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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 KullbackLeibler 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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 Fall '13
 F.Borrelli
 The Aeneid

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