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Reports tagged with polytopes:
TR04-070 | 22nd June 2004
Leonid Gurvits

Combinatorial and algorithmic aspects of hyperbolic polynomials

Let $p(x_1,...,x_n) =\sum_{ (r_1,...,r_n) \in I_{n,n} } a_{(r_1,...,r_n) } \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$
be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients.
The support of such polynomial $p(x_1,...,x_n)$
is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 ... more >>> TR09-144 | 24th December 2009 Prahladh Harsha, Adam Klivans, Raghu Meka An Invariance Principle for Polytopes Let$X$be randomly chosen from$\{-1,1\}^n$, and let$Y$be randomly chosen from the standard spherical Gaussian on$\R^n$. For any (possibly unbounded) polytope$P$formed by the intersection of$k$halfspaces, we prove that$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ... more >>> TR17-098 | 28th May 2017 Raman Arora, Amitabh Basu , Poorya Mianjy, Anirbit Mukherjee Understanding Deep Neural Networks with Rectified Linear Units Revisions: 2 In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give the first-ever polynomial time (in the size of data) algorithm to train to global optimality a ReLU DNN with one hidden layer, assuming the input dimension and number ... more >>> TR18-145 | 13th August 2018 Ryan O'Donnell, Rocco Servedio, Li-Yang Tan Fooling Polytopes We give a pseudorandom generator that fools$m$-facet polytopes over$\{0,1\}^n$with seed length$\mathrm{polylog}(m) \cdot \log n$. The previous best seed length had superlinear dependence on$m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any$\{0,1\}$-integer program. more >>> TR20-189 | 24th December 2020 Pavel Hrubes, Amir Yehudayoff Shadows of Newton polytopes We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of$P\$ to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity.

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