22 problems

Let $T$ be a support set and let $z$ be a fixed sign sequence on $T$. In the setting of Theorem, consider the random vector … where $v^0$ is as in the source and the signs are not…

Let $\mathrm{O}(n)$ be the group of real orthogonal $n\times n$ matrices, and let the least statistically-dependent basis (LSDB) be the basis obtained by optimizing statistical ind…

Strong lottery ticket conjecture at virtually all sparsity levels. There exists $N$ such that, with probability at least $\delta$, the network contains a mask $H$ with sparsity $s$…

Let $\theta\in\mathbb{R}^d$ be an $s$-sparse mean vector in the horizontal-split distributed mean-estimation model, where each of $K$ machines holds a subset of $m$ independent sam…

The partition characterisation conjecture. The graph $G$ is $(3,6)$-tight if and only if, for every edge $e\in E$, there exists a partition $(S_1,S_2,S_3)$ of $E$ such that

Deep neural networks are trained by gradient descent with small initialization; in this setting, the training dynamics and the notion of solution complexity are considered in the f…

Stronger edge bound conjecture. The number of edges satisfies

Dimensional sparsity-rate conjecture. In $d$ dimensions, the support of $\pi_\varepsilon$ should shrink at rate

For a polynomial $f\in\mathbb{Z}[X]$, let $P_n(H;f)$ denote the number of matrices $A\in\mathcal{M}_n(\mathbb{Z};H)$ with characteristic polynomial $f$. Characteristic-polynomial s…

Block-and-hole minimal-rigidity conjecture. The following statements are equivalent:

$(3,3)$-sparsity conjecture. Every $(3,3)$-tight simple graph is minimally rigid in $\ell_p^3$.

Let $G$) be a graph, where $\mathcal{R}_d$ denotes the $d$-dimensional rigidity matroid and an $\mathcal{R}_{d+1}$-bridge is an edge whose deletion lowers the rank in that matroid.…

Let $p$ be the number of covariates and $n$ the sample size. In the regime $p\leq n$, the paper's DCal estimator achieves $\sqrt{n}$-consistency under the minimal sparsity conditio…

The matrix $Q$ is the quadratic coefficient matrix, and MILO denotes the mixed-integer linear optimization formulation obtained by reformulating the convex quadratic optimization p…

Thurstone's conjecture. Simple structure resolves the rotational invariance of factor analysis.

Let $\boldsymbol a\in\mathbb Z^n$ be an arbitrary integer vector, let $b\in\mathbb Z$, and consider the knapsack feasibility problem … Write $\|\boldsymbol x\|_0$ for the number of…

N-1 redundancy sparsity conjecture. This sparsity is a natural property of the N-1 redundancy engineered into electrical networks.

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

Optimality conjecture. The lower rate $\psi^{(1)}$ is optimal for all admissible configurations of $n$, $p$, and sparsity parameters.

Unknown-variance extension conjecture. The results of Theorem concerning the ABOS property of mBIC1, mBIC2, and mBIC3 should also hold when $\sigma$ is unknown.

ABOS conjecture for mBIC1–mBIC3. In analogy with the known-variance case, mBIC1, mBIC2, and mBIC3 should be ABOS for a wide range of sparsity levels when $\sigma$ is unknown.

Let $A$ be an operator acting on a basis $(\psi_k)$, and consider the normalized pairwise values … The operator is said to act well on the basis when these values are not too large…