In statistics, the least absolute shrinkage and selection operator (Lasso) is a regression method that performs both variable selection and regularization. There is a lot of literature available, discussing the statistical properties of the regression coefficients estimated by the Lasso method. However, there lacks a comprehensive review discussing the algorithms to solve the optimization problem in Lasso. In this review, we summarize five representative algorithms to optimize the objective function in Lasso, including the iterative shrinkage threshold algorithm (ISTA), fast iterative shrinkage-thresholding algorithms (FISTA), coordinate gradient descent algorithm (CGDA), smooth L1 algorithm (SLA), and path following algorithm (PFA). Additionally, we also compare their convergence rate, as well as their potential strengths and weakness.

We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $\theta$ with the undesirable feature that its forward operator is generally non-compact (even if $\theta$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under H\"older-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.

pystacked implements stacked generalization (Wolpert, 1992) for regression and binary classification via Python's scikit-learn. Stacking combines multiple supervised machine learners -- the "base" or "level-0" learners -- into a single learner. The currently supported base learners include regularized regression, random forest, gradient boosted trees, support vector machines, and feed-forward neural nets (multi-layer perceptron). pystacked can also be used with as a `regular' machine learning program to fit a single base learner and, thus, provides an easy-to-use API for scikit-learn's machine learning algorithms.

We propose a new fast algorithm to estimate any sparse generalized linear model with convex or non-convex separable penalties. Our algorithm is able to solve problems with millions of samples and features in seconds, by relying on coordinate descent, working sets and Anderson acceleration. It handles previously unaddressed models, and is extensively shown to improve state-of-art algorithms. We release skglm, a flexible, scikit-learn compatible package, which easily handles customized datafits and penalties.

Abstract: Clustering is a widely used technique with a long and rich history in a variety of areas. However, most existing algorithms do not scale well to large datasets, or are missing theoretical guarantees of convergence. This paper introduces a provably robust clustering algorithm based on loss minimization that performs well on Gaussian mixture models with outliers. It provides theoretical guarantees that the algorithm obtains high accuracy with high probability under certain assumptions. Moreover, it can also be used as an initialization strategy for k-means clustering.

Azure Machine Learning is a one-stop solution for all of your machine learning projects that saves your time and cost. With Azure Machine Learning Studio, it is easy to develop and train machine learning models with complex algorithms. In this tutorial, we will learn more about Azure Machine Learning Studio and also, we will work on a small project to understand how the studio works? What is Azure Machine Learning Studio? Let's suppose that your organisation is trying to develop and train a machine learning model.

Aerodynamic performance evaluation is an important part of the aircraft aerodynamic design optimization process; however, traditional methods are costly and time-consuming. Despite the fact that various machine learning methods can achieve high accuracy, their application in engineering is still difficult due to their poor generalization performance and "black box" nature. In this paper, a knowledge-embedded meta learning model, which fully integrates data with the theoretical knowledge of the lift curve, is developed to obtain the lift coefficients of an arbitrary supercritical airfoil under various angle of attacks. In the proposed model, a primary network is responsible for representing the relationship between the lift and angle of attack, while the geometry information is encoded into a hyper network to predict the unknown parameters involved in the primary network. Specifically, three models with different architectures are trained to provide various interpretations. Compared to the ordinary neural network, our proposed model can exhibit better generalization capability with competitive prediction accuracy. Afterward, interpretable analysis is performed based on the Integrated Gradients and Saliency methods. Results show that the proposed model can tend to assess the influence of airfoil geometry to the physical characteristics. Furthermore, the exceptions and shortcomings caused by the proposed model are analysed and discussed in detail.

We propose a new strategy for anomaly detection at the LHC based on unsupervised quantum machine learning algorithms. To accommodate the constraints on the problem size dictated by the limitations of current quantum hardware we develop a classical convolutional autoencoder. The designed quantum anomaly detection models, namely an unsupervised kernel machine and two clustering algorithms, are trained to find new-physics events in the latent representation of LHC data produced by the autoencoder. The performance of the quantum algorithms is benchmarked against classical counterparts on different new-physics scenarios and its dependence on the dimensionality of the latent space and the size of the training dataset is studied. For kernel-based anomaly detection, we identify a regime where the quantum model significantly outperforms its classical counterpart. An instance of the kernel machine is implemented on a quantum computer to verify its suitability for available hardware. We demonstrate that the observed consistent performance advantage is related to the inherent quantum properties of the circuit used.

Most of the standard image and video codecs are block-based and depending upon the compression ratio the compressed images/videos suffer from different distortions. At low ratios, blurriness is observed and as compression increases blocking artifacts occur. Generally, in order to reduce blockiness, images are low-pass filtered which leads to more blurriness. Also, in bokeh mode images they are commonly seen: blurriness as a result of intentional blurred background while blocking artifact and global blurriness arising due to compression. Therefore, such visual media suffer from both blockiness and blurriness distortions. Along with this, noise is also commonly encountered distortion. Most of the existing works on quality assessment quantify these distortions individually. This paper proposes a methodology to blindly measure overall quality of an image suffering from these distortions, individually as well as jointly. This is achieved by considering the sum of absolute values of low and high-frequency Discrete Frequency Transform (DFT) coefficients defined as sum magnitudes. The number of blocks lying in specific ranges of sum magnitudes including zero-valued AC coefficients and mean of 100 maximum and 100 minimum values of these sum magnitudes are used as feature vectors. These features are then fed to the Machine Learning (ML) based Gaussian Process Regression (GPR) model, which quantifies the image quality. The simulation results show that the proposed method can estimate the quality of images distorted with the blockiness, blurriness, noise and their combinations. It is relatively fast compared to many state-of-art methods, and therefore is suitable for real-time quality monitoring applications.

The stochastic gradient noise (SGN) is a significant factor in the success of stochastic gradient descent (SGD). Following the central limit theorem, SGN was initially modeled as Gaussian, and lately, it has been suggested that stochastic gradient noise is better characterized using $S\alpha S$ L\'evy distribution. This claim was allegedly refuted and rebounded to the previously suggested Gaussian noise model. This paper presents solid, detailed empirical evidence that SGN is heavy-tailed and better depicted by the $S\alpha S$ distribution. Furthermore, we argue that different parameters in a deep neural network (DNN) hold distinct SGN characteristics throughout training. To more accurately approximate the dynamics of SGD near a local minimum, we construct a novel framework in $\mathbb{R}^N$, based on L\'evy-driven stochastic differential equation (SDE), where one-dimensional L\'evy processes model each parameter in the DNN. Next, we show that SGN jump intensity (frequency and amplitude) depends on the learning rate decay mechanism (LRdecay); furthermore, we demonstrate empirically that the LRdecay effect may stem from the reduction of the SGN and not the decrease in the step size. Based on our analysis, we examine the mean escape time, trapping probability, and more properties of DNNs near local minima. Finally, we prove that the training process will likely exit from the basin in the direction of parameters with heavier tail SGN. We will share our code for reproducibility.