scholarly journals Fast, robust and non-convex subspace recovery

2017 ◽  
Vol 7 (2) ◽  
pp. 277-336 ◽  
Author(s):  
Gilad Lerman ◽  
Tyler Maunu
Keyword(s):  
Computational Complexity ◽  
Stationary Point ◽  
Numerical Experiments ◽  
Global Minimum ◽  
Real Data ◽  
Dimensional Subspace ◽  
Iteration Complexity ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Speed And Accuracy

Abstract This work presents a fast and non-convex algorithm for robust subspace recovery. The datasets considered include inliers drawn around a low-dimensional subspace of a higher dimensional ambient space and a possibly large portion of outliers that do not lie nearby this subspace. The proposed algorithm, which we refer to as fast median subspace (FMS), is designed to robustly determine the underlying subspace of such datasets, while having lower computational complexity than existing accurate methods. We prove convergence of the FMS iterates to a stationary point. Further, under two special models of data, FMS converges to a point which is near to the global minimum with overwhelming probability. Under these models, we show that the iteration complexity is globally sublinear and locally $r$-linear. For one of the models, these results hold for any fixed fraction of outliers (< 1). Numerical experiments on synthetic and real data demonstrate its competitive speed and accuracy.

scholarly journals Augmented Arnoldi-Tikhonov Regularization Methods for Solving Large-Scale Linear Ill-Posed Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yiqin Lin ◽  
Liang Bao ◽  
Yanhua Cao
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Experiments ◽  
Large Scale ◽  
Krylov Subspace ◽  
Dimensional Subspace ◽  
Tikhonov Regularization Method ◽  
Rough Approximation ◽  
Ill Posed ◽  
Low Dimensional

We propose an augmented Arnoldi-Tikhonov regularization method for the solution of large-scale linear ill-posed systems. This method augments the Krylov subspace by a user-supplied low-dimensional subspace, which contains a rough approximation of the desired solution. The augmentation is implemented by a modified Arnoldi process. Some useful results are also presented. Numerical experiments illustrate that the augmented method outperforms the corresponding method without augmentation on some real-world examples.

scholarly journals An Adaptive Optimization Method Based on Learning Rate Schedule for Neural Networks

2021 ◽  
Vol 11 (2) ◽  
pp. 850
Author(s):  
Dokkyun Yi ◽  
Sangmin Ji ◽  
Jieun Park
Keyword(s):  
Artificial Intelligence ◽  
Cost Function ◽  
Numerical Experiments ◽  
Global Minimum ◽  
Optimization Method ◽  
Learning Method ◽  
Adaptive Optimization ◽  
The Cost ◽  
Proof Of Convergence ◽  
Learning Data

Artificial intelligence (AI) is achieved by optimizing the cost function constructed from learning data. Changing the parameters in the cost function is an AI learning process (or AI learning for convenience). If AI learning is well performed, then the value of the cost function is the global minimum. In order to obtain the well-learned AI learning, the parameter should be no change in the value of the cost function at the global minimum. One useful optimization method is the momentum method; however, the momentum method has difficulty stopping the parameter when the value of the cost function satisfies the global minimum (non-stop problem). The proposed method is based on the momentum method. In order to solve the non-stop problem of the momentum method, we use the value of the cost function to our method. Therefore, as the learning method processes, the mechanism in our method reduces the amount of change in the parameter by the effect of the value of the cost function. We verified the method through proof of convergence and numerical experiments with existing methods to ensure that the learning works well.

scholarly journals Supervised dimensionality reduction for big data

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Joshua T. Vogelstein ◽  
Eric W. Bridgeford ◽  
Minh Tang ◽  
Da Zheng ◽  
Christopher Douville ◽  
...  
Keyword(s):  
Dimensionality Reduction ◽  
Data Science ◽  
Real Data ◽  
Low Rank ◽  
Conditional Moment ◽  
Desktop Computer ◽  
Reduction Techniques ◽  
Reduction Methods ◽  
The Individual ◽  
Low Dimensional

