Lecture 10 & 11 : Real Stable and Hyperbolic Polynomials
نویسنده
چکیده
In this lecture we introduce hyperbolic and real stable polynomials as natural generalization of real-rootedness to multivariate polynomials. Then, we define strongly Rayleigh distributions as probability distribution whose generating function is a real stable polynomial. We will see that this is a generalization of random spanning tree distributions and determinantal distributions. We will finish this lecture by showing that strongly Rayleigh distributions satisfy strongest form negative dependence properties including negative association, stochastic dominance and log concavity. In the next lecture we will use the properties of strongly Rayleigh measures (extending the properties of random spanning tree distributions) to design a randomized rounding approximating algorithm for the symmetric traveling salesman problem. The material of this lecture are mostly based on [BB10; BBL09; Pem13; Vis13]. We use the notation R[z1, . . . , zd] to denote a degree d polynomial in z1, . . . , zd with real coefficients.
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Lecture 11 : Hyperbolic Polynomials and Hyperbolicity Cones
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