We present an unsupervised technique for detecting unusual activity in a large video set using many simple features. No complex activity models and no supervised feature selections are used. We divide the video into equal length segments and classify the extracted features into prototypes, from which a prototype-segment co-occurrence matrix is computed. Motivated by a similar problem in document-keyword analysis, we seek a correspondence relationship between prototypes and video segments which satisfies the transitive closure constraint. We show that an important sub-family of correspondence functions can be reduced to coembedding prototypes and segments to N-D Euclidean space. We prove that an efficient, globally optimal algorithm exists for the co-embedding problem. Experiments on various real-life videos have validated our approach. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences s of positive integers, which they called s-inversion sequences.Our object of study is the generating polynomial of the ascent statistic over the set of s-inversion sequences of length n. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the s-Eulerian polynomial. The main result of this paper is that, for any sequence s of positive integers, the s-Eulerian polynomial has only real roots.This result is first shown to generalize several existing results about the realrootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several q-analogs. We show that the MacMahon-Carlitz q-Eulerian polynomial has only real roots whenever q is a positive real number confirming a conjecture of Chow and Gessel. The same holds true for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively.Our results have interesting geometric consequences as well.

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