亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

This paper investigates the convergence time of log-linear learning to an $\epsilon$-efficient Nash equilibrium (NE) in potential games. In such games, an efficient NE is defined as the maximizer of the potential function. Existing results are limited to potential games with stringent structural assumptions and entail exponential convergence times in $1/\epsilon$. Unaddressed so far, we tackle general potential games and prove the first finite-time convergence to an $\epsilon$-efficient NE. In particular, by using a problem-dependent analysis, our bound depends polynomially on $1/\epsilon$. Furthermore, we provide two extensions of our convergence result: first, we show that a variant of log-linear learning that requires a factor $A$ less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.

We study the possibility of designing $N^{o(1)}$-round protocols for problems of substantially super-linear polynomial-time complexity on the congested clique with about $N^{1/2}$ nodes, where $N$ is the input size. We show that the exponent of the polynomial (if any) bounding the average time complexity of local computations performed at a node in such protocols has to be larger than that of the polynomial bounding the time complexity of the given problem.

We present a scalable machine learning (ML) force-field model for the adiabatic dynamics of cooperative Jahn-Teller (JT) systems. Large scale dynamical simulations of the JT model also shed light on the orbital ordering dynamics in colossal magnetoresistance manganites. The JT effect in these materials describes the distortion of local oxygen octahedra driven by a coupling to the orbital degrees of freedom of $e_g$ electrons. An effective electron-mediated interaction between the local JT modes leads to a structural transition and the emergence of long-range orbital order at low temperatures. Assuming the principle of locality, a deep-learning neural-network model is developed to accurately and efficiently predict the electron-induced forces that drive the dynamical evolution of JT phonons. A group-theoretical method is utilized to develop a descriptor that incorporates the combined orbital and lattice symmetry into the ML model. Large-scale Langevin dynamics simulations, enabled by the ML force-field models, are performed to investigate the coarsening dynamics of the composite JT distortion and orbital order after a thermal quench. The late-stage coarsening of orbital domains exhibits pronounced freezing behaviors which are likely related to the unusual morphology of the domain structures. Our work highlights a promising avenue for multi-scale dynamical modeling of correlated electron systems.

In this article, we combine Sweedler's classic theory of measuring coalgebras -- by which $k$-algebras are enriched in $k$-coalgebras for $k$ a field -- with the theory of W-types -- by which the categorical semantics of inductive data types in functional programming languages are understood. In our main theorem, we find that under some hypotheses, algebras of an endofunctor are enriched in coalgebras of the same endofunctor, and we find polynomial endofunctors provide many interesting examples of this phenomenon. We then generalize the notion of initial algebra of an endofunctor using this enrichment, thus generalizing the notion of W-type. This article is an extended version of arXiv:2303.16793, it adds expository introductions to the original theories of measuring coalgebras and W-types along with some improvements to the main theory and many explicitly worked examples.

We investigate the maximum cardinality and the mathematical structure of error-correcting codes endowed with the Kendall-$\tau$ metric. We establish an averaging bound for the cardinality of a code with prescribed minimum distance, discuss its sharpness, and characterize codes attaining it. This leads to introducing the family of $t$-balanced codes in the Kendall-$\tau$ metric. The results are based on novel arguments that shed new light on the structure of the Kendall-$\tau$ metric space.

In this paper, we propose Wasserstein proximals of $\alpha$-divergences as suitable objective functionals for learning heavy-tailed distributions in a stable manner. First, we provide sufficient, and in some cases necessary, relations among data dimension, $\alpha$, and the decay rate of data distributions for the Wasserstein-proximal-regularized divergence to be finite. Finite-sample convergence rates for the estimation in the case of the Wasserstein-1 proximal divergences are then provided under certain tail conditions. Numerical experiments demonstrate stable learning of heavy-tailed distributions -- even those without first or second moment -- without any explicit knowledge of the tail behavior, using suitable generative models such as GANs and flow-based models related to our proposed Wasserstein-proximal-regularized $\alpha$-divergences. Heuristically, $\alpha$-divergences handle the heavy tails and Wasserstein proximals allow non-absolute continuity between distributions and control the velocities of flow-based algorithms as they learn the target distribution deep into the tails.

Given an unconditional diffusion model $\pi(x, y)$, using it to perform conditional simulation $\pi(x \mid y)$ is still largely an open question and is typically achieved by learning conditional drifts to the denoising SDE after the fact. In this work, we express conditional simulation as an inference problem on an augmented space corresponding to a partial SDE bridge. This perspective allows us to implement efficient and principled particle Gibbs and pseudo-marginal samplers marginally targeting the conditional distribution $\pi(x \mid y)$. Contrary to existing methodology, our methods do not introduce any additional approximation to the unconditional diffusion model aside from the Monte Carlo error. We showcase the benefits and drawbacks of our approach on a series of synthetic and real data examples.

