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The only way that we can proceed to meaningfully learn from data such as this is by imposing some a priori prejudice on the nature of the complexity of functions we expect to elucidate. A common way of doing this is via ‘regularisation’. 2 Complexity Control: Regularisation A common, and generally very reasonable, assumption is that we typically expect that data is generated from smooth, rather than complex, functions. In a linear model framework, smoother functions typically have smaller weight magnitudes, so we can penalise complex functions by adding an appropriate penalty term to the cost function that we minimise: E(w) = ED (w) + λEW (w).

In fact, if b ∈ R(A), then z0 = A† b is k k k a solution, since Az0 = i=1 σi xi yi j=1 (1/σi )yj xj b = i=1 xi xi b = b, and the general solution is therefore z = A† b + N (A). Puzzle 6: How does this argument break down if b ∈ / R(A)? e. Az = b has no solution? One reasonable step would be to find that z that minimizes the Euclidean norm Az − b . However, adding any vector in N (A) to a solution z would also give a solution, so a reasonable second step is to require in addition that z is minimized.

Thus the right hand side is just |A|δki . Multiplying both sides on the right by (AT )−1 gives the result. 9 The name ‘tensor’ is sometimes incorrectly applied to arbitrary objects with more than one index. In factor a tensor is a generalization of the notion of a vector and is a geometrical object (has meaning independent of the choice of coordinate system); is a pseudo-tensor (transforms as a tensor, but changes sign upon inversion). C. Burges We can also use this to write the following closed form for the inverse: A−1 ij = 1 |A|(n − 1)!