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通过嵌入对齐实现链式多矩阵整合AI 解读

Motivated by the increasing demand for multi-source data integration in various scientific fields, in this article we study matrix completion in scenarios where the data exhibits certain block-wise missing structures—specifically, where only a few noisy submatrices representing (overlapping) parts of the full matrix are available. We propose the Chain-linked Multiple Matrix Integration (CMMI) procedure to efficiently combine the information that can be extracted from these…

随机化奇异值分解的扰动分析及统计应用AI 解读

Randomized singular value decomposition (RSVD) is a class of computationally efficient algorithms for computing the truncated SVD of large data matrices. Given an m×n matrix M̂, the prototypical RSVD algorithm outputs an approximation of the k leading left singular vectors of M̂ by computing the SVD of M̂(M̂⊤M̂)gG; here g≥1 is an integer and G∈Rn×k˜ is a random Gaussian sketching matrix with k˜≥k. In this paper we derive upper bounds for the l2 and l2,∞ distances between the…

谱嵌入的统计解释：广义随机点积图AI 解读

Abstract Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The generalisation is needed to model heterophilic connectivity (e.g. ‘opposites attract’) and to cope with negative eigenvalues more generally. We show that,…

二到无穷范数与奇异子空间几何及其高维统计应用AI 解读

The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary…

随机图归一化拉普拉斯矩阵特征向量的极限定理AI 解读

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and, furthermore, the mean and the covariance matrix of each row are functions of the associated vertex’s block…

潜在位置图上的普遍相合顶点分类AI 解读

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function $\kappa$, provided that the latent positions are i.i.d. from some distribution $F$. We then consider the exploitation task of vertex classification where the link function $\kappa$ belongs to the class of universal kernels and class labels are observed for a number of vertices tending to…

随机块模型图的一致邻接谱嵌入AI 解读

We present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel. We use an embedding procedure motivated by the random dot product graph model, a particular example of the latent position model. The embedding associates each node with a vector; these vectors are clustered via minimization of a square error criterion. We prove that this method is consistent for assigning nodes to blocks, as only a negligible number of nodes…