An official website of the United States government
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 a factor of $$n^{\alpha}$ for $$\frac{1}{2}\leq\alpha\leq1$$ in lower bounding $$\phi$$ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices $$R$$ with nonnegative entries, to obtain a two-sided \emph{gapped} refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such $$R$$, there is a nonnegative eigenvalue $$r$$ such that all eigenvalues of $$R$$ lie within the closed disk of radius $$r$$ about $$0$$. Further, if $$R$$ is irreducible, which means $$\phi(R)>0$$ (for suitably defined $$\phi$$), then $$r$$ is positive and all other eigenvalues lie within the \textit{open} disk, so (with eigenvalues sorted by real part), $$\re\lambda_{2}(R)
more »
« less
Analysis of stochastic Lanczos quadrature for spectrum approximation
The cumulative empirical spectral measure (CESM) $$\Phi[\mathbf{A}] : \mathbb{R} \to [0,1]$$ of a $$n\times n$$ symmetric matrix $$\mathbf{A}$$ is defined as the fraction of eigenvalues of $$\mathbf{A}$$ less than a given threshold, i.e., $$\Phi[\mathbf{A}](x) := \sum_{i=1}^{n} \frac{1}{n} {\large\unicode{x1D7D9}}[ \lambda_i[\mathbf{A}]\leq x]$$. Spectral sums $$\operatorname{tr}(f[\mathbf{A}])$$ can be computed as the Riemann–Stieltjes integral of $$f$$ against $$\Phi[\mathbf{A}]$$, so the task of estimating CESM arises frequently in a number of applications, including machine learning. We present an error analysis for stochastic Lanczos quadrature (SLQ). We show that SLQ obtains an approximation to the CESM within a Wasserstein distance of $$t \: | \lambda_{\text{max}}[\mathbf{A}] - \lambda_{\text{min}}[\mathbf{A}] |$$ with probability at least $$1-\eta$$, by applying the Lanczos algorithm for $$\lceil 12 t^{-1} + \frac{1}{2} \rceil$$ iterations to $$\lceil 4 ( n+2 )^{-1}t^{-2} \ln(2n\eta^{-1}) \rceil$$ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and Kolmogorov–Smirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments.
more »
« less
- Award ID(s):
- 1945652
- PAR ID:
- 10324443
- Date Published:
- Journal Name:
- Proceedings of the 38th International Conference on Machine Learning
- Volume:
- 139
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
For an $$r$$-uniform hypergraph $$H$$, let $$\nu^{(m)}(H)$$ denote the maximum size of a set $$M$$ of edges in $$H$$ such that every two edges in $$M$$ intersect in less than $$m$$ vertices, and let $$\tau^{(m)}(H)$$ denote the minimum size of a collection $$C$$ of $$m$$-sets of vertices such that every edge in $$H$$ contains an element of $$C$$. The fractional analogues of these parameters are denoted by $$\nu^{*(m)}(H)$$ and $$\tau^{*(m)}(H)$$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$$. In this paper we prove bounds on the ratio between the parameters $$\tau^{(m)}$$ and $$\nu^{(m)}$$, and their fractional analogues. Our main result is that, for every $$r$$-uniform hypergraph~$$H$$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $$r-1$.We also prove that, for every $$r$$-uniform hypergraph $$H$$, $$\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$$, where the Turán number $$\operatorname{ex}_r(n, k)$$ is the maximum number of edges in an $$r$$-uniform hypergraph on $$n$$ vertices that does not contain a copy of the complete $$r$$-uniform hypergraph on $$k$$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$.more » « less
-
Consider the linear transport equation in 1D under an external confining potential \begin{document}$$ \Phi $$\end{document}: \begin{document}$$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $$\end{document} For \begin{document}$$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $$\end{document} (with \begin{document}$$ \varepsilon >0 $$\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $$\end{document}, with an inverse polynomial decay rate \begin{document}$$ O({\langle} t{\rangle}^{-2}) $$\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $$\end{document}D under the external potential \begin{document}$$ \Phi $$\end{document}$.more » « less
-
Existing parallel algorithms for wavelet tree construction have a work complexity of $$O(n\log\sigma)$$. This paper presents parallel algorithms for the problem with improved work complexity. Our first algorithm is based on parallel integer sorting and has either $$O(n\log\log n\lceil\log\sigma/\sqrt{\log n\log\log n}\rceil)$$ work and polylogarithmic depth, or $$O(n\lceil\log\sigma/\sqrt{\log n}\rceil)$$ work and sub-linear depth. We also describe another algorithm that has $$O(n\lceil\log\sigma/\sqrt{\log n} \rceil)$$ work and $$O(\sigma+\log n)$$ depth. We then show how to use similar ideas to construct variants of wavelet trees (arbitrary-shaped binary trees and multiary trees) as well as wavelet matrices in parallel with lower work complexity than prior algorithms. Finally, we show that the rank and select structures on binary sequences and multiary sequences, which are stored on wavelet tree nodes, can be constructed in parallel with improved work bounds, matching those of the best existing sequential algorithms for constructing rank and select structures.more » « less
-
The spread of a graph $$G$$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $$G$$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $$n$$, the $$n$$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $$\lceil(2n-1)/3\rceil$$ vertices and $$\lfloor(n-2)/3\rfloor$$ isolated vertices. For planar graphs, we show that the extremal $$n$$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $$\lceil(2n-2)/3\rceil$$ vertices and $$\lfloor(n-4)/3\rfloor$$ isolated vertices.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
