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Nearly orthogonal lattices were formally defined in [4] , where their applications to image compression were also discussed. The idea of "near orthogonality" in 2-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in R n and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension n ≥ 3 does not have any local maxima on the nearly orthogonal set and has only one local minimum there: at the integer lattice Z n. Further, we show that the nearly orthogonal set cannot contain any perfect lattices for n ≥ 3 , although it contains multiple eutactic (and even strongly eutactic) lattices in every dimension. This implies that eutactic lattices, while always critical points of the packing density function, are not necessarily local maxima or minima even among the well-rounded lattices. We also prove that a (weakly) nearly orthogonal lattice in R n contains no more than 4 n − 2 minimal vectors (with any smaller even number possible) and establish some bounds on coherence of these lattices. [ABSTRACT FROM AUTHOR]

Abstract Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski’s Convex Body and Linear Forms theorems that exhibit some limitations of this framework. [ABSTRACT FROM AUTHOR]

This note supplies additions to our paper quoted in the title on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit ( 276 , 23 ) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three. [ABSTRACT FROM AUTHOR]

We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight ( k , n ) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k -dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2 k . We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect. [ABSTRACT FROM AUTHOR]

We say that a Euclidean lattice in R n is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group S n , i.e., if the lattice is closed under the action of some non-identity elements of S n . Given a fixed element τ ∈ S n , we study properties of the set of all lattices closed under the action of τ : we call such lattices τ -invariant . These lattices naturally generalize cyclic lattices introduced by Micciancio (2002, 2007), which we previously studied in Fukshansky and Sun (2014). Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ -invariant lattices in R n has positive co-dimension (and hence comprises zero proportion) for all τ different from an n -cycle. [ABSTRACT FROM AUTHOR]

Abstract: We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid. [ABSTRACT FROM AUTHOR]