Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition

• URL: http://arxiv.org/abs/2010.00351v2
• Date: Wed, 17 Feb 2021 03:02:59 GMT
• Title: Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition
• Authors: Dongkyu Kim and Dong-Hee Kim
• Abstract summary: We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output.
• Score: 0.7658140759553149
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. Proposing a minimal one-free-parameter neural network model, we analytically formulate the supervised learning problem for the canonical ensemble being used as a training data set. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output that is observed in a large neural network, theoretically validating its critical point prediction for unseen test data from different underlying lattices yet in the same universality class of the Ising criticality. We also numerically demonstrate the interpretation with the proposed one-parameter model by providing an example of finding a critical point with the learning of the Landau mean-field free energy being applied to the real data set from the uncorrelated random scale-free graph with a large degree exponent.

Related papers

• Revealing Decurve Flows for Generalized Graph Propagation [108.80758541147418]
  This study addresses the limitations of the traditional analysis of message-passing, central to graph learning, by defining em textbfgeneralized propagation with directed and weighted graphs. We include a preliminary exploration of learned propagation patterns in datasets, a first in the field.
  arXiv  Detail & Related papers  (2024-02-13T14:13:17Z)
• The twin peaks of learning neural networks [3.382017614888546]
  Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks. We explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks.
  arXiv  Detail & Related papers  (2024-01-23T10:09:14Z)
• The Decimation Scheme for Symmetric Matrix Factorization [0.0]
  Matrix factorization is an inference problem that has acquired importance due to its vast range of applications. We study this extensive rank problem, extending the alternative 'decimation' procedure that we recently introduced. We introduce a simple algorithm based on a ground state search that implements decimation and performs matrix factorization.
  arXiv  Detail & Related papers  (2023-07-31T10:53:45Z)
• Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network [59.79008107609297]
  We propose in this paper to approximate the joint posterior over the structure of a Bayesian Network. We use a single GFlowNet whose sampling policy follows a two-phase process. Since the parameters are included in the posterior distribution, this leaves more flexibility for the local probability models.
  arXiv  Detail & Related papers  (2023-05-30T19:16:44Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Dense Hebbian neural networks: a replica symmetric picture of unsupervised learning [4.133728123207142]
  We consider dense, associative neural-networks trained with no supervision. We investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations.
  arXiv  Detail & Related papers  (2022-11-25T12:40:06Z)
• Semi-Supervised Clustering of Sparse Graphs: Crossing the Information-Theoretic Threshold [3.6052935394000234]
  Block model is a canonical random graph model for clustering and community detection on network-structured data. No estimator based on the network topology can perform substantially better than chance on sparse graphs if the model parameter is below a certain threshold. We prove that with an arbitrary fraction of the labels feasible throughout the parameter domain.
  arXiv  Detail & Related papers  (2022-05-24T00:03:25Z)
• On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias [50.84569563188485]
  We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons. Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
  arXiv  Detail & Related papers  (2022-05-18T16:57:10Z)
• Data-heterogeneity-aware Mixing for Decentralized Learning [63.83913592085953]
  We characterize the dependence of convergence on the relationship between the mixing weights of the graph and the data heterogeneity across nodes. We propose a metric that quantifies the ability of a graph to mix the current gradients. Motivated by our analysis, we propose an approach that periodically and efficiently optimize the metric.
  arXiv  Detail & Related papers  (2022-04-13T15:54:35Z)
• Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration [0.0]
  We include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results show that the field can induce an order-disorder phase transition by breaking or restoring the symmetry. We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.
  arXiv  Detail & Related papers  (2020-09-30T18:44:18Z)
• Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
  We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
  arXiv  Detail & Related papers  (2020-02-20T15:43:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.