Detection thresholds in very sparse matrix completion

Bordenave, Charles; Coste, Simon; Nadakuditi, Raj Rao

doi:10.48550/arxiv.2005.06062

Cited by 9 publications

(26 citation statements)

References 44 publications

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“…Then, the minimal sufficient statistic becomes the n × p array of absolute values |X ij | and the signflip distribution is the conditional null distribution. See also Bordenave et al (2020) for different uses of signflips in sparse matrix completion. Section 4 analyzes Signflip PA under signal-plus-noise models by extending the framework developed in Dobriban (2020).…”

Section: Proposed Method: Signflip Parallel Analysismentioning

confidence: 99%

Selecting the number of components in PCA via random signflips

Hong¹,

Sheng²,

Dobriban³

2020

Preprint

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Dimensionality reduction via PCA and factor analysis is an important tool of data analysis. A critical step is selecting the number of components. However, existing methods (such as the scree plot, likelihood ratio, parallel analysis, etc) do not have statistical guarantees in the increasingly common setting where the data are heterogeneous. There each noise entry can have a different distribution. To address this problem, we propose the Signflip Parallel Analysis (Signflip PA) method: it compares data singular values to those of "empirical null" data generated by flipping the sign of each entry randomly with probability one-half. We show that Signflip PA consistently selects factors above the noise level in high-dimensional signal-plus-noise models (including spiked models and factor models) under heterogeneous settings. Here classical parallel analysis is no longer effective. To do this, we propose to leverage recent breakthroughs in random matrix theory, such as dimension-free operator norm bounds [Latala et al, 2018, Inventiones Mathematicae], and large deviations for the top eigenvalues of nonhomogeneous matrices [Husson, 2020]. To our knowledge, some of these results have not yet been used in statistics. We also illustrate that Signflip PA performs well in numerical simulations and on empirical data examples.

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Section: Proposed Method: Signflip Parallel Analysismentioning

confidence: 99%

Selecting the number of components in PCA via random signflips

Hong¹,

Sheng²,

Dobriban³

2020

Preprint

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“…The proofs of these lemmas can be easily adapted from the corresponding proofs of [12]. For the sake of brevity we only sketch the main ideas.…”

Section: Local Algorithms On the Factor Graphmentioning

confidence: 99%

The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Bresler¹,

Huang²

2021

Preprint

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Let Φ be a uniformly random k-SAT formula with n variables and m clauses. We study the algorithmic task of finding a satisfying assignment of Φ. It is known that a satisfying assignment exists with high probability at clause density m/n < 2 k log 2 − 1 2 (log 2 + 1) + o k (1), while the best polynomial-time algorithm known, the Fix algorithm of Coja-Oghlan [25], finds a satisfying assignment at the much lower clause density (1 − o k (1))2 k log k/k. This prompts the question: is it possible to efficiently find a satisfying assignment at higher clause densities?To understand the algorithmic threshold of random k-SAT, we study low degree polynomial algorithms, which are a powerful class of algorithms including Fix, Survey Propagation guided decimation (with bounded or mildly growing number of message passing rounds), and paradigms such as message passing and local graph algorithms. We show that low degree polynomial algorithms can find a satisfying assignment at clause density (1 − o k (1))2 k log k/k, matching Fix, and not at clause density (1 + o k (1))κ * 2 k log k/k, where κ * ≈ 4.911. This shows the first sharp (up to constant factor) computational phase transition of random k-SAT for a class of algorithms. Our proof establishes and leverages a new many-way overlap gap property tailored to random k-SAT.

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“…A proof sketch of this reduction was given already in Appendix A of [GJW20], but here we give the full details and determine the values of the parameters D, δ, γ, η. The main difficulty lies in establishing that the error probability δ is very small; for this we appeal to a result of [BCN20] that gives tail bounds for certain "local" functions on sparse random graphs.…”

Section: Proof Techniquesmentioning

confidence: 99%

Optimal Low-Degree Hardness of Maximum Independent Set

Wein¹

2020

Preprint

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We study the algorithmic task of finding a large independent set in a sparse Erdős-Rényi random graph with n vertices and average degree d. The maximum independent set is known to have size (2 log d/d)n in the double limit n → ∞ followed by d → ∞, but the best known polynomial-time algorithms can only find an independent set of half-optimal size (log d/d)n. We show that the class of low-degree polynomial algorithms can find independent sets of half-optimal size but no larger, improving upon a result of Gamarnik, Jagannath, and the author. This generalizes earlier work by Rahman and Virág, which proves the analogous result for the weaker class of local algorithms.

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Detection thresholds in very sparse matrix completion

Cited by 9 publications

References 44 publications

Selecting the number of components in PCA via random signflips

Selecting the number of components in PCA via random signflips

The Algorithmic Phase Transition of Random $k$-SAT for Low Degree Polynomials

Optimal Low-Degree Hardness of Maximum Independent Set

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