This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data. Theoretical error bounds are provided when the tangent spaces of the manifold satisfy certain incoherence conditions. We also provide a near optimal choice of the tuning parameters for the proposed optimization formulation with the help of a new curvature estimation method. The efficacy of our method is demonstrated on both synthetic and real datasets.

ABSTRACT Video sequences often contain structured noise and background artifacts that obscure dynamic content, posing challenges for accurate analysis and restoration. Robust principal component methods address this by decomposing data into low-rank and sparse components. Still, the sparsity assumption often fails to capture the rich variability present in real video data. To overcome this limitation, a hybrid framework that integrates low-rank temporal modeling with diffusion posterior sampling is proposed. The proposed method, Nuclear Diffusion, is evaluated on a real-world medical imaging problem, namely cardiac ultrasound dehazing, and demonstrates improved dehazing performance compared to traditional RPCA concerning contrast enhancement (gCNR) and signal preservation (KS statistic).

Recent significant advancements in Large Language Models (LLMs) have greatly propelled the development of Role-Playing Conversational Agents (RPCAs). These systems aim to create immersive user experiences through consistent persona adoption. However, current RPCA research faces dual limitations. First, existing work predominantly focuses on the textual modality, entirely overlooking critical paralinguistic features including intonation, prosody, and rhythm in speech, which are essential for conveying character emotions and shaping vivid identities. Second, the speech-based role-playing domain suffers from a long-standing lack of standardized evaluation benchmarks. Most current spoken dialogue datasets target only fundamental capability assessments, featuring thinly sketched or ill-defined character profiles. Consequently, they fail to effectively quantify model performance on core competencies like long-term persona consistency. To address this critical gap, we introduce VoxRole, the first comprehensive benchmark specifically designed for the evaluation of speech-based RPCAs. The benchmark comprises 13335 multi-turn dialogues, totaling 65.6 hours of speech from 1228 unique characters across 261 movies. To construct this resource, we propose a novel two-stage automated pipeline that first aligns movie audio with scripts and subsequently employs an LLM to systematically build multi-dimensional profiles for each character. Leveraging VoxRole, we conduct a multi-dimensional evaluation of contemporary spoken dialogue models, revealing crucial insights into their respective strengths and limitations in maintaining persona consistency.

Role-playing conversational agents (RPCAs) face persistent challenges in maintaining role consistency. To address this, we propose RAIDEN-R1, a novel reinforcement learning framework that integrates Verifiable Role-Awareness Reward (VRAR). The method introduces both singular and multi-term mining strategies to generate quantifiable rewards by assessing role-specific keys. Additionally, we construct a high-quality, role-aware Chain-of-Thought dataset through multi-LLM collaboration, and implement experiments to enhance reasoning coherence. Experiments on the RAIDEN benchmark demonstrate RAIDEN-R1's superiority: our 14B-GRPO model achieves 88.04% and 88.65% accuracy on Script-Based Knowledge and Conversation Memory metrics, respectively, outperforming baseline models while maintaining robustness. Case analyses further reveal the model's enhanced ability to resolve conflicting contextual cues and sustain first-person narrative consistency. This work bridges the non-quantifiability gap in RPCA training and provides insights into role-aware reasoning patterns, advancing the development of RPCAs.

Low-rank and sparse composite approximation is a natural idea to compress Large Language Models (LLMs). However, such an idea faces two primary challenges that adversely affect the performance of existing methods. The first challenge relates to the interaction and cooperation between low-rank and sparse matrices, while the second involves determining weight allocation across different layers, as redundancy varies considerably among them. To address these challenges, we propose a novel two-stage LLM compression method with the capability of global rank and sparsity optimization. It is noteworthy that the overall optimization space is vast, making comprehensive optimization computationally prohibitive. Therefore, to reduce the optimization space, our first stage utilizes robust principal component analysis to decompose the weight matrices of LLMs into low-rank and sparse components, which span the low dimensional and sparse spaces containing the resultant low-rank and sparse matrices, respectively. In the second stage, we propose a probabilistic global optimization technique to jointly identify the low-rank and sparse structures within the above two spaces. The appealing feature of our approach is its ability to automatically detect the redundancy across different layers and to manage the interaction between the sparse and low-rank components. Extensive experimental results indicate that our method significantly surpasses state-of-the-art techniques for sparsification and composite approximation.

The recently established RPCA [4] method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades.

The robust principal component analysis (RPCA) problem seeks to separate low-rank trends from sparse outlierswithin a data matrix, that is, to approximate a n\times d matrix D as the sum of a low-rank matrix L and a sparse matrix S .We examine the robust principal component analysis (RPCA) problem under data compression, wherethe data Y is approximately given by (L S)\cdot C, that is, a low-rank sparse data matrix that has been compressed to size n\times m (with m substantially smaller than the original dimension d) via multiplication witha compression matrix C . We give a convex program for recovering the sparse component S along with the compressed low-rank component L\cdot C, along with upper bounds on the error of this reconstructionthat scales naturally with the compression dimension m and coincides with existing results for the uncompressedsetting m d . Our results can also handle error introduced through additive noise or through missing data.The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration acrossa network of sensors, to test its performance in practice.

Robust Principal Component Analysis (RPCA) is a fundamental technique for decomposing data into low-rank and sparse components, which plays a critical role for applications such as image processing and anomaly detection. Traditional RPCA methods commonly use $\ell_1$ norm regularization to enforce sparsity, but this approach can introduce bias and result in suboptimal estimates, particularly in the presence of significant noise or outliers. Non-convex regularization methods have been proposed to mitigate these challenges, but they tend to be complex to optimize and sensitive to initial conditions, leading to potential instability in solutions. To overcome these challenges, in this paper, we propose a novel RPCA model that integrates adaptive weighted least squares (AWLS) and low-rank matrix factorization (LRMF). The model employs a {self-attention-inspired} mechanism in its weight update process, allowing the weight matrix to dynamically adjust and emphasize significant components during each iteration. By employing a weighted F-norm for the sparse component, our method effectively reduces bias while simplifying the computational process compared to traditional $\ell_1$-norm-based methods. We use an alternating minimization algorithm, where each subproblem has an explicit solution, thereby improving computational efficiency. Despite its simplicity, numerical experiments demonstrate that our method outperforms existing non-convex regularization approaches, offering superior performance and stability, as well as enhanced accuracy and robustness in practical applications.

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data.

The recently established RPCA [4] method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades.