8 problems

Hilbert-structure conjecture. The Hilbert structure is what enables the Eckart–Young-like results. The source explicitly notes that it has not established the link between the Hilb…

In the setting above, let $k=\sqrt{K(U(\mathcal{T}_{2r}))}$, so that the current estimates yield $k=m^{M/2}$ in one normalization and $k=1$ in the favorable case. Generic chaining…

Let an antisymmetric tensor be a tensor invariant under the appropriate sign changes induced by permutations of its indices, and consider tensors of multilinear rank $(r,r,r)$, wit…

Let $n-1,d-1\in\mathord{\mathbb N}$, let $k\in[d-1]$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb F}^n$. A best $k$-approximation of a tensor is an optimal orthogonal projecti…

Stabilization conjecture. In the Gaussian setting of Theorem 1, the expected number of critical points of $d_v$ on $X$ does not decrease when $n_p$ is replaced by $n_p-1$.

Generic uniqueness conjecture. Every $\mathcal{T}\in\mathbb{R}^{n_1\times\cdots\times n_d}$ has a unique rank-one approximation, unless $\mathcal{T}$ lies on a subvariety; moreover…

Let $\underline{\bf X}$ be a tensor in orbit $D_3$, and let $\underline{\bf Y}$ be a best rank-$1$ approximation of $\underline{\bf X}$. Orbit $D_3$ stability conjecture. Almost al…

Let $\underline{\bf X}$ be a generic real-valued $p\times p\times 2$ tensor, and let $\underline{\bf Y}$ be a best rank-$1$ approximation of $\underline{\bf X}$. Write…