Unitary Matrix Research Articles

Fine-Grained Self-Supervised Learning with Jigsaw puzzles for medical image classification

Classifying fine-grained lesions is challenging due to minor and subtle differences in medical images. This is because learning features of fine-grained lesions with highly minor differences is very difficult in training deep neural networks. Therefore, in this paper, we introduce Fine-Grained Self-Supervised Learning(FG-SSL) method for classifying subtle lesions in medical images. The proposed method progressively learns the model through hierarchical block such that the cross-correlation between the fine-grained Jigsaw puzzle and regularized original images is close to the identity matrix. We also apply hierarchical block for progressive fine-grained learning, which extracts different information in each step, to supervised learning for discovering subtle differences. Our method does not require an asymmetric model, nor does a negative sampling strategy, and is not sensitive to batch size. We evaluate the proposed fine-grained self-supervised learning method on comprehensive experiments using various medical image recognition datasets. In our experiments, the proposed method performs favorably compared to existing state-of-the-art approaches on the widely-used ISIC2018, APTOS2019, and ISIC2017 datasets.

GANN: Graph Alignment Neural Network for semi-supervised learning

Graph neural networks (GNNs) have been widely investigated in the field of semi-supervised graph machine learning. Most methods fail to exploit adequate graph information when labeled data is limited, leading to the problem of oversmoothing. To overcome this issue, we propose the Graph Alignment Neural Network (GANN), a simple and effective graph neural architecture. A unique learning algorithm with three alignment rules is proposed to thoroughly explore hidden information for insufficient labels. Firstly, to better investigate attribute specifics, we suggest the feature alignment rule to align the inner product of both the attribute and embedding matrices. Secondly, to properly utilize the higher-order neighbor information, we propose the cluster center alignment rule, which involves aligning the inner product of the cluster center matrix with the unit matrix. Finally, to get reliable prediction results with few labels, we establish the minimum entropy alignment rule by lining up the prediction probability matrix with its sharpened result. Extensive studies on graph benchmark datasets demonstrate that GANN can achieve considerable benefits in semi-supervised node classification and outperform state-of-the-art competitors.

A Heuristic Method for Solving Polynomial Matrix Equations

We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.

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.

Erasure Recovery Matrices for Data Erasures and Rearrangements

When studying signal reconstruction, the frames are often selected in advance as encoding tools. However, in practical applications, this encoding frame may be subject to attacks by intermediaries and generate errors. To solve this problem, in this paper, the erasure recovery matrices for data erasures and rearrangements are analyzed. Unlike the previous research, first of all, we introduce a kind of frame and its erasure recovery matrix M so that MI,Λ=Im×m, where Im×m is a unit matrix. In this case, we do not need to invert the matrix of the frame operator and the erasure recovery matrix, and this greatly simplifies reconstruction problems and calculations. Then three different construction algorithms of the above erasure recovery matrix M and the frame are proposed, and each of them has advantages. Furthermore, some restrictions on M so that the constructed frame and erasure recovery matrix M can recover coefficients from rearrangements are imposed. We prove that in some cases, the above M and frame can recover coefficients stably from m rearrangements.

Anchor graph-based multiview spectral clustering

Significant advances in graph-oriented clustering methods can be attributed to their effectiveness in leveraging relationships and complex structures within multiview data. However, several limitations persist in most existing graph-based multiview clustering approaches. First, quadratic or cubic complexity is required for graph construction or eigendecomposition of the Laplacian matrix in many existing methods. Second, certain methods overlook the differences between views and employ an identical indicator matrix, which can lead to over-learning in practical scenarios. Third, existing methods often neglect spatial structure and complementary information, focusing primarily on calculating error feature-by-feature using different norms. In order to tackle these drawbacks, we propose a new multiview spectral clustering model called Anchor Graph-based Multiview Spectral Clustering(AG-MSC). AG-MSC incorporates an adaptive weighting mechanism that assigns weights to each view, enhancing the robustness of the algorithm. Using a tensor Schatten p-norm constraint minimizes the discrepancy between indicator matrices obtained from different views, thereby preserving high-order information and spatial structure. To improve computational efficiency, we replace the full adjacency matrices of the corresponding views with anchor graphs. AG-MSC offers a distinct advantage over conventional spectral clustering by directly obtaining all sample categories without additional post-processing steps. We have validated the efficiency of our method through extensive experimental evaluations.

Robustly Train Normalizing Flows via KL Divergence Regularization

In this paper, we find that the training of Normalizing Flows (NFs) are easily affected by the outliers and a small number (or high dimensionality) of training samples. To solve this problem, we propose a Kullback–Leibler (KL) divergence regularization on the Jacobian matrix of NFs. We prove that such regularization is equivalent to adding a set of samples whose covariance matrix is the identity matrix to the training set. Thus, it reduces the negative influence of the outliers and the small sample number on the estimation of the covariance matrix, simultaneously. Therefore, our regularization makes the training of NFs robust. Ultimately, we evaluate the performance of NFs on out-of-distribution (OoD) detection tasks. The excellent results obtained demonstrate the effectiveness of the proposed regularization term. For example, with the help of the proposed regularization, the OoD detection score increases at most 30% compared with the one without the regularization.

