Robust Principal Component Analysis on Graphs

Shahid, Nauman; Kalofolias, Vassilis; Bresson, Xavier; Bronstein, Michael M.

doi:10.48550/arxiv.1504.06151

Cited by 4 publications

(19 citation statements)

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“…A. Clustering 1) Experimental Setup: Datasets: We perform our clustering experiments on 5 benchmark databases (as in [35], [36]): CMU PIE, ORL, YALE, MNIST and USPS. For the USPS and ORL datasets, we further run two types of experiments 1) on subset of datasets and 2) on full datasets.…”

Section: Resultsmentioning

confidence: 99%

“…Comparison with other methods: We compare the clustering performance of CPCA with 11 other models including: 1) k-means on original data 2) Laplacian Eigenmaps (LE) [4] 3) Locally Linear Embedding (LLE) [32] 4) Standard PCA 5) Graph Laplacian PCA (GLPCA) [14] 6) Manifold Regularized Matrix Factorization (MMF) [46] 7) Non-negative Matrix Factorization (NMF) [17] 8) Graph Regularized Nonnegative Matrix Factorization (GNMF) [6] 9) Robust PCA (RPCA) [7] 10) Robust PCA on Graphs (RPCAG) [35] and Table I: Summary of CPCA and its computational complexity for a dataset Y ∈ p×n . Throughout we assume that K, k, ρr, ρc, p n.…”

Section: Resultsmentioning

confidence: 99%

“…3. The time reported here corresponds to steps 2 to 6 of Table I, Algorithm 1 of [36] for FRPCAG, [7] for RPCA and [35] for RPCAG, excluding the construction of graphs G r , G c . The speed-up observed for these experiments from Table IX is 10 times over FRPCAG and 100 times over RPCA and RPCAG.…”

Section: Alternate Decodermentioning

confidence: 99%

“…} RPCAG [35] λ, γ γ ∈ {2 −3 , 2 −2 , • • • , 2 10 } FRPCAG [36] γ r , γ c γ r , γ c ∈ (0, 30) CPCA γ r , γ c γ r , γ c ∈ (0, 30) k (approximate decoder)…”

Section: Modelmentioning

confidence: 99%

“…1, however, RPCA just reduces the manifold to a point in the space. Extensions of RPCA and LRR such as Robust PCA on Graphs (RPCAG) [35] and Graph Regularized LRR (GLRR) [20] propose to incorporate graph regularization as a method to recover non-linear low-rank structures. These methods still suffer from the scalability problem for big datasets.…”

Section: Introductionmentioning

confidence: 99%

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Compressive PCA for Low-Rank Matrices on Graphs

Shahid

Perraudin

Puy

2017

IEEE Trans. on Signal and Inf. Process. over Networks

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Abstract-We introduce a novel framework for an approximate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherence of graph eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is suggested for the sampled data. Finally, several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for decoding the low-rank and cluster indicators for the full data matrix. Thus, we overcome the computational limitations of the standard linear low-rank recovery methods for big datasets. Our method can also be seen as a major step towards efficient recovery of nonlinear low-rank structures. For a matrix of size n × p, on a single core machine, our method gains a speed up of p 2 /k over Robust Principal Component Analysis (RPCA), where k p is the subspace dimension. Numerically, we can recover a low-rank matrix of size 10304 × 1000, 100 times faster than Robust PCA.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Alternate Decodermentioning

confidence: 99%

“…} RPCAG [35] λ, γ γ ∈ {2 −3 , 2 −2 , • • • , 2 10 } FRPCAG [36] γ r , γ c γ r , γ c ∈ (0, 30) CPCA γ r , γ c γ r , γ c ∈ (0, 30) k (approximate decoder)…”

Section: Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Compressive PCA for Low-Rank Matrices on Graphs

Shahid

Perraudin

Puy

2017

IEEE Trans. on Signal and Inf. Process. over Networks

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Fast Robust PCA on Graphs

Shahid,

Perraudin,

Kalofolias

et al. 2015

Preprint

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Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Shahid,

Perraudin,

Vandergheynst

2016

Preprint

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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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Robust Principal Component Analysis on Graphs

Cited by 4 publications

References 0 publications

Compressive PCA for Low-Rank Matrices on Graphs

Compressive PCA for Low-Rank Matrices on Graphs

Fast Robust PCA on Graphs

Low-Rank Matrices on Graphs: Generalized Recovery & Applications

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