8 problems

Let $\bar{f}$ be the population objective and $\bar{f}_{n}$ its empirical counterpart for the location parameter, and consider the empirical iterates of the ELU updates. The first…

Spurious fixed-point conjecture. There are only two spurious fixed points of $f(\theta)$ in $\mathbb{R}_{-}$. Moreover, there exists $\delta_{\mathrm{cr}}\in(0,1/2)$ such that the…

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…

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

Let $m$ be the number of mixture components and $D$ the number of features. A $k$-cluster region is a region in which the effective mixture behavior is concentrated on $k$ clusters…

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…

Consider the sample-based expectation-maximization algorithm for a mixture of two Gaussians in dimension $d=d_n$, with sample size $n$. The preceding analysis shows that the Euclid…