Trends in terrestrial temperature variability are perhaps more relevant for species viability than trends in mean temperature. In this paper, we develop methodology for estimating such trends using multi-resolution climate data from polar orbiting weather satellites. We derive two novel algorithms for computation that are tailored for dense, gridded observations over both space and time. We evaluate our methods with a simulation that mimics these data's features and on a large, publicly available, global temperature dataset with the eventual goal of tracking trends in cloud reflectance temperature variability.
Our methodology is similar in flavor to @cite_7 or related work in , but with several fundamental differences. These papers aim to discover the time-varying structure of a network. To achieve this goal, they use Gaussian likelihood with unknown precision matrix and introduce penalty terms which (1) encourage sparsity among the off-diagonal elements and (2) discourage changes in the estimated inverse covariance matrix from one time-step to the next. Our goal in the present work is to detect the temporal trend in the variance of each point in the network, but the network is known (corresponding to the grid over the earth) and fixed in time. To apply these methods in our context (e.g., HallacPark2017 , Eq. 2), we would enforce complete sparsity on the off-diagonal elements (since they are not estimated) and add a new penalty to enforce spatial behavior across the diagonal elements. Thus, is not simply a special case of these existing methods. Finally, these papers examine networks with hundreds of nodes and dozens to hundreds of time points. As discussed next, our data are significantly larger than these networks and attempting to estimate a full covariance would be prohibitive, were it necessary.
@cite_7
1988351624 2466376234 2949962875 2626325961
We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. We propose a deep recurrent belief network with distributed time delays for learning multivariate Gaussians. Learning long time delays in deep belief networks is difficult due to the problem of vanishing or exploding gradients with increase in delay. To mitigate this problem and improve the transparency of learning time-delays, we introduce the use of Gaussian networks with time-delays to initialize the weights of each hidden neuron. From our knowledge of time delays, it is possible to learn the long delays from short delays in a hierarchical manner. In contrast to previous works, here dynamic Gaussian Bayesian networks over training samples are evolved using Markov Chain Monte Carlo to determine the initial weights of each hidden layer of neurons. In this way, the time-delayed network motifs of increasing Markov order across layers can be modeled hierarchically using a deep model. To validate the proposed Variable-order Belief Network (VBN) framework, it is applied for modeling word dependencies in text. To explore the generality of VBN, it is further considered for a real-world scenario where the dynamic movements of basketball players are modeled. Experimental results obtained showed that the proposed VBN could achieve over 30 improvement in accuracy on real-world scenarios compared to the state-of-the-art baselines. In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.