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We discuss probabilistic neural networks for unsupervised learning with a fixed internal representation as models for machine understanding. Here understanding is intended as mapping data to an already existing representation which encodes an {\em a priori} organisation of the feature space. We derive the internal representation by requiring that it satisfies the principles of maximal relevance and of maximal ignorance about how different features are combined. We show that, when hidden units are binary variables, these two principles identify a unique model -- the Hierarchical Feature Model (HFM) -- which is fully solvable and provides a natural interpretation in terms of features. We argue that learning machines with this architecture enjoy a number of interesting properties, like the continuity of the representation with respect to changes in parameters and data, the possibility to control the level of compression and the ability to support functions that go beyond generalisation. We explore the behaviour of the model with extensive numerical experiments and argue that models where the internal representation is fixed reproduce a learning modality which is qualitatively different from that of more traditional models such as Restricted Boltzmann Machines.

Determining the memory capacity of two-layer neural networks with m hidden neurons and input dimension d (i.e., md+m total trainable parameters), which refers to the largest size of general data the network can memorize, is a fundamental machine-learning question. For non-polynomial real analytic activation functions, such as sigmoids and smoothed rectified linear units (smoothed ReLUs), we establish a lower bound of md/2 and optimality up to a factor of approximately 2. Analogous prior results were limited to Heaviside and ReLU activations, with results for smooth activations suffering from logarithmic factors and requiring random data. To analyze the memory capacity, we examine the rank of the network's Jacobian by computing the rank of matrices involving both Hadamard powers and the Khati-Rao product. Our computation extends classical linear algebraic facts about the rank of Hadamard powers. Overall, our approach differs from previous works on memory capacity and holds promise for extending to deeper models and other architectures.

We propose an automated nonlinear model reduction and mesh adaptation framework for rapid and reliable solution of parameterized advection-dominated problems, with emphasis on compressible flows. The key features of our approach are threefold: (i) a metric-based mesh adaptation technique to generate an accurate mesh for a range of parameters, (ii) a general (i.e., independent of the underlying equations) registration procedure for the computation of a mapping $\Phi$ that tracks moving features of the solution field, and (iii) an hyper-reduced least-square Petrov-Galerkin reduced-order model for the rapid and reliable estimation of the mapped solution. We discuss a general paradigm -- which mimics the refinement loop considered in mesh adaptation -- to simultaneously construct the high-fidelity and the reduced-order approximations, and we discuss actionable strategies to accelerate the offline phase. We present extensive numerical investigations for a quasi-1D nozzle problem and for a two-dimensional inviscid flow past a Gaussian bump to display the many features of the methodology and to assess the performance for problems with discontinuous solutions.

Linear regression and classification models with repeated functional data are considered. For each statistical unit in the sample, a real-valued parameter is observed over time under different conditions. Two regression models based on fusion penalties are presented. The first one is a generalization of the variable fusion model based on the 1-nearest neighbor. The second one, called group fusion lasso, assumes some grouping structure of conditions and allows for homogeneity among the regression coefficient functions within groups. A finite sample numerical simulation and an application on EEG data are presented.

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the domain decomposition approach, we derive an optimal control problem, for which we present the convergence analysis. The snapshots for the high-fidelity model are obtained with the Finite Element discretisation, and the model order reduction is then proposed both in terms of time and physical parameters, with a standard POD-Galerkin projection. We test the proposed methodology on two fluid dynamics benchmarks: the non-stationary backward-facing step and lid-driven cavity flow. Finally, also in view of future works, we compare the intrusive POD--Galerkin approach with a non--intrusive approach based on Neural Networks.

An established normative approach for understanding the algorithmic basis of neural computation is to derive online algorithms from principled computational objectives and evaluate their compatibility with anatomical and physiological observations. Similarity matching objectives have served as successful starting points for deriving online algorithms that map onto neural networks (NNs) with point neurons and Hebbian/anti-Hebbian plasticity. These NN models account for many anatomical and physiological observations; however, the objectives have limited computational power and the derived NNs do not explain multi-compartmental neuronal structures and non-Hebbian forms of plasticity that are prevalent throughout the brain. In this article, we unify and generalize recent extensions of the similarity matching approach to address more complex objectives, including a large class of unsupervised and self-supervised learning tasks that can be formulated as symmetric generalized eigenvalue problems or nonnegative matrix factorization problems. Interestingly, the online algorithms derived from these objectives naturally map onto NNs with multi-compartmental neurons and local, non-Hebbian learning rules. Therefore, this unified extension of the similarity matching approach provides a normative framework that facilitates understanding multi-compartmental neuronal structures and non-Hebbian plasticity found throughout the brain.

We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the correspondence-interleaving distance, a metric for hierarchical clusterings that we introduce. Taking certain one-parameter slices of degree-Rips recovers well-known methods for density-based clustering, but we show that these methods are unstable. However, we prove that degree-Rips, as a multiparameter object, is stable, and we propose an alternative approach for taking slices of degree-Rips, which yields a one-parameter hierarchical clustering algorithm with better stability properties. We prove that this algorithm is consistent, using the correspondence-interleaving distance. We provide an algorithm for extracting a single clustering from one-parameter hierarchical clusterings, which is stable with respect to the correspondence-interleaving distance. And, we integrate these methods into a pipeline for density-based clustering, which we call Persistable. Adapting tools from multiparameter persistent homology, we propose visualization tools that guide the selection of all parameters of the pipeline. We demonstrate Persistable on benchmark datasets, showing that it identifies multi-scale cluster structure in data.

The development of technologies for causal inference with the privacy preservation of distributed data has attracted considerable attention in recent years. To address this issue, we propose a data collaboration quasi-experiment (DC-QE) that enables causal inference from distributed data with privacy preservation. In our method, first, local parties construct dimensionality-reduced intermediate representations from the private data. Second, they share intermediate representations, instead of private data for privacy preservation. Third, propensity scores were estimated from the shared intermediate representations. Finally, the treatment effects were estimated from propensity scores. Our method can reduce both random errors and biases, whereas existing methods can only reduce random errors in the estimation of treatment effects. Through numerical experiments on both artificial and real-world data, we confirmed that our method can lead to better estimation results than individual analyses. Dimensionality-reduction loses some of the information in the private data and causes performance degradation. However, we observed that in the experiments, sharing intermediate representations with many parties to resolve the lack of subjects and covariates, our method improved performance enough to overcome the degradation caused by dimensionality-reduction. With the spread of our method, intermediate representations can be published as open data to help researchers find causalities and accumulated as a knowledge base.

We present Surjective Sequential Neural Likelihood (SSNL) estimation, a novel method for simulation-based inference in models where the evaluation of the likelihood function is not tractable and only a simulator that can generate synthetic data is available. SSNL fits a dimensionality-reducing surjective normalizing flow model and uses it as a surrogate likelihood function which allows for conventional Bayesian inference using either Markov chain Monte Carlo methods or variational inference. By embedding the data in a low-dimensional space, SSNL solves several issues previous likelihood-based methods had when applied to high-dimensional data sets that, for instance, contain non-informative data dimensions or lie along a lower-dimensional manifold. We evaluate SSNL on a wide variety of experiments and show that it generally outperforms contemporary methods used in simulation-based inference, for instance, on a challenging real-world example from astrophysics which models the magnetic field strength of the sun using a solar dynamo model.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.