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The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance. The optimality guarantee is enabled by the fact that the search space of our approach is not constrained to a generalization of bijections, unlike in other relaxations such as the Gromov--Wasserstein distance. We suggest the Frank--Wolfe algorithm with $O(n^3)$-time iterations for solving the relaxation and numerically demonstrate its performance on metric spaces of hundreds of points. In particular, we obtain a new upper bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric. Our approach is implemented as a Python package dGH.

The analysis of brain signals holds considerable importance in enhancing our comprehension of diverse learning techniques and cognitive mechanisms. Game-based learning is increasingly being recognized for its interactive and engaging educational approach. A pilot study of twelve participants divided into experimental and control groups was conducted to understand its effects on cognitive processes. Both groups were provided with the same contents regarding the basic structure of the graph. The participants in the experimental group engaged in a quiz-based game, while those in the control group watched a pre-recorded video. Functional Near-Infrared Spectroscopy (fNIRS) was employed to acquire cerebral signals, and a series of pre and post-tests were administered. The findings of our study indicate that the group engaged in the game activity displayed elevated levels of oxygenated hemoglobin compared to the group involved in watching videos. Conversely, the deoxygenated hemoglobin levels remained relatively consistent across both groups throughout the learning process. The aforementioned findings suggest that the use of game-based learning has a substantial influence on cognitive processes. Furthermore, it is evident that both the game and video groups exhibited higher neural activity in the Lateral Prefrontal cortex (PFC). The oxygenated hemoglobin ratio demonstrates that the game group had 2.33 times more neural processing in the Lateral PFC than the video group. This data is further supported by the knowledge gain analysis, which indicates that the game-based approach resulted in a 47.74% higher knowledge gain than the video group, as calculated from the difference in pre-and post-test scores.

We present a physics-inspired method for inferring dynamic rankings in directed temporal networks - networks in which each directed and timestamped edge reflects the outcome and timing of a pairwise interaction. The inferred ranking of each node is real-valued and varies in time as each new edge, encoding an outcome like a win or loss, raises or lowers the node's estimated strength or prestige, as is often observed in real scenarios including sequences of games, tournaments, or interactions in animal hierarchies. Our method works by solving a linear system of equations and requires only one parameter to be tuned. As a result, the corresponding algorithm is scalable and efficient. We test our method by evaluating its ability to predict interactions (edges' existence) and their outcomes (edges' directions) in a variety of applications, including both synthetic and real data. Our analysis shows that in many cases our method's performance is better than existing methods for predicting dynamic rankings and interaction outcomes.

We introduce graph width parameters, called $\alpha$-edge-crossing width and edge-crossing width. These are defined in terms of the number of edges crossing a bag of a tree-cut decomposition. They are motivated by edge-cut width, recently introduced by Brand et al. (WG 2022). We show that edge-crossing width is equivalent to the known parameter tree-partition-width. On the other hand, $\alpha$-edge-crossing width is a new parameter; tree-cut width and $\alpha$-edge-crossing width are incomparable, and they both lie between tree-partition-width and edge-cut width. We provide an algorithm that, for a given $n$-vertex graph $G$ and integers $k$ and $\alpha$, in time $2^{O((\alpha+k)\log (\alpha+k))}n^2$ either outputs a tree-cut decomposition certifying that the $\alpha$-edge-crossing width of $G$ is at most $2\alpha^2+5k$ or confirms that the $\alpha$-edge-crossing width of $G$ is more than $k$. As applications, for every fixed $\alpha$, we obtain FPT algorithms for the List Coloring and Precoloring Extension problems parameterized by $\alpha$-edge-crossing width. They were known to be W[1]-hard parameterized by tree-partition-width, and FPT parameterized by edge-cut width, and we close the complexity gap between these two parameters.

We propose theoretical analyses of a modified natural gradient descent method in the neural network function space based on the eigendecompositions of neural tangent kernel and Fisher information matrix. We firstly present analytical expression for the function learned by this modified natural gradient under the assumptions of Gaussian distribution and infinite width limit. Thus, we explicitly derive the generalization error of the learned neural network function using theoretical methods from eigendecomposition and statistics theory. By decomposing of the total generalization error attributed to different eigenspace of the kernel in function space, we propose a criterion for balancing the errors stemming from training set and the distribution discrepancy between the training set and the true data. Through this approach, we establish that modifying the training direction of the neural network in function space leads to a reduction in the total generalization error. Furthermore, We demonstrate that this theoretical framework is capable to explain many existing results of generalization enhancing methods. These theoretical results are also illustrated by numerical examples on synthetic data.

The Graphical House Allocation (GHA) problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics. Recent work has studied the computational aspects of GHA and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a nearly complete characterization of the approximability of GHA. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, and bounded-degree planar graphs. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we investigate the special case of bounded-degree trees in some detail. We first refute a conjecture by Hosseini et al. [2023] about the structural properties of exact optimal allocations on binary trees by means of a counterexample on a depth-$3$ complete binary tree. This refutation, together with our hardness results on trees, might suggest that approximating the optimal envy even on complete binary trees is infeasible. Nevertheless, we present a linear-time algorithm that attains a $3$-approximation on complete binary trees.

We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of $(\frac{3}{4} + \frac{1}{12n})$. Most of these results are based on complicated analyses, especially those providing better than $2/3$ factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.

Stratification in both the design and analysis of randomized clinical trials is common. Despite features in automated randomization systems to re-confirm the stratifying variables, incorrect values of these variables may be entered. These errors are often detected during subsequent data collection and verification. Questions remain about whether to use the mis-reported initial stratification or the corrected values in subsequent analyses. It is shown that the likelihood function resulting from the design of randomized clinical trials supports the use of the corrected values. New definitions are proposed that characterize misclassification errors as `ignorable' and `non-ignorable'. Ignorable errors may depend on the correct strata and any other modeled baseline covariates, but they are otherwise unrelated to potential treatment outcomes. Data management review suggests most misclassification errors are arbitrarily produced by distracted investigators, so they are ignorable or at most weakly dependent on measured and unmeasured baseline covariates. Ignorable misclassification errors may produce a small increase in standard errors, but other properties of the planned analyses are unchanged (e.g., unbiasedness, confidence interval coverage). It is shown that unbiased linear estimation in the absence of misclassification errors remains unbiased when there are non-ignorable misclassification errors, and the corresponding confidence intervals based on the corrected strata values are conservative.

Dimension reduction techniques have long been an important topic in statistics, and active subspaces (AS) have received much attention this past decade in the computer experiments literature. The most common approach towards estimating the AS is to use Monte Carlo with numerical gradient evaluation. While sensible in some settings, this approach has obvious drawbacks. Recent research has demonstrated that active subspace calculations can be obtained in closed form, conditional on a Gaussian process (GP) surrogate, which can be limiting in high-dimensional settings for computational reasons. In this paper, we produce the relevant calculations for a more general case when the model of interest is a linear combination of tensor products. These general equations can be applied to the GP, recovering previous results as a special case, or applied to the models constructed by other regression techniques including multivariate adaptive regression splines (MARS). Using a MARS surrogate has many advantages including improved scaling, better estimation of active subspaces in high dimensions and the ability to handle a large number of prior distributions in closed form. In one real-world example, we obtain the active subspace of a radiation-transport code with 240 inputs and 9,372 model runs in under half an hour.

Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.