Abstract: For a partition of [0,1] into intervals I1,...,In we prove the existence of a partition of ℤ into Λ1,..., Λn such that the complex exponential functions with frequencies in Λk form a Riesz basis for L2(Ik), and furthermore, that for any J⊆{1,2,...,n}, the exponential functions with frequencies in ⋃j∈J Λj form a Riesz basis for L2(I) for any interval I with length |I|=Σj∈J |Ij|. The construction extends to infinite partitions of [0,1], but with size limitations on the subsets J⊆ ℤ. The construction utilizes an interesting assortment of tools from analysis, probability, and number theory. This is joint work with Shauna Revay (GMU and Novetta), and Goetz Pfander (Catholic University of Eichstaett-Ingolstadt).
Abstract: The primary task of many applications is approximating/estimating a function through samples drawn from a probability distribution on the input space. The deep approximation is to approximate a function by compositions of many layers of simple functions, that can be viewed as a series of nested feature extractors. The key idea of deep learning network is to convert layers of compositions to layers of tuneable parameters that can be adjusted through a learning process, so that it achieves a good approximation with respect to the input data. In this talk, we shall discuss mathematical theory behind this new approach and approximation rate of deep network; we will also how this new approach differs from the classic approximation theory, and how this new theory can be used to understand and design deep learning network.
Abstract: The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in cryo-EM. Specifically, modern computational methods for three-dimensional map sharpening and atomic modeling of macromolecules by single particle cryo-EM are based on Wilson statistics. In this talk we use certain results about the decay rate of the Fourier transform to provide the first rigorous mathematical derivation of Wilson statistics. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher order spectra. These in turn provide theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation based methods in cryo-EM.
Abstract: Quantum field theory uses canonical quantization (CQ), and often fails, e.g., $\varphi^4_4$, etc. Affine quantization (AQ) - which will be introduced - can solve a variety of problems that CQ cannot. AQ can even be used to solve certain models regarded as nonrenormalizable. The specific procedures of AQ lead to a novel Fourier transformation that illustrates how AQ can create a generous contribution to quantum field theory.
Abstract: Quantum computers are moving out of physics labs and becoming generally programmable. In this talk, we start from quantum algorithms like magic state distillation and Shor factoring that make essential use of diagonal logical gates. The difficulty of reliably implementing these gates in some quantum error correcting code (QECC) is measured by their level in the Clifford hierarchy, a mathematical framework that was defined by Gottesman and Chuang when introducing the teleportation model of quantum computation. We describe a method of working backwards from a target logical diagonal gate at some level in the Clifford hierarchy to a quantum error correcting code (CSS code) in which the target logical can be implemented reliably.
This talk describes joint work with my graduate students Jingzhen Hu and Qingzhong Liang.
Abstract: Over the last decade, algorithms and systems based on deep learning and other data-driven methods have contributed to the reemergence of Artificial Intelligence-based systems with applications in national security, defense, medicine, intelligent transportation, and many other domains. However, another AI winter may be lurking around the corner if challenges due to bias, domain shift and lack of robustness to adversarial attacks are not considered while designing the AI systems. In this talk, I will present our approach to bias mitigation and designing AI systems that are robust to domain shift and a variety of adversarial attacks. Zoom Link for November 22nd Webinar: