Fast Robust PCA on Graphs

Shahid, Nauman; Perraudin, Nathanael; Kalofolias, Vassilis; Puy, Gilles; Vandergheynst, Pierre

doi:10.48550/arxiv.1507.08173

Cited by 3 publications

(15 citation statements)

References 37 publications

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“…Estimation of the brain connectivity graph using a Gaussian noise model has been proposed in [11]. On the other hand, focusing on low-rank estimation, some works have proposed to incorporate spectral graph regularization [12,13,14]. Their graphs are fixed in their algorithms, pre-computed from the noisy input data.…”

Section: Related Workmentioning

confidence: 99%

“…• Φ f given L, M: On the other hand, for a given estimate of the low-rank data L, tr(L T Φ f L) guides the refinement of Φ f and hence the underlying connectivity graph G. In particular, a graph G that is consistent with the signal variation in L is favored: large Wi,j if row i and j of L have similar values. In many problems, the given graph for a problem can be noisy (e.g., the graph is estimated from the noisy data itself [13,14]). The proposed formulation iteratively improves Φ f using the refined low-rank data.…”

Section: Simultaneous Low Rank and Graph Estimationmentioning

confidence: 99%

“…The estimation accuracies are evaluated by the reconstruction errors: L−L0 F / L0 F and Φf −Φ f F / Φ f F . All the methods are initialized with the same feature similarity graph (consider each row of X as a node) computed using the procedure in [14] with a K-nearest neighbor strategy (K = 10).…”

Section: Synthetic Experimentsmentioning

confidence: 99%

See 2 more Smart Citations

Simultaneous low-rank component and graph estimation for high-dimensional graph signals: Application to brain imaging

Liu

Nejati

Safavi

et al. 2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We propose an algorithm to uncover the intrinsic low-rank component of a high-dimensional, graph-smooth and grossly-corrupted dataset, under the situations that the underlying graph is unknown. Based on a model with a low-rank component plus a sparse perturbation, and an initial graph estimation, our proposed algorithm simultaneously learns the low-rank component and refines the graph. The refined graph improves the effectiveness of the graph smoothness constraint and increases the accuracy of the low-rank estimation. We derive the learning steps using ADMM. Our evaluations using synthetic and real brain imaging data in a supervised classification task demonstrate encouraging performance.

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Section: Related Workmentioning

confidence: 99%

Section: Simultaneous Low Rank and Graph Estimationmentioning

confidence: 99%

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Simultaneous low-rank component and graph estimation for high-dimensional graph signals: Application to brain imaging

Liu

Nejati

Safavi

et al. 2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…It is quite obvious that we need a de-noising setup for real world data. In the following therefore we motivate and discuss the second tool in depth which was introduced in [1]. The compression framework is presented in [3] and will not be repeated here for brevity.…”

Section: Data and Graph Compressionmentioning

confidence: 99%

“…"This techncial report is a detailed explanation of the novel low-rank recovery concepts introduced in the previous work, Fast Robust PCA on Graphs [1]. The readers can refer to [1] for experiments which are not repeated here for brevity. ".…”

Section: Introductionmentioning

confidence: 99%

Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Shahid,

Perraudin,

Vandergheynst

2016

Preprint

Self Cite

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Many real world datasets subsume a linear or non-linear low-rank structure in a very low-dimensional space. Unfortunately, one often has very little or no information about the geometry of the space, resulting in a highly under-determined recovery problem. Under certain circumstances, state-of-the-art algorithms provide an exact recovery for linear low-rank structures but at the expense of highly inscalable algorithms which use nuclear norm. However, the case of non-linear structures remains unresolved. We revisit the problem of low-rank recovery from a totally different perspective, involving graphs which encode pairwise similarity between the data samples and features. Surprisingly, our analysis confirms that it is possible to recover many approximate linear and non-linear low-rank structures with recovery guarantees with a set of highly scalable and efficient algorithms. We call such data matrices as Low-Rank matrices on graphs and show that many real world datasets satisfy this assumption approximately due to underlying stationarity. Our detailed theoretical and experimental analysis unveils the power of the simple, yet very novel recovery framework Fast Robust PCA on Graphs.2. Another idea is to use the manifold learning techniques such as LLE, laplacian eigenmaps or MDS. However, these techniques project the points into a new low dimensional space (projected space), which is not the task we would like to perform. We want the resulting points to remain in the data space.

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Joint Estimation of Low-Rank Components and Connectivity Graph in High-Dimensional Graph Signals: Application to Brain Imaging

Liu¹,

Nejati²,

Cheung³

2018

Preprint

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This paper presents a graph signal processing algorithm to uncover the intrinsic low-rank components and the underlying graph of a high-dimensional, graph-smooth and grosslycorrupted dataset. In our problem formulation, we assume that the perturbation on the low-rank components is sparse and the signal is smooth on the graph. We propose an algorithm to estimate the low-rank components with the help of the graph and refine the graph with better estimated low-rank components. We propose to perform the low-rank estimation and graph refinement jointly so that low-rank estimation can benefit from the refined graph, and graph refinement can leverage the improved low-rank estimation. We propose to address the problem with an alternating optimization. Moreover, we perform a mathematical analysis to understand and quantify the impact of the inexact graph on the low-rank estimation, justifying our scheme with graph refinement as an integrated step in estimating low-rank components. We perform extensive experiments on the proposed algorithm and compare with state-of-the-art lowrank estimation and graph learning techniques. Our experiments use synthetic data and real brain imaging (MEG) data that is recorded when subjects are presented with different categories of visual stimuli. We observe that our proposed algorithm is competitive in estimating the low-rank components, adequately capturing the intrinsic task-related information in the reduced dimensional representation, and leading to better performance in a classification task. Furthermore, we notice that our estimated graph indicates compatible brain active regions for visual activity as neuroscientific findings.

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Fast Robust PCA on Graphs

Cited by 3 publications

References 37 publications

Simultaneous low-rank component and graph estimation for high-dimensional graph signals: Application to brain imaging

Simultaneous low-rank component and graph estimation for high-dimensional graph signals: Application to brain imaging

Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Joint Estimation of Low-Rank Components and Connectivity Graph in High-Dimensional Graph Signals: Application to Brain Imaging

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