143 research outputs found
Recycling Randomness with Structure for Sublinear time Kernel Expansions
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
On the independence ratio of distance graphs
A distance graph is an undirected graph on the integers where two integers
are adjacent if their difference is in a prescribed distance set. The
independence ratio of a distance graph is the maximum density of an
independent set in . Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM
J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is
equal to the inverse of the fractional chromatic number, thus relating the
concept to the well studied question of finding the chromatic number of
distance graphs.
We prove that the independence ratio of a distance graph is achieved by a
periodic set, and we present a framework for discharging arguments to
demonstrate upper bounds on the independence ratio. With these tools, we
determine the exact independence ratio for several infinite families of
distance sets of size three, determine asymptotic values for others, and
present several conjectures.Comment: 39 pages, 12 figures, 6 table
An inertial lower bound for the chromatic number of a graph
Let ) and denote the chromatic and fractional chromatic
numbers of a graph , and let denote the inertia of .
We prove that:
1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{
and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right)
\le \chi_f(G)
We investigate extremal graphs for these bounds and demonstrate that this
inertial bound is not a lower bound for the vector chromatic number. We
conclude with a discussion of asymmetry between and , including some
Nordhaus-Gaddum bounds for inertia
Chromatic numbers of Cayley graphs of abelian groups: A matrix method
In this paper, we take a modest first step towards a systematic study of
chromatic numbers of Cayley graphs on abelian groups. We lose little when we
consider these graphs only when they are connected and of finite degree. As in
the work of Heuberger and others, in such cases the graph can be represented by
an integer matrix, where we call the dimension and the
rank. Adding or subtracting rows produces a graph homomorphism to a graph with
a matrix of smaller dimension, thereby giving an upper bound on the chromatic
number of the original graph. In this article we develop the foundations of
this method. In a series of follow-up articles using this method, we completely
determine the chromatic number in cases with small dimension and rank; prove a
generalization of Zhu's theorem on the chromatic number of -valent integer
distance graphs; and provide an alternate proof of Payan's theorem that a
cube-like graph cannot have chromatic number 3.Comment: 17 page
Long cycles and paths in distance graphs
AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result
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