TCS+ talk: Wednesday, October 9, Renato Ferreira Pinto Jr, U Waterloo
Clement Canonne
Our next talk will take place this coming Wednesday, October 9th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Renato Ferreira Pinto Jr from U Waterloo will speak about "Directed Isoperimetry and Monotonicity Testing: A Dynamical Approach" (abstract below).
Please sign up on the online form at https://sites.google.com/view/tcsplus/welcome/next-tcs-talk if you wish to join the talk as an individual or a group. Registration is /not/ required to attend the interactive talk, and the link will be posted on the website the day prior to the talk; however, by registering in the form, you will receive a reminder, along with the link. (The link to the recording will also be posted on our website afterwards.)
Hoping to see you all there,
The organizers
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Speaker: Renato Ferreira Pinto Jr (U Waterloo)
Title: Directed Isoperimetry and Monotonicity Testing: A Dynamical Approach
Abstract: This work explores the connection between classical isoperimetric inequalities, their directed analogues, and monotonicity testing. We study the setting of real-valued functions $f : [0,1]^d \to \mathbb{R}$ on the solid unit cube, where the goal is to test with respect to the $L^p$ distance. Our goals are twofold: to further understand the relationship between classical and directed isoperimetry, and to give a monotonicity tester with sublinear query complexity in this setting.
Our main results are 1) an $L^2$ monotonicity tester for $M$-Lipschitz functions with query complexity $\widetilde O(\sqrt{d} M^2 / \epsilon^2)$ and, behind this result, 2) the directed Poincaré inequality $\mathsf{dist}^{\mathsf{mono}}_2(f)^2 \le C \mathbb{E}[|\nabla^- f|^2]$, where the "directed gradient" operator $\nabla^-$ measures the local violations of monotonicity of $f$.
To prove the second result, we introduce a partial differential equation (PDE), the directed heat equation, which takes a one-dimensional function $f$ into a monotone function $f^*$ over time and enjoys many desirable analytic properties. We obtain the directed Poincaré inequality by combining convergence aspects of this PDE with the theory of optimal transport. Crucially for our conceptual motivation, this proof is in complete analogy with the mathematical physics perspective on the classical Poincaré inequality, namely as characterizing the convergence of the standard heat equation toward equilibrium.