Random permutation matrix models for graph products

We consider an exactly solvable random matrix model related to the random transfer matrix model for disordered conductors. In the conventional random matrix models the spacing distribution of nearest neighbor eigenvalues, when expressed in units of average spacing, has a universal behavior known generally as the Wigner distribution. In contrast, our model has a single parameter, as a function of which the spacing distribution crosses over from a Wigner to a distribution which is increasingly more Poisson-like, a feature common to a wide variety of physical systems including disorder and chaos.

The uniform random permutation hash (URP) algorithm has reliable irreversibility in the protection of biometrics template. However, by randomly permutation of feature vectors and calculation of dot product, URP method cannot completely retain the global information of the original feature vectors, it will affect the recognition rate of algorithm. Yet it’s important for recognition rate in biometrics cancelable template, therefore this paper proposed a new finger-vein cancelable template named random permutation projection (RPP). First, the random permutation matrix is generated by combining users’ token matrix with original feature vector. Second, the projection vector is obtained by multiplying original feature vector with random permutation matrix, we also record the maximum and sub-large index in projection vector. At last, the sub-large index is considered integrating with maximum index, this will significantly reduce error by only considering one index. The experimental results show that the RPP template improved the recognition rate of the algorithm on the PolyU and SDUMLA-FV data sets, RPP also meets the revocability standard of cancelable biometric templates.

Discrepancy of high-dimensional permutations, Discrete Analysis 2016:11, 8pp. A permutation matrix is a (necessarily square) 01-matrix with exactly one 1 in each row and column. A three-dimensional analogue of a permutation matrix is a three-dimensional grid of 0s and 1s -- more formally, a function $f:\{1,2,\dots,n\}^3\to\{0,1\}$ -- such that for every $x,y$ there is exactly one $z$ such that $f(x,y,z)=1$, and similarly for the other two coordinates. (An alternative generalization would be to say that for every $x$ there is exactly one pair $(y,z)$ such that $f(x,y,z)=1$, but this generalization is more interesting.) An equivalent, but less symmetric, definition is that it is a function $g:\{1,2,\dots,n\}^2\to\{1,2,\dots,n\}$ such that for each $x$ the function $y\mapsto g(x,y)$ is a bijection and for each $y$ the function $x\mapsto g(x,y)$ is a bijection. Such objects are called _Latin squares_: given a Latin square $g$ the corresponding three-dimensional permutation $f$ is defined by taking $f(x,y,z)$ to be 1 if and only if $g(x,y)=z$; in the other direction, given a three-dimensional permutation $f$, the corresponding Latin square is defined by setting $g(x,y)$ to be the unique $z$ such that $f(x,y,z)=1$. To put this more concisely, $f$ is the characteristic function of the graph of $g$. (Just to be clear, in the paper, the authors refer to a normal permutation as a one-dimensional permutation, since it permutes a one-dimensional array of objects, and in general a $d$-dimensional permutation has a $(d+1)$-dimensional array of 0s and 1s as its "matrix".) An important branch of combinatorics, with links to analysis, number theory and other areas, is discrepancy theory. A typical problem in discrepancy specifies two collections $\mathcal A$ and $\Sigma$ of a set $X$ and asks how "balanced" a set $S\in\Sigma$ can be with respect to the sets in $\mathcal A$. Given a set $A\in\mathcal A$, one defines the discrepancy of $S$ in $A$ to be the difference between $|S\cap A|$ and the "expected" size of $S\cap A$ (where this has some natural meaning that varies from problem to problem), and the discrepancy of $S$ with respect to $\mathcal A$ is the maximum of these discrepancies. We then try to minimize this quantity over all $S\in\Sigma$: this is the discrepancy of $\Sigma$ with respect to $\mathcal A$. This paper takes $\Sigma$ to be the set of three-dimensional permutation matrices , or rather the subsets $S\subset\{1,2,\dots,n\}^3$ of which they are the characteristic functions, and $\mathcal A$ is the set of all Cartesian products $A=X\times Y\times Z\subset\{1,2,\dots,n\}^3$. In this case, the "expected" size of an intersection $S\cap A$ is simply $|S||A|/n^3=|A|/n$, so the aim is to find a good estimate for the minimum over all sets $S$ that come from permutation matrices of the maximum over all Cartesian products $A$ of $\bigl||S\cap A|-|A|/n\bigr|$. The main conjecture in the paper is that for a random three-dimensional permutation matrix, this discrepancy is $O(|A|^{1/2})$, with a similar conjecture for higher dimensions. (Note that this a more precise statement than one obtains from the general formulation of discrepancy problems just given, since the bound depends on the size of $A$, but we could of course fix that size.) This would imply that for a random three-dimensional permutation matrix, the largest that $|X||Y||Z|$ can be if $S\cap (X\times Y\times Z)=\emptyset$ is $O(n^2)$. The authors prove that there exists a three-dimensional permutation matrix with this property, using ideas from Keevash's breakthrough results about designs. Besides the main conjecture, several other appealing open problems are given in the paper. One reason they are not straightforward is that it is significantly harder to analyse a random Latin square than it is a random permutation: we do not even have a satisfactory asymptotic estimate for their number when $n$ is large. ^{_{[Image by RainerTypke](https://en.wikipedia.org/wiki/Magic_square#/media/File:2152085cab.png)}}

