Spectral clustering is a popular unsupervised machine learning technique used to group similar objects or observations into clusters. In this article, we explore how the traditional method of building a similarity matrix using a natural exponential function can be improved upon by using an alternative approach that uses an exponential function with a base value. We also propose a new method for computing inclusion probabilities in pivotal sampling, which leads to better representation of species in the sample.
Eigenvalue Analysis
In this section, we examine the impact of using an exponential function with a base value on the eigenvalues of the Laplacian matrix. The results show that the eigenvalues reduce when using the alternative approach, indicating improved clustering performance. This is because the base value allows for more flexibility in the similarity matrix, enabling it to capture more subtle patterns and relationships between observations.
Sampling Analysis
In this section, we assess the quality of our samples via a sampling analysis. Our results demonstrate that the new method for computing inclusion probabilities leads to better representation of species in the sample, resulting in more accurate clustering. This is because the median function used in the new method provides more intuitive and realistic probabilities compared to the previous maximum function.
Conclusion
In conclusion, this article proposes a novel approach to building a similarity matrix using an exponential function with a base value, leading to improved clustering performance. Additionally, we introduce a new method for computing inclusion probabilities in pivotal sampling, which results in better representation of species in the sample. By demystifying complex concepts and providing engaging analogies, this summary aims to facilitate a deeper understanding of the article’s key findings and their implications for unsupervised machine learning techniques.
