Probabilistic Frames: An Overview
Finite frames can be viewed as mass points distributed in N-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of probabilistic fra
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Probabilistic Frames: An Overview Martin Ehler and Kasso A. Okoudjou
Abstract Finite frames can be viewed as mass points distributed in N-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of probabilistic frames, and we characterize one of their subclasses in terms of minimizers of some appropriate potential function. In addition, we survey a range of areas where probabilistic frames, albeit under different names, appear. These areas include directional statistics, the geometry of convex bodies, and the theory of t-designs. Keywords Probabilistic frame · POVM · Frame potential · Isotropic measure
12.1 Introduction Finite frames in RN are spanning sets that allow the analysis and synthesis of vectors in a way similar to basis decompositions. However, frames are redundant systems, and as such the reconstruction formula they provide is not unique. This redundancy plays a key role in many applications of frames which appear now in a range of areas that include, but are not limited to, signal processing, quantum computing, coding theory, and sparse representations; cf. [11, 22, 23] for an overview. By viewing the frame vectors as discrete mass distributions on RN , one can extend frame concepts to probability measures. This point of view was developed in [16] under the name of probabilistic frames and was further expanded in [18]. The goal of this chapter is to summarize the main properties of probabilistic frames and to bring forth their relationship to other areas of mathematics. M. Ehler () Institute of Biomathematics and Biometry, Helmholtz Zentrum München, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany e-mail: [email protected] K.A. Okoudjou Department of Mathematics, Norbert Wiener Center, University of Maryland, College Park, MD 20742, USA e-mail: [email protected] P.G. Casazza, G. Kutyniok (eds.), Finite Frames, 415 Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_12, © Springer Science+Business Media New York 2013
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M. Ehler and K.A. Okoudjou
The richness of the set of probability measures together with the availability of analytic and algebraic tools make it straightforward to construct many examples of probabilistic frames. For instance, by convolving probability measures, we have been able to generate new probabilistic frames from existing ones. In addition, the probabilistic framework considered in this chapter allows us to introduce a new distance on frames, namely the Wasserstein distance [35], also known as the Earth Mover’s distance [25]. Unlike standard frame distances in the literature such as the 2 -distance, the Wasserstein metric enables us to define a meaningful distance between two frames of different cardinalities. As we shall see later in Sect. 12.4, probabilistic frames are also tightly related to various notions that appeared in areas such as the theory of t-designs [15], positive operator valued measures (POVM) enco
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