Each substitution is order n2 to fairly compare the

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Unformatted text preview: tiable. Then the Bregman divergence associated with f is the function Df : Rn × Rn → R given by Df (x, y ) = f (x) − f (y ) − ∇f (y )T (x − y ). (a) Show that Df (x, y ) ≥ 0 for all x, y ∈ dom f . (b) Show that if f = · (c) Show that if f (x) = to be 0), then 2, 2 then Df (x, y ) = x − y 2 . 2 n i=1 xi log xi (negative entropy), with dom f = Rn (with 0 log 0 taken + n Df (x, y ) = i=1 (xi log(xi /yi ) − xi + yi ) , the Kullback-Leibler divergence between x and y . (d) Bregman projection. The previous parts suggest that Bregman divergences can be viewed as generalized ‘distances’, i.e., functions that measure how similar two vectors are. This suggests solving geometric problems that measure distance between vectors using a Bregman divergence rather than Euclidean distance. Explain whether minimize Df (x, y ) subject to x ∈ C , with variable x ∈ Rn , is a convex optimization problem (assuming C is convex). (e) Duality. Show that Dg (y ∗ , x∗ ) = Df (x, y ), where g = f ∗ and z ∗ = ∇f (z ). You can assume...
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