Mathematics (Faculty of)
http://hdl.handle.net/10012/9924
2023-02-02T15:43:13ZEfficient and Differentially Private Statistical Estimation via a Sum-of-Squares Exponential Mechanism
http://hdl.handle.net/10012/19141
Efficient and Differentially Private Statistical Estimation via a Sum-of-Squares Exponential Mechanism
Majid, Mahbod
As machine learning is applied to more privacy-sensitive data, it is becoming increasingly crucial to develop algorithms that maintain privacy. However, even the most basic high-dimensional statistical estimation tasks were not fully understood under differential privacy, specifically, there were no known efficient algorithms for mean estimation using the optimal number of samples under pure differential privacy.
We propose a new method for designing efficient and information-theoretically optimal algorithms for statistical estimation tasks that preserve privacy, using a combination of the Sum-of-Squares hierarchy and the exponential mechanism. The Sum-of-Squares hierarchy, a convex programming method, has been used to design efficient algorithms in robust statistics. The exponential mechanism, which has been widely used in differential privacy, is often used to design information-theoretically optimal algorithms, but can be inefficient. By combining these two approaches, we are able to create efficient algorithms that are also information-theoretically optimal. We apply this approach to mean estimation for heavy-tailed distributions and learning Gaussian distributions and achieve optimal results. We also show that this approach can be applied to other problems captured by the Sum-of-Squares hierarchy through a meta-theorem. Additionally, our algorithms highlight the strong connection between robustness and privacy.
We establish information-theoretical lower bounds to show the statistical optimality of our approaches. Technically we use packing lower bounds; however, the novelty of our lower bounds is in capturing the high probability setting.
2023-01-30T00:00:00ZA Stabilizer Formalism for Infinitely Many Qubits
http://hdl.handle.net/10012/19140
A Stabilizer Formalism for Infinitely Many Qubits
Kong, Xiangzhou
The study of infinite dimensional quantum systems has been an active area of discussion in quantum information theory, particularly in settings where certain properties are shown to be not attainable by any finite dimensional system (such as nonlocal correlations).
Similarly, the notion of stabilizer states has yielded interesting developments in areas like error correction, efficient simulation of quantum systems and its relation to graph states.
However, the commonly used model of tensor products of finite dimensional Hilbert spaces is not sufficiently general to capture infinite dimensional stabilizer states.
A more general framework quantum mechanical systems using C*-algebras has been instrumental in studying systems with an infinite number of discrete systems in quantum statistical mechanics and quantum field theory.
We propose a framework in the C*-algebra model (specifically, the CAR algebra) for the stabilizer formalism that extends to infinitely many qubits.
Importantly, the stabilizer states on the CAR algebra form a class of states that can attain unbounded entanglement and yet has a simple characterization through the group structure of its stabilizer.
In this framework, we develop a theory for the states, operations and measurements needed to study open questions in quantum information.
2023-01-27T00:00:00ZData-Driven Methods for System Identification and Lyapunov Stability
http://hdl.handle.net/10012/19139
Data-Driven Methods for System Identification and Lyapunov Stability
Quartz, Thanin
This thesis focuses on data-driven methods applied to system identification and stability analysis of dynamical systems. In the first major contribution of the theorem we propose a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing nonlinear controller. The SMT solver then verifies that the candidate Lyapunov function indeed satisfies the Lyapunov conditions. We provide theoretical guarantees of the proposed learning framework in terms of the
closed-loop stability for the unknown nonlinear system. We illustrate the effectiveness of the approach with a set of numerical experiments. We then examine another popular data driven method for system identification involving the Koopman operator. Methods based on the Koopman operator aim to approximate advancements of the state under the flow operator by a high-dimensional linear operator. This is accomplished by the extended mode decomposition (eDMD) algorithm which takes non-linear measurements of the state. Under the suitable conditions we have a result on the weak convergence of the eigenvalues and eigenfunctions of the eDMD operator that can serve as components of Lyapunov functions. Finally, we review methods for finding the region of attraction of an asymptotically stable fixed point and compare this method to the two methods mentioned above.
2023-01-27T00:00:00ZA Study of the Capabilities of Message-Oriented Middleware Systems
http://hdl.handle.net/10012/19137
A Study of the Capabilities of Message-Oriented Middleware Systems
Al-Manasrah, Wael
We present a comprehensive characterization study of open-source Message-Oriented Middleware (MOM) systems. We devised a rigorous methodology to select and study 10 popular and diverse MOM systems. For each system, we examine 42 features with a total of 134 different options. We found that MOM systems have evolved to provide a framework for modern cloud applications through high flexibility and configurability and by offering core building blocks for complex applications including transaction support, active messaging, resource management, flow control, and native support for multi-tenancy. A key result of our study, is that we believe there is an opportunity for the community to consolidate its efforts on fewer open-source projects.
We have also created an annotated data set that makes it easy to verify our findings, which can also be used to help practitioners and developers determine and understand the features of different systems. For a wider impact, our data set is publicly available at [https://docs.google.com/spreadsheets/d/1HrZ7ub19FuuBzA5z4aA6RfR5vnkdnm0bg3hxfADspEA/edit?usp=sharing].
2023-01-27T00:00:00Z