16 problems

Let $d$ be a positive integer, and let CPWL denote the class of continuous piecewise-linear functions on $\mathbb{R}^d$. Consider ReLU networks with unrestricted width and prescrib…

Let $\operatorname{\mathsf{CPWL}}$ denote the continuous piecewise linear functions on $\mathbb{R}^n$. The previously known upper bound is $\lceil \log_2(n+1)\rceil$ hidden layers.…

Let $\operatorname{\mathsf{CPWL}}_n$ denote the class of continuous piecewise-linear functions in $n$ variables, and let $\operatorname{\mathsf{ReLU}}_{n,k}$ denote the functions c…

Hertrich et al.'s geometric conjecture. There do not exist polytopes $P,Q\in\Delta(m)$ such that

Hertrich et al.'s equivalent conjecture. The function

For a ReLU neural network, its input domain is partitioned into linear regions, on each of which the represented function is linear. The search space formed by these regions may be…

Let $f$ be a continuous piecewise linear function, and consider shallow ReLU neural networks, allowing either finite width or infinite width with a representation cost. Ongie et al…

Invariance conjecture. The output of Algorithm is independent of how the set of constituents is split by Algorithm.

Piecewise linearity conjecture. The Seshadri function of $A$ is piecewise linear precisely in these cases.

Let $f\colon\mathbb{R}^4\to\mathbb{R}$ be … Let $H$ be the hyperplane arrangement consisting of the hyperplanes where two of $x_0=0,x_1,x_2,x_3,x_4$ are equal. A positively homogen…

For $n\boxin\mathbb{N}$, let … Here, $\operatorname{ReLU}_n(k)$ denotes the class of functions representable by ReLU neural networks with $n$ input variables and $k$ hidden layers,…

Grid triangulation branching conjecture. Any grid triangulation on $V$ admits an independent branching formulation with $\lceil\log_2(d_1)\rceil+\lceil\log_2(d_2)\rceil+3$ levels,…

Two-slope extremality conjecture. If the function has no uncovered components, then it is extreme.

Let $\pi\colon \mathbb R^k\to\mathbb R$ be an extreme function or a facet. Piecewise-linear approximation conjecture. Every extreme function, respectively facet, $\pi$ is either pi…

Let $\pi\colon \mathbb R\to\mathbb R$ be an absolutely continuous extreme function. A simple function takes only finitely many values almost everywhere. Generalized simple-derivati…

Let $\mu_{1}(m,n)$ be the maximum multiplicity of $z=0$ when $B(t)$ and $A(t)$ are polynomial functions of degrees $m$ and $n$, respectively. Let $\mu_{2}(m,n)$ be the maximum mult…