16 problems

Replica continuity conjecture. The analytic continuation of $f(\cdot)$ from $\mathbb{Z}^{+}$ to $\mathbb{R}^{+}$ equals the same function $f(m)$ for real $m$, and the limits with r…

Let MSRA denote Multi-sequence Spreading Random Access, and consider the associated multiple-measurement-vector (MMV) problem for activity detection. A well-conditioned MMV problem…

Let $N\ge 3$, and let $l$ and $m$ be positive integers such that $1\le l\le N$, $2\le m\le N-1$, and both $l$ and $m$ are relatively prime to $N$. Let $X_l(m,N)$ be the random vari…

Consider a communication system with an independent and identically distributed random matrix and a postulated minimum-mean-square-error estimator. The decoupling conjecture. The p…

In a compressive classification problem, suppose the signals from different classes span distinct linear subspaces, and compressive measurements are used for classification. Subspa…

Let ${\bf x}\in\mathbb{R}^n$ be a sparse signal whose non-zero values satisfy … for some $U\in\mathbb{R}^{+}$, and let ${\bf x}_{{\rm int}}\in\mathbb{Z}^n$ denote its integer appro…

Consider recovery from measurements for signals sparse in a redundant dictionary, in the setting where the current analysis incurs a logarithmic factor and the corresponding synthe…

Let $\bm{D}$ be the redundant dictionary and $\bm{z}$ a coefficient vector. In the relevant recovery analysis, approximate cancellations refer to cancellations affecting the entrie…

Let $L$ denote the parameter governing the current measurement bound and let $r=N/n$ be the redundancy of the dictionary, where $N$ is the number of dictionary elements and $n$ is…

Sparsity-promoting $l_1$-norm conjecture. The $l_1$ norm may promote sparsity in ways similar to those in compressive sensing.

In generalized distributed compressive sensing, partial common information may occur among multiple signals, and the proposed sequential correlation search attempts to identify the…

A check node is a node representing a measurement equation, and its degree is the number of neighboring variable nodes. Let $N$ check nodes have the same nonzero value. Modified Eq…

Let the MOE regularizer be the penalty used in the sparse coding formulation above, and let the $\ell_1$-term denote the standard sparsity penalty in compressive sensing. MOE regul…

Converse measurement-rate conjecture. The following conditions on the measurement rates are necessary for recovery with probability one:

$K+1$ measurements conjecture. Based on the stated observations, $K+1$ measurements per sensor should suffice as $J\to\infty$.

Let an object be represented in a basis in which it is compressible, meaning that it can be accurately described using only a few non-zero coefficients. Moravec et al.'s conjecture…