On the Practical Applicability of VC Dimension Bounds

This article addresses the question of whether some recent Vapnik-Chervonenkis (VC) dimension-based bounds on sample complexity can be regarded as a practical design tool. Specifically, we are interested in bounds on the sample complexity for the problem of training a pattern classifier such that we can expect it to perform valid generalization. Early results using the VC dimension, while being extremely powerful, suffered from the fact that their sample complexity predictions were rather impractical. More recent results have begun to improve the situation by attempting to take specific account of the precise algorithm used to train the classifier. We perform a series of experiments based on a task involving the classification of sets of vowel formant frequencies. The results of these experiments indicate that the more recent theories provide sample complexity predictions that are significantly more applicable in practice than those provided by earlier theories; however, we also find that the recent theories still have significant shortcomings.

What Size Net Gives Valid Generalization?

We address the question of when a network can be expected to generalize from m random training examples chosen from some arbitrary probability distribution, assuming that future test examples are drawn from the same distribution. Among our results are the following bounds on appropriate sample vs. network size. Assume 0 < ∊ ≤ 1/8. We show that if m ≥ O(W/∊ log N/∊) random examples can be loaded on a feedforward network of linear threshold functions with N nodes and W weights, so that at least a fraction 1 − ∊/2 of the examples are correctly classified, then one has confidence approaching certainty that the network will correctly classify a fraction 1 − ∊ of future test examples drawn from the same distribution. Conversely, for fully-connected feedforward nets with one hidden layer, any learning algorithm using fewer than Ω(W/∊) random training examples will, for some distributions of examples consistent with an appropriate weight choice, fail at least some fixed fraction of the time to find a weight choice that will correctly classify more than a 1 − ∊ fraction of the future test examples.

Resolution of life habits using multiple morphologic criteria: Shell form and life-mode in turritelliform gastropods

Shell form is not strictly linked to life habits in modern marine turritelliform gastropods. To test the usefulness of various morphological characters in determining life-mode, I present a set of predictions giving the expected distribution of characters occurring in turritelliform snails with three different life-modes. Burrowing species should lack sculpture, possess columellar folds and a flat whorl profile, and have an orthocline or prosocline aperture. Mobile epifaunal forms should have sculpture, a rounded whorl profile, a displaced tangential aperture and a smooth columella. Sedentary forms should resemble epifaunal forms but have non-tangential apertures. These predictions were tested with a sample of 105 Recent marine species. Each hypothesis was found to be a statistically valid generalization and in 92 of the species (88%) the life habits were correctly predicted. Accuracy may be further improved by considering additional features such as ratchet sculpture and disjunct or open coiling. These patterns of shell form can be used to interpret fossil species as burrowers, or as sedentary or active epifaunal forms. For example, the unusual Devonian murchisoniid gastropod Ptychocaulus verneuili is interpreted as an active burrower.The relatively imperfect relationship between shell form and life-mode in turritelliform gastropods, as compared to the Bivalvia, apparently results in part from the behavioral complexity of the Gastropoda. Gastropods have a repertoire of activities which would place them in different life-modes at different times; snail morphology reflects this complexity.