Abstract: While much of quantum cryptography requires communication of quantum states, recent research has proved major theoretical advantages in cryptography using quantum algorithms and classical communication only. This is an active and mathematically rich field which incorporates tools from lattice-based cryptography. The goal of this seminar series is to get a broad view of this area (including ongoing research at QuICS) with the hope of inspiring new ideas and collaboration. We welcome talks on related research directions, including proofs of quantumness and post-quantum cryptography. The seminar homepage is at https://qcrypto.umiacs.io .
Abstract: High dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves, results in empirically observed "stickiness" and poor theoretical mixing properties -- lack geometric ergodicity. In this talk, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the mixing time decreases with dimension.
(join work with Krzysztof Łatuszyński and Gareth O. Roberts)
Abstract: Many machine learning algorithms have been developed to elicit information from large datasets, however, techniques for quantifying the uncertainty in the estimates and for conducting statistical inference are not as well developed. We discuss resampling inferential techniques for these tasks, and present a unified way of studying the theoretical properties of these techniques using a framework involving random resampling weights. Depending on distributional properties of these random weights, the framework we propose can be used for consistent inference under very general conditions involving dependent data, high or infinite dimensional parameters and estimators obtained as approximate solutions to optimization problems, or alternatively for very precise and accurate inference in several other problems. Thus, our framework can balance the desirable robustness and efficiency goals of statistical inference. We further extend the framework for computational efficiency in big data applications, and prove that the extended framework is both uniformly consistent as well as computationally efficient under mild conditions. Illustrative examples from climate sciences, biomedical applications and ethical data sciences are discussed.
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