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The mathematical magic behind support vector machines: how to look at them from a Bayesian perspective?
Within the Bayesian statistical framework of machine learning, kernel methods arise from assumptions about the inner product space or similarity structure of the input. The original formation and regularization of some methods such as support vector machines (SVM) are not the essence of Bayesian, so understanding these methods from a Bayesian perspective will be of great help to our learning.
Many kernel methods are used for supervised learning problems, where the input space is usually a vector space and the output space is a scalar. Recently, these methods have been extended to handle problems with multiple outputs, such as multi-task learning.
The learning process of support vector machines actually hides deep mathematical connotations. This is not only a technical problem, but also an interesting challenge on how to deal with uncertainty. The elegance of support vector machines lies in their ability to automatically select the most informative features while remaining computationally efficient. As our understanding of support vector machines grows, we might as well consider: How does this mathematical wizardry change how we understand machine learning?
Basic concepts of supervised learning problems
Traditional supervised learning problems require us to learn a scalar-valued estimator based on a training set to predict the output of a new input point. These input-output pairs are formed into a training set, called S, which consists of n input-output pairs. In fact, our goal is to create an estimation function that predicts the output of these input points well.
In this process, a symmetric and positive binary function is called a kernel. For a very important estimator in machine learning, the generation of the kernel matrix is crucial.
Formalization perspective
In the regularization perspective, the main assumption is that the set of functions F belongs to a reborn kernel Hilbert space Hk. This framework allows us to model the problem from multiple aspects and improve the predictive performance of the model by effectively incorporating the established functions into the auxiliary learning process.
Reborn Kernel Hilbert Space (RKHS) is a set of functions based on symmetric and positive definite functions, which has some attractive properties, including the ability to generate energy minimization of functions.
This is based on two basic constraints: first, the control of the kernel to ensure the reliability of prediction, and second, regularization to obtain a balanced prediction ability and model complexity. At this time, the role of the regularizer becomes particularly important. It is responsible for controlling the complexity of the function, which is crucial to prevent overfitting.
Derivatives of estimators
By introducing the correlation of the Hilbert space of the regenerated kernel, we can understand how the estimator of the support vector machine is derived. This relies on a key theory - the performer theorem, which states that the optimal solution can be expressed as a linear combination of kernels in the training set. Such a conclusion not only provides theoretical support, but also makes this method practical.
We can express this function as a linear combination of kernel functions in the training set, and obtain the best prediction effect by minimizing the actual value.
Connection of Bayesian perspectives
From a Bayesian perspective, the kernel method is the core component of the Gaussian process, and the kernel function is also called the covariance function. Through this understanding, we can also reveal the mathematical equivalence between the regularization method and the Bayesian perspective. In many cases, the predictors they provide are essentially the same, providing an opportunity to explore correlations between different models.
In terms of understanding support vector machines, this immediate versatility in various models makes them an extremely attractive choice, affecting the development of today's machine learning more broadly. Through the in-depth analysis of mathematical structures in this article, perhaps we can't help but think about how future data analysis will continue to evolve to adapt to the growing complexity and needs?
The charm of mathematics lies in its profound logical and expressive abilities, especially in the field of machine learning. How can we continue to tap their potential?
