The meeting focused on various dimensionality reduction techniques and their applications in neuroscience, including linear methods like PCA and ICA, as well as nonlinear approaches such as t-SNE and UMAP. Chen presented on different mathematical concepts and algorithms for handling low-dimensional data representation, including T-distribution, attribute topology, and simplicial sets, while discussing their practical applications and limitations in neuroscience research. The discussion concluded with an exploration of signal separation methods, clustering techniques, and the importance of dimensionality reduction in analyzing neural data, particularly for understanding neural representations in multi-modal environments.
Frank Chen presented on dimensionality reduction techniques, starting with linear methods like PCA and ICA, and then introducing nonlinear methods such as t-SNE and UMAP. He explained how PCA finds directions of greatest variance in data, while ICA separates statistically independent sources. Frank also discussed the application of these methods in neuroscience, including a study involving brain imaging and virtual reality. He concluded by introducing t-SNE, a nonlinear dimensionality reduction technique that preserves local structures in data and is used for single-cell clustering and neuropopulation visualization.
陈骁 explained the concept of using a T-distribution to address the crowding problem in low-dimensional data representation, highlighting its advantages over Gaussian distribution. They discussed the limitations of pairwise conditional probability methods, including computational inefficiency and distortion of inter-cluster distances. Chen also introduced the UMAP algorithm, which aims to preserve both local and global structures by constructing a graph that captures high-dimensional data in low dimensions. The discussion concluded with Chen explaining how UMAP handles outliers and constructs a connected global graph, though some participants expressed skepticism about the practicality of this approach.
陈骁 explained the concept of attribute topology and simplicial sets, describing them as abstract mathematical structures. He explained that a simplicial set can be thought of as a skeleton of a graph, with simplexes representing the basic building blocks in different dimensions. Chen discussed the relationship between two functions, explaining that they form a homomorphism if the relations between the two spaces are preserved. He also mentioned that the concept connects to his previous work, though he did not elaborate on the specific connection.
The discussion focused on a mathematical paper involving dimensionality reduction and fuzzy sets. Chen explained that while the paper uses category theory and discusses dimensionality, the core concept is about finding equivalents between different types of homeomorphism in metric spaces and fuzzy sets. The paper introduces a technique called global k-nearest neighborhood graph, which creates a network by combining local graphs of data points. Chen emphasized that while the paper is mathematically rigorous, it may be difficult for data scientists to understand due to its complex presentation. The discussion concluded with an explanation of how the technique works with pseudometrics and its advantages over other methods, including faster computation and easier addition of new points.
陈骁 discussed the comparison between UMAP and t-SNE, highlighting that UMAP better captures the patterns of clusters with a flow, unlike t-SNE's splash-like appearance. Chen explained that PCA cannot handle temporal dynamics in neural data, as it only considers overall distribution and ignores the sequence of data points. Chen also illustrated an example using two groups of data to show that PCA cannot separate clusters that appear linearly mixed, emphasizing the need to consider local structure and manifold properties for better results.