Norm Of Vector Research Articles

Regularised Layerwise Weight Norm Based Skin Lesion Features Extraction and Classification

Melanoma is the most lethal malignant tumour, and its prevalence is increasing. Early detection and diagnosis of skin cancer can alert patients to manage precautions and dramatically improve the lives of people. Recently, deep learning has grown increasingly popular in the extraction and categorization of skin cancer features for effective prediction. A deep learning model learns and co-adapts representations and features from training data to the point where it fails to perform well on test data. As a result, overfitting and poor performance occur. To deal with this issue, we proposed a novel Consecutive Layerwise weight Constraint MaxNorm model (CLCM-net) for constraining the norm of the weight vector that is scaled each time and bounding to a limit. This method uses deep convolutional neural networks and also custom layer-wise weight constraints that are set to the whole weight matrix directly to learn features efficiently. In this research, a detailed analysis of these weight norms is performed on two distinct datasets, International Skin Imaging Collaboration (ISIC) of 2018 and 2019, which are challenging for convolutional networks to handle. According to the findings of this work, CLCM-net did a better job of raising the model’s performance by learning the features efficiently within the size limit of weights with appropriate weight constraint settings. The results proved that the proposed techniques achieved 94.42% accuracy on ISIC 2018, 91.73% accuracy on ISIC 2019 datasets and 93% of accuracy on combined dataset.

Three High-Rate Beamforming Methods for Active IRS-Aided Wireless Network

Due to its ability of breaking the double-fading effect experienced by passive intelligent reflecting surface (IRS), active IRS is evolving a potential technique for future 6 G wireless networks. To fully exploit the amplifying gain achieved by active IRS, two high-rate methods, maximum-ratio-reflecting (MRR) and selective-ratio-reflecting (SRR), are presented, which are motivated by maximum ratio combining and selective ratio combining. Moreover, both MRR and SRR are in closed-form expressions. To further improve the rate, a maximum approximate-signal-to-noise ratio (Max-ASNR) is first proposed with an alternately iterative infrastructure between adjusting the norm of beamforming vector and its normalized vector. This may make a substantial rate enhancement over existing equal-gain reflecting (EGR). Simulation results show that the proposed three methods perform much better than existing method EGR in terms of rate. They are in decreasing order of rate performance: Max-ASNR, MRR, SRR, and EGR.

Overparameterized Linear Regression Under Adversarial Attacks

We study the error of linear regression in the face of adversarial attacks. In this framework, an adversary changes the input to the regression model in order to maximize the prediction error. We provide bounds on the prediction error in the presence of an adversary as a function of the parameter norm and the error in the absence of such an adversary. We show how these bounds make it possible to study the adversarial error using analysis from non-adversarial setups. The obtained results shed light on the robustness of overparameterized linear models to adversarial attacks. Adding features might be either a source of additional robustness or brittleness. On the one hand, we use asymptotic results to illustrate how double-descent curves can be obtained for the adversarial error. On the other hand, we derive conditions under which the adversarial error can grow to infinity as more features are added, while at the same time, the test error goes to zero. We show this behavior is caused by the fact that the norm of the parameter vector grows with the number of features. It is also established that $\ell_\infty$ and $\ell_2$-adversarial attacks might behave fundamentally differently due to how the $\ell_1$ and $\ell_2$-norms of random projections concentrate. We also show how our reformulation allows for solving adversarial training as a convex optimization problem. This fact is then exploited to establish similarities between adversarial training and parameter-shrinking methods and to study how the training might affect the robustness of the estimated models.

Quaternion Scalar and Vector Norm Decomposition: Quaternion PCA for Color Face Recognition.

This paper proposes a decomposition called quaternion scalar and vector norm decomposition (QSVND) for approximation problems in color image processing. Different from traditional quaternion norm approximations that are always the single objective models (SOM), QSVND is adopted to transform the SOM into the bi-objective model (BOM). Furthermore, regularization is used to solve the BOM problem as a common scalarization method, which converts the BOM into a more reasonable SOM. This can handle over-fitting or under-fitting problems neglected in this kind of research for quaternion representation (QR) in color image processing. That is how to treat redundancy caused by the extra scalar part when the vector part of a quaternion is used to represent a color pixel. We apply QSVND to quaternion principal component analysis (QPCA) for color face recognition (FR), which can deal with the phenomenon of under-fitting of vector part norm approximation. Comparisons with the competing approaches on AR, FERET, FEI, and KDEF&AKDEF databases consistently show the superiority of the proposed approach for color FR.

Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems

<abstract><p>Consider a linear system $ Ax = b $ where the coefficient matrix $ A $ is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices $ W $. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when $ W = I $. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.</p></abstract>

Edge Devices Clustering for Federated Visual Classification: A Feature Norm Based Framework.

