Consistent Estimation of Mixed Memberships with Successive Projections

Panov, Maxim; Slavnov, Konstantin; Ushakov, Roman

doi:10.1007/978-3-319-72150-7_5

Cited by 15 publications

(25 citation statements)

References 18 publications

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“…The mixed membership learning is a core task in overlapped community detection (OCD) [16]. In the OCD frameworks proposed in [17][18][19], the mixed membership was provably learned using a fully observed A. However, OCD under partially observed edges is of great interest for applications like field survey based OCD [15] and hidden edge-robust network analysis [10].…”

Section: Problem Statementmentioning

confidence: 99%

“…such that M (:, n k ) = e k , where e k ∈ R K is the kth unit vector. The existence of Λ translates to the existence of the so-called pure nodes (i.e., nodes belong to a single cluster) [18,19]. This assumption is considered reasonable when the graph is large.…”

Section: Main Idea: Block Subspace Stitchingmentioning

confidence: 99%

“…Remark 2. Under the EQP, another solution for estimating M is to apply existing mixed membership learning algorithms, e.g., those 1.12 ×10 −13 0.101 0.0066 0.0261 0.0924 in [17][18][19] on the small blocks A ,m and individually learn the corresponding parts of M . Then, the entire M can be recovered by unifying the intrinsic column permutation ambiguity between blocks of M .…”

Section: Performance Characterization Under Binary Observationsmentioning

confidence: 99%

“…This is doable, but may have relatively poor identifiability guarantees. The reason is that learning M from A ,m via the methods in [17][18][19] requires that the convex hull of M to be well spread in the probability simplex (e.g., the existence of pure nodes [18,19] [27]. When the methods are applied on small subblocks A ,m , this assumption is less likely to hold by the corresponding submatrix M or Mm [28].…”

Section: Performance Characterization Under Binary Observationsmentioning

confidence: 99%

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Learning Mixed Membership from Adjacency Graph Via Systematic Edge Query: Identifiability and Algorithm

Ibrahim

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Graph clustering is a core technique for network analysis problems, e.g., community detection. This work puts forth a node clustering approach for largely incomplete adjacency graphs. Under the considered scenario, instead of having access to the complete graph, only a small amount of queries about the graph edges can be made for node clustering. This task is well-motivated in many large-scale network analysis problems, where complete graph acquisition is prohibitively costly. Prior work tackles this problem under the setting that the nodes only admit single membership and the clusters are disjoint, yet multiple membership nodes and overlapping clusters often arise in practice. Existing approaches also rely on random edge query patterns and convex optimization-based formulations, which give rise to a number of implementation and scalability challenges. This work offers a framework that provably learns the mixed membership of nodes from overlapping clusters using limited edge information. Our method is equipped with a systematic edge query pattern, which is arguably easier to implement relative to the random counterparts in certain applications, e.g., field survey based graph analysis. A lightweight scalable algorithm is proposed, and its performance characterizations are presented. Numerical experiments are used to showcase the effectiveness of our method.

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Section: Problem Statementmentioning

confidence: 99%

Section: Main Idea: Block Subspace Stitchingmentioning

confidence: 99%

Section: Performance Characterization Under Binary Observationsmentioning

confidence: 99%

Section: Performance Characterization Under Binary Observationsmentioning

confidence: 99%

See 2 more Smart Citations

Learning Mixed Membership from Adjacency Graph Via Systematic Edge Query: Identifiability and Algorithm

Ibrahim

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…When H does not have sum-to-one columns, this assumption can be "enforced" through column normalization of X, under the condition that W is nonnegative; see [1], [18]. We should mention that although our interest lies in NMF, the proposed method can also be applied to the so-called simplex-structured matrix factorization, where W is not required to be nonnegative; see, e.g., [8], [9], [25], [37].…”

Section: Problem Statementmentioning

confidence: 99%

Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach

Nguyen,

Fu,

2021

Preprint

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Nonnegative matrix factorization (NMF) often relies on the separability condition for tractable algorithm design. Separability-based NMF is mainly handled by two types of approaches, namely, greedy pursuit and convex programming. A notable convex NMF formulation is the so-called selfdictionary multiple measurement vectors (SD-MMV), which can work without knowing the matrix rank a priori, and is arguably more resilient to error propagation relative to greedy pursuit. However, convex SD-MMV renders a large memory cost that scales quadratically with the problem size. This memory challenge has been around for a decade, and a major obstacle for applying convex SD-MMV to big data analytics. This work proposes a memory-efficient algorithm for convex SD-MMV. Our algorithm capitalizes on the special update rules of a classic algorithm from the 1950s, namely, the Frank-Wolfe (FW) algorithm. It is shown that, under reasonable conditions, the FW algorithm solves the noisy SD-MMV problem with a memory cost that grows linearly with the amount of data. To handle noisier scenarios, a smoothed group sparsity regularizer is proposed to improve robustness while maintaining the low memory footprint with guarantees. The proposed approach presents the first linear memory complexity algorithmic framework for convex SD-MMV based NMF. The method is tested over a couple of unsupervised learning tasks, i.e., text mining and community detection, to showcase its effectiveness and memory efficiency.

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