$D^2$-sampling is a fundamental component of sampling-based clustering algorithms such as $k$-means++. Given a dataset $V \subset \mathbb{R}^d$ with $N$ points and a center set $C \subset \mathbb{R}^d$, $D^2$-sampling refers to picking a point from $V$ where the sampling probability of a point is proportional to its squared distance from the nearest center in $C$. Starting with empty $C$ and iteratively $D^2$-sampling and updating $C$ in $k$ rounds is precisely $k$-means++ seeding that runs in $O(Nkd)$ time and gives $O(\log{k})$-approximation in expectation for the $k$-means problem. We give a quantum algorithm for (approximate) $D^2$-sampling in the QRAM model that results in a quantum implementation of $k$-means++ that runs in time $\tilde{O}(\zeta^2 k^2)$. Here $\zeta$ is the aspect ratio (i.e., largest to smallest interpoint distance), and $\tilde{O}$ hides polylogarithmic factors in $N, d, k$. It can be shown through a robust approximation analysis of $k$-means++ that the quantum version preserves its $O(\log{k})$ approximation guarantee. Further, we show that our quantum algorithm for $D^2$-sampling can be 'dequantized' using the sample-query access model of Tang (PhD Thesis, Ewin Tang, University of Washington, 2023). This results in a fast quantum-inspired classical implementation of $k$-means++, which we call QI-$k$-means++, with a running time $O(Nd) + \tilde{O}(\zeta^2k^2d)$, where the $O(Nd)$ term is for setting up the sample-query access data structure. Experimental investigations show promising results for QI-$k$-means++ on large datasets with bounded aspect ratio. Finally, we use our quantum $D^2$-sampling with the known $ D^2$-sampling-based classical approximation scheme (i.e., $(1+\varepsilon)$-approximation for any given $\varepsilon>0$) to obtain the first quantum approximation scheme for the $k$-means problem with polylogarithmic running time dependence on $N$.

We consider a wide class of generalized Radon transforms $\mathcal R$, which act in $\mathbb{R}^n$ for any $n\ge 2$ and integrate over submanifolds of any codimension $N$, $1\le N\le n-1$. Also, we allow for a fairly general reconstruction operator $\mathcal A$. The main requirement is that $\mathcal A$ be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data $g_{j,k} = (\mathcal R f)_{j,k} + \eta_{j,k}$. We show that the reconstruction error $N_\epsilon^{\text{rec}}=\mathcal A \eta_{j,k}$ satisfies $N^{\text{rec}}(\check x;x_0)=\lim_{\epsilon\to0}N_\epsilon^{\text{rec}}(x_0+\epsilon\check x)$, $\check x\in D$. Here $x_0$ is a fixed point, $D\subset\mathbb{R}^n$ is a bounded domain, and $\eta_{j,k}$ are independent, but not necessarily identically distributed, random variables. $N^{\text{rec}}$ and $N_\epsilon^{\text{rec}}$ are viewed as continuous random functions of the argument $\check x$ (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of $\eta_{j,k}$ (and some other not very restrictive conditions on $x_0$ and $\mathcal A$), we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in $\mathbb{R}^3$, which shows an excellent match between theoretical predictions and simulated reconstructions.

Similar to the notion of h-adaptivity, where the discretization resolution is adaptively changed, I propose the notion of model adaptivity, where the underlying model (the governing equations) is adaptively changed in space and time. Specifically, this work introduces a hybrid and adaptive coupling of a 3D bulk fluid flow model with a 2D thin film flow model. As a result, this work extends the applicability of existing thin film flow models to complex scenarios where, for example, bulk flow develops into thin films after striking a surface. At each location in space and time, the proposed framework automatically decides whether a 3D model or a 2D model must be applied. Using a meshless approach for both 3D and 2D models, at each particle, the decision to apply a 2D or 3D model is based on the user-prescribed resolution and a local principal component analysis. When a particle needs to be changed from a 3D model to 2D, or vice versa, the discretization is changed, and all relevant data mapping is done on-the-fly. Appropriate two-way coupling conditions and mass conservation considerations between the 3D and 2D models are also developed. Numerical results show that this model adaptive framework shows higher flexibility and compares well against finely resolved 3D simulations. In an actual application scenario, a 3 factor speed up is obtained, while maintaining the accuracy of the solution.