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We study the problem of algorithmically optimizing the Hamiltonian $H_N$ of a spherical or Ising mixed $p$-spin glass. The maximum asymptotic value $\mathsf{OPT}$ of $H_N/N$ is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize $H_N/N$ up to a value $\mathsf{ALG}$ given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property. However, $\mathsf{ALG} < \mathsf{OPT}$ can also occur, and no efficient algorithm producing an objective value exceeding $\mathsf{ALG}$ is known. We prove that for mixed even $p$-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than $\mathsf{ALG}$ with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of $H_N$. It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving $\mathsf{ALG}$ mentioned above. To prove this result, we substantially generalize the overlap gap property framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.

Bayesian Likelihood-Free Inference (LFI) approaches allow to obtain posterior distributions for stochastic models with intractable likelihood, by relying on model simulations. In Approximate Bayesian Computation (ABC), a popular LFI method, summary statistics are used to reduce data dimensionality. ABC algorithms adaptively tailor simulations to the observation in order to sample from an approximate posterior, whose form depends on the chosen statistics. In this work, we introduce a new way to learn ABC statistics: we first generate parameter-simulation pairs from the model independently on the observation; then, we use Score Matching to train a neural conditional exponential family to approximate the likelihood. The exponential family is the largest class of distributions with fixed-size sufficient statistics; thus, we use them in ABC, which is intuitively appealing and has state-of-the-art performance. In parallel, we insert our likelihood approximation in an MCMC for doubly intractable distributions to draw posterior samples. We can repeat that for any number of observations with no additional model simulations, with performance comparable to related approaches. We validate our methods on toy models with known likelihood and a large-dimensional time-series model.

Data mining techniques offer great opportunities for developing ethics lines, tools for communication, participation and innovation whose main aim is to ensure improvements and compliance with the values, conduct and commitments making up the code of ethics. The aim of this study is to suggest a process for exploiting the data generated by the data generated and collected from an ethics line by extracting rules of association and applying the Apriori algorithm. This makes it possible to identify anomalies and behaviour patterns requiring action to review, correct, promote or expand them, as appropriate. Finally, I offer a simulated application of the Apriori algorithm, supplying it with synthetic data to find out its potential, strengths and limitations.

Functional graphical models explore dependence relationships of random processes. This is achieved through estimating the precision matrix of the coefficients from the Karhunen-Loeve expansion. This paper deals with the problem of estimating functional graphs that consist of the same random processes and share some of the dependence structure. By estimating a single graph we would be shrouding the uniqueness of different sub groups within the data. By estimating a different graph for each sub group we would be dividing our sample size. Instead, we propose a method that allows joint estimation of the graphs while taking into account the intrinsic differences of each sub group. This is achieved by a hierarchical penalty that first penalizes on a common level and then on an individual level. We develop a computation method for our estimator that deals with the non-convex nature of the objective function. We compare the performance of our method with existing ones on a number of different simulated scenarios. We apply our method to an EEG data set that consists of an alcoholic and a non-alcoholic group, to construct brain networks.

Optimal $k$-thresholding algorithms are a class of sparse signal recovery algorithms that overcome the shortcomings of traditional hard thresholding algorithms caused by the oscillation of the residual function. In this paper, we provide a novel theoretical analysis for the data-time tradeoffs of optimal $k$-thresholding algorithms. Both the analysis and numerical results demonstrate that when the number of measurements is small, the algorithms cannot converge; when the number of measurements is suitably large, the number of measurements required for successful recovery has a negative correlation with the number of iterations and the algorithms can achieve linear convergence. Furthermore, the theory presents that the transition point of the number of measurements is on the order of $k \log({en}/{k})$, where $n$ is the dimension of the target signal.

Gaussian Processes (GPs) provide powerful probabilistic frameworks for interpolation, forecasting, and smoothing, but have been hampered by computational scaling issues. Here we investigate data sampled on one dimension (e.g., a scalar or vector time series sampled at arbitrarily-spaced intervals), for which state-space models are popular due to their linearly-scaling computational costs. It has long been conjectured that state-space models are general, able to approximate any one-dimensional GP. We provide the first general proof of this conjecture, showing that any stationary GP on one dimension with vector-valued observations governed by a Lebesgue-integrable continuous kernel can be approximated to any desired precision using a specifically-chosen state-space model: the Latent Exponentially Generated (LEG) family. This new family offers several advantages compared to the general state-space model: it is always stable (no unbounded growth), the covariance can be computed in closed form, and its parameter space is unconstrained (allowing straightforward estimation via gradient descent). The theorem's proof also draws connections to Spectral Mixture Kernels, providing insight about this popular family of kernels. We develop parallelized algorithms for performing inference and learning in the LEG model, test the algorithm on real and synthetic data, and demonstrate scaling to datasets with billions of samples.

The availability of large microarray data has led to a growing interest in biclustering methods in the past decade. Several algorithms have been proposed to identify subsets of genes and conditions according to different similarity measures and under varying constraints. In this paper we focus on the exclusive row biclustering problem for gene expression data sets, in which each row can only be a member of a single bicluster while columns can participate in multiple ones. This type of biclustering may be adequate, for example, for clustering groups of cancer patients where each patient (row) is expected to be carrying only a single type of cancer, while each cancer type is associated with multiple (and possibly overlapping) genes (columns). We present a novel method to identify these exclusive row biclusters through a combination of existing biclustering algorithms and combinatorial auction techniques. We devise an approach for tuning the threshold for our algorithm based on comparison to a null model in the spirit of the Gap statistic approach. We demonstrate our approach on both synthetic and real-world gene expression data and show its power in identifying large span non-overlapping rows sub matrices, while considering their unique nature. The Gap statistic approach succeeds in identifying appropriate thresholds in all our examples.

Importance sampling is one of the most widely used variance reduction strategies in Monte Carlo rendering. In this paper, we propose a novel importance sampling technique that uses a neural network to learn how to sample from a desired density represented by a set of samples. Our approach considers an existing Monte Carlo rendering algorithm as a black box. During a scene-dependent training phase, we learn to generate samples with a desired density in the primary sample space of the rendering algorithm using maximum likelihood estimation. We leverage a recent neural network architecture that was designed to represent real-valued non-volume preserving ('Real NVP') transformations in high dimensional spaces. We use Real NVP to non-linearly warp primary sample space and obtain desired densities. In addition, Real NVP efficiently computes the determinant of the Jacobian of the warp, which is required to implement the change of integration variables implied by the warp. A main advantage of our approach is that it is agnostic of underlying light transport effects, and can be combined with many existing rendering techniques by treating them as a black box. We show that our approach leads to effective variance reduction in several practical scenarios.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

During recent years, active learning has evolved into a popular paradigm for utilizing user's feedback to improve accuracy of learning algorithms. Active learning works by selecting the most informative sample among unlabeled data and querying the label of that point from user. Many different methods such as uncertainty sampling and minimum risk sampling have been utilized to select the most informative sample in active learning. Although many active learning algorithms have been proposed so far, most of them work with binary or multi-class classification problems and therefore can not be applied to problems in which only samples from one class as well as a set of unlabeled data are available. Such problems arise in many real-world situations and are known as the problem of learning from positive and unlabeled data. In this paper we propose an active learning algorithm that can work when only samples of one class as well as a set of unlabelled data are available. Our method works by separately estimating probability desnity of positive and unlabeled points and then computing expected value of informativeness to get rid of a hyper-parameter and have a better measure of informativeness./ Experiments and empirical analysis show promising results compared to other similar methods.