2024

1. 

   Dimers and M-curves

   Alexander I. Bobenko , Nikolai Bobenko, and Yuri B. Suris

   Feb 2024

   Abs arXiv Bib Slides

   In this paper we develop a general approach to dimer models analogous to Krichever’s scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to obtain integrable dimer models. These are dimer models on doubly periodic bipartite graphs with quasi-periodic positive weights. Dimer models with periodic weights and Harnack curves are recovered as a special case. This generalization from Harnack curves to general M-curves leads to transparent algebro-geometric structures. In particular explicit formulas for the Ronkin function and surface tension as integrals of meromorphic differentials on M-curves are obtained. Furthermore we describe the variational principle for the height function in the quasi-periodic case. Based on Schottky uniformizations of Riemann surfaces we present concrete computational results including computing the weights and sampling dimer configurations with them. The computational results are in complete agreement with the theoretical predictions.

   @article{dimers_and_MCurves_2024_BBS,
     title = {Dimers and M-curves},
     author = {Bobenko, Alexander I. and Bobenko, Nikolai and Suris, Yuri B.},
     year = {2024},
     month = feb,
     primaryclass = {math-ph},
   }

2. 

   Dimers and M-curves: Limit Shapes from Riemann Surfaces

   Alexander I. Bobenko , and Nikolai Bobenko

   Jul 2024

   Abs arXiv Bib Video Slides

   We present a general approach for the study of dimer model limit shape problems via variational and integrable systems techniques. In particular we deduce the limit shape of the Aztec diamond and the hexagon for quasi-periodic weights through purely variational techniques. Putting an M-curve at the center of the construction allows one to define weights and algebro-geometric structures describing the behavior of the corresponding dimer model. We extend the quasi-periodic setup of our previous paper [7] to include a diffeomorphism from the spectral data to the liquid region of the dimer. Our novel method of proof is purely variational and exploits a duality between the dimer height function and its dual magnetic tension minimizer and applies to dimers with gas regions. We apply this to the Aztec diamond and hexagon domains to obtain explicit expressions for the complex structure of the liquid region of the dimer as well as the height function and its dual. We compute the weights and the limit shapes numerically using the Schottky uniformization technique. Simulations and predicted results match completely.

   @article{dimers_and_MCurves_Limit_Shapes_2024_BB,
     title = {Dimers and M-curves: Limit Shapes from Riemann Surfaces},
     author = {Bobenko, Alexander I. and Bobenko, Nikolai},
     year = {2024},
     month = jul,
     primaryclass = {math-ph},
   }

2022

1. Stat&Comp

   

   A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks

   Xiaoyu Ma, Sylvain Sardy, Nick Hengartner , Nikolai Bobenko , and Yen Ting Lin

   Statistics and Computing, Jul 2022

   Abs arXiv Bib

   To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (Biometrika 81(3):425–455, 1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform simulated and real data Monte Carlo experiments to quantify the effectiveness of our approach.

   @article{lasso_phase_transition_2022,
     title = {A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks},
     author = {Ma, Xiaoyu and Sardy, Sylvain and Hengartner, Nick and Bobenko, Nikolai and Lin, Yen Ting},
     year = {2022},
     journal = {Statistics and Computing},
     volume = {32},
     number = {99},
     doi = {10.1007/s11222-022-10169-0},
     url = {https://doi.org/10.1007/s11222-022-10169-0},
   }