Related papers: Group Frames with Few Distinct Inner Products and …
Many problems in areas such as compressive sensing and coding theory seek to design a set of equal-norm vectors with large angular separation. This idea is essentially equivalent to constructing a frame with low coherence. The elements of…
In the field of compressed sensing, a key problem remains open: to explicitly construct matrices with the restricted isometry property (RIP) whose performance rivals those generated using random matrix theory. In short, RIP involves…
We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic…
We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic…
We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is…
We investigate the theory of affine groups in the context of designing radar waveforms that obey the desired wideband ambiguity function (WAF). The WAF is obtained by correlating the signal with its time-dilated, Doppler-shifted, and…
Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two…
For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze…
This article examines group ring codes over finite fields and finite groups. We also present a section on two-dimensional cyclic codes in the quotient ring $\mathbb{F}_q[x, y] / \langle x^{l} - 1, y^{m} - 1 \rangle$. These two-dimensional…
We introduce a projective Riesz $s$-kernel for the unit sphere $\mathbb{S}^{d-1}$ and investigate properties of $N$-point energy minimizing configurations for such a kernel. We show that these configurations, for $s$ and $N$ sufficiently…
Symmetry plays a fundamental role in understanding natural phenomena and mathematical structures. This work develops a comprehensive theory for studying the persistent symmetries and degree of asymmetry of finite point configurations over…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the…
Coding schemes for several problems in network information theory are constructed starting from point-to-point channel codes that are designed for symmetric channels. Given that the point-to-point codes satisfy certain properties pertaining…
The physics of complex systems stands to greatly benefit from the qualitative changes in data availability and advances in data-driven computational methods. Many of these systems can be represented by interacting degrees of freedom on…
A new notion in frame theory has been introduced recently that called woven frames. %From the perspective of others, Woven and weaving frames are powerful tools for pre-processing signals and distributed data processing. The purpose of…
A common criterion in the design of finite Hilbert space frames is minimal coherence, as this leads to error reduction in various signal processing applications. Frames that achieve minimal coherence relative to all unit-norm frames are…
Special functions, coding theory and $t$-designs have close connections and interesting interplay. A standard approach to constructing $t$-designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the…
Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation…