19 problems

Fix a subspace dimension $d$ and distortion parameters $alpha = \beta \in (0,1)$. A random sparse matrix $\bm{\Phi}: \mathbb{R}^n \to \mathbb{R}^k$ has at most $\zeta$ nonzero entr…

Let $\mathcal{A}$ be a unital C-algebra, let $0<\varepsilon<1$, and let $M,N\in\mathbb{N}$ and $\mathbf{x}_1,\dots,\mathbf{x}_M\in\mathcal{A}^N$. Modular Johnson–Lindenstrauss flat…

Pressure-independence conjecture.

The framework models a data set $\{x_1,\dots,x_N\}\subset\mathbb{R}^d$ using a tangent subbundle consisting of affine subspaces of $\mathbb{R}^d$, with a kernel range parameter con…

Nonlinear $p$-stable projection conjecture. With probability at least $1-N^{-(c-2)}$, for all $x,y\in P$, if $\lVert x-y\rVert_p=O(\epsilon^2)$ then

Let $P\subset\mathbb{R}^D$ be a set of $N$ points, let $0<\epsilon\leq 1/4$, and let $c\geq 3$. Let $F:\mathbb{R}^D\to\mathbb{R}^k$ be the Cauchy random projection and let $\rho$ a…

Low-dimensional special linear group conjecture. For every prime power $\mathfrak{q}$ there exist $D=D(\mathfrak{q})\geqslant1$ and positive constants $c=c(\mathfrak{q})$,…

Let $(\mathcal{M},d_\mathcal{M})$ be an $n$-point metric space. For $p,D\geqslant 1$, a bi-Lipschitz distortion-$D$ embedding of $\mathcal{M}$ into a metric space means an embeddin…

For $p\in[2,\infty)$, let $\upalpha_p$ be the distortion parameter furnished by the average metric dimension-reduction theorem for $X=\ell_p$. The conjecture. … The source notes th…

Let $\mathsf{k}_n^\upalpha(\ell_\infty)$ denote the least dimension, in the relevant metric dimension-reduction problem, for which every $n$-point metric space admits distortion at…

Let a random vector have a non-Gaussian subspace whose marginals in every non-Gaussian direction are far from Gaussian in terms of moments, and consider the Reweighted PCA algorith…

Conjecture on removing the zero mean requirement. The zero mean requirement on $\mathcal{G}_1$ can be removed altogether. This would extend the dimension-reduction theory for thin…

Let (\text{\cursive{\begin{scriptsize}\,b\,\end{scriptsize}}}^b_\varepsilon)_\varepsilon denote the sequence of scaled transverse gradients in the membrane component, and let…

Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in\mathbb{R}^n$ have Euclidean norm at most $1$, let $0<\varepsilon<1$, and set … where $C$ is the absolute constant from Theorem … , the…

Let $f(n,d,\varepsilon)$ denote the smallest $m$ such that every $n$-point subset of $\ell_2^d$ can be embedded into $\ell_2^m$ with distortion at most $1+\varepsilon$. Dimension-r…

Let $\Omega$ be the elastic structure's domain, let $u$ denote the macroscopic displacement, let $v$ denote the fiber displacement, let ${\Greekmath 011B}_{ij}(u)$ be the stress te…

Let $X(t)$ be a general elliptically contoured process, and let (ii) of Theorem be the asserted result for the Gaussian case. Extension conjecture. The conclusion of (ii) of Theore…

The $ll_2$ flattening lemma of Johnson and Lindenstrauss gives dimension reduction by mapping data into a lower-dimensional Euclidean space. The intrinsic dimensionality conjecture…

Let $S$ be a midsurface, let $E^h(u^h)$ denote the elastic energy of a shell of thickness $h$, and suppose … For $i\geq 1$, write … when $i>1$, and let $\mathcal{V}_N$ denote the s…