报告题目：Stein-Log-Sobolev inequalities for the continuous Stein variational gradient descent method

报告摘要：The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between H⁻¹(ℝⁿ) and H¹(ℝⁿ), which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.

报告人简介：何塞・安东尼奥・卡里略，牛津大学数学研究所非线性偏微分方程讲席教授、牛津大学王后学院应用数学院士，欧洲科学院院士、欧洲科学院数学学部主任，国际应用数学、动力系统与数理生物领域顶尖学者。连续多年入选科睿唯安全球高被引科学家，成果刊发于 CPAM、Inventiones Mathematicae 等顶级数学期刊。曾获西班牙皇家科学院埃切加赖奖章、欧洲青年应用数学奖、英国皇家学会沃尔夫森研究奖等重要学术荣誉，当选 SIAM、AMS 会士，2023 年受邀在国际工业与应用数学大会做大会特邀报告。历任欧洲数学会应用数学委员会主席、国际工业与应用数学理事会理事，长期主持欧洲研究委员会高级基金项目，频繁在全球高校开设暑期学校、主持国际学术会议，推动偏微分方程在生命科学、工程领域的交叉应用，培养了大批欧美青年数学人才。

报告时间：2026年7月25日 9:30-10:30

报告地点：武之楼412教室