Large Order Behavior Near the AD Point: The Case of 𝒩 = 2, su(2), Nf = 2

Abstract A non-perturbative effect in κ (renormalized string coupling) obtained from the large order behavior in the vicinity of the prototypical Argyres–Douglas critical point of su(2), Nf = 2, $\mathcal {N} =2$ supersymmetric gauge theory can be studied in the Gross–Witten–Wadia unitary matrix model with the log term: one as the work done against the barrier of the effective potential by a single eigenvalue lifted from the sea and the other as a non-perturbative function contained in the solutions of the nonlinear differential Painlevé II equation that goes beyond the asymptotic series. The leading behaviors are of the form $\exp \{[-({4}/{3})({1}/{\kappa })] \, [1, \left({s}/{K}\right)^{{3}/{2}}]\}$. We make comments on their agreement.

Riemannian quantum circuit optimization for Hamiltonian simulation

Hamiltonian simulation, i.e. simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation algorithm.

Scalable machine learning-assisted clear-box characterization for optimally controlled photonic circuits

Photonic integrated circuits offer a compact and stable platform for generating, manipulating, and detecting light. They are instrumental for classical and quantum applications. Imperfections stemming from fabrication constraints, tolerances, and operation wavelength impose limitations on the accuracy and thus utility of current photonic integrated devices. Mitigating these imperfections typically necessitates a model of the underlying physical structure and the estimation of parameters that are challenging to access. Direct solutions are currently lacking for mesh configurations extending beyond trivial cases. We introduce a scalable and innovative method to characterize photonic chips through an iterative machine learning-assisted procedure. Our method is based on a clear-box approach that harnesses a fully modeled virtual replica of the photonic chip to characterize. The process is sample-efficient and can be carried out with a continuous-wave laser and powermeters. The model estimates individual passive phases, crosstalk, beamsplitter reflectivity values, and relative input/output losses. Building upon the accurate characterization results, we mitigate imperfections to enable enhanced control over the device. We validate our characterization and imperfection mitigation methods on a 12-mode Clements-interferometer equipped with 126 phase shifters, achieving beyond state-of-the-art chip control with an average 99.77% amplitude fidelity on 100 implemented Haar-random unitary matrices.

Solution to a Conjecture on the Permanental Sum

Let G be a graph with n vertices and m edges. A(G) and I denote, respectively, the adjacency matrix of G and an n by n identity matrix. For a graph G, the permanent of matrix (I+A(G)) is called the permanental sum of G. In this paper, we give a relation between the Hosoya index and the permanental sum of G. This implies that the computational complexity of the permanental sum is NP-complete. Furthermore, we characterize the graphs with the minimum permanental sum among all graphs of n vertices and m edges, where n+3≤m≤2n−3.

Persymmetric detection based on asymptotically optimal convex linear combination

Persymmetric structure has been utilized in space-time adaptive processing for heterogeneous environment, which leads to some detection methods based on persymmetric structure, such as persymmetric adaptive matched filter (PS-AMF). However, when the sample support is extremely limited, these methods still suffer the serious degradation in detection performance due to the large error in estimating covariance matrix. In this paper, an asymptotically optimal convex linear combination between transformed sample covariance matrix and identity matrix is considered herein to improve PS-AMF. The asymptotically optimal convex linear combination is conducive to diminishing the estimate error in estimating the transformed covariance matrix by weighting the transformed sample covariance matrix and identity matrix. Furthermore, the asymptotically optimal convex linear combination is improved by the reutilization of the convex linear combination, and the corresponding coefficients are derived. Then, the detection performance of the proposed method is approximately analyzed. At last, numerical simulations show that the proposed method performs well, compared with its counterparts, when the sample support is extremely limited.

Power-efficient programmable integrated multiport photonic interferometer in CMOS-compatible silicon nitride

Silicon nitride (SiN x ) is an appealing waveguide material choice for large-scale, high-performance photonic integrated circuits (PICs) due to its low optical loss. However, SiN x PICs require high electric power to realize optical reconfiguration via the weak thermo-optic effect, which limits their scalability in terms of device density and chip power dissipation. We report a 6-mode programmable interferometer PIC operating at the wavelength of 1550 nm on a CMOS-compatible low-temperature inductance coupled plasma chemical vapor deposition (ICP-CVD) silicon nitride platform. By employing suspended thermo-optic phase shifters, the PIC achieves 2× improvement in compactness and 10× enhancement in power efficiency compared to conventional devices. Reconfigurable 6-dimensional linear transformations are demonstrated including cyclic transformations and arbitrary unitary matrices. This work demonstrates the feasibility of fabricating power-efficient large-scale reconfigurable PICs on the low-temperature ICP-CVD silicon nitride platform.

