# Distributed Consensus for Global Matrix Formation in the Principal Component Pursuit Scenario
## 1 Introduction  
Consensus algorithms [1] have been studied as powerful tools for distributed networks [2] in multiple applications [3] such as control [4], estimation [5] and the internet of vehicles [6]. They are based on a graph that describes the interconnections of the agents in the network, and the goal of the network is to achieve agreement upon some data, policy or action. Different approaches have been developed, including static, directed [7], undirected--meaning that the communication between two agents over an edge of the graph is bidirectional--and time-varying graphs [8]. Recent developments in consensus algorithms emphasize their adaptability and scalability in dynamic networks, facilitating robust, distributed decision-making processes [9, 10]. These advancements have led to enhanced methodologies for reaching a consensus in networks with complex interconnections and varying topologies, demonstrating their effectiveness in a broader range of applications beyond traditional settings. Moreover, the utility of PCA extends into the realm of big data and high-dimensional data analysis, where it plays a crucial role in dimensionality reduction, feature extraction, and data compression [11]. The ability to distill essential features from large datasets while preserving the integrity of the underlying structure has profound implications for computational efficiency and the interpretability of data in machine learning and data science. There are many applications where PCA is relevant; some examples are video surveillance, face recognition [12], latent semantic indexing [13], ranking and collaborative filtering to anticipate states, image analysis including fluid dynamics [14], machine learning and food identification [15], among others. Examples of PCA applied to face image processing and fluid flows are shown in Figures 1 and 2. The video processing is similar, where each frame is re-shaped as a column vector, and they are stacked together in the new matrix, as shown in Figure 2.  
Integrating consensus algorithms with PCA in a distributed setting is a novel approach to tackling the challenges of large-scale data analysis. By leveraging the strengths of consensus algorithms to achieve a unified view from multiple distributed agents, combined with the PCA's capacity for dimensionality reduction, this methodology offers a compelling solution for decentralized data processing and analysis. The consensus-based PCA optimization framework enables distributed agents to collaboratively compute PCA decompositions, thereby overcoming limitations related to data privacy, storage, and computational bottlenecks [17; 18; 19; 20; 21; 22]. In this article, an intersection of consensus algorithms and the optimization problem of principal component analysis (PCA) [23; 24] is addressed. A consensus is reached along with a particular type of relaxation called convexification. It allows the network to solve the optimization with a low computational cost. The PCA problem addresses the following case. Suppose it is given a large data matrix \(M\in\mathcal{R}^{n\times n}\) with missing and noisy data, such that this matrix can be decomposed into a low-rank matrix \(L\in\mathcal{R}^{n\times n}\) and a sparse one \(S\in\mathcal{R}^{n\times n}\), where both matrix entries are of an arbitrary magnitude. The PCA question is how these components can be recovered with enough accuracy or even exactly recovered. Now consider there are multiple agents, which can be sensors, where each agent measures part of a global matrix \(M\), possibly with redundant data between agents. This aims for a local rule such that each sensor will find the same decomposition of the global \(M\). This situation can emerge, for instance, in a vertical or horizontal arrangement of cameras, where they cover a bigger image than what can be covered by a single agent.