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Register a new userOptimizing Photonic Nanostructures via Multi-fidelity Gaussian Processes
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We apply numerical methods in combination with finite-difference-time-domain (FDTD) simulations to optimize transmission properties of plasmonic mirror color filters using a multi-objective figure of merit over a five-dimensional parameter space by utilizing novel multi-fidelity Gaussian processes approach. We compare these results with conventional derivative-free global search algorithms, such as (single-fidelity) Gaussian Processes optimization scheme, and Particle Swarm Optimization---a commonly used method in nanophotonics community, which is implemented in Lumerical commercial photonics software. We demonstrate the performance of various numerical optimization approaches on several pre-collected real-world datasets and show that by properly trading off expensive information sources with cheap simulations, one can more effectively optimize the transmission properties with a fixed budget.
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How can we efficiently gather information to optimize an unknown function, when presented with multiple, mutually dependent information sources with different costs? For example, when optimizing a robotic system, intelligently trading off computer simulations and real robot testings can lead to significant savings. Existing methods, such as multi-fidelity GP-UCB or Entropy Search-based approaches, either make simplistic assumptions on the interaction among different fidelities or use simple heuristics that lack theoretical guarantees. In this paper, we study multi-fidelity Bayesian optimization with complex structural dependencies among multiple outputs, and propose MF-MI-Greedy, a principled algorithmic framework for addressing this problem. In particular, we model different fidelities using additive Gaussian processes based on shared latent structures with the target function. Then we use cost-sensitive mutual information gain for efficient Bayesian global optimization. We propose a simple notion of regret which incorporates the cost of different fidelities, and prove that MF-MI-Greedy achieves low regret. We demonstrate the strong empirical performance of our algorithm on both synthetic and real-world datasets.
We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We propose the harmonic kernel decomposition (HKD), which uses Fourier series to decompose a kernel as a sum of orthogonal kernels. Our variational approximation exploits this orthogonality to enable a large number of inducing points at a low computational cost. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections, and it significantly outperforms standard variational methods in scalability and accuracy. Notably, our approach achieves state-of-the-art results on CIFAR-10 among pure GP models.
Recommender systems are often designed based on a collaborative filtering approach, where user preferences are predicted by modelling interactions between users and items. Many common approaches to solve the collaborative filtering task are based on learning representations of users and items, including simple matrix factorization, Gaussian process latent variable models, and neural-network based embeddings. While matrix factorization approaches fail to model nonlinear relations, neural networks can potentially capture such complex relations with unprecedented predictive power and are highly scalable. However, neither of them is able to model predictive uncertainties. In contrast, Gaussian Process based models can generate a predictive distribution, but cannot scale to large amounts of data. In this manuscript, we propose a novel approach combining the representation learning paradigm of collaborative filtering with multi-output Gaussian processes in a joint framework to generate uncertainty-aware recommendations. We introduce an efficient strategy for model training and inference, resulting in a model that scales to very large and sparse datasets and achieves competitive performance in terms of classical metrics quantifying the reconstruction error. In addition to accurately predicting user preferences, our model also provides meaningful uncertainty estimates about that prediction.
We propose a method (TT-GP) for approximate inference in Gaussian Process (GP) models. We build on previous scalable GP research including stochastic variational inference based on inducing inputs, kernel interpolation, and structure exploiting algebra. The key idea of our method is to use Tensor Train decomposition for variational parameters, which allows us to train GPs with billions of inducing inputs and achieve state-of-the-art results on several benchmarks. Further, our approach allows for training kernels based on deep neural networks without any modifications to the underlying GP model. A neural network learns a multidimensional embedding for the data, which is used by the GP to make the final prediction. We train GP and neural network parameters end-to-end without pretraining, through maximization of GP marginal likelihood. We show the efficiency of the proposed approach on several regression and classification benchmark datasets including MNIST, CIFAR-10, and Airline.
Refining low-resolution (LR) spatial fields with high-resolution (HR) information is challenging as the diversity of spatial datasets often prevents direct matching of observations. Yet, when LR samples are modeled as aggregate conditional means of HR samples with respect to a mediating variable that is globally observed, the recovery of the underlying fine-grained field can be framed as taking an inverse of the conditional expectation, namely a deconditioning problem. In this work, we introduce conditional mean processes (CMP), a new class of Gaussian Processes describing conditional means. By treating CMPs as inter-domain features of the underlying field, a posterior for the latent field can be established as a solution to the deconditioning problem. Furthermore, we show that this solution can be viewed as a two-staged vector-valued kernel ridge regressor and show that it has a minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate its proficiency in a synthetic and a real-world atmospheric field downscaling problem, showing substantial improvements over existing methods.
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