10 problems

Let $\mathbf{L}^{\rm spat}$ and $\mathbf{L}^{\rm temp}$ denote the spatial and temporal components of the inflated dynamic Laplacian, let $a$ be the coupling parameter, and let…

Let $K$ be fixed. Let $U$ be the population eigenvector matrix and let $\widehat{U}$ be the corresponding sample eigenvector matrix. For a matrix $M$, write…

Conjecture on CHIME's performance. This phenomenon occurs because the differences between $\boldsymbol{\mu}_1$ and $\boldsymbol{\mu}_2$ are small, while…

Largest-entry conjecture. The largest entry of $\mathbf{v_2}$ satisfies

The CPQR procedure is applied to structured matrices arising in this spectral clustering setting. While matrices exist for which the worst-case CPQR bounds used are achieved, the s…

Let $\mathrm{P}$ be a probability measure whose support is a union of compact topological connected components. For $T>0$, let $\mathcal C_T$ denote the connected components of the…

Generalized geodesic-distance conjecture. Assumptions (A1)--(A3) imply that these equations have asymptotic solutions, and that the conclusions of the pointwise geodesic-distance t…

Let a kernelized spectral clustering method depend on a bandwidth parameter, and consider its population-level performance as a function of that bandwidth. Optimal-bandwidth conjec…

In a community-detection problem, compare regular spectral clustering with spectral clustering followed by $K$-means, under uniform and nonuniform node weights. K-means limitation…

Let $\rho$ be the eigenvalue scaling factor and let $R$ be the spectral decimation polynomial for the relevant Vicsek set. If $t$ is the largest fixed point of $R$, the paper uses…