AbstractTo solve key biomedical problems, experimentalists now routinely measure millions or billions of features (dimensions) per sample, with the hope that data science techniques will be able to build accurate data-driven inferences. Because sample sizes are typically orders of magnitude smaller than the dimensionality of these data, valid inferences require finding a low-dimensional representation that preserves the discriminating information (e.g., whether the individual suffers from a particular disease). There is a lack of interpretable supervised dimensionality reduction methods that scale to millions of dimensions with strong statistical theoretical guarantees. We introduce an approach to extending principal components analysis by incorporating class-conditional moment estimates into the low-dimensional projection. The simplest version, Linear Optimal Low-rank projection, incorporates the class-conditional means. We prove, and substantiate with both synthetic and real data benchmarks, that Linear Optimal Low-Rank Projection and its generalizations lead to improved data representations for subsequent classification, while maintaining computational efficiency and scalability. Using multiple brain imaging datasets consisting of more than 150 million features, and several genomics datasets with more than 500,000 features, Linear Optimal Low-Rank Projection outperforms other scalable linear dimensionality reduction techniques in terms of accuracy, while only requiring a few minutes on a standard desktop computer.

A Determinant Elimination Method for Bottleneck Assignment and Generalized Assignment Problems

2012 ◽  
Vol 239-240 ◽  
pp. 1522-1527
Author(s):  
Wen Bo Wu ◽  
Yu Fu Jia ◽  
Hong Xing Sun
Keyword(s):  
Computational Complexity ◽  
Optimization Algorithm ◽  
Assignment Problem ◽  
Numerical Experiments ◽  
High Efficiency ◽  
Synthesis Method ◽  
Assignment Problems ◽  
Generalized Assignment ◽  
The Cost ◽  
Time And Space Complexity

The bottleneck assignment (BA) and the generalized assignment (GA) problems and their exact solutions are explored in this paper. Firstly, a determinant elimination (DE) method is proposed based on the discussion of the time and space complexity of the enumeration method for both BA and GA problems. The optimization algorithm to the pre-assignment problem is then discussed and the adjusting and transformation to the cost matrix is adopted to reduce the computational complexity of the DE method. Finally, a synthesis method for both BA and GA problems is presented. The numerical experiments are carried out and the results indicate that the proposed method is feasible and of high efficiency.

scholarly journals Numerical Valuation of American Basket Options via Partial Differential Complementarity Problems

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1498
Author(s):  
Karel J. in’t Hout ◽  
Jacob Snoeijer
Keyword(s):  
Principal Component Analysis ◽  
Numerical Experiments ◽  
Complementarity Problems ◽  
Principal Component ◽  
Partial Differential ◽  
Basket Options ◽  
Basket Option ◽  
Option Values ◽  
Convergence Behaviour ◽  
Low Dimensional

We study the principal component analysis based approach introduced by Reisinger and Wittum (2007) and the comonotonic approach considered by Hanbali and Linders (2019) for the approximation of American basket option values via multidimensional partial differential complementarity problems (PDCPs). Both approximation approaches require the solution of just a limited number of low-dimensional PDCPs. It is demonstrated by ample numerical experiments that they define approximations that lie close to each other. Next, an efficient discretisation of the pertinent PDCPs is presented that leads to a favourable convergence behaviour.

An Improved and Low-dimensional Fingerprint-based Localization Method in Collocated Massive MIMO-OFDM Systems

2021 ◽  
Author(s):  
Seyedeh Samira Moosavi ◽  
Paul Fortier
Keyword(s):  
Computational Complexity ◽  
Gaussian Process ◽  
Massive Mimo ◽  
High Reliability ◽  
Gaussian Process Regression ◽  
Location Estimation ◽  
Ofdm Systems ◽  
Process Data ◽  
Localization Method ◽  
Low Dimensional

Abstract Localization has drawn significant attention in 5G due to the fast-growing demand for location-based service (LBS). Massive multiple-input multiple-output (M-MIMO) has been introduced in 5G as a powerful technology due to its evident potentials for communication performance enhancement and localization in complicated environments. Fingerprint-based (FP) localization are promising methods for rich scattering environments thanks to their high reliability and accuracy. The Gaussian process regression (GPR) method could be used as an FP-based localization method to facilitate localization and provide high accuracy. However, this method has high computational complexity, especially in large-scale environments. In this study, we propose an improved and low-dimensional FP-based localization method in collocated massive MIMO orthogonal frequency division multiplexing (OFDM) systems using principal component analysis (PCA), the affinity propagation clustering (APC) algorithm, and Gaussian process regression (GPR) to estimate the user's location. Fingerprints are first extracted based on instantaneous channel state information (CSI) by taking full advantage of the high-resolution angle and delay domains. First, PCA is used to pre-process data and reduce the feature dimension. Then, the training fingerprints are clustered using the APC algorithm to increase prediction accuracy and reduce computation complexity. Finally, each cluster's data distribution is accurately modelled using GPR to provide support for further localization. Simulation results reveal that the proposed method improves localization performance significantly by reducing the location estimation error. Additionally, it reduces the matching complexity and computational complexity.

