This paper introduces Laplacian Eigenmaps, a geometrically motivated algorithm for nonlinear dimensionality reduction and clustering. The algorithm constructs a representation for data sampled from a low-dimensional manifold embedded in a higher-dimensional space. It is based on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the heat equation. The algorithm is computationally efficient, has locality-preserving properties, and has a natural connection to clustering. The algorithm involves three steps: constructing a weighted graph based on the proximity of data points, choosing edge weights, and computing eigenvalues and eigenvectors of the Laplacian matrix. The graph is constructed by connecting nearby points, with two variations: ε-neighborhoods and n nearest neighbors. Edge weights are chosen using either a heat kernel or a simple-minded approach. The embedding is obtained by projecting the data onto the eigenvectors of the Laplacian matrix. The justification for the algorithm is based on the role of the Laplacian operator in providing an optimal embedding. The Laplacian of the graph is an approximation to the Laplace-Beltrami operator on the manifold. The embedding maps for the data come from approximations to a natural map defined on the entire manifold. The algorithm is shown to be robust to outliers and noise, and implicitly emphasizes natural clusters in the data. It is connected to spectral clustering algorithms and has applications in data representation, image processing, and speech recognition. The algorithm is demonstrated on several examples, including a toy vision example, words in the Brown corpus, and speech data. The results show that the algorithm effectively captures the intrinsic structure of the data. The paper also discusses the connection between the Laplacian and the heat kernel, and how this enables the selection of graph weights in a principled manner. The algorithm is shown to be related to the Laplace-Beltrami operator on manifolds and has applications in various areas of artificial intelligence, information retrieval, and data mining.This paper introduces Laplacian Eigenmaps, a geometrically motivated algorithm for nonlinear dimensionality reduction and clustering. The algorithm constructs a representation for data sampled from a low-dimensional manifold embedded in a higher-dimensional space. It is based on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the heat equation. The algorithm is computationally efficient, has locality-preserving properties, and has a natural connection to clustering. The algorithm involves three steps: constructing a weighted graph based on the proximity of data points, choosing edge weights, and computing eigenvalues and eigenvectors of the Laplacian matrix. The graph is constructed by connecting nearby points, with two variations: ε-neighborhoods and n nearest neighbors. Edge weights are chosen using either a heat kernel or a simple-minded approach. The embedding is obtained by projecting the data onto the eigenvectors of the Laplacian matrix. The justification for the algorithm is based on the role of the Laplacian operator in providing an optimal embedding. The Laplacian of the graph is an approximation to the Laplace-Beltrami operator on the manifold. The embedding maps for the data come from approximations to a natural map defined on the entire manifold. The algorithm is shown to be robust to outliers and noise, and implicitly emphasizes natural clusters in the data. It is connected to spectral clustering algorithms and has applications in data representation, image processing, and speech recognition. The algorithm is demonstrated on several examples, including a toy vision example, words in the Brown corpus, and speech data. The results show that the algorithm effectively captures the intrinsic structure of the data. The paper also discusses the connection between the Laplacian and the heat kernel, and how this enables the selection of graph weights in a principled manner. The algorithm is shown to be related to the Laplace-Beltrami operator on manifolds and has applications in various areas of artificial intelligence, information retrieval, and data mining.