Outside an opulent retreat in the mountains of Utah, the world is going to hell. Thanks to disinformation-spreading tools on the world's largest social media platform, people are being executed by bloodthirsty mobs and machine-gunned by their neighbors, politicians assassinated and governments crumbling. But inside Mountainhead, the billionaire tech moguls responsible for the chaos are smoking cigars and shooting the breeze, debating whether the eruption of global chaos is a crisis to be managed or a surge of "creative destruction" that will help usher humanity into a brighter future. If the fictional setting of Mountainhead, the debut feature by Jesse Armstrong, seems a little too close to reality, that's because it's meant to be. The movie, which stars Steve Carell, Jason Schwartzman, Ramy Youssef, and Cory Michael Smith, was conceived, written, cast, shot, edited, and released in about six months, an astonishingly short timeline for any director, let alone a first-timer.

It's late morning on a Monday in March and I am, for reasons I will explain momentarily, in a private bowling alley deep in the bowels of a 65 million mansion in Utah. Jesse Armstrong, the showrunner of HBO's hit series Succession, approaches me, monitor headphones around his neck and a wide grin on his face. "I take it you've seen the news," he says, flashing his phone and what appears to be his X feed in my direction. Everyone had: An hour earlier, my boss Jeffrey Goldberg had published a story revealing that U.S. national-security leaders had accidentally added him to a Signal group chat where they discussed their plans to conduct then-upcoming military strikes in Yemen. "Incredibly fucking depressing," Armstrong said.

We present the first mini-batch kernel $k$-means algorithm, offering an order of magnitude improvement in running time compared to the full batch algorithm. A single iteration of our algorithm takes $\widetilde{O}(kb^2)$ time, significantly faster than the $O(n^2)$ time required by the full batch kernel $k$-means, where $n$ is the dataset size and $b$ is the batch size. Extensive experiments demonstrate that our algorithm consistently achieves a 10-100x speedup with minimal loss in quality, addressing the slow runtime that has limited kernel $k$-means adoption in practice. We further complement these results with a theoretical analysis under an early stopping condition, proving that with a batch size of $\widetilde{\Omega}(\max \{\gamma^{4}, \gamma^{2}\} \cdot \epsilon^{-2})$, the algorithm terminates in $O(\gamma^2/\epsilon)$ iterations with high probability, where $\gamma$ bounds the norm of points in feature space and $\epsilon$ is a termination threshold. Our analysis holds for any reasonable center initialization, and when using $k$-means++ initialization, the algorithm achieves an approximation ratio of $O(\log k)$ in expectation. For normalized kernels, such as Gaussian or Laplacian it holds that $\gamma=1$. Taking $\epsilon = O(1)$ and $b=\Theta(\log n)$, the algorithm terminates in $O(1)$ iterations, with each iteration running in $\widetilde{O}(k)$ time.

Gus, a 19-year-old homeschooled Christian from Joliet, Illinois, is trawling Facebook. He's just recovered from a debilitating bout of depression, and he's looking for someone to talk to. Through an online personality test, he finds a match: Jiyun, a 20-year-old from Korea, who moved to New York City with her family for her brother's cancer treatment. Gus messages her, and they begin chatting. "I started to fall for him when I saw these tagged videos on Facebook," Jiyun reveals in Nancy Schwartzman's short documentary, xoxosms.

The Normal Means problem plays a fundamental role in many areas of modern high-dimensional statistics, both in theory and practice. And the Empirical Bayes (EB) approach to solving this problem has been shown to be highly effective, again both in theory and practice. However, almost all EB treatments of the Normal Means problem assume that the observations are independent. In practice correlations are ubiquitous in real-world applications, and these correlations can grossly distort EB estimates. Here, exploiting theory from Schwartzman (2010), we develop new EB methods for solving the Normal Means problem that take account of unknown correlations among observations. We provide practical software implementations of these methods, and illustrate them in the context of large-scale multiple testing problems and False Discovery Rate (FDR) control. In realistic numerical experiments our methods compare favorably with other commonly-used multiple testing methods.