Learning and Testing Variable Partitions

$$Let $F$ be a multivariate function from a product set $\Sigma^n$ to an Abelian group $G$. A $k$-partition of $F$ with cost $\delta$ is a partition of the set of variables $\mathbf{V}$ into $k$ non-empty subsets $(\mathbf{X}_1, \dots, \mathbf{X}_k)$ such that $F(\mathbf{V})$ is $\delta$-close to $F_1(\mathbf{X}_1)+\dots+F_k(\mathbf{X}_k)$ for some $F_1, \dots, F_k$ with respect to a given error metric. We study algorithms for agnostically learning $k$ partitions and testing $k$-partitionability over various groups and error metrics given query access to $F$. In particular we show that $1.$ Given a function that has a $k$-partition of cost $\delta$, a partition of cost $\mathcal{O}(k n^2)(\delta + \epsilon)$ can be learned in time $\tilde{\mathcal{O}}(n^2 \mathrm{poly} (1/\epsilon))$ for any $\epsilon > 0$. In contrast, for $k = 2$ and $n = 3$ learning a partition of cost $\delta + \epsilon$ is NP-hard. $2.$ When $F$ is real-valued and the error metric is the 2-norm, a 2-partition of cost $\sqrt{\delta^2 + \epsilon}$ can be learned in time $\tilde{\mathcal{O}}(n^5/\epsilon^2)$. $3.$ When $F$ is $\mathbb{Z}_q$-valued and the error metric is Hamming weight, $k$-partitionability is testable with one-sided error and $\mathcal{O}(kn^3/\epsilon)$ non-adaptive queries. We also show that even two-sided testers require $\Omega(n)$ queries when $k = 2$. This work was motivated by reinforcement learning control tasks in which the set of control variables can be partitioned. The partitioning reduces the task into multiple lower-dimensional ones that are relatively easier to learn. Our second algorithm empirically increases the scores attained over previous heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202

Analyticity for the (generalized) Navier-Stokes equations with rough initial data

We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in critical Besov spaces $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$ with $1< p<\infty, \ 1\le q\le \infty$ and the solution is global if $u_0$ is sufficiently small in $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$. In the case $p=\infty$, the analyticity for the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in modulation space $M^{-1}_{\infty,1}(\mathbb{R}^n)$ is obtained. We prove the global well-posedness for a fractional Navier-stokes equation ($\alpha=1/2$) with small data in critical Besov spaces $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p\leq\infty)$ and show the analyticity of solutions with small initial data either in $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p<\infty)$ or in $\dot{B}^0_{\infty,1} (\mathbb{R}^n)\cap {M}^0_{\infty,1}(\mathbb{R}^n)$. Similar results also hold for all $\alpha\in (1/2,1)$.Comment: 31 page

Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schr\"odinger Equations in Multi-dimensional Spaces

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u})_{\abs{\alpha}\ls 2}),\quad t\in \Real, x\in\Real^n, where \eps\in\{-1,0,1\}, n\gs 2 denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.Comment: 28 page