17 problems

Consider the Gaussian multi-index model with hidden indices $\{\boldsymbol{w}_1^\star,\ldots,\boldsymbol{w}_M^\star\}$ satisfying the separation condition required in Theorem 1 of…

Let $X_1,X_2,\ldots$ be a stationary ergodic process taking values in $\{1,\ldots,k\}$, and let $Y_1,Y_2,\ldots$ be conditionally independent given $X_1,X_2,\ldots$, with $Y_t$ con…

Let $V_{\mu^0}$ be the variational information matrix at the true parameter $\mu^0$. Consider models with unequal component weights or non-identity covariance matrices, and suppose…

Consider correlated Gaussian mixture models with $n$ nodes, attribute dimension $d$, and correlation parameter $\rho$. Let ${\boldsymbol \sigma}$ denote the community assignment an…

Davis et al.'s conjecture. There does not exist a computationally efficient algorithm for general GMM that achieves the optimal misclassification rate with a linear sample complexi…

Consider the two-mean Gaussian mixture model … with homogeneity null hypothesis $H_0:p=0$ or $t_1=t_2$ and alternative $H_1:p\in(0,1)$, $t_1\ne t_2$. Let the likelihood-ratio stati…

Let $X_1,\dots,X_n$ be i.i.d. standard normal random variables under the homogeneity null hypothesis $H_0:p=0$ or $t=0$ in the contaminated Gaussian mixture model … where…

Consider the contaminated Gaussian mixture model … with unknown mixture weight $p$ and mean parameter $t$, and let the quadratic approximation and the likelihood-ratio statistic be…

Polynomial-time computational conjecture. No polynomial-time method can achieve a better sample complexity than the spectral algorithm for this clustering problem.

Let a Gaussian mixture model have $K$ components with minimum mixing weight $pi_{\min}$, and let $R_{\min}$ denote the minimum separation and $R_i$ the separation associated with c…

Instance-specific rate conjecture. A more nuanced characterization of these polynomial systems should allow instance-specific rates of convergence for parameter estimation.

Let $X_i$ be observations from a Gaussian mixture model with noise covariance matrix $\boldsymbol\varepsilon_i\sim N(0,\Sigma)$, where the positive-definite covariance matrix…

Constant-norm initialization conjecture. The same result should apply when

Let $\mu_0$ and $\mu_1$ be Gaussian mixture models on $\mathbb{R}^d$, let $\gamma$ be an optimal transport plan between them, and let $(\mathrm{P}_t)_{t\in[0,1]}$ denote the interp…

Let the EM algorithm be applied to a mixture of spherical Gaussians, and call a critical point a strict saddle point when the Hessian of the log-likelihood has at least one strictl…

Consider a uniformly weighted mixture of $M$ isotropic Gaussian distributions, with population log-likelihood obtained in the infinite-sample limit. A local maximum is called bad i…

Automatic regularization conjecture. By automatically deciding how much regularized NMF should rely on the prior, it should not be necessary to manually change the regularization-p…