James H. Curry

Improving non-negative matrix factorizations through structured initialization

In this paper we explore a recent iterative compression technique called non-negative matrix factorization (NMF). Several special properties are obtained as a result of the constrained optimization problem of NMF. For facial images, the additive nature of NMF results in a basis of features, such as eyes, noses, and lips. We explore various methods for efficiently computing NMF, placing particular emphasis on the initialization of current algorithms. We propose using Spherical K-Means clustering to produce a structured initialization for NMF. We demonstrate some of the properties that result from this initialization and develop an efficient way of choosing the rank of the low-dimensional NMF representation.

A generalized Lorenz system

A 14-dimensional generalized Lorenz system of ordinary differential equations is constructed and its bifurcation sequence is then studied numerically. Several fundamental differences are found which serve to distinguish this model from Lorenzs original one, the most unexpected of which is a family of invariant two-tori whose ultimate bifurcation leads to a strange attractor. The strange attractor seems to have many of the gross features observed in Lorenzs model and therefore is an excellent candidate for a higher dimensional analogue.

On the iteration of a rational function: computer experiments with Newton's method

Using Newtons method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newtons method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.

On the Hénon transformation

In [4] Hénon studied a transformation which maps the plane into itself and appears to have an attractor with locally the structure of a Cantor set cross an interval. By making use of the characteristic exponent, frequency spectrum, and a theorem of Smale, our numerical experiments provide evidence for the existence of two distinct strange attractors for some parameter values, an exponential rate of mixing for the parameter values studied by Hénon, and an argument that there is a Cantor set in the trapping region of Hénon.

Non-negative matrix factorization: Ill-posedness and a geometric algorithm

Non-negative matrix factorization (NMF) has been proposed as a mathematical tool for identifying the components of a dataset. However, popular NMF algorithms tend to operate slowly and do not always identify the components which are most representative of the data. In this paper, an alternative algorithm for performing NMF is developed using the geometry of the problem. The computational costs of the algorithm are explored, and it is shown to successfully identify the components of a simulated dataset. The development of the geometric algorithm framework illustrates the ill-posedness of the NMF problem and suggests that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems.

On computing the entropy of the Henon attractor

In a recent article D. Ruelle [inLecture Notes in Physics, No. 80 (Springer, Berlin, 1978)] has conjectured that for the Hénon attractor its measure theoretic entropy should be equal to its characteristic exponent. This result is known to be true for systems which satisfy Smales Axiom A. In this article we report the results of our computations which suggest that Ruelles conjecture may be true for the Hénon attractor. Further, in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view.

Bounded Solutions of Finite Dimensional Approximations to the Boussinesq Equations

A class of finite systems of nonlinear ordinary differential equations are derived which yield finite dimensional approximation to the solutions of the Boussinesq equations. Such finite solutions, to these evolution equations, can be associated with trajectories in a phase space defined by the amplitudes of the components occurring in the differential equations. Once we make a passage to a phase space description of a system, a natural question which arises concerns the long time behavior of trajectories. In this paper we give sufficient conditions for a large class of approximate solutions to the Boussinesq equations to remain bounded for all time.

On the rate of approach to homoclinic tangency

Abstract It is well-known fact, observed by Poincare, that the crossing of stable and unstable manifolds of a saddle-type periodic point leads to quite complicated dynamics. Newhouse has shown that when the stable and unstable manifolds are tangent there will also be wild behavior. In this note we report on the rate of approach to homoclinic tangency.

Vertical spectral representation in primitive equation models of the atmosphere

Abstract Attempts to represent the vertical structure in primitive equation models of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small time steps were required for computational stability near the upper boundary with a vertical spectral method using Laguerre polynomials. Machenhauer and Daley used Legendre polynomials in their vertical spectral representation and found it necessary to use an artificial constraint to force temperature to zero when pressure was zero to control the upper-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presented. By formulating the model in terms of velocity and geopotential and then using the hydrostatic equation to calculate temperature from geopotential, temperature is necessarily zero when pressure is zero. This strategy works provided the multiplicative inverse of the first vertical derivative of the vertical...

On the dynamics of Laguerre's iteration: Z n -1

Abstract The Laguerre algorithm for computing the zeros of polynomial equations is well known. Many proofs of the global convergence of this method for the class of real polynomials with all real roots and starting from real initial conditions exist. In this article the global behavior of the Laguerre iteration is studied when applied to Zn-1. For this class of polynomials an almost complete description of the dynamics of Laguerre is rendered.