In the last lecture, we were looking at clustering sparse graphs in the stochastic block model with parameters p > q, where p = a n , and q = b n , we saw that we could not do perfect clustering as w.h.p, there will be a constant fraction of isolated vertices. However, we shall see that we can still do weak recovery, i.e., by allowing a small fraction of vertices to be misplaced, we will recove...

We demonstrate our prototype of the DBO database system. DBO is designed to facilitate scalable analytic processing over large data archives. DBO’s analytic processing performance is competitive with other database systems; however, unlike any other existing research or industrial system, DBO maintains a statistically meaningful guess to the final answer to a query from start to finish during q...

Consider a flock of birds that fly interacting between them. The interactions are modelled through a hierarchical system in which each bird, at each time step, adjusts its own velocity according to his past velocity and a weighted mean of the relative velocities of its superiors in the hierarchy. We consider the additional fact, that each of the birds can fail to see any of its superiors with c...

We have already seen in previous lectures that a concept class C is PAC-learnable by the hypothesis class H if there exists an algorithm A such that for all c ∈ C, for all distributions D over X, for all > 0, for all δ > 0, A takes m = finite number of examples given in the form S = 〈 (x1, c(x1)), ..., (xm, c(xm)) 〉 , where each xi chosen from the space X at random according to the target distr...

If this is was what we had from the start then the task of clustering would be trivial. However, usually, the graphs that must be clustered are not this perfect and contain edges between S1 and S2. These edges can be considered noise in the representation of G. Therefore, a clustering algorithm would be attempting to cluster a noisy representation, G ′ , of the perfect graph G. More explicitly,...

1 Rectangular matrix multiplication These notes are based on the algorithm and analysis from this paper [1]. We would like to partition the m × n × k cube of matrix multiplications to minimize the surface area of the maximum partition. Using a P processors we can define a p 1 × p 2 × p 3 grid of processors with p 1 p 2 p 3 = P , that yields an optimal bandwidth cost W , p 1 p 2 p 3 =P mn p 1 p ...