90 problems

Consider the high-dimensional global testing procedures developed in the paper, including tests based on studentized $L_q$-norm statistics and their monotonically indexed U-statist…

Let $W_\rho$ be the Wigner function of a quantum state, and let $\mathcal{A}(\alpha,r,L)$ be the smoothness class considered in the paper. Consider testing … against alternatives s…

Suppose $L$ i.i.d. observations $X_1,cldots,X_L$ are drawn either from the null distribution $\mathbb{Q}_n$ or from the alternative distribution $\mathbb{P}_n$. Let…

The nonparametric Kiefer--Weiss problem considers sequential testing with error probabilities tending to zero, equivalently as $c\to\infty$, where $c$ is the parameter governing th…

Low-degree conjecture. For many average-case high-dimensional testing problems, low-degree polynomials of degree

Consider a hypothesis-testing problem with input size $N$ and testing advantage as the performance measure. Strengthened low-degree conjecture. Low-degree polynomials should perfor…

Constrained moment-matching lower-bound conjecture.

Let $M_p(\epsilon,L)$ denote the constrained moment-matching quantity used in the paper, with tolerance parameter $\epsilon$ and moment parameter $L$. Constrained moment-matching c…

The interpolation regime concerns testing under non-smooth $\ell_p$ norms, with the rates given by the upper bounds for fixed odd $p\geq 1$. Tightness conjecture. The interpolation…

Adaptive weighted test power dominance conjecture. The power of $\phi_{\textnormal{wtd+}}$ dominates that of $\phi_{\textnormal{wtd}}$ across the full range of signal strengths.

Let $n^*$ denote the classical sample complexity of simple binary hypothesis testing, let $n^*_{\mathrm{id}}$ denote the sample complexity when each channel has an output alphabet…

Graphon quasirandomness conjecture. If there exists a constant-degree polynomial test distinguishing $\mathbb{G}(n,1/2)$ from any graphon, then the two distributions can also be di…

Quasirandomness conjecture. There exists a constant-degree polynomial test that distinguishes these hypotheses with high probability if and only if one of the signed subgraph count…

Let $G_n$ be the final snapshot of a preferential attachment network on $n$ vertices, and suppose the changepoint is in the late-change regime $\tau_n=n-\Delta$ with…

Let $\mathcal{Q}$ and $\mathcal{P}$ be sufficiently nice distributions, and let $\operatorname{Adv}_{\leq D}(\mathcal{Q},\mathcal{P})$ denote the degree-$D$ low-degree advantage fo…

Let $m$ denote the number of queries, and consider protocols for distinguishing between T-symmetry and Z-symmetry unitaries. Quadratic decay conjecture. There exist protocols for w…

Wasserstein-based testing conjecture. The testing procedure for determining which credal set between $\mathcal{M}(\underline{P}_1)$ and $\mathcal{M}(\underline{P}_2)$ contains…

Consider the statistical detection limits and the efficient algorithmic thresholds for detecting dependency in weighted random graphs. Inherent-gap conjecture. The gap between the…

Asymptotic power and detection-threshold conjecture. The asymptotic power against local alternatives and the detection thresholds of the tests based on $T_n(a)$ are conjectured to…

Conjecture 8.1. Proposition 4.2, namely the EPV criterion for the corresponding single-observation problem, remains valid for repeated observations.

Let the two distributions in a detection problem be given, and call a test statistic a low-degree polynomial if it is a polynomial of degree $O(\log n)$ in the underlying input var…

Let $(\mathcal{X}_n,\mathcal{P})$ be a sequence of CLVMs, giving rise to sequences of measures $\mathbb{P}_n$ and $\mathbb{Q}_n$ on $\Omega^N$, where $N=N(n)$. Let…

Let ${\mathcal Z}=\mathbb R$, let $Z$ be the observation, and let the null family be … Let $F_+$ denote the nonnegative measurable functions, and let ${\mathcal E}$ be the set of a…

Let $Z$ be the observation, let $\mu>0$ be the null mean bound, let $F_+$ denote the nonnegative measurable functions, and let ${\mathcal E}$ be the set of e-variables. The candida…

Generalized quantum Stein's lemma. For any such family,