Các mô hình tuyến tính giới thiệu trước đó trong văn bản cung cấp một điểm khởi đầu quan trọng cho sự phát triển của các công cụ học máy nhưng có thể giới hạn quan trọng mà chúng ta cần phải vượt qua để trang trải một quang phổ ứng dụng rộng hơn. Thứ nhất, dữ liệu trong nhiều trường hợp thú vị là không tuyến tính phân chia, sự cần thiết cho một bề mặt tách phức tạp không phải là một tạo tác do tiếng ồn nhưng là một hệ quả tự nhiên của đại diện | 8 Neural Networks and Kernel Machines for Vector and Structured Data PAOLO FRASCONI Dipartimento di Sistemi e Informatica Universita degli Studi di Firenze Firenze Italy 1. INTRODUCTION The linear models introduced earlier in the text offer an important starting point for the development of machine learning tools but are subject to important limitations that we need to overcome in order to cover a wider application spectrum. Firstly data in many interesting cases are not linearly separable the need for a complex separation surface is not an artifact due to noise but rather a natural consequence of the representation. For the sake of illustration let us 255 2005 by Taylor Francis Group LLC 256 Frasconi Figure 1 Artificial problems illustrating linear and nonlinear separability. Here the instance space X is R2 and the function f is realized by a hyperplane h that divides X into a positive and a negative semispace. If positive and negative points are arranged like they are in diagram b then no separating hyperplane exists. construct an artificial problem involving nonlinear separation our example is actually a rephrasing of the famous XOR problem 1 . Suppose that we are given a problem involving the discrimination between active and nonactive chemical compounds and suppose that we are using as features that characterize each compound two physico-chemical properties expressed as real numbers say charge and hydrophobicity . In this way each compound is represented by a two-dimensional real vector as in Fig. 1. Now suppose that active compounds points marked by have either low charge and high hydrophobicity or low hydrophobicity and high charge while nonactive compounds have either high charge and low hydrophobicity or high hydrophobicity and low charge as in Fig. 1b. It is easy to realize that in this situation there is no possible linear separation between active and nonactive instances. By contrast if active compounds had both high charge and high hydrophobicity as in
