10 problems

Let $Y=AX+E$ with $Y\in\mathbb{R}^{n\times p}$, $A\in\mathbb{R}^{m\times n}$, and $X\in\mathbb{R}^{m\times p}$, where $p\geq\max\\{n,m\\}$ and typically $m\geq n$. In the square no…

Let $Z_i$ ($i\in[d_z]$) be latent variables and let $A$ be a set of observed variables whose support sets $I_j$ satisfy the sufficient diversity condition: for each latent variable…

The paper studies matrix denoising and symmetric and non-symmetric dictionary learning in the Bayesian-optimal setting, with matrix signals of linear rank. Its solutions are expres…

Higher-order norm recovery conjecture. The dictionary can be recovered via maximizing any $ℓ_p$-norm with $p>2$.

Competing conjectures. In this regime, one conjecture is that spurious local minimizers exist, but descent methods with random initializations implicitly regularize themselves so t…

ITKM basin-of-attraction conjecture. The basin of attraction is conjectured to have size

Let $S\geq 1$ and consider the K-SVD criterion … Suppose the observations are generated by sparse random mixtures of an underlying dictionary. K-SVD tightness conjecture. The K-SVD…

Incoherence requirement conjecture. The requirement that the synthesis dictionary ${\bfD}$ be incoherent, and in particular have no linear dependencies among small numbers of its c…

Let $N$ denote the system size, let $\theta$ and $\rho$ be the model parameters, and let $\boldsymbol{D}^0$ and $\boldsymbol{X}^0$ be the planted dictionary and coefficients. For…

Let $\mathbf{\Phi}_0$ be an overcomplete dictionary and let $X_0$ be a coefficient matrix, with the recovery condition involving $u_k$ and $\bar{X}_k^{\star}$ as defined in the sur…