Orie 6334 Spectral Graph Theory Lecture 21
نویسندگان
چکیده
Just like matrix Chernoff bounds were a generalization of scalar Chernoff bounds, the multiplicative weights algorithm can be generalized to matrices. Recall that in the setup for the multiplicative weight update algorithm, we had a sequence of time steps t = 1, . . . , T ; in each time step t, we made a decision i ∈ {1...N} and got a value vt(i) ∈ [0, 1]. After we made a decision in time step t, we got to see all the values In matrix multiplicative weights, we make a decision u ∈ Rn, ||u|| = 1 and get a value uMtu where 0 Mt I, Mt ∈ Rn×n, so that uMtu ∈ [0, 1]. As with multiplicative weights, we make a randomized decision for the vector u based on some weights. We now maintain a weight matrix Wt ∈ Rn×n, Wt 0. Let Pt = Wt tr(Wk) so that tr(Pt) = 1 and Pt 0. If λit are eigenvalues of Pt, and xit are the corresponding orthonormal eigenectors, then Pt = ∑n i=1 λitxitx T it, λit ≥ 0, ∑n i=1 λit = 1; that is, Pt is a discrete distribution over the vectors xit, and we will choose the vector xit with probability λit.
منابع مشابه
Orie 6334 Spectral Graph Theory Lecture 8
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