The non-ergodic extended (NEE) regime in physical and random matrix (RM) models has attracted a lot of attention in recent years. Formally, NEE regime is characterized by its fractal wavefunctions and long-range spectral correlations such as number variance or spectral form factor. More recently, it’s proposed that this regime can be conveniently revealed through the eigenvalue spectra by means of singular-value-decomposition (SVD), whose results display a super-Poissonian behavior that reflects the minibands structure of NEE regime. In this work, we employ SVD to a number of RM models, and show it not only qualitatively reveals the NEE regime, but also quantitatively locates the ergodic-NEE transition point. With SVD, we further suggest the NEE regime in a new RM model–the sparse RM model.

Traditional groundwater models do not consider the influence of random factors, so their results of simulation and prediction cannot reflect the actual dynamics of ongoing groundwater changes. By comparing the statistical properties of random multidimensional time series, the random matrix theory can reflect the degree of deviation from randomness in actual data and reveal the behavioral characteristics of the overall correlation in actual data. On the basis of summarizing and analyzing the previous literature, this study expounded the research status and significance of groundwater impact analysis models, elaborated on the development background, current status, and future challenges of the random matrix model, introduced the methods and principles of the finite isometric properties and observation feature dimension, conducted the spectral analysis of the random matrix of groundwater impact analysis data, analyzed the eigenvalue and the eigenvector distribution of the random matrix of the groundwater impact analysis data, constructed a groundwater impact analysis model based on the random matrix model, designed the hydrodynamic model for the groundwater impact analysis, discussed the parameter variation mechanism of the groundwater impact analysis, and finally carried out a case application and its result analysis. The study results show that in the membership calculation formula of the random matrix model, it is found that there is an inverse relationship between the expected value of the target index and its membership value. According to the initial fuzzy membership function value, the random matrix model uses the normalization formula and evaluation criteria of the catastrophe model to carry out a comprehensive quantitative recursive operation and finally determines the concept, characteristics, influencing factors, evaluation content, purpose, and scope of the groundwater impact analysis model. The random matrix model can not only reflect the spatial and temporal differences of groundwater-influencing factors but also reflect the dynamic mechanism of aquifer or groundwater system or the inherent law of groundwater movement, so it is more applicable and convenient to evaluate groundwater resources.

For colocated multiple-input multiple-output (MIMO) radar transmitting orthogonal polyphase coded waveform, a low range, angular and Doppler sidelobe is desirable. In this paper, we propose a MIMO radar sidelobe suppression method based on random space-time coding (STC) method. In detail, at each pulse repetition period, a random STC matrix, composed of a diagonal random phase matrix multiplied by a random permutation matrix, is modulated to waveforms before transmission. Such a kind of STC matrix can keep the constant modulus property of transmitting waveforms and can decrease range, angular and Doppler sidelobe significantly. (4 pages)

We introduce a random interaction matrix model (RIMM) for finite-size strongly interacting fermionic systems whose single-particle dynamics is chaotic. The model is applied to Coulomb blockade quantum dots with irregular shape to describe the crossover of the peak spacing distribution from a Wigner-Dyson to a Gaussian-like distribution. The crossover is universal within the random matrix model and is shown to depend on a single parameter: a scaled fluctuation width of the interaction matrix elements. The crossover observed in the RIMM is compared with the results of an Anderson model with Coulomb interactions.

Random matrix model for eigenvalue statistics in random spin systems

We study the distribution of entries of a random permutation matrix under a "randomized basis," i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from Haar measure. It is shown that under certain conditions, the linear combination of entries of a random permutation matrix under a "randomized basis" converges to a sum of independent variables $sY + Z$ where $Y$ is Poisson distributed, $Z$ is normally distributed, and $s$ is a constant.