Lives and sustenance of migrant Indian women in Indenture and COVID-19

In the early nineteenth century, several indentured laborers and their descendants were exposed to political, social, and cultural shifts and instabilities that perpetuated over the decades. Simultaneously, in the pandemic era of COVID-19 and the colonial period, women workers had to overcome numerous hurdles, as they migrated without proper sustenance. The main objective of this study was to examine the working conditions, physical and psychological conditions, and sustenance of contracted laborers, especially migrant women, during the COVID-19 pandemic. Some of them are inequality in the labor markets, domestic workloads and workspaces, sexual abuse, mobility crises, child abuse, xenophobic attitudes, and sexual and gender-based violence (SGBV). By adopting a qualitative approach and critical examination, it is observed that the vulnerabilities of low-skilled women, the exploitation of migrant labor, as a matrix of identities, behaviors and power relationships are discernible in capitalist expansion, and the disparities that they induce while taking into account how the market mimics the prevalent sexual division of labor.

The G-invariant graph Laplacian Part I: Convergence rate and eigendecomposition

Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of G on the data set. We deem the latter construction the “G-invariant Graph Laplacian” (G-GL). We show that the G-GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the G-GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group SU(2).

Patterns among square roots of the 2 × 2 identity matrix

The 2 × 2 identity matrix, $${I_2} = \left( \begin{gathered}{\rm{1 \,\,\,0}} \hfill \\ {\rm{0 \,\,\,1}} \hfill \\ \end{gathered} \right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.

Interface fluctuations associated with split Fermi seas

We consider the asymptotic behaviour of a family of unidimensional lattice fermion models, which are in exact correspondence with certain probability laws on partitions and on unitary matrices. These models exhibit limit shapes, and in the case where the bulk of these shapes are described by analytic functions, the fluctuations around their interfaces have been shown to follow a universal Tracy–Widom distribution or its higher-order analogue. Non-differentiable bulk limit shape functions arise when a gap appears in some quantum numbers of the model, in other words when the Fermi sea is split. We show that split Fermi seas give rise to new interface fluctuations, governed by integer powers of universal distributions. This breakdown in universality is analogous to the behaviour of a random Hermitian matrix when the support of its limiting eigenvalue distribution has multiple cuts, with oscillations appearing in the limit of the two-point correlation function. We show that when the Fermi sea is split in the lattice fermion model, there are multiple cuts in the eigenvalue support of the corresponding unitary matrix model.

Sex and ancestry patterning of residual correlations in human dental development: Cooperative genetic interaction and phenotypic plasticity.

Most research in human dental age estimation has focused on point estimates of age, and most research on dental development theories has focused on morphology or eruption. Correlations between developing teeth using ordinal staging have received less attention. The effect of demographic variables on these correlations is unknown. I tested the effect of reference sample demographic variables on the residual correlation matrix using the lens of cooperative genetic interaction (CGI). The sample consisted of Moorrees et al., Journal of Dental Research, 1963, 42, 1490-1502, scores of left mandibular permanent teeth from panoramic radiographs of 880 London children 3-22.99 years of age stratified by year of age, sex, and Bangladeshi or European ancestry. A multivariate cumulative probit model was fit to each sex/ancestry group (n = 220), each sex or ancestry (n = 440), and all individuals (n = 880). Residual correlation matrices from nine reference sample configurations were compared using Bartlett's tests of between-sample difference matrices against the identity matrix, hierarchical cluster analysis, and dendrogram cophenetic correlations. Bartlett's test results were inconclusive. Cluster analysis showed clustering by tooth class, position within class, and developmental timing. Clustering patterns and dendrogram correlations showed similarity by sex but not ancestry. Expectations of CGI were supported for developmental staging. This supports using CGI as a model for explaining patterns of variation within the dentition. Sex was found to produce consistent patterns of dental correlations, whereas ancestry did not. Clustering by timing of development supports phenotypic plasticity in the dentition and suggests shared environment over genetic ancestry to explain population differences.

Eigenphase distributions of unimodular circular ensembles

Abstract Motivated by the study of Polyakov lines in gauge theories, Hanada and Watanabe recently presented a conjectured formula for the distribution of eigenphases of Haar-distributed random SU(N) matrices (β = 2), supported by explicit examples at small N and by numerical samplings at larger N. In this letter, I spell out a concise proof of their formula, and present its orthogonal and symplectic counterparts, i.e. the eigenphase distributions of Haar-random unimodular symmetric (β = 1) and selfdual (β = 4) unitary matrices parametrizing SU(N)/SO(N) and SU(2N)/Sp(2N), respectively.

Vanishing of quartic and sextic twists of L-functions

Let E be an elliptic curve over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Q}}}$$\\end{document}. We conjecture asymptotic estimates for the number of vanishings of L(E,1,χ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L(E,1,\\chi )$$\\end{document} as χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document} varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} twisted elliptic curve L-functions, with ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} even and composite, is a mixed family with both unitary and orthogonal aspects.