The theory of illumination subspaces is well developed and has been tested extensively on the Yale Face Database B (YDB) and CMU-PIE (PIE) data sets. This paper shows that if face recognition under varying illumination is cast as a problem of matching sets of images to sets of images, then the minimal principal angle between subspaces is sufficient to perfectly separate matching pairs of image sets from nonmatching pairs of image sets sampled from YDB and PIE. This is true even for subspaces estimated from as few as six images and when one of the subspaces is estimated from as few as three images if the second subspace is estimated from a larger set (10 or more). This suggests that variation under illumination may be thought of as useful discriminating information rather than unwanted noise.

New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes "with vanishing one-point invariants," the relations determine many-point invariants in terms of one-point invariants. ᭧ IntroductionTwo recent innovations coming out of string theory have led to some effective algorithms for counting rational curves in a given homology class on a projective manifold X. The expected numbers of rational curves are now usually referred to as "zero-point genus-zero Gromov-Witten invariants," while the more general m-point (genus-zero) invariants count the expected numbers of rational curves meeting m submanifolds (or cohomology classes) in general position.The first innovation consists of a set of relations, known as the WDVV (after Witten, Dijkgraaf, Verlinde and Verlinde), which express a hidden symmetry in the m-point invariants for m 3. The symmetry is used to prove the associativity of quantum cohomogy rings and also leads to a Frobenius manifold structure on the cohomology space of X [23]. Kontsevich and Manin used this symmetry in [16] to express all m-point invariants in terms of two-point invariants, when the (relevant) cohomology of X is generated by

Recent work has established that digital images of a human face, collected under various illumination conditions, contain discriminatory information that can be used in classification. In this paper we demonstrate that sufficient discriminatory information persists at ultralow resolution to enable a computer to recognize specific human faces in settings beyond human capabilities. For instance, we utilized the Haar wavelet to modify a collection of images to emulate pictures from a 25pixel camera. From these modified images, a low-resolution illumination space was constructed for each individual in the CMU-PIE database. Each illumination space was then interpreted as a point on a Grassmann manifold. Classification that exploited the geometry on this manifold yielded error-free classification rates for this data set. This suggests the general utility of a low-resolution illumination camera for set-based image recognition problems.

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We study configurations of 2-planes in P 4 that are combinatorially described by the Petersen graph. We discuss conditions for configurations to be locally Cohen-Macaulay and describe the Hilbert scheme of such arrangements. An analysis of the homogeneous ideals of these configurations leads, via linkage, to a class of smooth, general type surfaces in P 4 . We compute their numerical invariants and show that they have the unusual property that they admit (multiple) 7-secants. Finally, we demonstrate that the construction applied to Petersen arrangements with additional symmetry leads to surfaces with exceptional automorphism groups.