A coreset of a dataset with $n$ examples and $d$ features is a weighted subset of examples that is sufficient for solving downstream data analytic tasks. Nearly optimal constructions of coresets for least squares and $\ell_p$ linear regression with a single response are known in prior work. However, for multiple $\ell_p$ regression where there can be $m$ responses, there are no known constructions with size sublinear in $m$. In this work, we construct coresets of size $\tilde O(\varepsilon^{-2}d)$ for $p<2$ and $\tilde O(\varepsilon^{-p}d^{p/2})$ for $p>2$ independently of $m$ (i.e., dimension-free) that approximate the multiple $\ell_p$ regression objective at every point in the domain up to $(1\pm\varepsilon)$ relative error. If we only need to preserve the minimizer subject to a subspace constraint, we improve these bounds by an $\varepsilon$ factor for all $p>1$. All of our bounds are nearly tight. We give two application of our results. First, we settle the number of uniform samples needed to approximate $\ell_p$ Euclidean power means up to a $(1+\varepsilon)$ factor, showing that $\tilde\Theta(\varepsilon^{-2})$ samples for $p = 1$, $\tilde\Theta(\varepsilon^{-1})$ samples for $1 < p < 2$, and $\tilde\Theta(\varepsilon^{1-p})$ samples for $p>2$ is tight, answering a question of Cohen-Addad, Saulpic, and Schwiegelshohn. Second, we show that for $1<p<2$, every matrix has a subset of $\tilde O(\varepsilon^{-1}k)$ rows which spans a $(1+\varepsilon)$-approximately optimal $k$-dimensional subspace for $\ell_p$ subspace approximation, which is also nearly optimal.
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We present a novel data-driven strategy to choose the hyperparameter $k$ in the $k$-NN regression estimator without using any hold-out data. We treat the problem of choosing the hyperparameter as an iterative procedure (over $k$) and propose using an easily implemented in practice strategy based on the idea of early stopping and the minimum discrepancy principle. This model selection strategy is proven to be minimax-optimal over some smoothness function classes, for instance, the Lipschitz functions class on a bounded domain. The novel method often improves statistical performance on artificial and real-world data sets in comparison to other model selection strategies, such as the Hold-out method, 5-fold cross-validation, and AIC criterion. The novelty of the strategy comes from reducing the computational time of the model selection procedure while preserving the statistical (minimax) optimality of the resulting estimator. More precisely, given a sample of size $n$, if one should choose $k$ among $\left\{ 1, \ldots, n \right\}$, and $\left\{ f^1, \ldots, f^n \right\}$ are the estimators of the regression function, the minimum discrepancy principle requires the calculation of a fraction of the estimators, while this is not the case for the generalized cross-validation, Akaike's AIC criteria, or Lepskii principle.
We investigate shift-invariant vectorial Boolean functions on $n$~bits that are lifted from Boolean functions on $k$~bits, for $k\leq n$. We consider vectorial functions that are not necessarily permutations, but are, in some sense, almost bijective. In this context, we define an almost lifting as a Boolean function for which there is an upper bound on the number of collisions of its lifted functions that does not depend on $n$. We show that if a Boolean function with diameter $k$ is an almost lifting, then the maximum number of collisions of its lifted functions is $2^{k-1}$ for any $n$. Moreover, we search for functions in the class of almost liftings that have good cryptographic properties and for which the non-bijectivity does not cause major security weaknesses. These functions generalize the well-known map $\chi$ used in the Keccak hash function.
We develop a novel clustering method for distributional data, where each data point is regarded as a probability distribution on the real line. For distributional data, it has been challenging to develop a clustering method that utilizes the mode of variation of data because the space of probability distributions lacks a vector space structure, preventing the application of existing methods for functional data. In this study, we propose a novel clustering method for distributional data on the real line, which takes account of difference in both the mean and mode of variation structures of clusters, in the spirit of the $k$-centres clustering approach proposed for functional data. Specifically, we consider the space of distributions equipped with the Wasserstein metric and define a geodesic mode of variation of distributional data using geodesic principal component analysis. Then, we utilize the geodesic mode of each cluster to predict the cluster membership of each distribution. We theoretically show the validity of the proposed clustering criterion by studying the probability of correct membership. Through a simulation study and real data application, we demonstrate that the proposed distributional clustering method can improve cluster quality compared to conventional clustering algorithms.
Clustering is one of the staples of data analysis and unsupervised learning. As such, clustering algorithms are often used on massive data sets, and they need to be extremely fast. We focus on the Euclidean $k$-median and $k$-means problems, two of the standard ways to model the task of clustering. For these, the go-to algorithm is $k$-means++, which yields an $O(\log k)$-approximation in time $\tilde O(nkd)$. While it is possible to improve either the approximation factor [Lattanzi and Sohler, ICML19] or the running time [Cohen-Addad et al., NeurIPS 20], it is unknown how precise a linear-time algorithm can be. In this paper, we almost answer this question by presenting an almost linear-time algorithm to compute a constant-factor approximation.
In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of the primal formulation and the dual formulation of the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.
In this note we propose a new algorithm for checking whether two counting functions on a free monoid $M_r$ of rank $r$ are equivalent modulo a bounded function. The previously known algorithm has time complexity $O(n)$ for all ranks $r>2$, however in case $r=2$ it was estimated only as $O(n^2)$. Here we apply a new approach, based on explicit basis expansion and weighted rectangles summation, which allows us to construct a much simpler algorithm with time complexity $O(n)$ for any $r\geq 2$.
We provide, for each natural number $n$ and each class among $D_n(\Sigma^0_1)$, $\bar D_n(\Sigma^0_1)$ and $D_{2n+1}(\Sigma^0_1)\oplus\bar D_{2n+1}(\Sigma^0_1)$, a regular language whose associated omega-power is complete for this class.
The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval $[-1, 1]$ are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.
Graph neural networks form a class of deep learning architectures specifically designed to work with graph-structured data. As such, they share the inherent limitations and problems of deep learning, especially regarding the issues of explainability and trustworthiness. We propose $\mu\mathcal{G}$, an original domain-specific language for the specification of graph neural networks that aims to overcome these issues. The language's syntax is introduced, and its meaning is rigorously defined by a denotational semantics. An equivalent characterization in the form of an operational semantics is also provided and, together with a type system, is used to prove the type soundness of $\mu\mathcal{G}$. We show how $\mu\mathcal{G}$ programs can be represented in a more user-friendly graphical visualization, and provide examples of its generality by showing how it can be used to define some of the most popular graph neural network models, or to develop any custom graph processing application.
Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{$k$-times bin-packing ($k$BP)}, where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by $k$-times bin-packing for some finite $k$. We also show that $k$-times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem.
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