 Award ID(s):
 1945652
 Publication Date:
 NSFPAR ID:
 10324443
 Journal Name:
 Proceedings of the 38th International Conference on Machine Learning
 Volume:
 139
 Sponsoring Org:
 National Science Foundation
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The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ \emph{symmetric} doubly stochastic matrix, then \[ \frac{1\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1\lambda_{2}(M))} \] where $\phi(M)=\min_{S\subseteq[n],S\leq n/2}\left(\frac{1}{S}\sum_{i\in S,j\not\in S}M_{i,j}\right)$ is the edge expansion of $M$, and $\lambda_{2}(M)$ is the second largest eigenvalue of $M$. We study the relationship between $\phi(A)$ and the spectral gap $1\re\lambda_{2}(A)$ for \emph{any} doubly stochastic matrix $A$ (not necessarily symmetric), where $\lambda_{2}(A)$ is a nontrivial eigenvalue of $A$ with maximum real part. Fiedler showed that the upper bound on $\phi(A)$ is unaffected, i.e., $\phi(A)\leq\sqrt{2\cdot(1\re\lambda_{2}(A))}$. With regards to the lower bound on $\phi(A)$, there are known constructions with \[ \phi(A)\in\Theta\left(\frac{1\re\lambda_{2}(A)}{\log n}\right), \] indicating that at least a mild dependence on $n$ is necessary to lower bound $\phi(A)$. In our first result, we provide an \emph{exponentially} better construction of $n\times n$ doubly stochastic matrices $A_{n}$, for which \[ \phi(A_{n})\leq\frac{1\re\lambda_{2}(A_{n})}{\sqrt{n}}. \] In fact, \emph{all} nontrivial eigenvalues of our matrices are $0$, even though the matrices are highly \emph{nonexpanding}. We further show that this bound is in the correct range (up to the exponent of $n$), by showing that for any doubly stochastic matrix $A$, \[ \phi(A)\geq\frac{1\re\lambda_{2}(A)}{35\cdot n}. \] As a consequence, unlike the symmetric case, there is a (necessary) loss of amore »

Consider the linear transport equation in 1D under an external confining potential
:\begin{document}$ \Phi $\end{document} For
(with\begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} small), we prove phase mixing and quantitative decay estimates for\begin{document}$ \varepsilon >0 $\end{document} , with an inverse polynomial decay rate\begin{document}$ {\partial}_t \varphi : =  \Delta^{1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document} . In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in\begin{document}$ O({\langle} t{\rangle}^{2}) $\end{document} D under the external potential\begin{document}$ 1 $\end{document} .\begin{document}$ \Phi $\end{document} 
A longstanding conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree $T$ with $n$~edges, it is conjectured that there exists a labeling $f\colon V(T) \to \{0,1,\ldots,n\}$ such that the set of induced edge labels $\bigl\{ f(u)f(v) : \{u,v\}\in E(T) \bigr\}$ is exactly $\{1,2,\ldots,n\}$. We extend this concept to allow for multigraphs with edge multiplicity at most~$2$. A \emph{2fold graceful labeling} of a graph (or multigraph) $G$ with $n$~edges is a onetoone function $f\colon V(G) \to \{0,1,\ldots,n\}$ such that the multiset of induced edge labels is comprised of two copies of each element in $\bigl\{ 1,2,\ldots, \lfloor n/2 \rfloor \bigr\}$ and, if $n$ is odd, one copy of $\bigl\{ \lceil n/2 \rceil \bigr\}$. When $n$ is even, this concept is similar to that of 2equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length $n \not\equiv 1 \pmod{4}$, and complete bipartite graphs admit 2fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is $2$fold graceful.

Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i N P does not haveE M A J ∘A C C ^{0}∘T H R circuits of polynomialmore » 
Abstract Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \ x_i  x_j \^2} {\sigma ^2} ) $ is widely used in graphbased geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $\sigma $, and a common practice called selftuned kernel adaptively sets a $\sigma _i$ at each point $x_i$ by the $k$nearest neighbor (kNN) distance. When $x_i$s are sampled from a $d$dimensional manifold embedded in a possibly highdimensional space, unlike with fixedbandwidth kernels, theoretical results of graph Laplacian convergence with selftuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted)Laplacian for a new family of kNN selftuned kernels $W^{(\alpha )}_{ij} = k_0( \frac{ \ x_i  x_j \^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $, where $\hat{\rho }$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $\alpha $. When $\alpha = 1$, the limiting operator is the weighted manifold Laplacian $\varDelta _p$. Specifically, we prove the pointwise convergence of $L_N f $ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$more »