| High-dimensional data exists in many fields,such as computer vision,natural language processing,reinforcement learning,and condensed matter physics.With the increase of data dimensionality,the bottleneck of traditional data processing methods has gradually emerged,mainly reflected in the time or space complexity,or the amount of data required,which increases significantly with the dimensionality of data.Then the development of new algorithms becomes necessary to deal with such high-dimensional scenarios.In practice,we observe that high-dimensional data usually does not fill the entire high-dimensional space,but instead often has certain low-complexity features;for example,high-dimensional vectors are sparse,matrices have the low-rank property and general high-dimensional data often concentrates on some low-dimensional manifolds.Therefore,the development of algorithms utilizing special low dimensional properties of high-dimensional data can improve the efficiency of high-dimensional data processing.This dissertation will focus on mining low dimensional features of high-dimensional data in common physical scenarios and developing the corresponding special algorithms to deal with them.This dissertation can be divided into three parts with respect to three different physical scenarios,and each of them can be solved by exploring the lowdimensional structure in the corresponding scene.The first scene is the analytical continuation problem,which can be tackled with the help of sparsity.And this problem often hinders the wide application of the quantum Monte Carlo algorithm.Because the imaginary time Green function obtained in the Monte Carlo calculation needs to be converted into the real frequency spectral function through analytical continuation,and the bottleneck of this problem is the identification of the physical solution among many candidate ones.We notice that the original problem becomes a linear inverse problem after discretization,and the physical spectral function often have sparsity.Therefore,we can quantify the hardness with the help of compressed sensing theory and design corresponding neural networks to solve it.The second scenario is the image processing problem in the field of angular resolution photoemission spectroscopy(ARPES),for which we are inspired by low-rankness to design algorithms to handle.However,the experimental data is inevitably influenced by noise or grid with respect to the instrument,and the expensiveness of experimental data often hinders the construction of a large dataset required in common deep learning algorithms.We notice that the band information itself is interrelated and has a weak correlation with the additional mesh and the noise signal,so we try to separate both the band signal and the additional mesh or noise from the noisy observations.Motivated by the low-rank approximation,we implement a more complex autoencoder to model the energy band signal.The results show that a single image,rather than a training set,is enough to achieve this separation.In the third part,we come to the problem of finding the order parameter in manybody physics which has implicit low-dimensional structures.The many-body problem naturally has a high-dimensional space of configurations.Configurations respecting physical properties are often concentrated in some certain corners of the whole space.But it is difficult to identify the corresponding low dimensional structure.Thus,we apply the information bottleneck theory implemented by autoencoder to express such structure implicitly.The results about Ising model show the consistency between the low dimensional structure and the physical order parameter,and both of them are able to indicate phase transition.In addition,we note that different decoders are closely related to the form of the Hamiltonian.We can then design a decoder from the perspective of solving the Hamiltonian. |
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