Near-Optimal Column-Based Matrix Reconstruction

Boutsidis, Christos; Drineas, Petros; Magdon‐Ismail, Malik

doi:10.1137/12086755x

Cited by 142 publications

(260 citation statements)

References 24 publications

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“…holds with a probability of at least 0.6 − δ. For modified Nyström approximation [36] the bound is derived by combining the near-optimal sampling [57] and the error-driven adaptive sampling [58],…”

Section: B Approximate Consistency Of Nyström Approximationmentioning

confidence: 99%

Approximate Kernel Selection via Matrix Approximation

Ding

Liao

Liu

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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Kernel selection is of fundamental importance for the generalization of kernel methods. This paper proposes an approximate approach for kernel selection by exploiting the approximability of kernel selection and the computational virtue of kernel matrix approximation. We define approximate consistency to measure the approximability of the kernel selection problem. Based on the analysis of approximate consistency, we solve the theoretical problem of whether, under what conditions and at what speed, the approximate criterion is close to the accurate one, establishing the foundations of approximate kernel selection. We introduce two selection criteria based on error estimation and prove the approximate consistency of multilevel circulant matrix (MCM) approximation and Nyström approximation under these criteria. Under the theoretical guarantees of the approximate consistency, we design approximate algorithms for kernel selection, which exploit the computational advantages of MCM and Nyström approximations to conduct kernel selection in a linear or quasi-linear complexity. We experimentally validate the theoretical results for the approximate consistency and evaluate the effectiveness of the proposed kernel selection algorithms.

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Section: B Approximate Consistency Of Nyström Approximationmentioning

confidence: 99%

Approximate Kernel Selection via Matrix Approximation

Ding

Liao

Liu

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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“…Wang et al proposed an adaptive sampling method for CUR decomposition [12]. They first employed the near-optimal column selection algorithm [13] to select c columns to form both C and the first c rows in R. Then, they selected the remaining r − c rows according to the residual. They iteratively performed such column-row picking step for a given number of times and showed that the expected error is proportional to K − K c , where K c is the best rank-c approximation of K from SVD decomposition.…”

Section: Cur Approximationmentioning

confidence: 99%

Incremental Spectral Clustering via Fastfood Features and Its Application to Stream Image Segmentation

Zhang

et al. 2018

Symmetry

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We propose an incremental spectral clustering method for stream data clustering and apply it to stream image segmentation. The main idea in our work consists of generating the data points in the kernel space by Fastfood features and iteratively calculating the eigendecomposition of data. Compared with the popular Nyström-based approximation, our work accesses each data point only once while Nyström, in particular the sampling scheme, will go through the entire dataset first and calculate the embeddings of data points with a second visit. As a result, our method is able to learn data partitions incrementally and improve eigenvector approximation with more and more data seen from a stream. By contrast, the performance of the standard Nyström is fixed when the sample set is selected. Experimental results show the superiority of our method.

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“…La idea de sketch también ha sido utilizada en (Pedarsani et al, 2015) para la estimación de matrices dispersas mediante códigos de grafos dispersos, en (Bahmani y Romberg, 2015) para la estimación de matrices de bajo rango dispersas, o en el producto de tensores expuesto anteriormente. Otra forma de aproximarse a sketching es encontrar un pequeño subconjunto de columnas o filas de la matriz que aproximen la matriz entera, esto es conocido como Problema de Selección de Subconjunto Columna (Boutsidis et al, 2009(Boutsidis et al, , 2014Drineas y Kannan, 2003). Un último enfoque ha sido Direcciones Frecuentes, el cual se realiza de forma determinística (Ghashami et al, 2016).…”

Section: Métodos Aleatoriosunclassified

Una Revisión de los Estimadores de Matrices de Bajo Rango y Matrices Dispersas

Hoyos

Jojoa

2018

Inf. tecnol.

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Se presenta una revisión selectiva de los desarrollos más recientes en la estimación de la matriz de covarianza con un número de muestras menores a la dimensión ambiente. En particular se consideran estimadores para las estructuras bajo rango y dispersa, dando especial atención a los algoritmos más utilizados en la práctica. Se presentan estimadores que recurren al uso de técnicas clásicas como umbrales y descomposición de valor único (SVD), así como estimadores más novedosos basados en matrices aleatorias y optimización convexa. Una de las principales conclusiones del estudio es que a pesar de los grandes avances en el desarrollo de nuevos estimadores, temas relacionados al tiempo de cómputo de los programas convexos que utilizan los estimadores han sido poco explorados. Además, se observa la falta de trabajos sobre el desarrollo de estimadores para matrices con estructura conjuntamente dispersa y bajo rango.

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Near-Optimal Column-Based Matrix Reconstruction

Cited by 142 publications

References 24 publications

Approximate Kernel Selection via Matrix Approximation

Approximate Kernel Selection via Matrix Approximation

Incremental Spectral Clustering via Fastfood Features and Its Application to Stream Image Segmentation

Una Revisión de los Estimadores de Matrices de Bajo Rango y Matrices Dispersas

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