19 problems

Let $P$) and $Q$ be two colored paths on $m$ vertices with concentration matrices $K_P$ and $K_Q$ satisfying … A reflection of a colored path is the path obtained by reversing its…

Let $\mathcal{G}$ be a colored graph, with concentration matrix $K$ and vanishing ideal $I_{\mathcal{G}}$. A symmetry of $\mathcal{G}$ is a permutation matrix $P$ satisfying … For…

Let $P$ and $Q$ be colored paths, and consider the determinant of their concentration matrices. A non-trivial path color configuration is a pair of colorings, other than the config…

Let $\mathcal{G}$ be an undirected colored cycle, and let $I_{\mathcal{G}}$ denote its vanishing ideal. A linear binomial is a binomial of degree one in the generators of this idea…

Let $C_n$ be the cycle graph on $n$ vertices, and let $\MLdeg(C_n)$ denote the number of critical points of the Gaussian log-likelihood function for generic sample covariance data,…

Gaussoid-axioms sufficiency conjecture. If $a \perp\!\!\!\perp b \mid C$ holds for every covariance matrix $\Sigma \in \mathcal{M}_{\mathcal{G}, i \perp\!\!\!\perp j\mid K}$, then…

Sturmfels–Uhler conjecture. The ideal $I_n$ is the radical ideal defining $X_{L_{C_n}}$.

Let $G$ be a directed acyclic graph (DAG) with toric vanishing ideal $I_G$. The relevant constructions are safe gluing, gluing at sinks, and adding a sink. Recursive classification…

Let $G$ be a directed acyclic graph (DAG), with vanishing ideal $I_G$ and shortest trek ideal $ST_G$. Shortest-trek characterization conjecture. The vanishing ideal $I_G$ is toric…

Let $G$ be a directed acyclic graph (DAG), and let $I_G$ denote its vanishing ideal. The operations considered are safe gluing, gluing at sinks, and adding a new sink; a complete D…

Let $G_1$ and $G_2$ be directed acyclic graphs (DAGs) with toric vanishing ideals. A safe gluing combines them along a common clique, subject to the safe-gluing conditions describe…

Let $L_n$ be the linear space associated to the Gaussian $n$-cycle model, and let $L_n^{-1}$ denote its reciprocal variety and $L_n^\perp$ its polar space. A scheme is monomial if…

Uhler's volume inequality conjecture. In the situation of the figure,

Let $C_p$ be the cycle graph of length $p\geq 3$, and let its maximum likelihood degree (ML degree) be the number of solutions to the likelihood equations for generic data. Exponen…

Elementwise error-rate conjecture. For general $\mathcal{E}^*$, the rates of convergence of the elementwise $\ell_{\infty}$ error are at least comparable to those achieved by…

Uh­ler's conjecture. In each of these three models, the MLE does not exist for $n=1$.

Let $G^*$ be a true Gaussian graphical model and let $\mathrm{BIC}_{\gamma}(G)$ denote the extended Bayesian information criterion defined above. The theorem establishes consistenc…

Let $\mathcal{C}_{\mathcal{G}}$ denote the cone associated with a graph $\mathcal{G}$. A basic open semialgebraic set is an open semialgebraic set defined by finitely many strict p…

Adaptive-Lasso covariance-selection conjecture. The conditions imposed for the adaptive-Lasso pursuit are more general than the conditions required for the GLasso.