Diagonal Map Research Articles

Cellular approximations to the diagonal map

We describe an elementary algorithm for recursively constructing diagonal approximations on those finite regular CW-complexes for which the closure of each cell can be explicitly collapsed to a point. The algorithm is based on the standard proof of the acyclic carrier theorem, made constructive through the use of explicit contracting homotopies. It can be used as a theoretical tool for constructing diagonal approximations on families of polytopes in situations where the diagonals are required to satisfy certain coherence conditions. We compare its output to existing diagonal approximations for the families of simplices, cubes, associahedra and permutahedra. The algorithm yields a new explanation of a magical formula for the associahedron derived by Markl and Shnider [Trans. Amer. Math. Soc. 358 (2006), pp. 2353–2372] and Masuda, Thomas, Tonks, and Vallette [J. Éc. polytech. Math. 8 (2021), pp. 121–146] and Theorem 4.1 provides a magical formula for other polytopes. We also describe a computer implementation of the algorithm and illustrate it on a range of practical examples including the computation of cohomology rings for some low-dimensional manifolds. To achieve some of these examples the paper includes two approaches to generating a regular CW-complex structure on closed compact 3 3 -manifolds, one using an implementation of Dehn surgery on links and the other using an implementation of pairwise identifications of faces in a tessellated boundary of the 3 3 -ball. The latter is illustrated in Proposition 8.1 with a topological classification of all closed orientable 3 3 -manifolds arising from pairwise identifications of faces of the cube.

The Hochschild cohomology ring of monomial algebras

We give an explicit description of a diagonal map on the Bardzell resolution for any monomial algebra, and we use this diagonal map to describe the cup product on Hochschild cohomology. Then, we prove that the cup product is zero in positive degrees for triangular monomial algebras. Our proof uses the graded-commutativity of the cup product on Hochschild cohomology and does not rely on explicit computation of the Hochschild cohomology modules.

Pentagram Rigidity for Centrally Symmetric Octagons

Abstract In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the $3$-diagonal map acting on affine equivalence classes of centrally symmetric octagons. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

On the diagonal product of special unitary matrices

We study the image of SU(n) under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary.

Approximation of Almost Diagonal Non-linear Maps by Lattice Lipschitz Operators

Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the n-dimensional case, such a map can be represented as a vector of size n of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.

Topological complexity of $S^3/Q_8$ as fibrewise L-S category

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\proj_{1} \colon X \times X \to X$ with the diagonal map $\Delta \colon X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

Poincaré complex diagonals and the Bass trace conjecture

For a finitely dominated Poincaré duality space $M$ , we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$ . We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class

We develop new machinery for producing decomposability tests for involutive solutions to the Yang–Baxter equation. It is based on the seminal decomposability theorem of Rump and on “cabling” operations on solutions and their effect on the diagonal map T . Our machinery yields an elementary proof of a recent decomposability theorem of Camp-Mora and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the language of braces) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garsidetheoretic approach to involutive solutions.

Efficient Spectral Graph Convolutional Network Deployment on Memristive Crossbars

Graph Neural Networks (GNNs) have attracted increasing research interest for their remarkable capability to model graph-structured knowledge. However, GNNs suffer from intensive data exchange and poor data locality, which will cause critical performance and energy bottlenecks under conventional complementary metal oxide semiconductor (CMOS)-based von-Neumann computing architectures (graphics processing unit (GPU), central processing unit (CPU)) for the “Memory Wall” issue. Fortunately, memristive crossbar-based computation has emerged as one of the most promising neuromorphic computing architectures, which has been widely studied as the computing platform for convolutional neural network (CNNs), recurrent neural network (RNNs), spiking neural network (SNNs), etc. This paper proposes the deployment of spectral graph convolutional networks (GCNs) on memristive crossbars. Further, based on the structure of GCNs (extremely high sparsity and unbalanced non-zero data distribution) and the neuromorphic characteristics of memristive crossbar circuit, we propose the acceleration method that consists of Sparse Laplace Matrix Reordering and Diagonal Block Matrix Multiplication. The simulated experiment on memristor crossbars achieves 90.3% overall accuracy on the supervised learning graph dataset (QM7), and compared with the original computation, the proposed acceleration computing framework (with half-size diagonal blocks) achieves a 27.3% reduction of memristor number. Additionally, on the unsupervised learning dataset (Karate club), our method shows no loss of accuracy with half-size diagonal block mapping and reaches a 32.2% reduction of memristor number.

Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.

Twisted Segre products

We introduce the notion of the twisted Segre product A∘ψB of Z-graded algebras A and B with respect to a twisting map ψ. It is proved that if A and B are noetherian Koszul Artin-Schelter regular algebras and ψ is a twisting map such that the twisted Segre product A∘ψB is noetherian, then A∘ψB is a noncommutative graded isolated singularity. To prove this result, the notion of densely (bi-)graded algebras is introduced. Moreover, we show that the twisted Segre product A∘ψB of A=k[u,v] and B=k[x,y] with respect to a diagonal twisting map ψ is a noncommutative quadric surface (so in particular it is noetherian), and we compute the stable category of graded maximal Cohen-Macaulay modules over it.

FDG-PET combined with learning vector quantization allows classification of neurodegenerative diseases and reveals the trajectory of idiopathic REM sleep behavior disorder

Background and Objectives18F-fluorodeoxyglucose (FDG) positron emission tomography (PET) combined with principal component analysis (PCA) has been applied to identify disease-related brain patterns in neurodegenerative disorders such as Parkinson’s disease (PD), Dementia with Lewy Bodies (DLB) and Alzheimer’s disease (AD). These patterns are used to quantify functional brain changes at the single subject level. This is especially relevant in determining disease progression in idiopathic REM sleep behavior disorder (iRBD), a prodromal stage of PD and DLB. However, the PCA method is limited in discriminating between neurodegenerative conditions. More advanced machine learning algorithms may provide a solution. In this study, we apply Generalized Matrix Learning Vector Quantization (GMLVQ) to FDG-PET scans of healthy controls, and patients with AD, PD and DLB. Scans of iRBD patients, scanned twice with an approximate 4 year interval, were projected into GMLVQ space to visualize their trajectory. MethodsWe applied a combination of SSM/PCA and GMLVQ as a classifier on FDG-PET data of healthy controls, AD, DLB, and PD patients. We determined the diagnostic performance by performing a ten times repeated ten fold cross validation. We analyzed the validity of the classification system by inspecting the GMLVQ space. First by the projection of the patients into this space. Second by representing the axis, that span this decision space, into a voxel map. Furthermore, we projected a cohort of RBD patients, whom have been scanned twice (approximately 4 years apart), into the same decision space and visualized their trajectories. ResultsThe GMLVQ prototypes, relevance diagonal, and decision space voxel maps showed metabolic patterns that agree with previously identified disease-related brain patterns. The GMLVQ decision space showed a plausible quantification of FDG-PET data. Distance traveled by iRBD subjects through GMLVQ space per year (i.e. velocity) was correlated with the change in motor symptoms per year (Spearman’s rho =0.62, P=0.004). ConclusionIn this proof-of-concept study, we show that GMLVQ provides a classification of patients with neurodegenerative disorders, and may be useful in future studies investigating speed of progression in prodromal disease stages.

On the string topology coproduct for Lie groups

The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a so-called based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.

Blind Dual Watermarking Scheme Using Stucki Kernel and SPIHT for Image Self-Recovery

In this paper we propose a blind dual watermarking scheme using Set Partitioning in Hierarchical Trees (SPIHT) and Stucki Kernel halftone technique for the tamper detection and image self-recovery. The watermark consists of authentication bits for tampering area location and recovery bits for image restoration. We generate two recovery bits to ensure the high-quality recovery of the tampered image. The primary recovery bit is generated by the SPIHT encoding, and the secondary recovery bit is generated by the Stucki Kernel halftone technique. Then the authentication bit is generated based on the recovery bits. Before embedding the watermark, we shuffle the watermark bits through Arnold cat mapping and diagonal mapping to improve the security and quality of the restored image. LSB-based watermarking technique is used to embed the watermark into the original image to ensure the invisibility of the watermarked image. Experiments have been conducted on two datasets, BOW2 and USC-SIPI, and results show that the proposed scheme can achieve high restoration quality. Comparison with the existing works demonstrate the good performance and superiority of the proposed scheme.

