Provable deterministic leverage score sampling

Abstract

We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". In this work, we provide a novel theoretical analysis of deterministic leverage score sampling. We show that such sampling can be provably as accurate as its randomized counterparts, if the leverage scores follow a moderately steep power-law decay. We support this power-law assumption by providing empirical evidence that such decay laws are abundant in real-world data sets. We then demonstrate empirically the performance of deterministic leverage score sampling, which many times matches or outperforms the state-of-the-art techniques.

Dynamic tree-structured sparse RPCA via column subset selection for background modeling and foreground detection

“…We used the same frames as [24], [13], [25], [26], [27], and [28], for qualitative comparison. [29] .0903 (10) .2574 (9) .4473 (10) .4344 (10) .3602 (10) .6554 (8) .5713 (7) .3561 (10) .2751 (9) .3830 (10) Stauffer [30] .7570 (5) .6854 (6) .7948 (7) .7580 (8) .6519 (6) .5363 (10) .3335 (10) .3838 (9) .1388 (10) .4842 (9) Culibrk [27] .5256 (7) .4636 (8) .7540 (8) .7368 (9) .6276 (9) .5696 (9) .3923 (9) .4779 (8) .4928 (8) .5600 (8) DECOLOR [7] .3416 (9) .2075 (10) .9022 (5) .8700 (4) .646 (8) .6822 (5) .8169 (3) .6589 (4) .7480 (3) .6525 (7) Maddalena [28] .6960 (6) .6554 (7) .8247 …”

“…To alleviate the curse of dimensionality and scale with an RPCA-based problem, we must leverage on the fact that such data have in fact low intrinsic dimensionality. We approach this problem as a Column Subset Selection Problem (CSSP) [4], [5] by which means it is possible to select a handful of the most representative and important columns of a matrix. Assuming that we have a long video of a scene at our disposal with hundreds or even thousands of frames, only a handful of these frames determine a model of the background; the rest will either contaminate the background or will be redundant to process.…”

“…The theoretical computer science community has come up with randomized [17] and deterministic [14], [18] algorithms to solve this problem [5]. The CSSP is defined as: Let A ∈ R m×n and let c < n be a sampling parameter.…”

Foreground Segmentation with Tree-Structured Sparse RPCA

“…As the resolution of the images or the length of the video increase, RPCA becomes progressively computationally inefficient. There exist deterministic algorithms for solving the Column Subset Selection Problem (CSSP), that use probability distributions to find the most representative columns in a matrix [3]. The CSSP is defined as: Let A ∈ R m×n and let c n be a sampling parameter.…”

“…To alleviate the dimensionality and the curse of scale with an RPCA-based problem we use the Column Subset Selection Problem (CSSP) [3] that selects a handful of the most representative and important columns of a matrix. Assuming that we have a long video of a scene at our disposal with hundreds or even thousands of frames, only a handful of these frames determine a model of the background; the rest will either contaminate the background or will be redundant to process.…”