Sparse random tensors: Concentration, regularization and applications

Zhu, Yizhe

doi:10.1214/21-ejs1838

Cited by 8 publications

(5 citation statements)

References 52 publications

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“…where we use the fact that P ≤ q − 1 and w = 1, and the last inequality is from (80). Since (q − 1)µ i > (q − 1)d for i ∈ [r 0 ], from (82),…”

Section: Structure Of Near Eigenvectorsmentioning

confidence: 99%

“…Sparse random hypergraphs are fundamental objects in probabilistic combinatorics, but their spectral properties (their adjacency tensors or adjacency matrices) have not been well understood yet. Eigenvalues and the spectral norm for adjacency tensors were considered in [46,55,82,62,31], and the eigenvalue distributions and concentration of the adjacency matrices were studied in [44,65,42,41,61,40]. We believe the non-backtracking operator for hypergraphs could be a promising tool in the study of sparse random tensors and random hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

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Sparse random hypergraphs: Non-backtracking spectra and community detection

Stephan¹,

Zhu²

2022

Preprint

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We consider the community detection problem in a sparse q-uniform hypergraph G, assuming that G is generated according to the so-called Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al. in [12]. We characterize the spectrum of the non-backtracking operator for the sparse HSBM, and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on n vertices can be reduced to an eigenvector problem of a 2n × 2n non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with k blocks generated according to a general symmetric probability tensor.

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“…where we use the fact that P ≤ q − 1 and w = 1, and the last inequality is from (80). Since (q − 1)µ i > (q − 1)d for i ∈ [r 0 ], from (82),…”

Section: Structure Of Near Eigenvectorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sparse random hypergraphs: Non-backtracking spectra and community detection

Stephan¹,

Zhu²

2022

Preprint

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“…The recent paper [60] analyzed non-uniform hypergraph community detection by using hypergraph embedding and optimization algorithms and obtained weak consistency when the expected degrees are of ω(log n), again a complementary regime to ours. Results on spectral norm concentration of sparse random tensors were obtained in [19,48,29,36,62], but no provable tensor algorithm in the bounded expected degree is known. Testing for the community structure for non-uniform hypergraphs was studied in [56,30], which is a problem different from community detection.…”

Section: 3mentioning

confidence: 99%

“…Hypergraphs can represent more complex relationships among data [9,8], including recommendation systems [11,38], computer vision [26,55], and biological networks [42,53], and they have been shown empirically to have advantages over graphs [61]. Besides community detection problems, sparse hypergraphs and their spectral theory have also found applications in data science [29,62,27], combinatorics [20,22,52], and statistical physics [12,50].…”

Section: Introductionmentioning

confidence: 99%

Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

Dumitriu¹,

Wang²,

Zhu³

2021

Preprint

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We consider the community detection problem in sparse random hypergraphs under the nonuniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a γ fraction of the vertices classified correctly, where γ ∈ (0.5, 1) depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in [24] for non-uniform HSBMs.Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors;(3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

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“…Although random tensors appear frequently in data science problems [5,19,26,36,48,37,41,16,14,28,24,7,46,12,54], a systematic theory of random tensors is still in its infancy.…”

Section: Relaxing Independence?mentioning

confidence: 99%

Marchenko-Pastur law with relaxed independence conditions

Bryson

Vershynin

Zhao

2019

Preprint

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We prove the Marchenko-Pastur law for the eigenvalues of sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario -the block-independent model -the coordinates of the data are partitioned into n blocks each of length d in such a way that the entries in different blocks are independent but the entries from the same block may be dependent. In the second scenario -the random tensor model -the data is the homogeneous random tensor of order d, i.e. the coordinates of the data are all n d different products of d variables chosen from a set of n independent random variables. We show that Marchenko-Pastur law holds for the block-independent model as long as d = o(n) and for the random tensor model as long as d = o(n 1/3 ). Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

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Sparse random tensors: Concentration, regularization and applications

Cited by 8 publications

References 52 publications

Sparse random hypergraphs: Non-backtracking spectra and community detection

Sparse random hypergraphs: Non-backtracking spectra and community detection

Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

Marchenko-Pastur law with relaxed independence conditions

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