Multi-view clustering (MVC) seeks to uncover the intrinsic group structures embedded in multi-view data, which has attracted considerable attention in recent years. Existing approaches predominantly concentrate on incorporating suitable model priors to capture consistency across views. However, these explicit constraints often fail to hold in scenarios involving significant modal differences between views or the presence of noise, thereby limiting the efficacy of these methods in more complex contexts. To address these issues, this paper introduces BONE, a lightweight and interpretable MVC framework that Bridges Optimization and Neural networks for Efficient MVC. By leveraging learnable parameters to extract high-level features from low-level features derived through classical optimization, BONE integrates the consistency information across views without the need for explicit prior constraints, while eliminating the necessity for pre-training or post-processing. Extensive experiments show that BONE achieves clustering performance comparable to or even better than existing deep MVC methods, while using only 1% of the parameters, offering a new perspective for designing efficient MVC algorithms.

Sparse Principal Component Analysis (SPCA) is a powerful technique for dimensionality reduction and feature extraction in high-dimensional data, with applications spanning various fields such as computer vision, pattern recognition, and data mining. However, the computational intensity of SPCA presents a significant challenge, necessitating the development of efficient and robust algorithms. In this paper, we shed light on the SPCA problem and uncover intriguing structures that enable us to design an efficient algorithm, which we have named SPCA_ACC. Firstly, we identify a separable structure in this problem, which prompts us to draw on the Variable Projection (VP) strategy and generalize it to separable nonlinear problem in Stiefel manifold. This strategy projects out part of the parameters to obtain a reduced problems, allowing the SPCA_ACC algorithm to optimize in a lower-dimensional parameter space. Secondly, we resolve the coupling between different parameters of the SPCA problem in the optimization process on a fixed coordinate-sparsity manifold, which opens the way to the use of second-order Riemannian accelerated VP strategy. Moreover, we systematically analyze the advantages of using VP to solve the SPCA problem from a theoretical perspective, and confirm the local quadratic convergence of our algorithm. Numerical experiments on datasets of different sizes and types demonstrate that our method achieves rapid convergence and significantly reduces computational costs.

We propose an online learning algorithm tailored for a class of machine learning models within a separable stochastic approximation framework. The central idea of our approach is to exploit the inherent separability in many models, recognizing that certain parameters are easier to optimize than others. This paper focuses on models where some parameters exhibit linear characteristics, which are common in machine learning applications. In our proposed algorithm, the linear parameters are updated using the recursive least squares (RLS) algorithm, akin to a stochastic Newton method. Subsequently, based on these updated linear parameters, the nonlinear parameters are adjusted using the stochastic gradient method (SGD). This dual-update mechanism can be viewed as a stochastic approximation variant of block coordinate gradient descent, where one subset of parameters is optimized using a second-order method while the other is handled with a first-order approach. We establish the global convergence of our online algorithm for non-convex cases in terms of the expected violation of first-order optimality conditions. Numerical experiments demonstrate that our method achieves significantly faster initial convergence and produces more robust performance compared to other popular learning algorithms. Additionally, our algorithm exhibits reduced sensitivity to learning rates and outperforms the recently proposedslimTrainalgorithm (Newman et al. 2022). For validation, the code has been made available on GitHub.