Illumination estimation via random vector functional link based on improved arithmetic optimization algorithm

AbstractIn order to solve the problem that the existing illumination estimation methods have low accuracy in image restoration, a random vector functional link illumination estimation algorithm based on whale optimization arithmetic algorithm is proposed in this article. First, in order to improve the optimization accuracy of arithmetic optimization algorithm, the whale optimization algorithm is applied to optimize the initial population of the arithmetic optimization algorithm. Then, we utilize the improved arithmetic optimization algorithm to improve the input weight and hidden layer offset of random vector functional link, and exclude the uncertainty caused by the random generation of random vector functional link's input weight and hidden layer bias. Finally, the proposed whale arithmetic optimization algorithm‐random vector functional link model is applied to SFU lab dataset and Gehler‐Shi dataset for illumination estimation. Compared with the other earlier illumination estimation algorithms, the proposed illumination estimation model in this article has better effect and higher stability. In addition, due to the diagonal mapping method for color correction and image restoration, the proposed illumination estimation algorithm can also get better results in image restoration.

Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory

We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even though the measurement record is not informationally complete. We then compare this with the information generated and fidelities obtained by application of a different Haar random unitary at each time step. We give an upper bound on the maximal information that can be obtained in tomography and show that a covariance matrix taken from the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution saturates this bound. We find that physically, this corresponds to an application of a different Haar random unitary at each time step. We show that repeated application of random diagonal unitaries gives a covariance matrix in tomographic estimation that corresponds to a new ensemble of random matrices. We analytically and numerically estimate eigenvalues of this ensemble and show the information gain to be bounded from below by the Porter-Thomas distribution.

Decomposable Pauli diagonal maps and tensor squares of qubit maps

It is a well-known result due to Størmer [Acta Math. 110, 233–278 (1963)] that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by Filippov and Magadov [J. Phys. A: Math. Theor. 50(5), 055301 (2017)]. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.

A novel diagnostic map for computer‐aided diagnosis of skin cancer

In teledermoscopy, images are transmitted through a communication channel to the medical facility for medical consultation. This yields to bandwidth congestion and consumption of large storage size which impairs the transmission of the high-resolution dermoscopy image. This study proposes a novel technique by generating a diagnostic feature map, named diagonal compressive sensing (CS) features map. This proposed map is generated using the significant diagnostic features by aligning the feature vector in a diagonal map, which is then compressed using CS technique. Eventually, the recovered feature map at the receiver side is applied to the classification system for decision-making. In addition, physicians at the receiver side who were trained to read the feature maps can verify the classification decision, and then provide feedback to the patient. The results demonstrated that the proposed diagnostic map achieved less transmission time due to the small size of the feature map along with the compression process. Furthermore, the feature map has drastically improved the classification performance metrics, including the accuracy, which increased, for example, from 88.6% to 98% at 80% compression ratio compared to the traditional method on the compressed whole image.

The diagonal of the associahedra

This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.

Adaptive Feature Calculation and Diagonal Mapping for Successive Recovery of Tampered Regions

This article proposes an adaptive scheme for image tampered region localization and content recovery. To generate the watermark information comprised of the authentication data and recovery data, we firstly propose the Adaptive Authentication Feature Calculation algorithm to obtain the authentication data, which includes the information of block location and block feature. The DWT-based Block Feature Calculation method is then proposed to calculate the block feature, and the quantization method is employed to calculate the block location. The recovery data is composed of self-recovery bits and mapped-recovery bits. The self-recovery bits are obtained by the Set Partitioning in Hierarchical Trees encoding algorithm. For retrieving the damaged codes caused by tampering, we propose the Diagonal Mapping algorithm and apply it to the self-recovery bits, thus generating the mapped-recovery bits, to provide a guarantee of recovery data. Experimental results show the superior performance of the proposed scheme in terms of tamper detection and image recovery, by comparing with the state-of-the-art works. The results demonstrate that the proposed method shows efficiency in the adaptiveness, well localization, strong capability for image recovery, and the effectiveness of attack resistance.