By Johan A. K. Suykens

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24) and S* — Ufc Sk- An admissible structure is one that satisfies the following three properties: 1. The set S* is everywhere dense in S. 2. The VC-dimension hk of each set Sk of functions is finite. 3. Any element Sk of the structure contains totally bounded functions 0 < Q(z, a) < Bk, a e A f c . 20) is minimal. The SRM principle actually suggests a trade-off between the quality of the approximation and the complexity of the approximating function. 20)) increases. The SRM principle takes both factors into account.

Cucker, S. Smale Choosing the optimal 7 We now focus on the approximation error £(/7). g. , V(\\f II*' P II Since the minimum above is attained at /7 we deduce A basic result in [CS] (Proposition 1, Chapter I) states that, for all / € -/p) 2 + <7p (2-4) where d^ is a non-negative quantity depending only on p. Therefore the approximation error £(/7) is bounded by £^(7) + <72 where PROOF OF THE MAIN RESULT. 4) for / = /7iZ, This proves the first part of the Main Result. Note that this is actually a family of bounds parameterized by t < 2 and 0 < 0 < I and depends on m, S, K and fp.

Unfortunately, the set of separating hyperplanes is not flexible enough to provide low empirical risk for many real life problems [13]. Two opportunities were considered to increase the flexibility of the sets of functions: 1. to use a richer set of indicator functions which are superpositions of linear indicator functions 2. to map the input vectors into a high dimensional feature space and construct in this space a A-margin separating hyperplane. The first idea corresponds to the neural network.