13,054 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
New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix
The purpose of this article is to improve existing lower bounds on the
chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency
matrix sorted in non-increasing order.
First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / -
sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower
bound which only involves the maximum and minimum eigenvalues, i.e., the case
. We provide several examples for which the new bound exceeds the {\sc
Hoffman} lower bound.
Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^-
are the sums of the squares of positive and negative eigenvalues, respectively.
To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We
show that the conjectured lower bound is tight for several families of graphs.
We also performed various searches for a counter-example, but none was found.
Our proofs rely on a new technique of converting the adjacency matrix into
the zero matrix by conjugating with unitary matrices and use majorization of
spectra of self-adjoint matrices.
We also show that the above bounds are actually lower bounds on the
normalized orthogonal rank of a graph, which is always less than or equal to
the chromatic number. The normalized orthogonal rank is the minimum dimension
making it possible to assign vectors with entries of modulus one to the
vertices such that two such vectors are orthogonal if the corresponding
vertices are connected.
All these bounds are also valid when we replace the adjacency matrix A by W *
A where W is an arbitrary self-adjoint matrix and * denotes the Schur product,
that is, entrywise product of W and A
Three notions of tropical rank for symmetric matrices
We introduce and study three different notions of tropical rank for symmetric
and dissimilarity matrices in terms of minimal decompositions into rank 1
symmetric matrices, star tree matrices, and tree matrices. Our results provide
a close study of the tropical secant sets of certain nice tropical varieties,
including the tropical Grassmannian. In particular, we determine the dimension
of each secant set, the convex hull of the variety, and in most cases, the
smallest secant set which is equal to the convex hull.Comment: 23 pages, 3 figure
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
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