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Register a new userWind Field Reconstruction with Adaptive Random Fourier Features
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Added by
Eld Emanuel Str\\\"om
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging and Inverse distance weighting. Random Fourier features is a linear model $beta(pmb x) = sum_{k=1}^K beta_k e^{iomega_k pmb x}$ approximating the velocity field, with frequencies $omega_k$ randomly sampled and amplitudes $beta_k$ trained to minimize a loss function. We include a physically motivated divergence penalty term $| abla cdot beta(pmb x)|^2$, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.
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Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.
The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the loss gradient based on these classes, known as the sampled softmax method. However, the sampled softmax provides a biased estimate of the gradient unless the samples are drawn from the exact softmax distribution, which is again expensive to compute. Therefore, a widely employed practical approach involves sampling from a simpler distribution in the hope of approximating the exact softmax distribution. In this paper, we develop the first theoretical understanding of the role that different sampling distributions play in determining the quality of sampled softmax. Motivated by our analysis and the work on kernel-based sampling, we propose the Random Fourier Softmax (RF-softmax) method that utilizes the powerful Random Fourier Features to enable more efficient and accurate sampling from an approximate softmax distribution. We show that RF-softmax leads to low bias in estimation in terms of both the full softmax distribution and the full softmax gradient. Furthermore, the cost of RF-softmax scales only logarithmically with the number of classes.
The method of random projection (RP) is the standard technique in machine learning and many other areas, for dimensionality reduction, approximate near neighbor search, compressed sensing, etc. Basically, RP provides a simple and effective scheme for approximating pairwise inner products and Euclidean distances in massive data. Closely related to RP, the method of random Fourier features (RFF) has also become popular, for approximating the Gaussian kernel. RFF applies a specific nonlinear transformation on the projected data from random projections. In practice, using the (nonlinear) Gaussian kernel often leads to better performance than the linear kernel (inner product), partly due to the tuning parameter $(gamma)$ introduced in the Gaussian kernel. Recently, there has been a surge of interest in studying properties of RFF. After random projections, quantization is an important step for efficient data storage, computation, and transmission. Quantization for RP has also been extensive studied in the literature. In this paper, we focus on developing quantization algorithms for RFF. The task is in a sense challenging due to the tuning parameter $gamma$ in the Gaussian kernel. For example, the quantizer and the quantized data might be tied to each specific tuning parameter $gamma$. Our contribution begins with an interesting discovery, that the marginal distribution of RFF is actually free of the Gaussian kernel parameter $gamma$. This small finding significantly simplifies the design of the Lloyd-Max (LM) quantization scheme for RFF in that there would be only one LM quantizer for RFF (regardless of $gamma$). We also develop a variant named LM$^2$-RFF quantizer, which in certain cases is more accurate. Experiments confirm that the proposed quantization schemes perform well.
In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the oft-considered set of block frequency sparse functions of the form $$f(x) = sum^{n}_{j=1} sum^{B-1}_{k=0} c_{omega_j + k} e^{i(omega_j + k)x},~~{ omega_1, dots, omega_n } subset left(-leftlceil frac{N}{2}rightrceil, leftlfloor frac{N}{2}rightrfloorright]capmathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.
This paper proposes a novel, rigorous and simple Fourier-transformation approach to study resonances in a perfectly conducting slab with finite number of subwavelength slits of width $hll 1$. Since regions outside the slits are variable separated, by Fourier transforming the governing equation, we could express field in the outer regions in terms of field derivatives on the aperture. Next, in each slit where variable separation is still available, wave field could be expressed as a Fourier series in terms of a countable basis functions with unknown Fourier coefficients. Finally, by matching field on the aperture, we establish a linear system of infinite number of equations governing the countable Fourier coefficients. By carefully asymptotic analysis of each entry of the coefficient matrix, we rigorously show that, by removing only a finite number of rows and columns, the resulting principle sub-matrix is diagonally dominant so that the infinite dimensional linear system can be reduced to a finite dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn provides a simple, asymptotic formula describing resonance frequencies with accuracy ${cal O}(h^3log h)$. We emphasize that such a formula is more accurate than all existing results and is the first accurate result especially for slits of number more than two to our best knowledge. Moreover, this asymptotic formula rigorously confirms a fact that the imaginary part of resonance frequencies is always ${cal O}(h)$ no matter how we place the slits as long as they are spaced by distances independent of width $h$.
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