In this work, we present our reproducibility study of Label-Free Explainability for Unsupervised Models, a paper that introduces two post‐hoc explanation techniques for neural networks: (1) label‐free feature importance and (2) label‐free example importance. Our study focuses on the reproducibility of the authors’ most important claims: (i) perturbing features with the highest importance scores causes higher latent shift than perturbing random pixels, (ii) label‐free example importance scores help to identify training examples that are highly related to a given test example, (iii) unsupervised models trained on different tasks show moderate correlation among the highest scored features and (iv) low correlation in example scores measured on a fixed set of data points, and (v) increasing the disentanglement with β in a β‐VAE does not imply that latent units will focus on more different features.

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h = \lceil \frac{n}{2} \rceil$ for each $n \leq 1$. Studying the piercing problem, we show that $0.

In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb{F}_p$, with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call $p$-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property.

We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host–Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra.