Facing Up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes

By Thomas Zaslavsky. 1975

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In Algorithms & Complexity

Citations in the corpus

On the Number of Linear Regions of Deep Neural Networks (2014, methodology): “By Zaslavsky (1975), the latter is ∑n j=0 ( k2m j ) , which behaves as O((k2m)n), i.e., polynomially in k and in m. Proof of Theorem 8.”
The Face Semigroup Algebra of a Hyperplane Arrangement (2005, methodology): “Zaslavsky’s Theorem applied to (AX)Y gives the number of faces of support Y in AX is ∑ W∈[X,Y ] |µ(W,Y )| since the intersection lattice of (AX)Y is the interval [X, Y ] in L.”
Neural network approximation (2020, background): “However, when d > 1, Zaslavsky’s theorem shows that the number of cells in P can grow as fast as CW d when W ≥ d.”
COMPUTING THE TUTTE POLYNOMIAL OF A HYPERPLANE ARRAGEMENT (2004, background): “1 (Zaslavsky, [45]) Let A be a hyperplane arrangement in R.”
A Simple Explanation for the Existence of Adversarial Examples with Small Hamming Distance (2019, background): “The maximal possible number of cells is =0 (Zaslavsky, 1975).”
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups (2006, background): “of Orlik and Solomon [97] and Zaslavsky’s theorem [134] (which implies that the number of chambers of the Catalan arrangement is equal to (−1)nφ(−1)) establishes the formula.”
On the number of response regions of deep feed forward networks with piece-wise linear activations (2013, background): “The general answer to Problem 1 is given by Zaslavsky’s theorem (Zaslavsky, 1975, Theorem A), which is one of the central results from the theory of hyperplane arrangements.”
Bounding and Counting Linear Regions of Deep Neural Networks (2017, methodology): “Zaslavsky (1975) shows that an arrangement of n hyperplanes divides a d -dimensional space into at most =0 regions, a bound that is attained when they are in general position.”
Deformations of Coxeter hyperplane arrangements and their characteristic polynomials (2000, unknown): “(Zaslavsky [69]).”
SYZYGIES OF ORIENTED MATROIDS (2000, unknown): “(Zaslavsky’s Formula) [21], [6, Theorem 4.”
Inside-out polytopes (2003, background): “Steiner wished to count the number of regions of R 2 or R 3 dissected by an arrangement, a project that was not truly completed, and then only for hyperplanes, until 150 years later in [34].”