Iterative Algorithm for Discrete Structure Recovery

Gao, Chao; Zhang, Anderson Y.

doi:10.48550/arxiv.1911.01018

Cited by 10 publications

(30 citation statements)

References 51 publications

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“…We also impose the following "balanced cluster size" assumption in order to make the presentation convenient. Such an assumption is widely used in the literature of mixture model clustering (Löffler et al, 2019;Gao and Zhang, 2019;Wu et al, 2020).…”

Section: Assumptionsmentioning

confidence: 99%

“…Remark 2 (Comparison with theory for iterative algorithm with single discrete structure). Recently, Gao and Zhang (2019) developed a framework for the convergence analysis in iterative algorithms with single discrete structure, including the Lloyd algorithm for Gaussian mixture model as a special instance. In comparison, our tensor block model admits multiple discrete structures (i.e., clusters in each mode) and their techniques do not directly apply.…”

Section: Algorithmic Theoretical Guaranteesmentioning

confidence: 99%

See 1 more Smart Citation

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Han¹,

Luo²,

Wang³

et al. 2020

Preprint

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High-order clustering aims to identify heterogeneous substructure in multiway dataset that arises commonly in neuroimaging, genomics, and social network studies. The non-convex and discontinuous nature of the problem poses significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, high-order Lloyd algorithm (HLloyd) and high-order spectral clustering (HSC), for highorder clustering in tensor block model. The convergence of the proposed procedure is established, and we show that our method achieves exact clustering under reasonable assumptions. We also give the complete characterization for the statistical-computational trade-off in high-order clustering based on three different signal-to-noise ratio regimes. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.

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Section: Assumptionsmentioning

confidence: 99%

Section: Algorithmic Theoretical Guaranteesmentioning

confidence: 99%

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Han¹,

Luo²,

Wang³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Lloyd's algorithm) to obtain minimax recovery bounds. This approach turned out to be fruitful in various latent space problems with discrete structure [Chen andLei, 2018, Gao andZhang, 2019] and our procedure can be interpreted as one in-stance of this strategy in a non-parametric setting with a continuous latent space.…”

Section: Related Workmentioning

confidence: 99%

Localization in 1D non-parametric latent space models from pairwise affinities

Giraud¹,

Issartel²,

Verzélen³

2021

Preprint

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We consider the problem of estimating latent positions in a one-dimensional torus from pairwise affinities. The observed affinity between a pair of items is modeled as a noisy observation of a function f (x * i , x * j ) of the latent positions x * i , x * j of the two items on the torus. The affinity function f is unknown, and it is only assumed to fulfill some shape constraints ensuring that f (x, y) is large when the distance between x and y is small, and vice-versa. This non-parametric modeling offers a good flexibility to fit data. We introduce an estimation procedure that provably localizes all the latent positions with a maximum error of the order of log(n)/n, with highprobability. This rate is proven to be minimax optimal. A computationally efficient variant of the procedure is also analyzed under some more restrictive assumptions. Our general results can be instantiated to the problem of statistical seriation, leading to new bounds for the maximum error in the ordering.

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“…The model (1.1) is an instance of the more general MRA problem that was studied thoroughly in recent years [9,14,8,3,2,18,4,34,31,11,10,38,1,27,6,30,23,25,24,19]. In its generalized version, the MRA model is formulated as (1.1), but the signal x may lie in an arbitrary vector space (not necessarily R L ), the dihedral group D 2L is replaced by an arbitrary group G, and g ∼ ρ is a distribution over G (in some cases, an additional fixed linear operator acting on the signal is also considered, e.g., [10,8,16,12]).…”

Section: Introductionmentioning

confidence: 99%

Super-resolution multi-reference alignment

Bendory

Jaffe

Leeb

et al. 2021

Information and Inference: A Journal of the IMA

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We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in ${\mathbb{R}}^M$ is uniquely determined when the number $L$ of samples per observation is of the order of the square root of the signal’s length ($L=O(\sqrt{M})$). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to $1/\textrm{SNR}^3$. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ($L=M$). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.

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Iterative Algorithm for Discrete Structure Recovery

Cited by 10 publications

References 51 publications

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

Localization in 1D non-parametric latent space models from pairwise affinities

Super-resolution multi-reference alignment

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