Noise-resistant pulse oximetry using a synthetic reference signal.

We describe a noise-resistant pulse oximetry algorithm suited to both signal reconstruction and oxygen saturation estimation. The algorithm first detects relatively clean signal sections from which the heart rate is estimated. The heart rate is used to construct a synthetic reference signal that matches an idealized pulse signal. An adaptive filter continuously processes the sensor signals, reconstructing signals in a linear subspace defined by the reference signal. A projective subspace algorithm is then applied to find the oxygenation level of the blood. We show that under specific circumstances this algorithm solves the sufficiency condition for signal reconstruction in linear saturation estimators. The core principle of using a frequency modulated synthetic reference signal can be applied to adaptive filtering of other physiological signals controlled by the heartbeat, such as blood pressure and electrocardiogram.

On a natural homotopy between linear and nonlinear single-layer networks.

In this paper we formulate a homotopy approach for solving for the weights of a network by smoothly transforming a linear single layer network into a nonlinear perceptron network. While other researchers have reported potentially useful numerical results based on heuristics related to this approach, the work presented here provides the first rigorous exposition of the deformation process. Results include a complete description of how the weights relate to the data space, a proof of the global convergence and validity of the method, and a rigorous formulation of the generalized orthogonality theorem to provide a geometric perspective of the solution process. This geometric interpretation clarifies conditions resulting in the appearance of local minima and infinite weights in network optimization procedures, and the similarities of and differences between optimizing the weights in a nonlinear network and optimizing the weights in a linear network. The results provide a strong theoretical foundation for quantifying performance bounds on finite neural networks and for constructing globally convergent optimization approaches on finite data sets.

On the uniqueness of weights in single-layer perceptrons.

In this paper the geometric formulation of the single layer perceptron weight optimization problem previously described by Coetzee et al. (1993, 1996) is combined with results from other researchers on nonconvex set projections to describe sufficient conditions for uniqueness of weight solutions. It is shown that the perceptron data surface is pseudoconvex and has infinite folding, allowing for the specification of a region of desired vectors having unique projections purely in terms of the local curvature of the data surface. No information is therefore required regarding the global curvature or size of the data surface. These results in principle allow for a posteriori evaluation of whether a weight solution is unique or globally optimal, and for a priori scaling of desired vector values to ensure uniqueness, through analysis of the input data. The practical applicability of these results from a numerical perspective is evaluated on some carefully chosen examples.