Grassmannian fusion frames and its use in block sparse recovery N. Mukund Sriram a,n , B.S. Adiga b , K.V.S. Hari a a Statistical Signal Processing Lab, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, India b Tata Consultancy Services: Innovation Labs, Bangalore 560066, India article info Article history: Received 4 April 2013 Received in revised form 8 July 2013 Accepted 17 July 2013 Available online 26 July 2013 Keywords: Fusion frame Optimal Grassmannian packing Grassmannian fusion frame Block sparsity Block coherence Simplex bound abstract Tight fusion frames which form optimal packings in Grassmannian manifolds are of interest in signal processing and communication applications. In this paper, we study optimal packings and fusion frames having a specific structure for use in block sparse recovery problems. The paper starts with a sufficient condition for a set of subspaces to be an optimal packing. Further, a method of using optimal Grassmannian frames to construct tight fusion frames which form optimal packings is given. Then, we derive a lower bound on the block coherence of dictionaries used in block sparse recovery. From this result, we conclude that the Grassmannian fusion frames considered in this paper are optimal from the block coherence point of view. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Frames provide redundant representation of signals. The concept of frames occurs when we are dealing with overcomplete spanning systems [1,2]. In recent years, the concept of fusion frames which are frames of subspaces has been introduced [3]. Frames represent a signal by a collection of scalars, while fusion frames represent a signal by a collection of vectors which are projections of the signal onto the fusion frame subspaces. Fusion frames occur in many applications like packet encoding, distributed sensing, error correction, sparse signal processing and neurology [3–5]. In these applications, it is required that the frames have some desirable properties. One such property is that the fusion frame is an equidi- mensional, equidistant tight fusion frame. These frames also form optimal packings and have been called as Grassman- nian fusion frames [4,6]. The Grassmannian packing problem is concerned with arrangement of n subspaces of dimension d in K dimen- sional Euclidean space so that the subspaces are as far apart as possible [7]. It is important to note that Grassmannian fusion frames exist only for certain values of n, K and d. The construction of Grassmannian fusion frames is a complex problem and only a few methods are available in the literature. Grassmannian fusion frames can be con- structed using either algebraic methods or numerical methods. Some examples of algebraic methods are using Hadamard matrix [6], partial Fourier matrix [5], special groups like Clifford groups and lattices [4]. Though these methods are accurate, they are applicable only for certain sizes and dimensions. The alternating projection algorithm is a numerical method to construct Grassmannian fusion frames of arbitrary size and dimension [8]. However, this algorithm may give suboptimal results which are not very accurate and have convergence issues. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/sigpro Signal Processing 0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.07.016 n Corresponding author. Tel.: +91 9731093578, +91 23489915. E-mail addresses: mukundns@ece.iisc.ernet.in (N. Mukund Sriram), bs.adiga@tcs.com (B.S. Adiga), hari@ece.iisc.ernet.in (K.V.S. Hari). Signal Processing 94 (2014) 498–502