Blind Multilinear Identification

We discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible: (i) closely located or highly correlated sources in antenna array processing, (ii) highly correlated spreading codes in CDMA radio communication, (iii) nearly dependent spectra in fluorescent spectroscopy. This has important implications --- in the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescent spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that specialize to the usual ones for matrices and operators but apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor

Efficient text fingerprinting via Parikh mapping

AbstractWe consider the problem of fingerprinting text by sets of symbols. Specifically, if S is a string, of length n, over a finite, ordered alphabet Σ, and S′ is a substring of S, then the fingerprint of S′ is the subset φ of Σ of precisely the symbols appearing in S′. In this paper we show efficient methods of answering various queries on fingerprint statistics. Our preprocessing is done in time O(n|Σ|lognlog|Σ|) and enables answering the following queries: (1)Given an integer k, compute the number of distinct fingerprints of size k in time O(1).(2)Given a set φ⊆Σ, compute the total number of distinct occurrences in S of substrings with fingerprint φ in time O(|Σ|logn)

Detecting One-variable Patterns

Given a pattern $p = s_1x_1s_2x_2\cdots s_{r-1}x_{r-1}s_r$ such that $x_1,x_2,\ldots,x_{r-1}\in\{x,\overset{{}_{\leftarrow}}{x}\}$, where $x$ is a variable and $\overset{{}_{\leftarrow}}{x}$ its reversal, and $s_1,s_2,\ldots,s_r$ are strings that contain no variables, we describe an algorithm that constructs in $O(rn)$ time a compact representation of all $P$ instances of $p$ in an input string of length $n$ over a polynomially bounded integer alphabet, so that one can report those instances in $O(P)$ time.Comment: 16 pages (+13 pages of Appendix), 4 figures, accepted to SPIRE 201

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $\epsilon \in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $\epsilon$ and the case that it is at most $a_d \epsilon$, with query complexity and running time $c_d \epsilon^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns

Terahertz Communications and Sensing for 6G and Beyond: A Comprehensive View

The next-generation wireless technologies, commonly referred to as the sixth generation (6G), are envisioned to support extreme communications capacity and in particular disruption in the network sensing capabilities. The terahertz (THz) band is one potential enabler for those due to the enormous unused frequency bands and the high spatial resolution enabled by both short wavelengths and bandwidths. Different from earlier surveys, this paper presents a comprehensive treatment and technology survey on THz communications and sensing in terms of the advantages, applications, propagation characterization, channel modeling, measurement campaigns, antennas, transceiver devices, beamforming, networking, the integration of communications and sensing, and experimental testbeds. Starting from the motivation and use cases, we survey the development and historical perspective of THz communications and sensing with the anticipated 6G requirements. We explore the radio propagation, channel modeling, and measurements for THz band. The transceiver requirements, architectures, technological challenges, and approaches together with means to compensate for the high propagation losses by appropriate antenna and beamforming solutions. We survey also several system technologies required by or beneficial for THz systems. The synergistic design of sensing and communications is explored with depth. Practical trials, demonstrations, and experiments are also summarized. The paper gives a holistic view of the current state of the art and highlights the issues and challenges that are open for further research towards 6G.Comment: 55 pages, 10 figures, 8 tables, submitted to IEEE Communications Surveys & Tutorial