Modern surgery has gone from long incisions to tiny cuts guided by robots and AI. In the process, however, surgeons have lost something vital: the chance to feel inside the body directly. Without palpation, it becomes harder to detect tissue abnormalities during an operation. A group of surgeons and engineers across Europe is now trying to bring back this vital aspect of surgery. Working within an EU-funded research collaboration called PALPABLE, they are developing a soft robotic "fingertip" that can sense how firm or soft tissue is during minimally invasive and robotic surgery.

Heavy-tailed distributions are ubiquitous in real-world data, where rare but extreme events dominate risk and variability. However, standard Variational Autoencoders (VAEs) employ simple decoder distributions (e.g., Gaussian) that fail to capture heavy-tailed behavior, while existing heavy-tail-aware extensions remain restricted to predefined parametric families whose tail behavior is fixed a priori. We propose the Phase-Type Variational Autoencoder (PH-VAE), whose decoder distribution is a latent-conditioned Phase-Type (PH) distribution defined as the absorption time of a continuous-time Markov chain (CTMC). This formulation composes multiple exponential time scales, yielding a flexible and analytically tractable decoder that adapts its tail behavior directly from the observed data. Experiments on synthetic and real-world benchmarks demonstrate that PH-VAE accurately recovers diverse heavy-tailed distributions, significantly outperforming Gaussian, Student-t, and extreme-value-based VAE decoders in modeling tail behavior and extreme quantiles. In multivariate settings, PH-VAE captures realistic cross-dimensional tail dependence through its shared latent representation. To our knowledge, this is the first work to integrate Phase-Type distributions into deep generative modeling, bridging applied probability and representation learning.

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Sparse recovery is among the most well-studied problems in learning theory and high-dimensional statistics. In this work, we investigate the statistical and computational landscapes of sparse recovery with $\ell_\infty$ error guarantees. This variant of the problem is motivated by \emph{variable selection} tasks, where the goal is to estimate the support of a $k$-sparse signal in $\mathbb{R}^d$. Our main contribution is a provable separation between the \emph{oblivious} (``for each'') and \emph{adaptive} (``for all'') models of $\ell_\infty$ sparse recovery. We show that under an oblivious model, the optimal $\ell_\infty$ error is attainable in near-linear time with $\approx k\log d$ samples, whereas in an adaptive model, $\gtrsim k^2$ samples are necessary for any algorithm to achieve this bound. This establishes a surprising contrast with the standard $\ell_2$ setting, where $\approx k \log d$ samples suffice even for adaptive sparse recovery. We conclude with a preliminary examination of a \emph{partially-adaptive} model, where we show nontrivial variable selection guarantees are possible with $\approx k\log d$ measurements.

Two key features make correlation clustering quite suitable in real-world applications.

This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science.