15,192 research outputs found
Quantum automorphisms of folded cube graphs
We show that the quantum automorphism group of the Clebsch graph is
. This answers a question by Banica, Bichon and Collins from 2007.
More general for odd , the quantum automorphism group of the folded -cube
graph is . Furthermore, we show that if the automorphism group of a
graph contains a pair of disjoint automorphisms this graph has quantum
symmetry.Comment: 19 pages, corrected a mistake in the proof of Theorem 2.2 + minor
change
On the Complexity of the Mis\`ere Version of Three Games Played on Graphs
We investigate the complexity of finding a winning strategy for the mis\`ere
version of three games played on graphs : two variants of the game
, introduced by Stockmann in 2004 and the game on both directed and undirected graphs. We show that on general
graphs those three games are -Hard or Complete. For one
-Hard variant of , we find an algorithm to compute
an effective winning strategy in time when
is a bipartite graph
A simpler approach to obtaining an O(1/t) convergence rate for the projected stochastic subgradient method
In this note, we present a new averaging technique for the projected
stochastic subgradient method. By using a weighted average with a weight of t+1
for each iterate w_t at iteration t, we obtain the convergence rate of O(1/t)
with both an easy proof and an easy implementation. The new scheme is compared
empirically to existing techniques, with similar performance behavior.Comment: 8 pages, 6 figures. Changes with previous version: Added reference to
concurrently submitted work arXiv:1212.1824v1; clarifications added; typos
corrected; title changed to 'subgradient method' as 'subgradient descent' is
misnome
A New Game Invariant of Graphs: the Game Distinguishing Number
The distinguishing number of a graph is a symmetry related graph
invariant whose study started two decades ago. The distinguishing number
is the least integer such that has a -distinguishing coloring. A
distinguishing -coloring is a coloring
invariant only under the trivial automorphism. In this paper, we introduce a
game variant of the distinguishing number. The distinguishing game is a game
with two players, the Gentle and the Rascal, with antagonist goals. This game
is played on a graph with a set of colors. Alternately,
the two players choose a vertex of and color it with one of the colors.
The game ends when all the vertices have been colored. Then the Gentle wins if
the coloring is distinguishing and the Rascal wins otherwise. This game leads
to define two new invariants for a graph , which are the minimum numbers of
colors needed to ensure that the Gentle has a winning strategy, depending on
who starts. These invariants could be infinite, thus we start by giving
sufficient conditions to have infinite game distinguishing numbers. We also
show that for graphs with cyclic automorphisms group of prime odd order, both
game invariants are finite. After that, we define a class of graphs, the
involutive graphs, for which the game distinguishing number can be
quadratically bounded above by the classical distinguishing number. The
definition of this class is closely related to imprimitive actions whose blocks
have size . Then, we apply results on involutive graphs to compute the exact
value of these invariants for hypercubes and even cycles. Finally, we study odd
cycles, for which we are able to compute the exact value when their order is
not prime. In the prime order case, we give an upper bound of
Block-Coordinate Frank-Wolfe Optimization for Structural SVMs
We propose a randomized block-coordinate variant of the classic Frank-Wolfe
algorithm for convex optimization with block-separable constraints. Despite its
lower iteration cost, we show that it achieves a similar convergence rate in
duality gap as the full Frank-Wolfe algorithm. We also show that, when applied
to the dual structural support vector machine (SVM) objective, this yields an
online algorithm that has the same low iteration complexity as primal
stochastic subgradient methods. However, unlike stochastic subgradient methods,
the block-coordinate Frank-Wolfe algorithm allows us to compute the optimal
step-size and yields a computable duality gap guarantee. Our experiments
indicate that this simple algorithm outperforms competing structural SVM
solvers.Comment: Appears in Proceedings of the 30th International Conference on
Machine Learning (ICML 2013). 9 pages main text + 22 pages appendix. Changes
from v3 to v4: 1) Re-organized appendix; improved & clarified duality gap
proofs; re-drew all plots; 2) Changed convention for Cf definition; 3) Added
weighted averaging experiments + convergence results; 4) Clarified main text
and relationship with appendi
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