Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX

Scalable Uncertainty Quantification via GenerativeBootstrap Sampler

It has been believed that the virtue of using statistical procedures is on uncertainty quantification in statistical decisions, and the bootstrap method has been commonly used for this purpose. However, nowadays as the size of data massively increases and statistical models become more complicated, the implementation of bootstrapping turns out to be practically challenging due to its repetitive nature in computation. To overcome this issue, we propose a novel computational procedure called {\it Generative Bootstrap Sampler} (GBS), which constructs a generator function of bootstrap evaluations, and this function transforms the weights on the observed data points to the bootstrap distribution. The GBS is implemented by one single optimization, without repeatedly evaluating the optimizer of bootstrapped loss function as in standard bootstrapping procedures. As a result, the GBS is capable of reducing computational time of bootstrapping by hundreds of folds when the data size is massive. We show that the bootstrapped distribution evaluated by the GBS is asymptotically equivalent to the conventional counterpart and empirically they are indistinguishable. We examine the proposed idea to bootstrap various models such as linear regression, logistic regression, Cox proportional hazard model, and Gaussian process regression model, quantile regression, etc. The results show that the GBS procedure is not only accelerating the computational speed, but it also attains a high level of accuracy to the target bootstrap distribution. Additionally, we apply this idea to accelerate the computation of other repetitive procedures such as bootstrapped cross-validation, tuning parameter selection, and permutation test

Regularized Linear Discriminant Analysis Using a Nonlinear Covariance Matrix Estimator

Linear discriminant analysis (LDA) is a widely used technique for data classification. The method offers adequate performance in many classification problems, but it becomes inefficient when the data covariance matrix is ill-conditioned. This often occurs when the feature space's dimensionality is higher than or comparable to the training data size. Regularized LDA (RLDA) methods based on regularized linear estimators of the data covariance matrix have been proposed to cope with such a situation. The performance of RLDA methods is well studied, with optimal regularization schemes already proposed. In this paper, we investigate the capability of a positive semidefinite ridge-type estimator of the inverse covariance matrix that coincides with a nonlinear (NL) covariance matrix estimator. The estimator is derived by reformulating the score function of the optimal classifier utilizing linear estimation methods, which eventually results in the proposed NL-RLDA classifier. We derive asymptotic and consistent estimators of the proposed technique's misclassification rate under the assumptions of a double-asymptotic regime and multivariate Gaussian model for the classes. The consistent estimator, coupled with a one-dimensional grid search, is used to set the value of the regularization parameter required for the proposed NL-RLDA classifier. Performance evaluations based on both synthetic and real data demonstrate the effectiveness of the proposed classifier. The proposed technique outperforms state-of-art methods over multiple datasets. When compared to state-of-the-art methods across various datasets, the proposed technique exhibits superior performance.Comment: \c{opyright} 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other work

Lecture notes on ridge regression

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapter introduces its lasso counterpart

When “time varying” volatility meets “transaction cost” in portfolio selection

We propose a new strategy for mean–variance portfolio selection that tackles transaction costs and change detection in covariance matrix simultaneously. The new strategy solely rebalances the portfolio when change points are detected in the covariance matrix, striking an optimal trade-off between rebalancing the portfolio to capturing the recent information in return data and avoiding excessive trading. Our empirical results suggest favorable out-of-sample performance of the new strategy in terms of portfolio variance, portfolio turnovers and portfolio sharpe ratio with transaction cost. We also show that these gains come from the improved accuracy for covariance matrix prediction and the ability for tracking significant changes in covariance matrix

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

Information Theory and Its Application in Machine Condition Monitoring

Condition monitoring of machinery is one of the most important aspects of many modern industries. With the rapid advancement of science and technology, machines are becoming increasingly complex. Moreover, an exponential increase of demand is leading an increasing requirement of machine output. As a result, in most modern industries, machines have to work for 24 hours a day. All these factors are leading to the deterioration of machine health in a higher rate than before. Breakdown of the key components of a machine such as bearing, gearbox or rollers can cause a catastrophic effect both in terms of financial and human costs. In this perspective, it is important not only to detect the fault at its earliest point of inception but necessary to design the overall monitoring process, such as fault classification, fault severity assessment and remaining useful life (RUL) prediction for better planning of the maintenance schedule. Information theory is one of the pioneer contributions of modern science that has evolved into various forms and algorithms over time. Due to its ability to address the non-linearity and non-stationarity of machine health deterioration, it has become a popular choice among researchers. Information theory is an effective technique for extracting features of machines under different health conditions. In this context, this book discusses the potential applications, research results and latest developments of information theory-based condition monitoring of machineries