Federated learning is a privacy-preserving distributed learning paradigm where multiple devices collaboratively train a model, which is applicable to edge computing environments. However, the non-IID data distributed in multiple devices degrades the performance of the federated model due to severe weight divergence. This paper presents a clustered federated learning framework named cFedFN for visual classification tasks in order to reduce the degradation. Especially, this framework introduces the computation of feature norm vectors in the local training process and divides the devices into multiple groups by the similarities of the data distributions to reduce the weight divergences for better performance. As a result, this framework gains better performance on non-IID data without leakage of the private raw data. Experiments on various visual classification datasets demonstrate the superiority of this framework over the state-of-the-art clustered federated learning frameworks.

Fixed‐time fault‐tolerant control of uncertain MIMO switched interconnected systems subject to state delay, unknown control directions, and quantized nonlinear inputs

SummaryThis paper investigates design of an adaptive fixed‐time fault‐tolerant decentralized controller for a class of uncertain multi‐input multi‐output (MIMO) switched large‐scale non‐strict interconnected systems under arbitrary switching subject to unknown control directions, quantized nonlinear inputs, actuator failures unknown external disturbances, and unmodeled dynamics. In addition to interconnected terms, time‐varying delayed interconnected terms have been considered in the system model which makes it more general than previous works in the literature. The proposed controller can handle switched systems with unknown switching signal and different types of input nonlinearities including, saturation, backlash, and dead‐zone. The singularity problem in designing the fixed time controller has been solved. The quantizer and actuators fault parameters are assumed to be unknown. The Razumikhin lemma has been used to deal with the delayed interconnected terms. To cope with the system unknown dynamics, neural networks (NNs) have been applied and by updating the maximum norms of the networks weight vectors the computational load has been reduced. The explosion of complexity occurring in the traditional back‐stepping technique has been avoided by applying dynamic surface control (DSC). Finally, by defining an appropriate common Lyapunov function (CLF), fixed‐time convergence of system outputs and the closed‐loop system stability have been established. The effectiveness of the proposed controller has been shown via simulation study.

An abstract sub-supersolution method in partially ordered spaces

We present a general framework in which sub-supersolution techniques can be applied. Specifically, we consider a normed vector space endowed with a partial order. This allows us to introduce a notion of sub- and supersolutions relative to an equation driven by operators defined from the space to its topological dual. We provide sufficient conditions to guarantee that the interval determined by an ordered pair of sub-supersolutions contains a solution of the initial problem. We also study existence of extremal positive or negative solutions and location of nodal solutions. Examples of applications to quasilinear boundary value problems are given.

Norms of Dual Complex Vectors and Dual Complex Matrices

Norms of Dual Complex Vectors and Dual Complex Matrices

Transmit Beam Control in Low-Altitude Slow-Moving Small Targets Detection Based on Peak to Average Power Ratio Constraint

When the radar system detects low-altitude, small, slow-moving (LSS) targets, the strong clutter interference from the ground will cause false alarms and affect the detection performance. In this paper, a phased array radar transmit beam steering algorithm is proposed to minimize strong interference from ground radiation. By minimizing the weighted vector norm and choosing variable sidelobe levels, the beam pattern can achieve deep notches in the ground-related area and maintain good main lobe detection performance. Furthermore, the designed beam should be insensitive to array mismatch and be robust. In addition, a peak-to-average power ratio (PAPR) constraint is introduced to fully utilize the transmitted energy. This optimization problem can be transformed into a second-order cone programming (SOCP) problem and solved using an off-the-shelf solver. The simulation results verify that the transmit beam synthesized by this method can meet the requirements of minimizing the main lobe loss and low side lobes on the ground side.

Minimizing cycles in tournaments and normalized \(q\)-norms

Akin to the Erd\H{o}s-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any $d\in (0,1]$, among all $n$-vertex tournaments with $d\binom{n}{3}$ many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Kr\'al' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for $d\geq 1/36$. In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers $\ell\not\equiv 2\mod 4$, there exists some constant $c_\ell>0$ such that if $d\geq 1-c_\ell$, then the number of $\ell$-cycles is also asymptotically minimized by the same family of extremal examples for $4$-cycles. In doing so, we answer a question of Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector with given $p$-norm for any integers $q>p>1$. For integers $\ell\equiv 2\mod 4$, however the same phenomena do not hold for $\ell$-cycles, for which we can construct an explicit family of tournaments containing fewer $\ell$-cycles for any given number of $3$-cycles. We conclude by proposing two conjectures on the minimization problem for general cycles in tournaments.

