We investigate the signal reconstruction performance of sparse linear regression in the presence of noise when piecewise continuous nonconvex penalties are used. Among such penalties, we focus on the smoothly clipped absolute deviation (SCAD) penalty. The contributions of this study are three-fold: We first present a theoretical analysis of a typical reconstruction performance, using the replica method, under the assumption that each component of the design matrix is given as an independent and identically distributed (i.i.d.) Gaussian variable. This clarifies the superiority of the SCAD estimator compared with $\ell_1$ in a wide parameter range, although the nonconvex nature of the penalty tends to lead to solution multiplicity in certain regions. This multiplicity is shown to be connected to replica symmetry breaking in the spin-glass theory, and associated phase diagrams are given. We also show that the global minimum of the mean square error between the estimator and the true signal is located in the replica symmetric phase. Second, we develop an approximate formula efficiently computing the cross-validation error without actually conducting the cross-validation, which is also applicable to the non-i.i.d. design matrices. It is shown that this formula is only applicable to the unique solution region and tends to be unstable in the multiple solution region. We implement instability detection procedures, which allows the approximate formula to stand alone and resultantly enables us to draw phase diagrams for any specific dataset. Third, we propose an annealing procedure, called nonconvexity annealing, to obtain the solution path efficiently. Numerical simulations are conducted on simulated datasets to examine these results to verify the consistency of the theoretical results and the efficiency of the approximate formula and nonconvexity annealing.
We study the following three fundamental problems about ridge regression: (1) what is the structure of the estimator? (2) how to correctly use cross-validation to choose the regularization parameter? and (3) how to accelerate computation without losing too much accuracy? We consider the three problems in a unified large-data linear model. We give a precise representation of ridge regression as a covariance matrix-dependent linear combination of the true parameter and the noise. We study the bias of $K$-fold cross-validation for choosing the regularization parameter, and propose a simple bias-correction. We analyze the accuracy of primal and dual sketching for ridge regression, showing they are surprisingly accurate. Our results are illustrated by simulations and by analyzing empirical data.
This article is intended for beginners in deep learning who wish to gain knowledge about probability and statistics and also as a reference for practitioners. In my previous article, I wrote about the concepts of linear algebra for deep learning in a top down approach ( link for the article) (If you do not have enough idea about linear algebra, please read that first).The same top down approach is used here.Providing the description of use cases first and then the concepts. All the example code uses python and numpy.Formulas are provided as images for reuse. Probability is the science of quantifying uncertain things.Most of machine learning and deep learning systems utilize a lot of data to learn about patterns in the data.Whenever data is utilized in a system rather than sole logic, uncertainty grows up and whenever uncertainty grows up, probability becomes relevant. By introducing probability to a deep learning system, we introduce common sense to the system.Otherwise the system would be very brittle and will not be useful.In deep learning, several models like bayesian models, probabilistic graphical models, hidden markov models are used.They depend entirely on probability concepts.
Incorporating semantic features from the WordNet lexical database is among one of the many approaches that have been tried to improve the predictive performance of text classification models. The intuition behind this is that keywords in the training set alone may not be extensive enough to enable generation of a universal model for a category, but if we incorporate the word relationships in WordNet, a more accurate model may be possible. Other researchers have previously evaluated the effectiveness of incorporating WordNet synonyms, hypernyms, and hyponyms into text classification models. Generally, they have found that improvements in accuracy using features derived from these relationships are dependent upon the nature of the text corpora from which the document collections are extracted. In this paper, we not only reconsider the role of WordNet synonyms, hypernyms, and hyponyms in text classification models, we also consider the role of WordNet meronyms and holonyms. Incorporating these WordNet relationships into a Coordinate Matching classifier, a Naive Bayes classifier, and a Support Vector Machine classifier, we evaluate our approach on six document collections extracted from the Reuters-21578, USENET, and Digi-Trad text corpora. Experimental results show that none of the WordNet relationships were effective at increasing the accuracy of the Naive Bayes classifier. Synonyms, hypernyms, and holonyms were effective at increasing the accuracy of the Coordinate Matching classifier, and hypernyms were effective at increasing the accuracy of the SVM classifier.