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1. How Deep Are Deep Gaussian Processes?

   Matthew M. Dunlop, Mark A. ( September 2018 , Journal of machine learning research)

   Recent research has shown the potential utility of deep Gaussian processes. These deep structures are probability distributions, designed through hierarchical construction, which are conditionally Gaussian. In this paper, the current published body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples generated from a deep Gaussian process have a Markovian structure with respect to the depth parameter, and the effective depth of the resulting process is interpreted in terms of the ergodicity, or non-ergodicity, of the resulting Markov chain. For the classes of deep Gaussian processes introduced,more »we provide results concerning their ergodicity and hence their effective depth. We also demonstrate how these processes may be used for inference; in particular we show how a Metropolis-within-Gibbs construc-tion across the levels of the hierarchy can be used to derive sampling tools which are robust to the level of resolution used to represent the functions on a computer. For illustration, we consider the effect of ergodicity in some simple numerical examples.« less
2. Block-randomized Stochastic Proximal Gradient for Constrained Low-rank Tensor Factorization

   https://doi.org/10.1109/ICASSP.2019.8682465  

   Fu, Xiao ; Gao, Cheng ; Wai, Hoi-To ; Huang, Kejun ( May 2019 , ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   This work focuses on canonical polyadic decomposition (CPD) for large-scale tensors. Many prior works rely on data sparsity to develop scalable CPD algorithms, which are not suitable for handling dense tensor, while dense tensors often arise in applications such as image and video processing. As an alternative, stochastic algorithms utilize data sampling to reduce per-iteration complexity and thus are very scalable, even when handling dense tensors. However, existing stochastic CPD algorithms are facing some challenges. For example, some algorithms are based on randomly sampled tensor entries, and thus each iteration can only updates a small portion of the latent factors.more »This may result in slow improvement of the estimation accuracy of the latent factors. In addition, the convergence properties of many stochastic CPD algorithms are unclear, perhaps because CPD poses a hard nonconvex problem and is challenging for analysis under stochastic settings. In this work, we propose a stochastic optimization strategy that can effectively circumvent the above challenges. The proposed algorithm updates a whole latent factor at each iteration using sampled fibers of a tensor, which can quickly increase the estimation accuracy. The algorithm is flexible-many commonly used regularizers and constraints can be easily incorporated in the computational framework. The algorithm is also backed by a rigorous convergence theory. Simulations on large-scale dense tensors are employed to showcase the effectiveness of the algorithm.« less
3. Anderson acceleration for seismic inversion

   https://doi.org/10.1190/geo2020-0462.1  

   Yang, Yunan ( January 2021 , GEOPHYSICS)

   State-of-the-art seismic imaging techniques treat inversion tasks such as full-waveform inversion (FWI) and least-squares reverse time migration (LSRTM) as partial differential equation-constrained optimization problems. Due to the large-scale nature, gradient-based optimization algorithms are preferred in practice to update the model iteratively. Higher-order methods converge in fewer iterations but often require higher computational costs, more line-search steps, and bigger memory storage. A balance among these aspects has to be considered. We have conducted an evaluation using Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate the steepest-descent algorithm, which we innovatively treat as amore »fixed-point iteration. Independent of the unknown parameter dimensionality, the computational cost of implementing the method can be reduced to an extremely low dimensional least-squares problem. The cost can be further reduced by a low-rank update. We determine the theoretical connections and the differences between AA and other well-known optimization methods such as L-BFGS and the restarted generalized minimal residual method and compare their computational cost and memory requirements. Numerical examples of FWI and LSRTM applied to the Marmousi benchmark demonstrate the acceleration effects of AA. Compared with the steepest-descent method, AA can achieve faster convergence and can provide competitive results with some quasi-Newton methods, making it an attractive optimization strategy for seismic inversion.« less
4. PIVE: Per-Iteration Visualization Environment for Real-Time Interactions with Dimension Reduction and Clustering

   Hannah Kim, Jaegul Choo ( February 2017 , Proceedings of the ... AAAI Conference on Artificial Intelligence)

   One of the key advantages of visual analytics is its capability to leverage both humans's visual perception and the power of computing. A big obstacle in integrating machine learning with visual analytics is its high computing cost. To tackle this problem, this paper presents PIVE (Per-Iteration Visualization Environment) that supports real-time interactive visualization with machine learning. By immediately visualizing the intermediate results from algorithm iterations, PIVE enables users to quickly grasp insights and interact with the intermediate output, which then affects subsequent algorithm iterations. In addition, we propose a widely-applicable interaction methodology that allows efficient incorporation of user feedback intomore »virtually any iterative computational method without introducing additional computational cost. We demonstrate the application of PIVE for various dimension reduction algorithms such as multidimensional scaling and t-SNE and clustering and topic modeling algorithms such as k-means and latent Dirichlet allocation.« less
5. Efficient One-Shot Technique for Adjoint-Based Unsteady Optimization

   https://doi.org/10.2514/1.J060142  

   Djeddi, Reza ; Ekici, Kivanc ( June 2021 , AIAA Journal)

   A computationally efficient one-shot approach with a low memory footprint is presented for unsteady optimization. The proposed technique is based on a novel and unique approach that combines local-in-time and fixed-point iteration methods to advance the unconverged primal and adjoint solutions forward and backward in time to evaluate the sensitivity of the globally time-integrated objective function. This is in some ways similar to the piggyback iterations in which primal and adjoint solutions are evaluated at a frozen design. During each cycle, the primal, adjoint, and design update problems are solved to advance the optimization problem. This new coupled approach ismore »shown to provide significant savings in the memory footprint while reducing the computational cost of primal and adjoint evaluations per design cycle. The method is first applied to an inverse design problem for the unsteady lid-driven cavity. Following this, vortex suppression and mean drag reduction for a circular cylinder in crossflow is considered. Both of these objectives are achieved by optimizing the rotational speeds for steady or periodically oscillating excitations. For all cases presented in this work, the proposed technique is shown to provide significant reductions in memory as well as computational time. It is also shown that the unsteady optimization problem converges to the same optimal solution obtained using a conventional approach.« less