68 problems

Consider the $[P/P]$-Padé approximant $R_P(z)=T_P(z)/Q_P(z)$, where $T_P$ and $Q_P$ have degree $P$ and are computed from the first $2P+1$ Taylor coefficients of $f$. Let $t$ denot…

The spectral method is applied to a multi-arc configuration whose solution has singular exponents along the arcs. Optimal-parameter conjecture. The optimal choice of parameters for…

Let a non-uniform hypergraph stochastic block model consist of multiple independent uniform hypergraph layers sharing a vertex set and community assignment. In the binary case, let…

Consider the spectral Galerkin space discretization discussed in the paper, with projection $P_h$ onto the retained Fourier modes, and the numerical schemes tested for the SPDE. Th…

Maillard et al.'s threshold conjecture. This condition is conjectured to be the computationally optimal weak recovery threshold: if it is violated, no polynomial-time algorithm sho…

General superiority conjecture. The superior performance of dual Laguerre sets over a single Hermite set reflects a general principle: for the relevant class of analytic functions,…

Let be the finite-domain discrete problem with fixed ends or zero Dirichlet boundary conditions, and let $\mathcal{L}_h$ be a regular Sturm–Liouville operator. A dispersion relatio…

Let the transformed/rescaled eigenvalue problem be the differential eigenvalue problem in, and let the sine-transform-based differentiation be the discretisation defined in. For a…

FSD extension conjecture. Incorporating the FSD framework into the classical analysis of the population excess risk of PCR should extend both Theorem and Theorem to spectral method…

The Gaussian Planted Submatrix problem has signal parameter $\beta$, while the Planted Clique problem is obtained from a centered Erdős–Rényi adjacency matrix with an inserted cliq…

Consider the Gaussian Planted Submatrix model in the setting of Theorem, with signal parameter $\beta$, signal vector $\bm{x}$, and leading eigenvector…

Let $\bm{L}_{\sigma}(\widehat{\bm{Y}})$ be a $\sigma$-Laplacian, and suppose it has an outlier eigenvalue associated with a signal vector. Eigenvector-correlation conjecture. The c…

Fix even $r$ and $\ell$ with $n\gg\ell\geq r/2$, and let $Z_S\sim\mathcal N(0,1)$ independently for each $r$-subset $S\subset[n]$. Define the Kikuchi matrix $M(\lambda)$, indexed b…

Let $N\in\mathbb{N}$ and let $X_N$ be the finite-dimensional polynomial space with orthogonal projection $\Pi_{X_N}$. Let $\Omega$, $\partial_x$, and $\partial_x^*$ be the operator…

Consider a multi-index model with Gaussian design, and spectral estimators based on the linearization of approximate message passing. Optimality conjecture. Such spectral estimator…

NASM is a neural adaptive spectral method whose architecture explicitly disentangles coefficients from basis functions. NASM generalization conjecture. We conjecture that NASM's ab…

Spectral-gap condition conjecture. The condition $\sigma_2^*<1$ is implied by the criticality condition.

Let $\vb{C}$ be the covariance matrix governing the biased spatial directions, let $q_c$ be the first value of the measurement ratio $q$ at which the spectral overlap $\rho(q)$ van…

Let $\bm{x}$ be random as in the paper's spiked-prior assumption, let $p(y)$ be a density defining an additive noise channel $\mathcal{P}$, and let $F_{\mathcal{P}}$ denote its Fis…

Aliasing estimate. There exists $c(\lambda)>0$ such that

Consider the spectral estimator formed from the data $(y,X)$, with aspect ratio $\delta$ and associated matrix $D$. Let $a^*$ and $a^\circ$ be the quantities defining the explicit,…

Let $F(\delta, \overline{\Sigma},\mathcal T)$ denote the criticality quantity for the spectral estimator, where $\delta$ is the aspect ratio, $\overline{\Sigma}$ is the normalized…

Let $A$ be the observed matrix associated with a tensor-completion problem, let $M$ denote the underlying signal matrix, and let $B$ be the matrix whose eigenvectors are embedded i…

BBP transition conjecture. The instability of the identity-denoiser state-evolution fixed point when $\lambda(\bm{\Delta})>1$ corresponds to the BBP transition for $\tilde{\bm{Y}}$…

Poisson tau-method solvability conjecture. If $f\in\Pi_N^d$ and the tau polynomials $P_k$ span $\Pi_{N-2,N}^{(d)}$, then the tau-modified equation