \emph{first} convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD.

We study the theory of random bipartite graph whose adjacency matrix is generated according to a connectivity matrix $M$. We consider the bipartite graph to be sparse, i.e., the entries of $M$ are upper bounded by certain sparsity parameter. We show that the performance of estimating the connectivity matrix $M$ depends on the sparsity of the graph. We focus on two measurement of performance of estimation: the error of estimating $M$ and the error of estimating the column space of $M$. In the first case, we consider the operator norm and Frobenius norm of the difference between the estimation and the true connectivity matrix. In the second case, the performance will be measured by the difference between the estimated projection matrix and the true projection matrix in operator norm and Frobenius norm. We will show that the estimators we propose achieve the minimax optimal rate.

How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question.Recently, there has been a surge of interest in reducing LP sizes using *random projections*, which can accelerate solving LPs independently of improving LP solvers. This paper explores a new direction of *data-driven projections*, which use projection matrices learned from data instead of random projection matrices.Given training data of $n$-dimensional LPs, we learn an $n\times k$ projection matrix with $n > k$. When addressing a future LP instance, we reduce its dimensionality from $n$ to $k$ via the learned projection matrix, solve the resulting LP to obtain a $k$-dimensional solution, and apply the learned matrix to it to recover an $n$-dimensional solution.On the theoretical side, a natural question is: how much data is sufficient to ensure the quality of recovered solutions? We address this question based on the framework of *data-driven algorithm design*, which connects the amount of data sufficient for establishing generalization bounds to the *pseudo-dimension* of performance metrics. We obtain an $\tilde{\mathrm{O}}(nk^2)$ upper bound on the pseudo-dimension, where $\tilde{\mathrm{O}}$ compresses logarithmic factors. We also provide an $\Omega(nk)$ lower bound, implying our result is tight up to an $\tilde{\mathrm{O}}(k)$ factor. On the practical side, we explore two simple methods for learning projection matrices: PCA-and gradient-based methods. While the former is relatively efficient, the latter can sometimes achieve better solution quality. Experiments demonstrate that learning projection matrices from data is indeed beneficial: it leads to significantly higher solution quality than the existing random projection while greatly reducing the time for solving LPs.

Vision-Language Models (VLMs) inherit significant social biases from their training data, notably in gender representation. Current fairness interventions often adopt a difference-unaware perspective that enforces uniform treatment across demographic groups. These approaches, however, fail to distinguish between contexts where neutrality is required and those where group-specific attributes are legitimate and must be preserved. Building upon recent advances in difference-aware fairness for text-only models, we extend this concept to the multimodal domain and formalize the problem of difference-aware gender fairness for image captioning and text-to-image generation. We advocate for selective debiasing, which aims to mitigate unwanted bias in neutral contexts while preserving valid distinctions in explicit ones. To achieve this, we propose BioPro (Bias Orthogonal Projection), an entirely training-free framework. BioPro identifies a low-dimensional gender-variation subspace through counterfactual embeddings and applies projection to selectively neutralize gender-related information. Experiments show that BioPro effectively reduces gender bias in neutral cases while maintaining gender faithfulness in explicit ones, thus providing a promising direction toward achieving selective fairness in VLMs. Beyond gender bias, we further demonstrate that BioPro can effectively generalize to continuous bias variables, such as scene brightness, highlighting its broader applicability.

Large Language Models (LLMs) face a significant bottleneck during autoregressive inference due to the massive memory footprint of the Key-Value (KV) cache. Existing compression techniques like token eviction, quantization, or other low-rank methods often risk information loss, have fixed limits, or introduce significant computational overhead from explicit decompression steps. In this work, we introduce SWAN, a novel, fine-tuning-free framework that eliminates this overhead. Our method uses an offline orthogonal matrix to rotate and prune the KV-cache, which is then used directly in the attention computation without any reconstruction. Our extensive experiments demonstrate that SWAN, augmented with a small dense buffer, offers a robust trade-off, maintaining performance close to the uncompressed baseline even at aggressive 50-60% memory savings per-token on KV-cache. A key advantage is its runtime-tunable compression level, allowing operators to dynamically adjust the memory footprint, a flexibility absent in methods requiring fixed offline configurations. This combination of a decompression-free design, high performance under compression, and adaptability makes SWAN a practical and efficient solution for serving LLMs with long contexts.

Formally, we define differential privacy as follows.

Feature hashing, also known as the hashing trick, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms.