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In cut sparsification, all cuts of a hypergraph $H=(V,E,w)$ are approximated within $1\pm\epsilon$ factor by a small hypergraph $H'$. This widely applied method was generalized recently to a setting where the cost of cutting each $e\in E$ is provided by a splitting function, $g_e: 2^e\to\mathbb{R}_+$. This generalization is called a submodular hypergraph when the functions $\{g_e\}_{e\in E}$ are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work focused on the setting where $H'$ is a reweighted sub-hypergraph of $H$, and measured size by the number of hyperedges in $H'$. We study such sparsification, and also a more general notion of representing $H$ succinctly, where size is measured in bits. In the sparsification setting, where size is the number of hyperedges, we present three results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in $n=|V|$; (ii) monotone-submodular hypergraphs admit sparsifiers of size $O(\epsilon^{-2} n^3)$; and (iii) we propose a new parameter, called spread, to obtain even smaller sparsifiers in some cases. In the succinct-representation setting, we show that a natural family of splitting functions admits a succinct representation of much smaller size than via reweighted subgraphs (almost by factor $n$). This large gap is surprising because for graphs, the most succinct representation is attained by reweighted subgraphs. Along the way, we introduce the notion of deformation, where $g_e$ is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions.

Extended Reality (XR) is one of the most important 5G/6G media applications that will fundamentally transform human interactions. However, ensuring low latency, high data rate, and reliability to support XR services poses significant challenges. This letter presents a novel AI-assisted service provisioning scheme that leverages predicted frames for processing rather than relying solely on actual frames. This method virtually increases the network delay budget and consequently improves service provisioning, albeit at the expense of minor prediction errors. The proposed scheme is validated by extensive simulations demonstrating a multi-fold increase in supported XR users and also provides crucial network design insights.

In generative modeling, numerous successful approaches leverage a low-dimensional latent space, e.g., Stable Diffusion models the latent space induced by an encoder and generates images through a paired decoder. Although the selection of the latent space is empirically pivotal, determining the optimal choice and the process of identifying it remain unclear. In this study, we aim to shed light on this under-explored topic by rethinking the latent space from the perspective of model complexity. Our investigation starts with the classic generative adversarial networks (GANs). Inspired by the GAN training objective, we propose a novel "distance" between the latent and data distributions, whose minimization coincides with that of the generator complexity. The minimizer of this distance is characterized as the optimal data-dependent latent that most effectively capitalizes on the generator's capacity. Then, we consider parameterizing such a latent distribution by an encoder network and propose a two-stage training strategy called Decoupled Autoencoder (DAE), where the encoder is only updated in the first stage with an auxiliary decoder and then frozen in the second stage while the actual decoder is being trained. DAE can improve the latent distribution and as a result, improve the generative performance. Our theoretical analyses are corroborated by comprehensive experiments on various models such as VQGAN and Diffusion Transformer, where our modifications yield significant improvements in sample quality with decreased model complexity.

With the fast development of reconfigurable intelligent surface (RIS), the network topology becomes more complex and varied, which makes the network design and analysis extremely challenging. Most of the current works adopt the binary system stochastic geometric, missing the coupling relationships between the direct and reflected paths caused by RISs. In this paper, we first define the typical triangle which consists of a base station (BS), a RIS and a user equipment (UE) as the basic ternary network unit in a RIS-assisted ultra-dense network (UDN). In addition, we extend the Campbell's theorem to the ternary system and present the ternary probability generating functional (PGFL) of the stochastic geometry. Based on the ternary stochastic geometry theory, we derive and analyze the coverage probability, area spectral efficiency (ASE), area energy efficiency (AEE) and energy coverage efficiency (ECE) of the RIS-assisted UDN system. Simulation results show that the RISs can improve the system performances, especially for the UE who has a high signal to interference plus noise ratio (SINR), as if the introduced RIS brings in Matthew effect. This phenomenon of RIS is appealing for guiding the design of complex networks.

The problem $\textrm{PosSLP}$ involves determining whether an integer computed by a given straight-line program is positive. This problem has attracted considerable attention within the field of computational complexity as it provides a complete characterization of the complexity associated with numerical computation. However, non-trivial lower bounds for $\textrm{PosSLP}$ remain unknown. In this paper, we demonstrate that $\textrm{PosSLP} \in \textrm{BPP}$ would imply that $\textrm{NP} \subseteq \textrm{BPP}$, under the assumption of a conjecture concerning the complexity of the radical of a polynomial proposed by Dutta, Saxena, and Sinhababu (STOC'2018). Our proof builds upon the established $\textrm{NP}$-hardness of determining if a univariate polynomial computed by an SLP has a real root, as demonstrated by Perrucci and Sabia (JDA'2005). Therefore, our lower bound for $\textrm{PosSLP}$ represents a significant advancement in understanding the complexity of this problem. It constitutes the first non-trivial lower bound for $\textrm{PosSLP}$ , albeit conditionally. Additionally, we show that counting the real roots of an integer univariate polynomial, given as input by a straight-line program, is $\#\textrm{P}$-hard.

This work addresses a version of the two-armed Bernoulli bandit problem where the sum of the means of the arms is one (the symmetric two-armed Bernoulli bandit). In a regime where the gap between these means goes to zero as the number of prediction periods approaches infinity, i.e., the difficulty of detecting the gap increases as the sample size increases, we obtain the leading order terms of the minmax optimal regret and pseudoregret for this problem by associating each of them with a solution of a linear heat equation. Our results improve upon the previously known results; specifically, we explicitly compute these leading order terms in three different scaling regimes for the gap. Additionally, we obtain new non-asymptotic bounds for any given time horizon. Although optimal player strategies are not known for more general bandit problems, there is significant interest in considering how regret accumulates under specific player strategies, even when they are not known to be optimal. We expect that the methods of this paper should be useful in settings of that type.

Wheeler automata were introduced in 2017 as a tool to generalize existing indexing and compression techniques based on the Burrows-Wheeler transform. Intuitively, an automaton is said to be Wheeler if there exists a total order on its states reflecting the co-lexicographic order of the strings labeling the automaton's paths; this property makes it possible to represent the automaton's topology in a constant number of bits per transition, as well as efficiently solving pattern matching queries on its accepted regular language. After their introduction, Wheeler automata have been the subject of a prolific line of research, both from the algorithmic and language-theoretic points of view. A recurring issue faced in these studies is the lack of large datasets of Wheeler automata on which the developed algorithms and theories could be tested. One possible way to overcome this issue is to generate random Wheeler automata. Motivated by this observation, in this paper we initiate the theoretical study of random Wheeler automata, focusing on the deterministic case (Wheeler DFAs -- WDFAs). We start by extending the Erd\H{o}s-R\'enyi random graph model to WDFAs, and proceed by providing an algorithm generating uniform WDFAs according to this model. Our algorithm generates a uniform WDFA with $n$ states, $m$ transitions, and alphabet's cardinality $\sigma$ in $O(m)$ expected time ($O(m\log m)$ worst-case time w.h.p.) and constant working space for all alphabets of size $\sigma \le m/\ln m$. As a by-product, we also give formulas for the number of distinct WDFAs and obtain that $ n\sigma + (n - \sigma) \log \sigma$ bits are necessary and sufficient to encode a WDFA with $n$ states and alphabet of size $\sigma$, up to an additive $\Theta(n)$ term. We present an implementation of our algorithm and show that it is extremely fast in practice, with a throughput of over 8 million transitions per second.

Automata operating on infinite objects feature prominently in the theory of the modal $\mu$-calculus. One such application concerns the tableau games introduced by Niwi\'{n}ski & Walukiewicz, of which the winning condition for infinite plays can be naturally checked by a nondeterministic parity stream automaton. Inspired by work of Jungteerapanich and Stirling we show how determinization constructions of this automaton may be used to directly obtain proof systems for the $\mu$-calculus. More concretely, we introduce a binary tree construction for determinizing nondeterministic parity stream automata. Using this construction we define the annotated cyclic proof system $\mathsf{BT}$, where formulas are annotated by tuples of binary strings. Soundness and Completeness of this system follow almost immediately from the correctness of the determinization method.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.