On the (non)existence of best low-rank approximations of generic IxJx2 arrays

Several conjectures and partial proofs have been formulated on the (non)existence of a best low-rank approximation of real-valued IxJx2 arrays. We analyze this problem using the Generalized Schur Decomposition and prove (non)existence of a best rank-R approximation for generic IxJx2 arrays, for all values of I,J,R. Moreover, for cases where a best rank-R approximation exists on a set of positive volume only, we provide easy-to-check necessary and sufficient conditions for the existence of a best rank-R approximation

On the generic and typical ranks of 3-tensors

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.Comment: 24 page

Array operators using multiple dispatch: a design methodology for array implementations in dynamic languages

Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large low-level library. Defining this functionality at the user level instead provides greater flexibility for application domains not envisioned by the language designer. Only a few languages, such as C++ and Haskell, provide the necessary power to define $n$-dimensional arrays, but these systems rely on compile-time abstraction, sacrificing some flexibility. In contrast, dynamic languages make it straightforward for the user to define any behavior they might want, but at the possible expense of performance. As part of the Julia language project, we have developed an approach that yields a novel trade-off between flexibility and compile-time analysis. The core abstraction we use is multiple dispatch. We have come to believe that while multiple dispatch has not been especially popular in most kinds of programming, technical computing is its killer application. By expressing key functions such as array indexing using multi-method signatures, a surprising range of behaviors can be obtained, in a way that is both relatively easy to write and amenable to compiler analysis. The compact factoring of concerns provided by these methods makes it easier for user-defined types to behave consistently with types in the standard library.Comment: 6 pages, 2 figures, workshop paper for the ARRAY '14 workshop, June 11, 2014, Edinburgh, United Kingdo

Matrix Completion in Colocated MIMO Radar: Recoverability, Bounds & Theoretical Guarantees

It was recently shown that low rank matrix completion theory can be employed for designing new sampling schemes in the context of MIMO radars, which can lead to the reduction of the high volume of data typically required for accurate target detection and estimation. Employing random samplers at each reception antenna, a partially observed version of the received data matrix is formulated at the fusion center, which, under certain conditions, can be recovered using convex optimization. This paper presents the theoretical analysis regarding the performance of matrix completion in colocated MIMO radar systems, exploiting the particular structure of the data matrix. Both Uniform Linear Arrays (ULAs) and arbitrary 2-dimensional arrays are considered for transmission and reception. Especially for the ULA case, under some mild assumptions on the directions of arrival of the targets, it is explicitly shown that the coherence of the data matrix is both asymptotically and approximately optimal with respect to the number of antennas of the arrays involved and further, the data matrix is recoverable using a subset of its entries with minimal cardinality. Sufficient conditions guaranteeing low matrix coherence and consequently satisfactory matrix completion performance are also presented, including the arbitrary 2-dimensional array case.Comment: 19 pages, 7 figures, under review in Transactions on Signal Processing (2013