arXiv:1111.5899v1 [cs.IT] 25 Nov 2011 SAMPLING, FILTERING AND SPARSE APPROXIMATIONS ON COMBINATORIAL GRAPHS Isaac Z. Pesenson 1 Meyer Z. Pesenson 2 Abstract. In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schrodinger’s group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denois- ing, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory. Keywords and phrases: Combinatorial Laplace operator, Poincare and Plancherel-Polya inequalities, Paley-Wiener spaces, best approx- imations, sparse approximations, Schr¨ odinger Semigroup, modulus of continuity, Hilbert frames. Subject classifications: Primary: 42C99, 05C99, 94A20; Secondary: 94A12 1. Introduction During the last years harmonic analysis on combinatorial graphs attracted con- siderable attention. The interest is stimulated in part by multiple existing and potential applications of analysis on graphs to information theory, signal analysis, image processing, computer sciences, learning theory, astronomy [2], [3], [5]–[8], [12], [17], [24]– [26]. Some of the approaches to large data sets or images consider them as graphs. However, for hyperspectral images, for example, this leads to graphs with too many vertices imbedded into high dimensional spaces, thus making dimension reduction necessary for effective data mining. It seems that one possible way to approach this problem is by using ideas from the classical sampling theory which has already proved very fruitful in various branches of applied mathematics. Let us remind the Classical Shannon-Nyquist sampling Theorem. It states that for all Paley-Wiener functions of a fixed bandwidth defined on Euclidean space one can find ”not very dense” sampling sets which can be used to represent all relevant Paley-Wiener functions. In some sense it allows to reduce the set of all points of Euclidean space to a countable set of points. Moreover, since the set of 1 Department of Mathematics, Temple University, Philadelphia, PA 19122; pesen- son@math.temple.edu. The author was supported in part by the National Geospatial-Intelligence Agency University Research Initiative (NURI), grant HM1582-08-1-0019. 2 Spitzer Science Center, California Institute of Technology, MC 314-6, Pasadena, CA 91125; misha@ipac.caltech.edu. The author was supported in part by the National Geospatial-Intelligence Agency University Research Initiative (NURI), grant HM1582-08-1-0019. 1