Minimum Rank Research Articles

Notions of Rank Efficiency for the Random Assignment Problem

ABSTRACTAn assignment is rank efficient if there is no other assignment where the expected number of agents who received one of their top choices is weakly higher. We introduce new notions of rank efficiency for the random assignment problem and illustrate a hierarchy between them. In a rank‐minimizing assignment, agents receive objects with a minimum rank on average. An ex‐post rank efficient random assignment has at least one lottery over only rank efficient deterministic assignments. Thus, it could still have another lottery with some rank‐dominated deterministic assignments in its support. If each deterministic assignment in any decomposition of a random assignment is rank efficient, we call it a robust ex‐post rank efficient assignment. We demonstrate that rank‐minimizing implies rank efficiency, which indicates (robust) ex‐post rank efficiency. Moreover, we introduce a mechanism that provides an ex‐post rank efficient random assignment. We also prove that ex‐post rank efficiency is incompatible with strategyproofness or fairness in the sense of weak envy‐freeness and equal division lower bound.

Minimal rank weighted weak Drazin inverses

The concept of a minimal rank weak Drazin inverse for square matrices is extended to rectangular matrices. Precisely, a minimal rank weighted weak Drazin inverse is introduced and its properties are investigated. Some known generalized inverses such as the weighted Drazin inverse, the weighted core-EP inverse, and the weighted $p$-WGI are particular cases of a minimal rank weighted weak Drazin inverse. Thus, a wider class of generalized inverses is proposed. General representation forms of a minimal rank weighted weak Drazin inverse are presented as well as its canonical form. Applying the minimal rank weighted weak Drazin inverse, corresponding systems of linear matrix equations are solved and their solutions are expressed. As consequences of our results, new properties of minimal rank weak Drazin inverse are obtained.

Triangular ranks do not bound minimum ranks in matrix completion

Triangular ranks do not bound minimum ranks in matrix completion

Quantum null-hypothesis device-independent Schmidt number witness

We investigate the dimensionality of bipartite quantum systems by construction of a device-independent null witness test. This test assesses whether a given bipartite state conforms with the expected quantum dimension, Schmidt number, and distinguishes between real and complex spaces. By employing local measurements on each party, the proposed method aims to determine the minimal rank. By performing an experimental demonstration on IBM Quantum devices, we prove the exceptional accuracy of the test and its usefulness in diagnostics beyond routine calibrations. One of the tests shows agreement with theoretical expectations within statistical errors. However, the second test failed by more than 6 standard deviations, indicating unspecified parasitic entanglements, with no known simple origin.

On the rank structure of the Moore-Penrose inverse of singular k-banded matrices

It is well-established that, for an n×n singular k-banded complex matrix B, the submatrices of the Moore-Penrose inverse B† of B located strictly below (resp. above) its kth superdiagonal (resp. kth subdiagonal) have a certain bounded rank s depending on n, k and rankB. In this case, B† is said to satisfy a semiseparability condition. In this paper our focus is on singular strictly k-banded complex matrices B, and we show that the Moore-Penrose inverse of such a matrix satisfies a stronger condition, called generator representability. This means that there exist two matrices of rank at most s whose parts strictly below the kth diagonal (resp. above the kth subdiagonal) coincide with the same parts of B†. When n≥3k, we prove that s is precisely the minimum rank of these two matrices. We also illustrate through examples that when n<3k those matrices may have rank less than s.

Design of a load balancing Objective Function for RPL

Routing protocols for Internet of Things (IoT) play a major role in the performance of the network. The standard Routing Protocol for Low-Power and Lossy Networks (RPL) suffers from a number of limitations including congestion of higher-level nodes and unbalanced topology. This paper proposes a novel Objective Function called Load Balanced Minimum Rank with Hysteresis Objective Function (LB_MRHOF), which assigns child nodes to the most suitable parent in the topology. The Objective Function utilizes a weight of the Expected Transmission Count (ETX) and number of children to calculate the Composite ETX and Number of Children (CENOC) which estimates the load on each node. The attained CENOC is used to select the optimum parent for each node in the topology, where nodes with high CENOC are avoided in the parent selection process. The proposed Objective Function has been evaluated under random and hierarchical network topologies. In addition, the evaluation has investigated the influence of the number of nodes by testing for small, medium and large-scale networks. Results have shown that the proposed Objective Function outperforms MRHOF, OF_FUZZY and OF-EC in terms of Packet Delivery Ratio (PDR) and reduces nodal hop-count under all tested scenarios, with no compromise in energy consumption. They have also revealed that the best performance achieved by LB_MRHOF is attained under large-scale networks. The resulting network topology which is formed by the proposed Objective Function has shown improved balance and more depth.

Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels

We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Second, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.

An efficient approach for discovering Graph Entity Dependencies (GEDs)

Graph entity dependencies (GEDs) are novel graph constraints, unifying keys and functional dependencies, for property graphs. They have been found useful in many real-world data quality and data management tasks, including fact checking on social media networks and entity resolution. In this paper, we study the discovery problem of GEDs—finding a minimal cover of valid GEDs in a given graph data. We formalise the problem, and propose an effective and efficient approach to overcome major bottlenecks in GED discovery. In particular, we leverage existing graph partitioning algorithms to enable fast GED-scope discovery, and employ effective pruning strategies over the prohibitively large space of candidate dependencies. Furthermore, we define an interestingness measure for GEDs based on the minimum description length principle, to score and rank the mined cover set of GEDs. Finally, we demonstrate the scalability and effectiveness of our GED discovery approach through extensive experiments on real-world benchmark graph data sets; and present the usefulness of the discovered rules in different downstream data quality management applications.

Weak MPD and DMP inverses

Based on a minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse, our aim is to solve weakened systems of matrix equations than already existing systems for defining the MPD and DMP inverses. These new systems have unique solutions called the weak MPD and DMP inverses, which present new types of generalized inverses involving MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Many characterizations and expressions for weak MPD and DMP inverses are established. Consequently, we obtain several new and some known properties of MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Furthermore, the weak MPD and DMP inverses are applied in solving linear equations. As a consequence, we get a fundamental result related to application of the Moore–Penrose inverse in solving linear equation.

Primitively 2-universal senary integral quadratic forms

For a positive integer m, a (positive definite integral) quadratic form is called primitively m-universal if it primitively represents all quadratic forms of rank m. It was proved in [9] that there are exactly 107 equivalence classes of primitively 1-universal quaternary quadratic forms. In this article, we prove that the minimal rank of primitively 2-universal quadratic forms is six, and there are exactly 201 equivalence classes of primitively 2-universal senary quadratic forms.

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations—often referred to as persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

Combinatorics of essential sets for positroids

Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows the study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic intervals. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals.

Spreading in graphs

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph G are contaminated initially, before the process starts. By the q-forcing rule, a contaminated vertex having at most q uncontaminated neighbors enforces all the neighbors to become contaminated, while by the p-percolation rule, an uncontaminated vertex becomes contaminated if at least p of its neighbors are contaminated. If given a set S of initially contaminated vertices all vertices eventually become contaminated when continuously applying the q-forcing rule (respectively the p-percolation rule), S is called a q-forcing set (respectively, a p-percolating set) in G. In this paper, we consider sets S that are at the same time q-forcing sets and p-percolating sets, and call them (p,q)-spreading sets. Given positive integers p and q, the minimum cardinality of a (p,q)-spreading set in G is a (p,q)-spreading number, σ(p,q)(G), of G. While q-forcing sets have been studied in a dozen of papers, the decision version of the corresponding graph invariant has not been considered earlier, and we fill the gap by proving its NP-completeness. This, in turn, enables us to prove the NP-completeness of the decision version of the (p,q)-spreading number in graphs for an arbitrary choice of p and q. Again, for every p∈N and q∈N∪{∞}, we find a linear-time algorithm for determining the (p,q)-spreading number of a tree, where in the case p≥2 we apply Riedl’s algorithm from [Largest and smallest minimal percolating sets in trees, Electron. J. Combin. 19 (2012) Paper 64] on p-percolation in trees. In addition, we present a lower and an upper bound on the (p,q)-spreading number of a tree and characterize extremal families of trees. In the case of square grids, we combine some results of Bollobás from [The Art of Mathematics: Coffee Time in Memphis. Cambridge Univ. Press, New York, 2006], and the AIM Minimum Rank-Special Graphs Work Group from [Zero forcing sets and the minimum rank of graphs, Linear algebra Appl. 428 2008 1628–1648], and new results on (2,1)-spreading and (4,q)-spreading to obtain σ(p,q)(Pm□Pn) for all (p,q)∈(N∖{3})×(N∪{∞}) and all m,n∈N.

Latent structure and reliability

There are numerous reliability coefficients and α is the most popular. It is known that different coefficients can be appropriate in specific conditions and that α should not be used indiscriminately. However, some coefficients and conditions, particularly regarding the latent structure, lacked attention in previous research. In four Monte Carlo simulations, this study compared α, λ2, maximized λ4, λ4 based on locally optimal splits, μ2, Gilmer-Feldt, Kaiser-Caffrey α, Heise-Bohrnstedt Ω, Joreskog's ρ, ωtotal, algebraic greatest lower bound, greatest lower bound based on minimum rank factor analysis in every condition and also hierarchical ω and asymptotic ω hierarchical in multidimensional conditions. Findings suggest each of these coefficients can be useful at least in some conditions. Most differences in performance were observed in congeneric conditions and conditions with up to moderate loadings. Some coefficients were found to be more useful than previously considered. Results are discussed in the context of existing theory and previous Monte Carlo studies.

Advancing Healthcare Networks: Optimizing Multi-Objective RPL for Diverse Traffic in Low-Power, Lossy Environments

RPL: Powering modern healthcare IoT Networks. The Routing Protocol for Low Power and Lossy Networks(RPL) is a protocol specifically designed for routing in networks characterized by low power and lossy connections built on IPv6. It is specifically designed for the growing use of Instantaneous IoT (Internet of Things) applications. These networks cater to the specialized requirements of Low Power (resource-constrained) and Lossy Networks(unstable networks), also known as LLNs, where routing data smoothly and prioritizing traffic types are the significant challenges. Studies have shown that standard RPL, which makes use of a set of routing rules, struggles to provide the level of performance that many IoT applications in modern-day healthcare demand. This limits RPL’s potential in scenarios where we have a mix of data traffic, which is very common in healthcare settings. To handle the diverse data traffic found in healthcare settings, our method involves distributing several instances of RPL settings throughout the network. The main objective of this strategy is to improve the effectiveness of important and critical healthcare applications. We have developed and thoroughly tested our system using Contiki, which is an open-source IoT operating system. We have three instances in our MultiInstance RPL setup, each of which is intended to handle different kinds of network traffic and provide varying Quality of Service (QoS) measures. Different sets of Objective Function (OF) rules are used by each instance to decide which route is optimal for the data it processes. We have made use of OF0 (Objective Function 0) and MRHOF (Minimum Rank with Hysteresis Objective Function). We have rigorously tested our solution. Our focus has been on metrics like delay, average energy consumption and Packet Delivery Ratio (PDR). Compared to standard RPL, our approach improved packet delivery rates and significantly reduced delays for high priority data packets. This research shows how to better address the complex data traffic needs of real-time healthcare IoT networks.

An intelligent fuzzy enabled parent node selection approach in low power networks

The Internet of Things lies in establishing low-power and lossy networks created by interconnecting many wireless devices with limited resources. Fascinatingly, an IPv6 routing protocol for low-power and lossy networks has become a common practice for these applications. Even though this protocol addresses the challenges of low-power networks, many issues concerning the quality of service and energy consumption are open to the research community. The protocol relies on a destination-oriented directed acyclic graph, and the root selection depends on some constraints and metrics associated with an objective function (OF). The conventional OFs select parents based on a single metric, such as the expected transmission count or the number of nodes to travel. The current paper proposes an enhancement to the OF metric, aiming to decrease node energy and enhance the quality of service. This improvement is achieved by the factors, including the received signal strength indicator, node distance, power, link quality indicator, and expected transmission count, to select reliable communication links. The minimum power needed for reliable communication is predicted from the received signal strength indicator, node distance, receiver power, and link quality indicator using a nonlinear support vector machine. The OF value of the candidate node is computed from the power level and expected transmission count combined using the Takagi-Sugeno fuzzy model. The proposed OF is implemented in the Cooja simulator and compared against minimum rank with hysteresis OF and OF zero. A considerable improvement in the packet delivery ratio and a 37.5% reduction in energy consumption is obtained.

Delegation-Relegation for Boolean Matrix Factorization

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.

RANKS OF SUBMATRICES IN THE REFLEXIVE SOLUTIONS OF SOME MATRIX EQUATIONS

Maximal and minimal ranks of the two submatrices X₁ and X₂ in the (skew-) Hermitian reflexive solution X=U[X₁ 00 X₂]U^{∗} of the matrix equation AXA^{∗}=C, in the reflexive solution of the matrix equation AXB=C are derived. Then necessary and sufficient conditions for these reflexive solutions to have special forms, and the general expressions of these reflexive solutions are achieved.

Bounded rank perturbations of a regular matrix pencil

In this paper we study the possible Kronecker invariants of an arbitrary matrix pencil obtained by bounded rank perturbation of a regular matrix pencil. We solve this problem in the case of a perturbation of minimal rank.

The rank of syzygies of canonical curves

Abstract We prove that the linear syzygy spaces of a general canonical curve are spanned by syzygies of minimal rank.