## 1 Introduction
A significant advantage of HDBSCAN over DBSCAN is its ability to manage clusters of varying densities. Agglomerative clustering [13] is another hierarchical clustering method. Employing a bottom-up strategy, it starts with each data point as a separate cluster and then merges them based on their proximity, forming a hierarchical structure. This process continues until all points are combined into a single cluster or a predetermined level is reached. Agglomerative clustering, while effective, is computationally intensive for larger datasets and is sensitive to noise and outliers, which can lead to misinterpretation of cluster structures. The Isotonic Splitting (ISO-SPLIT) clustering algorithm, developed by [PERSON] et al. [14] in 2015, addresses the specific challenges of clustering high-dimensional data. It uses isotonic regression to automatically determine the best number of clusters and adjusts them iteratively based on statistical tests that evaluate the likelihood of data points belonging to the same cluster. Algorithms like K-means and DBSCAN are well-established methodologies that have evolved over a significant period. Despite their long-standing presence, these state-of-the-art methods are still utilized in contemporary comparative studies of novel research [15]. In contrast to basic clustering algorithms, recent advancements in the field have led to the development of sophisticated spike-sorting algorithms, including KiloSort [16] and Spysort [17]. These innovative tools are capable of supporting large-scale analyses of multichannel neural recordings and have shown promising accuracy in spike-sorting tasks [18]. However, their potential impact and methodological advancements have not been extensively considered in literature reviews or comparative studies. The current focus tends to be narrowly centered on the clustering aspect of spike sorting, overlooking the broader potential of these advanced tools. Beyond traditional clustering methods that utilize statistical techniques such as hierarchical, probabilistic, fuzzy, and density-based methods, this paper makes the following contributions:  
* Introduces a novel neural signals clustering algorithm, referred to as G-NMS, which integrates geometric principles through the Gershgorin circle theorem [19] and incorporates the nonmaximum suppression (NMS) algorithm [20]. * Demonstrates the robustness and efficiency of G-NMS across various noisy neural recordings, ranging from low to high SNRs. * Highlights the effective integration of deep learning techniques and eigenvalue inclusion theorems in G-NMS, marking a significant advancement in the clustering domain and contributing significantly to the body of research in this field. Figure 1 outlines the comprehensive framework of the proposed G-NMS clustering algorithm. This paper is structured as follows: Section 2, Materials and Methods, provides a detailed explanation of each step of the proposed method and includes a description of the dataset used for testing and comparison. Section 3, Results and Discussion, presents a comprehensive explanation of the results and discusses how the performance of the proposed method varies with changing parameters. Finally, Section 4 concludes the paper by summarizing the findings, highlighting the limitations, and suggesting potential future research directions. Figure 1: Overview of G–NMS clustering framework for neural spike clustering. The workflow begins with raw signal processing (**left panel**) and covariance metric calculation. The (**central panel**) shows the projection of SPD matrices to target space \(\mathrm{T_{M}C_{p}}\) using Log-Euclidean and Riemannian metrics. The (**right middle panel**) depicts the vertical matrix transformation and Gershgorin Circle Theorem application. Non-Maximum Suppression is applied (**bottom right panel**) to enhance separation. Kernel Density Estimation (**bottom central panel**) visualizes estimated density for cluster identification. The (**bottom left panel**) illustrates the separation of GC’s into Class 1 (red) and Class 2 (blue).