Spectral Algorithms on Manifolds through Diffusion
The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper introduces a new perspective, asserting that input data are situated within a low-dimensional manifold embedded in a higher-dimensional Euclidean space. We study the convergence performance of spectral algorithms in the RKHSs, specifically those generated by the heat kernels, known as diffusion spaces. Incorporating the manifold structure of the input, we employ integral operator techniques to derive tight convergence upper bounds concerning generalized norms, which indicates that the estimators converge to the target function in strong sense, entailing the simultaneous convergence of the function itself and its derivatives. These bounds offer two significant advantages: firstly, they are exclusively contingent on the intrinsic dimension of the input manifolds, thereby providing a more focused analysis. Secondly, they enable the efficient derivation of convergence rates for derivatives of any k-th order, all of which can be accomplished within the ambit of the same spectral algorithms. Furthermore, we establish minimax lower bounds to demonstrate the asymptotic optimality of these conclusions in specific contexts. Our study confirms that the spectral algorithms are practically significant in the broader context of high-dimensional approximation.
Code (0)
등록된 구현이 없습니다.
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Large-scale spectral clustering using diffusion coordinates on landmark-based bipartite graphs
Spectral clustering has received a lot of attention due to its ability to separate nonconvex, non-intersecting manifolds, but its high computational complexity has significantly limited its applicability. Motivated by th…
ClusteringImage/Document ClusteringSemantic SegmentationDiffusion-based Semi-supervised Spectral Algorithm for Regression on Manifolds
We introduce a novel diffusion-based spectral algorithm to tackle regression analysis on high-dimensional data, particularly data embedded within lower-dimensional manifolds. Traditional spectral algorithms often fall sh…
regressionManifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion
In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. For this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score …
Manifold Diffusion Fields
We present Manifold Diffusion Fields (MDF), an approach that unlocks learning of diffusion models of data in general non-Euclidean geometries. Leveraging insights from spectral geometry analysis, we define an intrinsic c…
DiversityGenerative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on man…
Denoising