Efficient local search for L_1 and L_2 binary matrix factorization

Mirisaee, Hamid; Gaussier, Éric; Termier, Alexandre

doi:10.3233/ida-160832

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…We refer the reader to the tutorial http://people.mpi-inf.mpg.de/~pmiettin/bmf_tu and the references therein for more details. Although BMF is conjectured to be NP-hard [34,29], there is, to the best of our knowledge, no formal proof of this fact. We will prove in this paper that it is in fact NP-hard.…”

Section: Binary Matrix Factorization and Densest Bipartite Subgraphmentioning

confidence: 91%

“…BMF was used successfully to mine discrete patterns with applications for example to analyze gene expression data [34,41,29]. We refer the reader to the tutorial http://people.mpi-inf.mpg.de/ ~pmiettin/bmf_tu and the references therein for more details.…”

Section: Binary Matrix Factorization and Densest Bipartite Subgraphmentioning

confidence: 99%

“…We refer the reader to the tutorial http://people.mpi-inf.mpg.de/ ~pmiettin/bmf_tu and the references therein for more details. Although BMF is conjectured to be NP-hard [34,29],…”

Section: Binary Matrix Factorization and Densest Bipartite Subgraphmentioning

confidence: 99%

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On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation

Gillis

Vavasis

2018

Mathematics of OR

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The low-rank matrix approximation problem with respect to the component-wise ℓ 1 -norm (ℓ 1 -LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine learning. Robust PCA aims at recovering a low-rank matrix that was perturbed with sparse noise, with applications for example in foreground-background video separation. Although ℓ 1 -LRA is strongly believed to be NP-hard, there is, to the best of our knowledge, no formal proof of this fact. In this paper, we prove that ℓ 1 -LRA is NP-hard, already in the rank-one case, using a reduction from MAX CUT. Our derivations draw interesting connections between ℓ 1 -LRA and several other well-known problems, namely, robust PCA, ℓ 0 -LRA, binary matrix factorization, a particular densest bipartite subgraph problem, the computation of the cut norm of {−1, +1} matrices, and the discrete basis problem, which we all prove to be NP-hard. ., √ p > |E||V |. Therefore, we choose p > |E| 2 |V | 2 and also p a power of 2.Corollary 2. It is NP-hard to compute the cut norm (3.1) of {−1, +1}-matrices.Corollary 3. Rank-one ℓ 0 -LRA, that is, problem (1.3) with r = 1, is NP-hard.Corollary 4. Rank-one BMF (2.1) is NP-hard.Remark 1 (The discrete basis problem and boolean matrix factorization). The discrete basis problem (DSP) [28], also known as Boolean matrix factorization, is similar to BMF and can be formulated a follows: given a binary matrix M ∈ {0, 1} m×n and a rank r, solvewhere 0 0 = 0, 0 1 = 1 and 1 1 = 1. For r = 1, BMF and DSP coincide, therefore our result also implies that rank-one DSP is NP-hard. Note that DSP is closely related to the rectangle covering problem which is equivalent to the minimum biclique cover problem in bipartite graph; see Fiorini et al. [13].

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Section: Binary Matrix Factorization and Densest Bipartite Subgraphmentioning

confidence: 91%

Section: Binary Matrix Factorization and Densest Bipartite Subgraphmentioning

confidence: 99%

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On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation

Gillis

Vavasis

2018

Mathematics of OR

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“…More efficient algorithms are model based: they build a model from the visible ratings and compute all the missing ratings from the model. Widely used model-based rating prediction methods include PLSA [20], the Restricted Boltzmann Machine (RBM) [21], and a series of matrix factorization techniques [22][23][24][25].…”

Section: Rating Predictionmentioning

confidence: 99%

Exploiting Explicit and Implicit Feedback for Personalized Ranking

Chen

2016

Mathematical Problems in Engineering

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The problem of the previous researches on personalized ranking is that they focused on either explicit feedback data or implicit feedback data rather than making full use of the information in the dataset. Until now, nobody has studied personalized ranking algorithm by exploiting both explicit and implicit feedback. In order to overcome the defects of prior researches, a new personalized ranking algorithm (MERR_SVD++) based on the newest xCLiMF model and SVD++ algorithm was proposed, which exploited both explicit and implicit feedback simultaneously and optimized the well-known evaluation metric Expected Reciprocal Rank (ERR). Experimental results on practical datasets showed that our proposed algorithm outperformed existing personalized ranking algorithms over different evaluation metrics and that the running time of MERR_SVD++ showed a linear correlation with the number of rating. Because of its high precision and the good expansibility, MERR_SVD++ is suitable for processing big data and has wide application prospect in the field of internet information recommendation.

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“…This compensates for the greedy behavior of algorithms such as Asso, GreConD, and TopFiber. TERMIER, 2016) propose a neighborhood for searching improvements in each row. Additionally, the authors present different versions of the search by linearizing the objective function, which is an idea explored by us.…”

Section: Algorithmic Approachesmentioning

confidence: 99%

Binary Matrix Factorization Post-Processing and Applications

MIRANDA SPYRIDES

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Efficient local search for L_1 and L_2 binary matrix factorization

Cited by 4 publications

References 21 publications

On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation

On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation

Exploiting Explicit and Implicit Feedback for Personalized Ranking

Binary Matrix Factorization Post-Processing and Applications

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Efficient local search for L_1 and L_2 binary matrix factorization

Cited by 4 publications

References 21 publications

On the Complexity of Robust PCA and ℓ1-Norm Low-Rank Matrix Approximation

On the Complexity of Robust PCA and ℓ1-Norm Low-Rank Matrix Approximation

Exploiting Explicit and Implicit Feedback for Personalized Ranking

Binary Matrix Factorization Post-Processing and Applications

Contact Info

Product

Resources

About

On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation

On the Complexity of Robust PCA and ℓ₁-Norm Low-Rank Matrix Approximation