4,952 research outputs found
Numerical range for random matrices
We analyze the numerical range of high-dimensional random matrices, obtaining
limit results and corresponding quantitative estimates in the non-limit case.
For a large class of random matrices their numerical range is shown to converge
to a disc. In particular, numerical range of complex Ginibre matrix almost
surely converges to the disk of radius . Since the spectrum of
non-hermitian random matrices from the Ginibre ensemble lives asymptotically in
a neighborhood of the unit disk, it follows that the outer belt of width
containing no eigenvalues can be seen as a quantification the
non-normality of the complex Ginibre random matrix. We also show that the
numerical range of upper triangular Gaussian matrices converges to the same
disk of radius , while all eigenvalues are equal to zero and we prove
that the operator norm of such matrices converges to .Comment: 23 pages, 4 figure
Neutrino mixing, interval matrices and singular values
We study the properties of singular values of mixing matrices embedded within
an experimentally determined interval matrix. We argue that any physically
admissible mixing matrix needs to have the property of being a contraction.
This condition constrains the interval matrix, by imposing correlations on its
elements and leaving behind only physical mixings that may unveil signs of new
physics in terms of extra neutrino species. We propose a description of the
admissible three-dimensional mixing space as a convex hull over experimentally
determined unitary mixing matrices parametrized by Euler angles which allows us
to select either unitary or nonunitary mixing matrices. The unitarity-breaking
cases are found through singular values and we construct unitary extensions
yielding a complete theory of minimal dimensionality larger than three through
the theory of unitary matrix dilations. We discuss further applications to the
quark sector.Comment: Misprints correcte
Construction of aggregation operators with noble reinforcement
This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. <br /
Toward a probability theory for product logic: states, integral representation and reasoning
The aim of this paper is to extend probability theory from the classical to
the product t-norm fuzzy logic setting. More precisely, we axiomatize a
generalized notion of finitely additive probability for product logic formulas,
called state, and show that every state is the Lebesgue integral with respect
to a unique regular Borel probability measure. Furthermore, the relation
between states and measures is shown to be one-one. In addition, we study
geometrical properties of the convex set of states and show that extremal
states, i.e., the extremal points of the state space, are the same as the
truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal
logic for probabilistic reasoning on product logic events and prove soundness
and completeness with respect to probabilistic spaces, where the algebra is a
free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur
The Rate of Convergence of AdaBoost
The AdaBoost algorithm was designed to combine many "weak" hypotheses that
perform slightly better than random guessing into a "strong" hypothesis that
has very low error. We study the rate at which AdaBoost iteratively converges
to the minimum of the "exponential loss." Unlike previous work, our proofs do
not require a weak-learning assumption, nor do they require that minimizers of
the exponential loss are finite. Our first result shows that at iteration ,
the exponential loss of AdaBoost's computed parameter vector will be at most
more than that of any parameter vector of -norm bounded by
in a number of rounds that is at most a polynomial in and .
We also provide lower bounds showing that a polynomial dependence on these
parameters is necessary. Our second result is that within
iterations, AdaBoost achieves a value of the exponential loss that is at most
more than the best possible value, where depends on the dataset.
We show that this dependence of the rate on is optimal up to
constant factors, i.e., at least rounds are necessary to
achieve within of the optimal exponential loss.Comment: A preliminary version will appear in COLT 201
- …