It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one -1 entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. The proofs rely on solving polynomial systems of equations by Gröbner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice.

We consider learning a probabilistic classifier from partially-labelled supervision (inputs denoted with multiple possibilities) using standard neural architectures with a softmax as the final layer. We identify a bias phenomenon that can arise from the softmax layer in even simple architectures that prevents proper exploration of alternative options, making the dynamics of gradient descent overly sensitive to initialization. We introduce a novel loss function that allows for unbiased exploration within the space of alternative outputs.

We explore element-wise convex combinations of two permutation-aligned neural network parameter vectors $\Theta_A$ and $\Theta_B$ of size $d$. We conduct extensive experiments by examining various distributions of such model combinations parametrized by elements of the hypercube and its vicinity. Our findings reveal that broad regions of the hypercube form surfaces of low loss values, indicating that the notion of linear mode connectivity extends to a more general phenomenon which we call mode combinability.

This paper describes how we train BERT models to carry over a coding system developed on the paragraphs of a Hungarian literary journal to another. The aim of the coding system is to track trends in the perception of literary translation around the political transformation in 1989 in Hungary. To evaluate not only task performance but also the consistence of the annotation, moreover, to get better predictions from an ensemble, we use 10-fold crossvalidation.

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances is less than 1/4. Our argument implies the upper bound of 0.2470.

Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates useful lemmas for automated theorem provers, demonstrating improvement for several representative systems and solving a hard problem not solved by any system for twenty years. By focusing on condensed detachment problems we simplify the setting considerably, allowing us to get at the essence of lemmas and their role in proof search.

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.

Modern deep artificial neural networks have achieved great success in the domain of computer vision and beyond. However, their application to many real-world tasks is undermined by certain limitations, such as overconfident uncertainty estimates on out-of-distribution data or performance deterioration under data distribution shifts. Several types of deep learning models used for density estimation through probabilistic generative modeling have been shown to fail to detect out-of-distribution samples by assigning higher likelihoods to anomalous data.

Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm. Replacing the Kullback-Leibler divergence with a general $f$-divergence leads to a natural generalization. The case of divergences defined by superlinear functions was recently studied by Di Marino and Gerolin. Using convex analysis, we extend the theory developed so far to include all $f$-divergences defined by functions of Legendre type, and prove that under some mild conditions, strong duality holds, optimums in both the primal and dual problems are attained, the generalization of the $c$-transform is well-defined, and we give sufficient conditions for the generalized Sinkhorn algorithm to converge to an optimal solution.

We employ a toolset — dubbed Dr. Frankenstein — to analyse the similarity of representations in deep neural networks. With this toolset we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initialisations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer.

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art learning algorithm — a graph neural network (GNN) – into the plCoP theorem prover.

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art learning algorithm — a graph neural network (GNN) – into the plCoP theorem prover.

Building on previous work in the nilspace-theoretic approach to the study of Host–Kra factors of measure-preserving systems, we prove that every ergodic $\mathbb{F}_p^\omega$-system of order $k$ is a factor of an Abramov $\mathbb{F}_p^\omega$-system of order $k$. This answers a question of Jamneshan, Shalom and Tao.

Many automated theorem provers are based on backward chaining: reasoning starts from a goal statement which we aim to prove and each inference step reduces one of the goals to a (possibly empty) set of new subgoals. We thus maintain a set of open goals that need to be proven and the proof is complete once the open goal set becomes empty. For each goal, there can be several valid inferences, resulting in different successor goal sets and selecting the right inference constitutes the core problem of such theorem provers which has been thoroughly studied in the past half century.

World models represent the basic mechanisms of a system and can provide predictions about how transformations (actions) affect the state of the system. Such models have recently gained attention in Reinforcement Learning (RL) and in several domains model based learning systems performed similarly or better than highly tuned model free variants [1, 8, 12]. World models can increase sample efficiency since trajectories can be generated without interacting with the environment, and they can aid exploration by yielding a semantically meaningful latent structure that allows for identifying promising directions.

We introduce a notion of complexity for systems of linear forms called \emph{sequential Cauchy-Schwarz complexity}, which is parametrized by two positive integers $k,\ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,\ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-\ell}$-th power of the Gowers $U^{k+1}$-norms of the functions.

Our paper explores the game theoretic value of the 7-in-a-row game. We reduce the problem to solving a finite board game, which we target using Proof Number Search. We present a number of heuristic improvements to Proof Number Search and examine their effect within the context of this particular game. Although our paper does not solve the 7-in-a-row game, our experiments indicate that we have made significant progress towards it.

Variational representations of $f$-divergences are central to many machine learning algorithms, with Lipschitz constrained variants recently gaining attention. Inspired by this, we define the Moreau-Yosida approximation of $f$-divergences with respect to the Wasserstein-$1$ metric. The corresponding variational formulas provide a generalization of a number of recent results, novel special cases of interest and a relaxation of the hard Lipschitz constraint. Additionally, we prove that the so-called tight variational representation of $f$-divergences can be to be taken over the quotient space of Lipschitz functions, and give a characterization of functions achieving the supremum in the variational representation.