scholarly journals Stiffness Analysis to Predict the Spread Out of Fake Information

2021 ◽  
Vol 13 (9) ◽  
pp. 222
Author(s):  
Raffaele D'Ambrosio ◽  
Giuseppe Giordano ◽  
Serena Mottola ◽  
Beatrice Paternoster
Keyword(s):  
Numerical Experiments ◽  
Real Data ◽  
Sir Model ◽  
Stiffness Index ◽  
Fake News ◽  
Stiffness Analysis ◽  
Data Support

This work highlights how the stiffness index, which is often used as a measure of stiffness for differential problems, can be employed to model the spread of fake news. In particular, we show that the higher the stiffness index is, the more rapid the transit of fake news in a given population. The illustration of our idea is presented through the stiffness analysis of the classical SIR model, commonly used to model the spread of epidemics in a given population. Numerical experiments, performed on real data, support the effectiveness of the approach.

scholarly journals Neural Excursions from Low-Dimensional Manifold Structure Explain Intersubject Variation in Human Motor Learning

2021 ◽  
Author(s):  
Corson N Areshenkoff ◽  
Daniel J Gale ◽  
Joe Y Nashed ◽  
Dominic Standage ◽  
John Randall Flanagan ◽  
...  
Keyword(s):  
Motor Learning ◽  
Large Scale ◽  
Brain Activity ◽  
Dimensional Subspace ◽  
Learning Performance ◽  
Dimensional Manifold ◽  
Learning Abilities ◽  
Manifold Structure ◽  
Electrophysiological Studies ◽  
Low Dimensional

Humans vary greatly in their motor learning abilities, yet little is known about the neural mechanisms that underlie this variability. Recent neuroimaging and electrophysiological studies demonstrate that large-scale neural dynamics inhabit a low-dimensional subspace or manifold, and that learning is constrained by this intrinsic manifold architecture. Here we asked, using functional MRI, whether subject-level differences in neural excursion from manifold structure can explain differences in learning across participants. We had subjects perform a sensorimotor adaptation task in the MRI scanner on two consecutive days, allowing us to assess their learning performance across days, as well as continuously measure brain activity. We find that the overall neural excursion from manifold activity in both cognitive and sensorimotor brain networks is associated with differences in subjects' patterns of learning and relearning across days. These findings suggest that off-manifold activity provides an index of the relative engagement of different neural systems during learning, and that intersubject differences in patterns of learning and relearning across days are related to reconfiguration processes in cognitive and sensorimotor networks during learning.

Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

scholarly journals Discovering a sparse set of pairwise discriminating features in high-dimensional data

2020 ◽  
Author(s):  
Samuel Melton ◽  
Sharad Ramanathan
Keyword(s):  
Single Cell ◽  
Dimensional Space ◽  
Cell Types ◽  
Dimensional Subspace ◽  
Supplementary Information ◽  
High Dimensional ◽  
Technological Advances ◽  
Data Points ◽  
Low Dimensional ◽  
Sparse Set

Abstract Motivation Recent technological advances produce a wealth of high-dimensional descriptions of biological processes, yet extracting meaningful insight and mechanistic understanding from these data remains challenging. For example, in developmental biology, the dynamics of differentiation can now be mapped quantitatively using single-cell RNA sequencing, yet it is difficult to infer molecular regulators of developmental transitions. Here, we show that discovering informative features in the data is crucial for statistical analysis as well as making experimental predictions. Results We identify features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low-dimensional space. We propose an unsupervised method to identify the subset of features that define a low-dimensional subspace in which clustering can be conducted. This is achieved by averaging over discriminators trained on an ensemble of proposed cluster configurations. We then apply our method to single-cell RNA-seq data from mouse gastrulation, and identify 27 key transcription factors (out of 409 total), 18 of which are known to define cell states through their expression levels. In this inferred subspace, we find clear signatures of known cell types that eluded classification prior to discovery of the correct low-dimensional subspace. Availability and implementation https://github.com/smelton/SMD. Supplementary information Supplementary data are available at Bioinformatics online.

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