Best Subset Selection

	Best Subset Selection Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space X (i.e., the domain, space of features or explanatory variables). ''Sparsity regularization methods'' focus on selecting the input variables that best describe the output. ''Structured sparsity regularization methods'' generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in X . Common motivation for the use of structured sparsity methods are model interpretability, high-dimensional learning (where dimensionality of X may be higher than the number of observations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Statistical Learning Theory Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. Introduction The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input-output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Proximal Gradient Methods For Learning Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is \ell_1 regularization (also known as Lasso) of the form :\min_ \frac\sum_^n (y_i- \langle w,x_i\rangle)^2+ \lambda \, w\, _1, \quad \text x_i\in \mathbb^d\text y_i\in\mathbb. Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as ''sparsity'' (in the case of lasso) or ''group structure'' (in the case of group lasso). Relevant background Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form : \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sparse Approximation Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processing, signal processing, machine learning, medical imaging, and more. Sparse decomposition Noiseless observations Consider a linear system of equations x = D\alpha, where D is an underdetermined m\times p matrix (m < p) and $x \in \mathbb^m,\alpha \in \mathbb^p$ . The matrix $D$ (typically assumed to be full-rank) is referred to as the dictionary, and $x$ is a signal of interest. The core sparse representation problem is defined as the quest for the sparsest possible representation $\alpha$ satisfying $x = D\alpha$ . Due to the underdetermined nature of $D$ , this linear system admits in general infinitely many possible solutions, and among these we seek the one with the fewe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Statistical Learning Theory Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data. Statistical learning theory has led to successful applications in fields such as computer vision, speech recognition, and bioinformatics. Introduction The goals of learning are understanding and prediction. Learning falls into many categories, including supervised learning, unsupervised learning, online learning, and reinforcement learning. From the perspective of statistical learning theory, supervised learning is best understood. Supervised learning involves learning from a training set of data. Every point in the training is an input-output pair, where the input maps to an output. The learning problem consists of inferring the function that maps between the input and the output, such that the learned function can be used to predict t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sociolinguistics Sociolinguistics is the descriptive study of the effect of any or all aspects of society, including cultural norms, expectations, and context, on the way language is used, and society's effect on language. It can overlap with the sociology of language, which focuses on the effect of language on society. Sociolinguistics overlaps considerably with pragmatics and is closely related to linguistic anthropology. Sociolinguistics' historical interrelation with anthropology can be observed in studies of how language varieties differ between groups separated by social variables (e.g., ethnicity, religion, status, gender, level of education, age, etc.) and/or geographical barriers (a mountain range, a desert, a river, etc.). Such studies also examine how such differences in usage and differences in beliefs about usage produce and reflect social or socioeconomic classes. As the usage of a language varies from place to place, language usage also varies among social classes, and it is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Facial Recognition System A facial recognition system is a technology capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and works by pinpointing and measuring facial features from a given image. Development began on similar systems in the 1960s, beginning as a form of computer application. Since their inception, facial recognition systems have seen wider uses in recent times on smartphones and in other forms of technology, such as robotics. Because computerized facial recognition involves the measurement of a human's physiological characteristics, facial recognition systems are categorized as biometrics. Although the accuracy of facial recognition systems as a biometric technology is lower than iris recognition and fingerprint recognition, it is widely adopted due to its contactless process. Facial recognition systems have been deployed in advanced human–compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Magnetic Resonance Imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio waves to generate images of the organs in the body. MRI does not involve X-rays or the use of ionizing radiation, which distinguishes it from CT and PET scans. MRI is a medical application of nuclear magnetic resonance (NMR) which can also be used for imaging in other NMR applications, such as NMR spectroscopy. MRI is widely used in hospitals and clinics for medical diagnosis, staging and follow-up of disease. Compared to CT, MRI provides better contrast in images of soft-tissues, e.g. in the brain or abdomen. However, it may be perceived as less comfortable by patients, due to the usually longer and louder measurements with the subject in a long, confining tube, though "Open" MRI designs mostly relieve this. Additionally, implants and oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Compressed Sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined system, underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity, which requires the signal to be sparse in some domain. The second one is incoherence, which is applied through the isometric property, which is sufficient for sparse signals. Overview A common goal of the engineering field of signal processing is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Multiple Kernel Learning Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm. Reasons to use multiple kernel learning include a) the ability to select for an optimal kernel and parameters from a larger set of kernels, reducing bias due to kernel selection while allowing for more automated machine learning methods, and b) combining data from different sources (e.g. sound and images from a video) that have different notions of similarity and thus require different kernels. Instead of creating a new kernel, multiple kernel algorithms can be used to combine kernels already established for each individual data source. Multiple kernel learning approaches have been used in many applications, such as event recognition in video, object recognition in images, and biomedical data fusion. Algorithms Multiple kernel learning algorithms have been developed for supervised, semi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Loss Function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Loss Function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Differentiable Function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]