5,573 research outputs found
Quadratic Discriminant Analysis Revisited
In this thesis, we revisit quadratic discriminant analysis (QDA), a standard classification method. Specifically, we investigate the parameter estimation and dimension reduction problems for QDA.
Traditionally, the parameters of QDA are estimated generatively; that is the parameters are estimated by maximizing the joint likelihood of observations and their labels. In practice, classical QDA, though computationally efficient, often underperforms discriminative classifiers, such as SVM, Boosting methods, and logistic regression. Motivated by recent research on hybrid generative/discriminative learning, we propose to estimate the parameters of QDA by minimizing a convex combination of negative joint log-likelihood and negative conditional log-likelihood of observations and their labels. For this purpose, we propose an iterative majorize-minimize (MM) algorithm for classifiers of which conditional distributions are from the exponential family; in each iteration of the MM algorithm, a convex optimization problem needs to be solved. To solve the convex problem specially derived for QDA, we propose a block-coordinate descent algorithm that sequentially updates the parameters of QDA; in each update, we present a trust region method for solving optimal estimations, of which we have closed form solutions in each iteration. Numerical experiments show: 1) the hybrid approach to QDA is competitive with, and in some cases significant better than other approaches to QDA, SVM with polynomial kernel () and logistic regression with linear and quadratic features; 2) in many cases, our optimization method converges faster to equal or better optimums than the conjugate gradient method used in the literature.
Dimension reduction methods are commonly used to extract more compact features in the hope to build more efficient and possibly more robust classifiers. It is well known that Fisher\u27s discriminant analysis generates optimal lower dimensional features for linear discriminant analysis. However, ...for QDA, where so far there has been no universally accepted dimension-reduction technique in the literature\u27\u27, though considerable efforts have been made. To construct a dimension reduction method for QDA, we generalize the Fukunaga-Koontz transformation, and propose novel affine feature extraction (AFE) methods for binary QDA. The proposed AFE methods have closed-form solutions and thus can be solved efficiently. We show that 1) the AFE methods have desired geometrical, statistical and information-theoretical properties; and 2) the AFE methods generalize dimension reduction methods for LDA and QDA with equal means. Numerical experiments show that the new proposed AFE method is competitive with, and in some cases significantly better than some commonly used linear dimension reduction techniques for QDA in the literature
On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono
Recently, Bruinier and Ono found an algebraic formula for the partition
function in terms of traces of singular moduli of a certain non-holomorphic
modular function. In this paper we prove that the rational polynomial having
these singuar moduli as zeros is (essentially) irreducible, settling a question
of Bruinier and Ono. The proof uses careful analytic estimates together with
some related work of Dewar and Murty, as well as extensive numerical
calculations of Sutherland
Elliptic Curves and Hyperdeterminants in Quantum Gravity
Hyperdeterminants are generalizations of determinants from matrices to
multi-dimensional hypermatrices. They were discovered in the 19th century by
Arthur Cayley but were largely ignored over a period of 100 years before once
again being recognised as important in algebraic geometry, physics and number
theory. It is shown that a cubic elliptic curve whose Mordell-Weil group
contains a Z2 x Z2 x Z subgroup can be transformed into the degree four
hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and
coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2
hypermatrix of coefficients can be reduced to a quartic elliptic curve whose
J-invariant is expressed in terms of the hypermatrix and related invariants
including the degree 24 hyperdeterminant. These connections between elliptic
curves and hyperdeterminants may have applications in other areas including
physics.Comment: 7 page
Two-dimensional lattices with few distances
We prove that of all two-dimensional lattices of covolume 1 the hexagonal
lattice has asymptotically the fewest distances. An analogous result for
dimensions 3 to 8 was proved in 1991 by Conway and Sloane. Moreover, we give a
survey of some related literature, in particular progress on a conjecture from
1995 due to Schmutz Schaller.Comment: 21 pages, final version, accepted for publication in L'Enseignement
Math\'ematiqu
Class invariants for certain non-holomorphic modular functions
Inspired by prior work of Bruinier and Ono and Mertens and Rolen, we study
class polynomials for non-holomorphic modular functions arising from modular
forms of negative weight. In particular, we give general conditions for the
irreducibility of class polynomials. This allows us to easily generate
infintely many new class invariants
Edwards curves and CM curves
Edwards curves are a particular form of elliptic curves that admit a fast,
unified and complete addition law. Relations between Edwards curves and
Montgomery curves have already been described. Our work takes the view of
parameterizing elliptic curves given by their j-invariant, a problematic that
arises from using curves with complex multiplication, for instance. We add to
the catalogue the links with Kubert parameterizations of X0(2) and X0(4). We
classify CM curves that admit an Edwards or Montgomery form over a finite
field, and justify the use of isogenous curves when needed
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