Multiplier Operator on Framed Hilbert Spaces

Very often the operators that we study appear most naturally in highly non-diagonal representation. The main goal of spectral theory is to solve this problem by exhibiting for many operators a natural orthonormal basis with respect to which the operators have diagonal representations. However, this can be done only for certain classes of operators. The most important such class is probably the class of compact operators. The problem is that it is often hard to tell whether an operator is compact looking at its non-diagonal representation. In this thesis, we will study a class of operators for which we can determine all of their basic operator-theoretic properties from their original representation which is not diagonal in the classical sense. There are many important subclasses of operators which belong in our class, including Toeplitz operators on various function spaces, some pseudo-differential operators, some singular integral operators, etc

Uncertainty Principles in Framed Hilbert Spaces

The uncertainty principle is one of the most fundamental concepts in harmonic analysis. It has many facets and appears in many different forms. In the first two chapters of the thesis, we state the classical uncertainty principle and the Balian-Low theorem, and introduce the concept of framed Hilbert spaces as a general setting for studying problems of uncertainty principle type. These spaces could be viewed as a special type of reproducing kernel Hilbert spaces which include many function spaces that play an important role in harmonic analysis. The thesis mainly consists of Chapters 3-5 whose common theme is the uncertainty principle. In Chapter 3, we define a very general interpolation set called d-approximate interpolation set in framed Hilbert spaces. And we prove a necessary density condition for those sets similar to the one for usual interpolation sets. In Chapter 4, we focus on the problem of estimating the sampling constant in the space of multiband-limited functions. Comparing with the old results, by imposing suboptimal conditions on the sampling set, we obtain a much-improved sampling constant which depends linearly on the length of the multiband. In Chapter 5, we give a very general version of the Balian-Low theorem, which does not necessarily require the orthonormal basis to be the time-frequency shifts of a single function

FedPrompt: Communication-Efficient and Privacy Preserving Prompt Tuning in Federated Learning

Federated learning (FL) has enabled global model training on decentralized data in a privacy-preserving way by aggregating model updates. However, for many natural language processing (NLP) tasks that utilize pre-trained language models (PLMs) with large numbers of parameters, there are considerable communication costs associated with FL. Recently, prompt tuning, which tunes some soft prompts without modifying PLMs, has achieved excellent performance as a new learning paradigm. Therefore we want to combine the two methods and explore the effect of prompt tuning under FL. In this paper, we propose "FedPrompt" as the first work study prompt tuning in a model split learning way using FL, and prove that split learning greatly reduces the communication cost, only 0.01% of the PLMs' parameters, with little decrease on accuracy both on IID and Non-IID data distribution. This improves the efficiency of FL method while also protecting the data privacy in prompt tuning.In addition, like PLMs, prompts are uploaded and downloaded between public platforms and personal users, so we try to figure out whether there is still a backdoor threat using only soft prompt in FL scenarios. We further conduct backdoor attacks by data poisoning on FedPrompt. Our experiments show that normal backdoor attack can not achieve a high attack success rate, proving the robustness of FedPrompt.We hope this work can promote the application of prompt in FL and raise the awareness of the possible security threats