Pub Date : 2019-01-01DOI: 10.4310/amsa.2019.v4.n2.a6
T. Lai, Anna Choi, K. Tsang
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Pub Date : 2018-09-17DOI: 10.4310/amsa.2019.v4.n2.a3
V. Moncrief, A. Marini, R. Maitra
It has long been realized that the natural orbit space for non-abelian Yang-Mills dynamics is a positively curved (infinite dimensional) Riemannian manifold. Expanding on this result I.M. Singer proposed that strict positivity of the corresponding Ricci tensor (computable through zeta function regularization) could play a fundamental role in establishing that the associated Schroedinger operator admits a spectral gap. His argument was based on representing the (regularized) kinetic term in the Schroedinger operator as a Laplace-Beltrami operator on this positively curved orbit space. We revisit Singer's proposal and show how, when the contribution of the Yang-Mills potential energy is taken into account, the role of the original orbit space Ricci tensor is instead played by a Bakry-Emery Ricci tensor computable from the ground state wave functional of the quantum theory. We next review our ongoing Euclidean-signature-semi-classical program for deriving asymptotic expansions for such wave functionals and discuss how, by keeping the dynamical nonlinearities and non-abelian gauge invariances intact at each level of the analysis, our approach surpasses that of conventional perturbation theory for the generation of approximate wave functionals. Though our main focus is on Yang-Mills theory we derive the orbit space curvature for scalar electrodynamics and prove that, whereas the Maxwell factor remains flat, the interaction naturally induces positive curvature in the (charged) scalar factor of the resulting orbit space. This has led us to the conjecture that such orbit space curvature effects could furnish a source of mass for ordinary Klein-Gordon type fields provided the latter are (minimally) coupled to gauge fields, even in the abelian case. Finally we discuss the potential applicability of our Euclidean-signature program to the Wheeler-DeWitt equation of canonical quantum gravity.
{"title":"Orbit space curvature as a source of mass in quantum gauge theory","authors":"V. Moncrief, A. Marini, R. Maitra","doi":"10.4310/amsa.2019.v4.n2.a3","DOIUrl":"https://doi.org/10.4310/amsa.2019.v4.n2.a3","url":null,"abstract":"It has long been realized that the natural orbit space for non-abelian Yang-Mills dynamics is a positively curved (infinite dimensional) Riemannian manifold. Expanding on this result I.M. Singer proposed that strict positivity of the corresponding Ricci tensor (computable through zeta function regularization) could play a fundamental role in establishing that the associated Schroedinger operator admits a spectral gap. His argument was based on representing the (regularized) kinetic term in the Schroedinger operator as a Laplace-Beltrami operator on this positively curved orbit space. We revisit Singer's proposal and show how, when the contribution of the Yang-Mills potential energy is taken into account, the role of the original orbit space Ricci tensor is instead played by a Bakry-Emery Ricci tensor computable from the ground state wave functional of the quantum theory. We next review our ongoing Euclidean-signature-semi-classical program for deriving asymptotic expansions for such wave functionals and discuss how, by keeping the dynamical nonlinearities and non-abelian gauge invariances intact at each level of the analysis, our approach surpasses that of conventional perturbation theory for the generation of approximate wave functionals. Though our main focus is on Yang-Mills theory we derive the orbit space curvature for scalar electrodynamics and prove that, whereas the Maxwell factor remains flat, the interaction naturally induces positive curvature in the (charged) scalar factor of the resulting orbit space. This has led us to the conjecture that such orbit space curvature effects could furnish a source of mass for ordinary Klein-Gordon type fields provided the latter are (minimally) coupled to gauge fields, even in the abelian case. Finally we discuss the potential applicability of our Euclidean-signature program to the Wheeler-DeWitt equation of canonical quantum gravity.","PeriodicalId":42896,"journal":{"name":"Annals of Mathematical Sciences and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45122487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-05-22DOI: 10.4310/AMSA.2019.V4.N1.A1
Wenqing Hu, C. J. Li, Lei Li, Jian‐Guo Liu
We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.
{"title":"On the diffusion approximation of nonconvex stochastic gradient descent","authors":"Wenqing Hu, C. J. Li, Lei Li, Jian‐Guo Liu","doi":"10.4310/AMSA.2019.V4.N1.A1","DOIUrl":"https://doi.org/10.4310/AMSA.2019.V4.N1.A1","url":null,"abstract":"We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.","PeriodicalId":42896,"journal":{"name":"Annals of Mathematical Sciences and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45314559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.4310/amsa.2022.v7.n1.a3
Tie-xiang Li, Pei Chuang, M. Yueh
Facial expression recognition (FER) is an active topic that has many applications. The development of effective algorithms for FER has been a competitive research field in the last two decades. In this paper, we propose a fully automatic 3D FER method based on the sparse approximation of 2D feature images. For a prescribed feature defined on the 3D facial surface, we apply a parameterization that not only maps the facial surface onto the unit disk but also locally preserves the feature. To ensure the uniqueness of the solution, some aligning constraints are further taken into account while computing the desired parameterization. The facial surface associated with the feature is then converted into the 2D image of the parameter domain. To recognize the expression of a test facial image, we apply an existing 2D expression recognition model, which is built upon sparse representation. Numerical experiments indicate that the accuracy of the proposed FER algorithm reaches 71.42% on a benchmark facial expression database, which is promising for practical applications.
{"title":"An optimal transportation-based recognition algorithm for 3D facial expressions","authors":"Tie-xiang Li, Pei Chuang, M. Yueh","doi":"10.4310/amsa.2022.v7.n1.a3","DOIUrl":"https://doi.org/10.4310/amsa.2022.v7.n1.a3","url":null,"abstract":"Facial expression recognition (FER) is an active topic that has many applications. The development of effective algorithms for FER has been a competitive research field in the last two decades. In this paper, we propose a fully automatic 3D FER method based on the sparse approximation of 2D feature images. For a prescribed feature defined on the 3D facial surface, we apply a parameterization that not only maps the facial surface onto the unit disk but also locally preserves the feature. To ensure the uniqueness of the solution, some aligning constraints are further taken into account while computing the desired parameterization. The facial surface associated with the feature is then converted into the 2D image of the parameter domain. To recognize the expression of a test facial image, we apply an existing 2D expression recognition model, which is built upon sparse representation. Numerical experiments indicate that the accuracy of the proposed FER algorithm reaches 71.42% on a benchmark facial expression database, which is promising for practical applications.","PeriodicalId":42896,"journal":{"name":"Annals of Mathematical Sciences and Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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