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Optimum distance flag codes (ODFCs), as special flag codes, have received a lot of attention due to its application in random network coding. In 2021, Alonso-Gonz\'{a}lez et al. constructed optimal $(n,\mathcal{A})$-ODFC for $\mathcal {A}\subseteq \{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $k\in \mathcal A$ and $k|n$. In this paper, we introduce a new construction of $(n,\mathcal A)_q$-ODFCs by maximum rank-metric codes. It is proved that there is an $(n,\mathcal{A})$-ODFC of size $\frac{q^n-q^{k+r}}{q^k-1}+1$ for any $\mathcal{A}\subseteq\{1,2,\ldots,k,n-k,\ldots,n-1\}$ with $\mathcal A\cap \{k,n-k\}\neq\emptyset$, where $r\equiv n\pmod k$ and $0\leq r<k$. Furthermore, when $k>\frac{q^r-1}{q-1}$, this $(n,\mathcal A)_q$-ODFC is optimal. Specially, when $r=0$, Alonso-Gonz\'{a}lez et al.'s result is also obtained.

The topic of inverse problems, related to Maxwell's equations, in the presence of nonlinear materials is quite new in literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, starting from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. And the complexity exponentially grows when the focus is on nonlinear material properties. Recently, the Monotonicity Principle has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background given by this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The Monotonicity Principle is the foundation of a class of non-iterative algorithms for tomography of linear materials. It has been successfully applied to various problems, governed by different PDEs. In the linear case, MP based inversion methods ensure excellent performances and compatibility with real-time applications. We focus on problems governed by elliptical PDEs and, as an example of application, we treat the Magnetostatic Permeability Tomography problem, in which the aim is to retrieve the spatial behaviour of magnetic permeability through boundary measurements in DC operations. In this paper, we provide some preliminary results giving the foundation of our method and extended numerical examples.

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. Perhaps the most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices. In this paper, we propose a deflation varimax procedure that solves each row of an orthogonal matrix sequentially. In addition to its net computational gain and flexibility, we are able to fully establish theoretical guarantees for the proposed procedure in a broad context. Adopting this new varimax approach as the second step after PCA, we further analyze this two step procedure under a general class of factor models. Our results show that it estimates the factor loading matrix in the optimal rate when the signal-to-noise-ratio (SNR) is moderate or large. In the low SNR regime, we offer possible improvement over using PCA and the deflation procedure when the additive noise under the factor model is structured. The modified procedure is shown to be optimal in all SNR regimes. Our theory is valid for finite sample and allows the number of the latent factors to grow with the sample size as well as the ambient dimension to grow with, or even exceed, the sample size. Extensive simulation and real data analysis further corroborate our theoretical findings.

Weakly-supervised segmentation with label-efficient sparse annotations has attracted increasing research attention to reduce the cost of laborious pixel-wise labeling process, while the pairwise affinity modeling techniques play an essential role in this task. Most of the existing approaches focus on using the local appearance kernel to model the neighboring pairwise potentials. However, such a local operation fails to capture the long-range dependencies and ignores the topology of objects. In this work, we formulate the affinity modeling as an affinity propagation process, and propose a local and a global pairwise affinity terms to generate accurate soft pseudo labels. An efficient algorithm is also developed to reduce significantly the computational cost. The proposed approach can be conveniently plugged into existing segmentation networks. Experiments on three typical label-efficient segmentation tasks, i.e. box-supervised instance segmentation, point/scribble-supervised semantic segmentation and CLIP-guided semantic segmentation, demonstrate the superior performance of the proposed approach.

We consider two classes of natural stochastic processes on finite unlabeled graphs. These are Euclidean stochastic optimization algorithms on the adjacency matrix of weighted graphs and a modified version of the Metropolis MCMC algorithm on stochastic block models over unweighted graphs. In both cases we show that, as the size of the graph goes to infinity, the random trajectories of the stochastic processes converge to deterministic curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lov\'{a}sz and Szegedy in \cite{lovasz2010decorated}, are a refinement of the concept of graphons that can distinguish between two infinite exchangeable arrays that give rise to the same graphon limit. We introduce new metrics on this space which provide us with a natural notion of convergence for our limit theorems. This notion is equivalent to the convergence of infinite-exchangeable arrays. Under suitable assumptions and a specified time-scaling, the Metropolis chain admits a diffusion limit as the number of vertices go to infinity. We then demonstrate that, in an appropriately formulated zero-noise limit, the stochastic process of adjacency matrices of this diffusion converges to a deterministic gradient flow curve on the space of graphons introduced in\cite{Oh2023}. A novel feature of this approach is that it provides a precise exponential convergence rate for the Metropolis chain in a certain limiting regime. The connection between a natural Metropolis chain commonly used in exponential random graph models and gradient flows on graphons, to the best of our knowledge, is new in the literature as well.

The categorical Gini correlation, $\rho_g$, was proposed by Dang et al. to measure the dependence between a categorical variable, $Y$ , and a numerical variable, $X$. It has been shown that $\rho_g$ has more appealing properties than current existing dependence measurements. In this paper, we develop the jackknife empirical likelihood (JEL) method for $\rho_g$. Confidence intervals for the Gini correlation are constructed without estimating the asymptotic variance. Adjusted and weighted JEL are explored to improve the performance of the standard JEL. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals. The proposed methods are illustrated in an application on two real datasets.

This work studies the parameter identification problem of a generalized non-cooperative game, where each player's cost function is influenced by an observable signal and some unknown parameters. We consider the scenario where equilibrium of the game at some observable signals can be observed with noises, whereas our goal is to identify the unknown parameters with the observed data. Assuming that the observable signals and the corresponding noise-corrupted equilibriums are acquired sequentially, we construct this parameter identification problem as online optimization and introduce a novel online parameter identification algorithm. To be specific, we construct a regularized loss function that balances conservativeness and correctiveness, where the conservativeness term ensures that the new estimates do not deviate significantly from the current estimates, while the correctiveness term is captured by the Karush-Kuhn-Tucker conditions. We then prove that when the players' cost functions are linear with respect to the unknown parameters and the learning rate of the online parameter identification algorithm satisfies \mu_k \propto 1/\sqrt{k}, along with other assumptions, the regret bound of the proposed algorithm is O(\sqrt{K}). Finally, we conduct numerical simulations on a Nash-Cournot problem to demonstrate that the performance of the online identification algorithm is comparable to that of the offline setting.

Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

Suppose a gambler pays one coin per coup to play a two-armed Futurity slot machine, an antique casinos, and two coins are refunded for every two consecutive gambler losses. This payoff is called the Futurity award. The casino owner honestly advertises that each arm on his/her two-armed machine is fair in the sense that the asymptotic expected profit of both gambler and dealer is 0 if the gambler only plays either arm. The gambler is allowed to play either arm on each coup alternatively in some deterministic order or at random. For almost 90 years, since Futurity slot machines is designed in 1936, an open problem that has not been solved for a long time is whether the slot machine will obey the so-called "long bet will lose" phenomenon so common to casino games. Ethier and Lee [Ann. Appl. Proba. 20(2010), pp.1098-1125] conjectured that a player will also definitely lose in the long run by applying any non-random-mixture strategy. In this paper, we shall prove Ethier and Lee's conjecture. Our result with Ethier and Lee's conclusion straightforwardly demonstrates that players decide to use either random or non-random two-arm strategies before playing and then repeated without interruption, the casino owners are always profitable even when the Futurity award is taken into account. The contribution of this work is that it helps complete the demystification of casino profitability. Moreover, it paves the way for casino owners to improve casino game design and for players to participate effectively in gambling.