45 problems

A robust chaos conjecture asserts that chaos is a robust, high-probability behavior for high-dimensional, bounded, nonlinear dynamical systems. The paper reports near-unity experim…

Let $f$ be a mapping (neural network) and let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture. The periodic-window probability conjecture asserts that,…

Let $f$ be a mapping (neural network) as defined in the paper, with sufficiently high dimension $d$. A hyperbolicity violation conjecture asserts that there exists at least one bif…

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 $s$…

Asymptotic half-density conjecture. The arithmetic expressive capacity approaches $\frac{1}{2}$ as $p\to\infty$:

Let $f\colon \mathbb{R}^n \to \mathbb{R}$ be the output of a fully connected neural network with $\ell$ hidden layers of widths $n_1,\ldots,n_\ell$ and a Pfaffian activation functi…

Bubeck–Li–Nagaraj conjecture. The stated lower bound should hold for every Lipschitz activation and arbitrary weights, with no restriction on their magnitudes. The conjecture quant…

For a fixed ReLU-network architecture, let $\mathcal{P}$ denote the parameter space obtained from the space of weights by taking into account the trivial scaling and permutation sy…

Effective Lipschitz concentration. The effective Lipschitz constant satisfies $\Lambda_{\mathrm{eff}}\ll\Lambda(W)$, specifically $\Lambda_{\mathrm{eff}}=O(\mathrm{poly}(H))$, whil…

Width-one identifiability conjecture. 1. There exist architectures $\mathcal{A}=(n_0,\ldots,n_{L+1})$ with $n_\ell=1$ for some $\ell\leq L$ for which no identifiable parameters exi…

LDS implies GAS of LTN equilibria. If $W-D\in\mathcal{LDS}$, then for every $u\in\mathbb{R}^n$, the linear-threshold network admits a unique globally asymptotically stable equilibr…

Neural-network complexity conjecture. The complexity of the architecture, measured by the number of layers and its width, is controlled by $4\|\tau_{\Omega}\|_{\infty}4$ and…

Because the outputs are constrained by $\mathbf{1}^T x=1$, let $x'=x-\frac{1}{d}\mathbf{1}$, so that $\mathbf{1}^T x'=0$, and let $W\in\mathbb{R}^{m\times d}$ act on the input. Def…

MHA weak identifiability conjecture. For every parameter $\theta\in\Theta$ of an MHA model, the parameter $\theta$ is weakly identifiable with respect to the symmetry group

Weak identifiability conjecture. With mild assumptions on the activation function $\sigma$, for every parameter $\theta \in \Theta$ of an MLP, there exists a parameter…

Let $n\geq 1$, $L\geq 1$, and let $f:[0,1]^n\to\mathbb{R}$ or $f:[0,1]^n\to\mathbb{R}^m$ be continuous. Consider a Sprecher Network with architecture $n\to[d_1,\dots,d_L]\to 1$ or…

Let $n,m\in\mathbb{N}$ with $m>1$, and let $f:[0,1]^n\to\mathbb{R}^m$ be continuous. A Sprecher Network has architecture $n\to[d_1]\to m$ with one hidden block of width…

Let $d$, $k$, and $s$ be positive integers, and let $p_1,\ldots,p_k\in\mathbb{R}[x_1,\ldots,x_d]$ be homogeneous polynomials of the same degree $s$, no two of which are linearly de…

Completeness conjecture. Every reflection-positive process admits a representation as a deep NN-QM whose inputs $y_t^{(i)}$ are symmetric Markov processes.

Balanced-regime confinement and collapse conjecture. If the set $\mathcal{B}$ is nonempty, there exist asymptotic solutions confined to the balanced regime for all times. Moreover,…

Let normalized deep self-attention networks be parametrized via $(\mathbf{M},L)$, where these parameters encode the network as defined in the paper. The normalization removes layer…

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

ReLU approximation error conjecture. Under these assumptions, the displayed bound holds.

Realization-space degree conjecture.