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We demonstrate the utility of an unsupervised machine learning tool for the detection of phase transitions in off-lattice systems. We focus on the application of principal component analysis (PCA) to detect the freezing transitions of two-dimensional hard-disk and three-dimensional hard-sphere systems as well as liquid-gas phase separation in a patchy colloid model. As we demonstrate, PCA autonomously discovers order-parameter-like quantities that report on phase transitions, mitigating the need for a priori construction or identification of a suitable order parameter—thus streamlining the routine analysis of phase behavior. In a companion paper, we further develop the method established here to explore the detection of phase transitions in various model systems controlled by compositional demixing, liquid crystalline ordering, and non-equilibrium active forces.
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Minimax rates for sparse signal detection under correlation
Abstract We fully characterize the nonasymptotic minimax separation rate for sparse signal detection in the Gaussian sequence model with $$p$$ equicorrelated observations, generalizing a result of Collier, Comminges and Tsybakov. As a consequence of the rate characterization, we find that strong correlation is a blessing, moderate correlation is a curse and weak correlation is irrelevant. Moreover, the threshold correlation level yielding a blessing exhibits phase transitions at the $$\sqrt{p}$$ and $$p-\sqrt{p}$$ sparsity levels. We also establish the emergence of new phase transitions in the minimax separation rate with a subtle dependence on the correlation level. Additionally, we study group structured correlations and derive the minimax separation rate in a model including multiple random effects. The group structure turns out to fundamentally change the detection problem from the equicorrelated case and different phenomena appear in the separation rate.
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- PAR ID:
- 10474890
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- Volume:
- 12
- Issue:
- 4
- ISSN:
- 2049-8772
- Format(s):
- Medium: X Size: p. 2873-2969
- Size(s):
- p. 2873-2969
- Sponsoring Org:
- National Science Foundation
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