In statistics, linear regression is an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variable) denoted X. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression. (This term should be distinguished from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.)
In linear regression, data are modeled using linear predictor functions, and unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, linear regression refers to a model in which the conditional mean of y given the value of X is an affine function of X. Less commonly, linear regression could refer to a model in which the median, or some other quantile of the conditional distribution of y given X is expressed as a linear function of X. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of y given X, rather than on the joint probability distribution of y and X, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
· If the goal is prediction, or forecasting, or reduction, linear regression can be used to fit a predictive model to an observed data set of y and X values. After developing such a model, if an additional value of X is then given without its accompanying value of y, the fitted model can be used to make a prediction of the value of y.
· Given a variable y and a number of variables X1, …, Xp that may be related to y, linear regression analysis can be applied to quantify the strength of the relationship between y and the Xj, to assess which Xj may have no relationship with y at all, and to identify which subsets of the Xj contain redundant information about y.
Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the ‘lack of fit’ in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares loss function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms ‘least squares’ and ‘linear model’ are closely linked, they are not synonymous. …
We present a method for automatically organizing and evaluating the quality of different publishing venues in Computer Science. Since this method only requires paper publication data as its input, we can demonstrate our method on a large portion of the DBLP dataset, spanning 50 years, with millions of authors and thousands of publishing venues. By formulating venue authorship as a regression problem and targeting metrics of interest, we obtain venue scores for every conference and journal in our dataset. The obtained scores can also provide a per-year model of conference quality, showing how fields develop and change over time. Additionally, these venue scores can be used to evaluate individual academic authors and academic institutions. We show that using venue scores to evaluate both authors and institutions produces quantitative measures that are comparable to approaches using citations or peer assessment. In contrast to many other existing evaluation metrics, our use of large-scale, openly available data enables this approach to be repeatable and transparent. …
Graph Warp Module (GWM)
Recently, Graph Neural Networks (GNNs) are trending in the machine learning community as a family of architectures that specializes in capturing the features of graph-related datasets, such as those pertaining to social networks and chemical structures. Unlike for other families of the networks, the representation power of GNNs has much room for improvement, and many graph networks to date suffer from the problem of underfitting. In this paper we will introduce a Graph Warp Module, a supernode-based auxiliary network module that can be attached to a wide variety of existing GNNs in order to improve the representation power of the original networks. Through extensive experiments on molecular graph datasets, we will show that our GWM indeed alleviates the underfitting problem for various existing networks, and that it can even help create a network with the state-of-the-art generalization performance. …
Global Second-order Pooling Neural Network
Deep Convolutional Networks (ConvNets) are fundamental to, besides large-scale visual recognition, a lot of vision tasks. As the primary goal of the ConvNets is to characterize complex boundaries of thousands of classes in a high-dimensional space, it is critical to learn higher-order representations for enhancing non-linear modeling capability. Recently, Global Second-order Pooling (GSoP), plugged at the end of networks, has attracted increasing attentions, achieving much better performance than classical, first-order networks in a variety of vision tasks. However, how to effectively introduce higher-order representation in earlier layers for improving non-linear capability of ConvNets is still an open problem. In this paper, we propose a novel network model introducing GSoP across from lower to higher layers for exploiting holistic image information throughout a network. Given an input 3D tensor outputted by some previous convolutional layer, we perform GSoP to obtain a covariance matrix which, after nonlinear transformation, is used for tensor scaling along channel dimension. Similarly, we can perform GSoP along spatial dimension for tensor scaling as well. In this way, we can make full use of the second-order statistics of the holistic image throughout a network. The proposed networks are thoroughly evaluated on large-scale ImageNet-1K, and experiments have shown that they outperformed non-trivially the counterparts while achieving state-of-the-art results. …