Quadratics - CSC 411 / CSC D11 Quadratics 4 Quadratics The...

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CSC 411 / CSC D11 Quadratics 4 Quadratics The objective functions used in linear least-squares and regularized least-squares are multidimen- sional quadratics. We now analyze multidimensional quadratics further. We will see many more uses of quadratics further in the course, particularly when dealing with Gaussian distributions. The general form of a one-dimensional quadratic is given by: f ( x ) = w 2 x 2 + w 1 x + w 0 (1) This can also be written in a slightly different way (called standard form): f ( x ) = a ( x b ) 2 + c (2) where a = w 2 , b = w 1 / (2 w 2 ) , c = w 0 w 2 1 / 4 w 2 . These two forms are equivalent, and it is easy to go back and forth between them (e.g., given a, b, c , what are w 0 , w 1 , w 2 ?). In the latter form, it is easy to visualize the shape of the curve: it is a bowl, with minimum (or maximum) at b , and the “width” of the bowl is determined by the magnitude of a , the sign of a tells us which direction the bowl points ( a positive means a convex bowl, a negative means a concave bowl), and c tells us how high or low the bowl goes (at x = b ). We will now generalize these intuitions for higher-dimensional quadratics. The general form for a 2D quadratic function is: f ( x 1 , x 2 ) = w 1 , 1 x 2 1 + w 1 , 2 x 1 x 2 + w 2 , 2 x 2 2 + w 1 x 1 + w 2 x 2 + w 0 (3) and, for an N -D quadratic, it is: f ( x 1 , ...x N ) = s 1 i N, 1 j N w i,j x i x j + s 1 i N w i x i + w 0 (4) Note that there are three sets of terms: the quadratic terms ( w i,j x i x j ), the linear terms ( w i x i ) and the constant term (
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This note was uploaded on 11/09/2010 for the course CS CSCD11 taught by Professor Davidfleet during the Spring '10 term at University of Toronto- Toronto.

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Quadratics - CSC 411 / CSC D11 Quadratics 4 Quadratics The...

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