17 problems

Let $X$ be the matrix of normalized weekly returns for the S&P 500 companies, and suppose that data are available for at least $80\%$ of the index. The covariance matrix used for p…

Let $d$ be the PCA dimension and let $\alpha>1$ be the decay parameter in the setting of Theorem. The PCA-Net theorem currently requires a number of PCA samples scaling as…

Outlier-spectrum conjecture. A tree with outliers has more outlier $J_D$-eigenvalues.

Symmetry conjecture. Symmetry of $C_x$ is sufficient for its $J_D$-diagonalizability, even when $C_x$ has repeated $J_D$-eigenvalues.

PCA objective conjecture. The objective function of principal component analysis is a valuated supermatroid.

Existence conjecture. There exists a tropical Fermat–Weber point

Tropical PCA nesting conjecture. For every $s\geq 1$, there exists a tropical Fermat-Weber point $x^*\in\mathcal U_m$ of the sample that is contained in the $s$th-order tropical PC…

Let $n$ be a finite sample size, and let the empirical PCA solution denote the PCA solution computed from the sample. Finite-sample empirical PCA conjecture. From simulation result…

Phase-transition conjecture. If $A(\beta_i)\leq 0$, then

Zero-noise projection conjecture. In this limit, the model recovers projections to the latent space in a sense analogous to ordinary probabilistic principal component analysis, whe…

Utility-function conjecture. $\mathcal{U}$ is non-increasing and convex in $t$. Numerical evidence supports this stronger claim, which would describe not only when the information/…

Let $F_t$ be the limiting spectral distribution of the noisy sample covariance matrix at noise power $t\geq 0$, and let $N(t)$ denote the number of connected components of its supp…

Let $\eta_1,\dots,\eta_n$ be noise parameters whose empirical distribution satisfies … as $n\to\infty$, and let $\tau_{\max}$ be the supremum of the support of $\nu$. Assume that…

Let $n,d\to\infty$ with $n/d\to c>0$ and $n_\ell/n\to p_\ell$ for $\ell\in\{1,\dots,L\}$. Let $A$ and $\beta_i$ be the quantities defined in the paper, and let $\hat{u}_i$ be the…

Concentration conjecture. The concentration properties of this depth-weighted Tyler scatter matrix can be obtained by taking an approach similar to that of Soloveychik (2014). The…

Let $C$ be a positive semidefinite matrix and let … Consider high-dimensional empirical covariance matrices $YY'$ and rank-$r$ projections onto their leading singular spaces. Wisha…

Bias conjecture. The classical estimator $\hat{\bm{\tau}}^{(0)}$ is biased. The conjecture identifies a possible bias in the traditional estimator of population contribution rates,…