10 problems

Scaling conjecture for finite graphs. Consider the finite SIRG $\mathbb{G}^{(n)}$. If $\phi(k_n)=o(\psi(n))$, then

Let $D\subset\mathbb R^d$ be open, bounded, and convex, and let $\mathcal W$ be a renormalization class on $\overline D$. Assume that the asymptotic fixed-point equation has a uniq…

Let $D\subset\mathbb R^d$ be open, bounded, and convex, let ${\cal W}$ be a renormalization class on $\overline D$, and suppose that the asymptotic fixed-point equation in the zero…

Let $n$, $d$, and $K$ be positive integers with $d\geq K$, and let $\Delta_*$ denote the separation parameter for the Gaussian-mixture clustering model. Write $a\gtrsim_{\log} b$ w…

Let $Q$ and $K$ be arbitrary matrices, and let $V$ be a matrix whose largest eigenvalue $\lambda_{\max}>0$ is real, with eigenspace $L$ satisfying $\mathop{\rm \dim} L\geq2$. Assume…

Let $Q$ and $K$ be arbitrary matrices, and let $V$ be diagonalizable with $d$ different positive real eigenvalues. Denote its largest eigenvalue by $\lambda_{\max}$, and let $\xi$…

Generalized uniqueness conjecture. The ground-truth clustering ${\cal C}$ is the only valid clustering consistent with the entire query matrix.

Let $k=2$, let $\bfX\in\{0,1\}^n$ be the label vector, and let a querying scheme use $m$ same cluster' queries, each involving two elements ($\Delta=2$). A scheme is $(1-\delta)$-g…

SDP recovery conjecture. The SDP relaxation recovers the clusters with high probability whenever

Let $\psi^{\sharp}$ denote either spin field, let $\lambdaup>0$ be sufficiently small, and let $\Lambdaup\subset\mathbb{Z}^d$ be a finite volume. For even $n\geq 2$, write…