62 research outputs found

    Excluding pairs of tournaments

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    The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph HH there exists a constant c(H)>0c(H)>0 such that every graph GG that does not contain HH as an induced subgraph contains a clique or a stable set of size at least ∣V(G)∣c(H)|V(G)|^{c(H)}. The conjecture is still open. Its equivalent directed version states that for every given tournament HH there exists a constant c(H)>0c(H)>0 such that every HH-free tournament TT contains a transitive subtournament of order at least ∣V(T)∣c(H)|V(T)|^{c(H)}. We prove in this paper that {H1,H2}\{H_{1},H_{2}\}-free tournaments TT contain transitive subtournaments of size at least ∣V(T)∣c(H1,H2)|V(T)|^{c(H_{1},H_{2})} for some c(H1,H2)>0c(H_{1},H_{2})>0 and several pairs of tournaments: H1H_{1}, H2H_{2}. In particular we prove that {H,Hc}\{H,H^{c}\}-freeness implies existence of the polynomial-size transitive subtournaments for several tournaments HH for which the conjecture is still open (HcH^{c} stands for the \textit{complement of HH}). To the best of our knowledge these are first nontrivial results of this type

    Recycling Randomness with Structure for Sublinear time Kernel Expansions

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    We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features
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