We review recent progress in modeling credit risk for correlated assets. We employ a new interpretation of the Wishart model for random correlation matrices to model non-stationary effects. We then use the Merton model in which default events and losses are derived from the asset values at maturity. To estimate the time development of the asset values, the stock prices are used, the correlations of which have a strong impact on the loss distribution, particularly on its tails. These correlations are non-stationary, which also influences the tails. We account for the asset fluctuations by averaging over an ensemble of random matrices that models the truly existing set of measured correlation matrices. As a most welcome side effect, this approach drastically reduces the parameter dependence of the loss distribution, allowing us to obtain very explicit results, which show quantitatively that the heavy tails prevail over diversification benefits even for small correlations. We calibrate our random matrix model with market data and show how it is capable of grasping different market situations. Furthermore, we present numerical simulations for concurrent portfolio risks, i.e., for the joint probability densities of losses for two portfolios. For the convenience of the reader, we give an introduction to the Wishart random matrix model.

The Gaussian unitary ensemble is a random matrix model (RMM) for the Wigner law. While random matrices in this model are infinitely divisible, the Wigner law is infinitely divisible not in the classical but in the free sense. We prove that any variance mixture of Gaussian distributions -- whether infinitely divisible or not in the classical sense -- admits a RMM of non Gaussian infinitely divisible random matrices. More generally, it is shown that any mixture of the Wigner law admits a RMM. A key role is played by the fact that the Gaussian distribution is the mixture of Wigner law with the ]]> 2$-gamma distribution.

This study proposes a scheme for using modified coefficients of the DCT of an image to generate a lossless visible watermark. The major contribution of the proposed technique is the improved security against attack to remove watermarks under stricter assumption of Kerckhoffs' principle. After the host images and watermarks are decomposed into several frequencies, the DCT coefficients of the watermark are embedded into the DCT coefficients of the host image. Integer mapping is then used to perform 2-dimensional DCT. The major advantage of the method is the improved security achieved by using a random permutation matrix to factorize the transformation matrix. That is, since the embedding stage multiplies the transformation matrix by a random permutation matrix, illicit users, even under the stricter assumption of Kerckhoffs principle that the proposed embedding method is known by illicit users, cannot properly recover the host image without the correct permutation matrix. Unlike methods that embed the watermark in quantized frequency-domain coefficients, the watermarked image remains in raw lossless image form instead of some lossy form of quantized coefficients e.g., JPEG-formatted. Maintaining the lossless format of the watermarked image provides reversibility.

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Levy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Levy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫0∞e−1dΨtd, d ≥ 1, where Ψtd is a d × d matrix-valued Levy process satisfying an Ilog condition.

Single cell RNA-seq (scRNAseq) workflows typically start with a count matrix and end with the clustering of sampled cells. While a range of methods have been developed to cluster scRNAseq datasets, no theoretical tools exist to explain why a particular cluster exists or why a hypothesized cluster is missing. Recently, several authors have shown that eigenvalues of scRNAseq count matrices can be approximated using random matrix models. In this work, we extend these previous works to the study of a scRNAseq workflow. We model scaled count matrices using random matrices with normally distributed entries. Using these random matrix models, we quantify the differential expression of a cluster and develop predictions for the workflow, and in particular clustering, as a function of the differential expression. We also use results from random matrix theory (RMT) to develop predictive formulas for portions of the scRNAseq workflow. Using simulated and real datasets, we show that our predictions are accurate if certain conditions hold on differential expression, with our RMT based predictions requiring particularly stringent condition. We find that real datasets violate these conditions, leading to bias in our predictions, but our predictions are better than a naive estimator and we point out future work that can improve the predictions. To our knowledge, our formulas represents the first predictive results for scRNAseq workflows.

An accurate and efficient uncertainty quantification of the dynamic response of complex structural systems is crucial for their design and analysis. Among the many approaches proposed, the random matrix approach has received significant attention over the past decade. In this paper two new random matrix models, namely (1) generalized scalar Wishart distribution and (2) generalized diagonal Wishart distribution have been proposed. The central aims behind the proposition of the new models are to (1) improve the accuracy of the statistical predictions, (2) simplify the analytical formulations and (3) improve computational efficiency. Identification of the parameters of the newly proposed random matrix models has been discussed. Closed-form expressions have been derived using rigorous analytical approaches. It is considered that the dynamical system is proportionally damped and the mass and stiffness properties of the system are random. The newly proposed approaches are compared with the existing Wishart random matrix model using numerical case studies. Results from the random matrix approaches have been validated using an experiment on a vibrating plate with randomly attached spring-mass oscillators. One hundred nominally identical samples have been created and separately tested within a laboratory framework. Relative merits and demerits of different random matrix formulations are discussed and based on the numerical and experimental studies the recommendation for the best model has been given. A simple step-by-step method for implementing the new computational approach in conjunction with general purpose finite element software has been outlined.