The influence of pressure waves in tidal gravity records

For the reduction of atmospheric effects, observed gravity has initially been corrected by using the computed barometric admittance k of the in situ measured pressure, expressed in nms−2/hPa units and estimated by least squares method. However, the local pressure changes alone cannot account for the atmospheric mass attraction and loading when the coherent pressure field exceeds a specific size, i.e., with increasing periodicities. To overcome this difficulty, it is necessary to compute the total atmospheric effect at each station using the global pressure field. However, the direct subtraction of the total gravity effect, provided by the models of pressure correction, is not yet satisfactory for S2 and other tidal components, such as K2 and P1, which include solar heating pressure tides. This paper identifies the origin of the problem and presents strategies to obtain a satisfactory solution.First, we set up a difference vector between the tidal factors of M2 and S2 after correction of the pressure and ocean tides effects. This vector, hereafter denoted as RES, presents the advantage of being practically insensitive to calibration errors. The minimum discrepancy between the tidal parameters of M2 and S2 corresponds to the minimum of the RES vector norm d.Secondly we adopt the hybrid pressure correction method, separating the local and the global pressure contribution of the models and replacing the local contribution by the pressure measured at the station multiplied by an admittance kATM.We tested this procedure on 8 stations from the IGETS superconducting gravimeters network (former GGP network). For stations at an altitude lower than 1000 m, the value of dopt is always smaller than 0.0005. The discrepancy between the tidal parameters of the M2 and S2 waves is always lower than 0.05% on the amplitude factors and 0.025° on the phases. For these stations, a correlation exists between the altitude and the value kopt. The results at the three Central European stations Conrad, Pecny and Vienna are in excellent agreement (0.05%) with the DDW99NH model for all the main tidal waves.

Analysis on Norms of Word Embedding and Hidden Vectors in Neural Conversational Model Based on Encoder-Decoder RNN

Analysis on Norms of Word Embedding and Hidden Vectors in Neural Conversational Model Based on Encoder-Decoder RNN

Extendable orthogonal sets of integral vectors

Motivated by a model in quantum computation, we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the quaternions and other hypercomplex numbers.

Sequential multi-view subspace clustering

Self-representation based subspace learning has shown its effectiveness in many applications, but most existing methods do not consider the difference between different views. As a result, the learned self-representation matrix cannot well characterize the clustering structure. Moreover, some methods involve an undesired weighted vector of the tensor nuclear norm, which reduces the flexibility of the algorithm in practical applications. To handle these problems, we present a tensorized multi-view subspace clustering. Specifically, our method employs matrix factorization and decomposes the self-representation matrix to orthogonal projection matrix and affinity matrix. We also add ℓ1,2-norm regularization on affinity representation to characterize the cluster structure. Moreover, the proposed method uses weighted tensor Schatten p-norm to explore higher-order structure and complementary information embedded in multi-view data, which can allocate the ideal weight for each view automatically without additional weight and penalty parameters. We apply the adaptive loss function to the model to maintain the robustness to outliers and efficiently learn the data distribution. Extensive experimental results on different datasets reveal that our method is superior to other state-of-the-art multi-view subspace clustering methods.

The Robustness of Pencil Beam Synthesis Without Considering Sensor Uncertainties

Sensor uncertainties such as amplitude and phase errors may deteriorate the performance of array pattern synthesis. Therefore, several robust synthesis methods have been proposed. In this communication, we show that, for pencil beams, the direct synthesis method that does not take into account sensor errors also has robustness against sensor uncertainties. We first reformulate the robust synthesis method based on the worst-case performance criterion as a perturbation of the direct synthesis method. Then we prove that the direct and robust synthesis methods have the same solution up to a scaling factor. Thus, they produce the same array pattern (normalized). The condition for this result is that the error norms of steering vectors in different directions are equal to a small positive number, which is a reasonable assumption in practice. In addition, we derive a new optimization model for the robust synthesis method that can be solved much faster than the existing one.

Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

AbstractWe present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four‐moment theorem for the so‐called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost‐deterministic iteration count for the iterative methods in question. © 2022 Wiley Periodicals LLC.

Detecting bifurcations in dynamical systems with CROCKER plots.

Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems or they require carefully chosen input parameters and a significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology-a tool from topological data analysis-that utilizes Betti numbers and CROCKER plots. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter collection of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison with the Rösenstein algorithm for computing the maximum Lyapunov exponent.

Characterizations and Classification of Paraconvex Multimaps

Abstract Paraconvex multimaps are revisited in normed vector space setting. A parallel is provided with the studies conducted for real valued paraconvex functions on generalized convexities and monotonicities. Several characterizations are then obtained. The links with some generalized convexities for multimaps are examined and a first classification is achieved. In addition, two representation results for 2-paraconvex multimaps are given.

Data‐based iterative learning control for multiphase batch processes

AbstractIn order to cope with the problem of the robustness conditions dependence on system parameters information, this paper investigates a data‐based iteration learning control (ILC) for multiphase batch processes with different dimensions and system uncertainty. Firstly, by minimizing the residual between the actual subsystem output and the approximated subsystem output, a gradient‐type approximation law is designed to approximate the system lower triangular parameters matrix and initial state. Secondly, by minimizing the approximated tracking error between the desired trajectory and the approximated output, a data‐based ILC is constructed in an interactive mode with the approximation law. Finally, the boundedness of the approximation error of the real system parameters from the approximated parameters is derived by means of vector norm theory, while the unconditional robustness of the proposed data‐based ILC is proved. Simulation results illustrate the effectiveness and practicability of the proposed data‐based ILC.