316 problems

Mean-field equation conjecture. The dynamics of $\mu_t^*$ is given by $\mathcal L(X_0^*)=\lim_{N\rightarrow\infty}\mathcal L(X_0^{1,N})=\mu_0^*$, with $S_0^*$ distributed according…

Let $\Delta^n$ be the $n$-simplex, and let $h_{\Delta^n}$ denote its support function, defined by … The depth of a neural network is the number of hidden layers required to compute…

A neural network consists of neurons receiving excitatory and inhibitory inputs. A network is dynamically balanced when these inputs are sufficiently numerous that their network-wi…

Equality conjecture.

Input-adaptivity conjecture. The same linear improvement in approximation ability should hold even when the choice of inputs is allowed to depend on the function being approximated…

Stationary-distribution conjecture. For each $\theta>1$, the process $Z_t$ admits exactly two stationary distributions: the trivial one, the Dirac mass at $0$, which is reached onl…

Pennington–Worah conjecture. The asymptotic spectral distribution is preserved through multiple layers only by activation functions with $\theta_2(f)=0$, and it is given by the Mar…

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…

Let $\mathbf{x}(t)$ be the vector of electric potentials of $n$ neurons, and consider the linear system … where ${\mathbf{J}}'$ is a real symmetric $n\times n$ matrix with independ…

Dense-solution-region conjecture. Algorithmic success is associated with the existence of dense regions of solutions which, despite being rare and invisible to the equilibrium meas…

Basis-decomposition conjecture. The double-scoring procedure applies broadly to models admitting basis decompositions, with masking interpretable as projection onto learned subspac…

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 a network be formed by stacking individually contractive layers, and let gain denote the performance improvement attributed to the countercurrent multiplier operator. Compositi…

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…

Let $F\colon \mathbb{R}^n \to \mathbb{R}^m$ be a fully connected neural network classifier with $\ell$ hidden layers of widths $n_1,\ldots,n_\ell \geq n$, logistic sigmoid activati…

Mesoscopic multiplier estimate. There are absolute constants $C,c_0>0$ such that, in the width band above, if $d\ge\varepsilon^{-2}\log^C n$, then with probability at least $1-1/n$…

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…

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 ${\mathcal{B}}_L$ denote the unit ball at depth $L$ in the univariate ReLU setting. Depth-three stabilization conjecture. The unit balls stabilize after depth $L=3$, namely … C…

The state of neuron $j$ is $u^j_t\in\mathbb{R}^c$, and the state of the directed edge from neuron $k$ to neuron $j$ is $z^{jk}_t\in\mathbb{R}^d$. The network has separate inhibitor…

Let $X$ be the data manifold formed by a collection of token embeddings in a $1536$-dimensional embedding space, and let $N$ denote the number of sampled token embeddings. Full-dim…

Let $d$ be the ambient dimension, let $k$ be the sparsity level satisfying $d^{\iota}<k<o(d)$ for some small $\iota>0$, and let $s^\star=1$. Consider the sparse warm-up algorithm w…