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Dimension Reduction by Christopher J.C. Burges

By Christopher J.C. Burges

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1 Original Nystr¨ om The Nystr¨ om method originated as a method for approximating the solution of Fredholm integral equations of the second kind [71]. Let’s consider the homogeneous d-dimensional form with density p(x), x ∈ Rd . This family of equations has the form: k(x, y)u(y)p(y)dy = λu(x). 2) i=1 which when applied to the sample points becomes a matrix equation Kmm um = mλum (with components Kij ≡ k(xi , xj ) and ui ≡ u(xi )). 2), which gives a much better approximation using simple interpolation [71].

Then the principal submatrix A ∈ Sr of K (which itself is the Gram matrix of the first r rows of Z) has full rank. Now letting n ≡ m − r, write the matrix K as: Kmm ≡ Arr Bnr Brn . 3) Since A has full rank, the r rows Arr Brn are linearly independent, and since K has rank r, the n rows Bnr Cnn can be expanded in terms of them, that is, there exists Hnr such that: Bnr Cnn = Hnr Arr Brn . 4) The first r columns give H = B A−1 , and the last n columns then give C = B A−1 B. Thus K must be of the form2 : Kmm = A B A B = −1 B BA B A−1 rr A mr B rm .

1 The Nystr¨ om Method 331 an otherwise infeasible algorithm feasible [34]. In this section only, we adopt the notation that matrix indices refer to sizes unless otherwise stated, so that Amm means that A ∈ Mm . 1 Original Nystr¨ om The Nystr¨ om method originated as a method for approximating the solution of Fredholm integral equations of the second kind [71]. Let’s consider the homogeneous d-dimensional form with density p(x), x ∈ Rd . This family of equations has the form: k(x, y)u(y)p(y)dy = λu(x).

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