Linear Hyperplane Research Articles

Investigations on interpretable nuclear-grade hydraulic damper monitoring and diagnosis from multi-source high-dimensional profile data

Condition monitoring and fault diagnosis is important to ensure smooth machine operation, in which the use of multi-source high-dimensional profile data composed of multiple physical signals collected by multiple sensors for machine health monitoring and fault diagnosis has not been fully explored. In this paper, interpretable monitoring and diagnostic methods are proposed to analyze multi-source high-dimensional profile data and identify equipment operational status. Firstly, polynomials of segmenting profiles as well as a Lasso model are established to obtain sparse features. Then, for offline data with historical fault labels, an optimal linear hyperplane classifier is constructed to interpretably study the impact of different features on a decision. For online data with only normal samples, constructing an optimal hypersphere enables the measurement of operational status. Large-scale nuclear-grade hydraulic damper profile data are used to demonstrate how the proposed interpretable models work and show their physical interpretation for monitoring and fault diagnosis.

Ensemble One-Class Support Vector Machine for Sea Surface Target Detection Based on k-Means Clustering

Sea surface target detection is a key stage in a typical target detection system and directly influences the performance of the whole system. As an effective discriminator, the one-class support vector machine (OCSVM) has been widely used in target detection. In OCSCM, training samples are first mapped to the hypersphere in the kernel space with the Gaussian kernel function, and then, a linear classification hyperplane is constructed in each cluster to separate target samples from other classes of samples. However, when the distribution of the original data is complex, the transformed data in the kernel space may be nonlinearly separable. In this situation, OCSVM cannot classify the data correctly, because only a linear hyperplane is constructed in the kernel space. To solve this problem, a novel one-class classification algorithm, referred to as ensemble one-class support vector machine (En-OCSVM), is proposed in this paper. En-OCSVM is a hybrid model based on k-means clustering and OCSVM. In En-OCSVM, training samples are clustered in the kernel space with the k-means clustering algorithm, while a linear decision hyperplane is constructed in each cluster. With the combination of multiple linear classification hyperplanes, a complex nonlinear classification boundary can be achieved in the kernel space. Moreover, the joint optimization of the k-means clustering model and OCSVM model is realized in the proposed method, which ensures the linear separability of each cluster. The experimental results based on the synthetic dataset, benchmark datasets, IPIX datasets, and SAR real data demonstrate the better performance of our method over other related methods.

Analysis of the integration path of contemporary costume culture and clothing design innovation in the context of big data

Abstract This paper uses feature fusion methods to classify the characteristics of dress culture, focusing on the application scope of pixel-level, feature-level, and decision-level. A system for recognizing dress culture classification is constructed by combining machine vision technology, and the support vector machine algorithm is used to solve the nonlinear mapping problem in two-dimensional space. The range of values of support vectors is established by using decision functions to construct a linear hyperplane. The results show that integrating the color system of costume culture and clothing design can lead to new design paths. 70% of the clothing design works have natural color matching, and 60% have color matching that can alter body shape. The research in this paper provides a new reference path for clothing design innovation.

On the triviality of a family of linear hyperplanes

Let k be a field, m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form x1r1⋯xmrmy=F(x1,…,xm,z,t), A the coordinate ring of V and G=X1r1⋯XmrmY−F(X1,…,Xm,Z,T). In [13], the second author had studied the case m=1 and had obtained several necessary and sufficient conditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1,Y,Z,T].In this paper, we study the general higher-dimensional variety V for each m⩾1 and obtain analogous conditions for V to be isomorphic to Am+2 and G to be a coordinate in k[X1,…,Xm,Y,Z,T], under a certain hypothesis on F. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds.We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each d⩾3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic.

Prime power residue and linear coverings of vector space over [formula omitted

Let q be an odd prime and B={bj}j=1l be a finite set of nonzero integers that does not contain a perfect qth power. We show that B has a qth power modulo every prime p≠q and not dividing ∏b∈Bb if and only if B corresponds to a linear hyperplane covering of Fqk. Here, k is the number of distinct prime factors of the q-free part of elements of B. Consequently: (i) a set B⊂Z∖{0} with cardinality less than q+1 cannot have a qth power modulo almost every prime unless it contains a perfect qth power and (ii) For every set B={bj}j=1l⊂Z∖{0} and for every (cj)j=1l∈(Fq∖{0})l the set B contains a qth power modulo every prime p≠q and not dividing ∏j=1l if and only if the set {bjcj}j=1l does so.

Global Solar Radiation Modelling using an Artificial Neural Network for Kazaure, Jigawa State, Nigeria

This research presents an algorithm based on Artificial Neural Networks (ANN), for estimating monthly mean daily and hourly values of solar global radiation. To effectively investigate solar energy consumption and estimate solar renewable energy resources, the Hourly Global Solar Radiation measurements are necessary. In order to predict monthly average daily global sun irradiance on a horizontal area of Kazaure- Nigeria, this study creates a model utilizing ANN to solve the problem of solar energy distribution. Five empirical correlations are developed using the data from 42 months to aid in the prediction of the solar energy distribution pattern. The software is constructed around the Multilayer Perceptron under categorized tabs, with Multilayer perception in neural network Toolbox in MATLAB 9.7 version as a feed forward ANN that maps sets of input data into a set of suitable output. It differs from conventional linear perception by employing three or more layers of neurons (nodes) with nonlinear activation functions. It is also more effective than perceptrons in identifying input that is not linearly separable by a linear hyper-plane. Results obtained utilizing the suggested structure reveals good agreement between the calculated and measured levels of global solar irradiation. The ANN model is shown to be superior when compared to empirical models, due to negligible noise margin.

Artificial Intelligence-Assisted Peer Review in Radiation Oncology

<h3>Purpose/Objective(s)</h3> Peer review is reflective of the entire patient treatment process with clinical information, therapeutic parameters and potential treatment variations. Our hypothesis is that using artificial intelligence (AI), we can enhance the efficacy of the peer review process by screening cases with a potential for treatment interruption. <h3>Materials/Methods</h3> From 3,899 radiotherapy patients (7,168 plans) treated from 2014-2021 in our department, 36 features of clinical and therapeutic parameters were used as input for two AI models: multivariable least absolute shrinkage and selection operator (LASSO) logistic regression model and pattern recognition feed forward neural network (NN). LASSO is a shrinkage regularization method that assigns coefficients to significantly useful features. NN passes input features through multiple hidden nodes to assign weights and biases for feature selection. Each method results in continuous probability of treatment interruption. Performance metrics of accuracy, sensitivity and specificity were calculated for evaluations. <h3>Results</h3> Overall, 8.1% of all cases had treatment interruptions, most commonly in head and neck (18.8%) compared to other sites (19% vs 7-9%, p<0.01). For LASSO model, testing set sensitivity, specificity and accuracy ranged from 65-89%, 31-53% and 35-54%, respectively, with higher sensitivity than specificity. Spine/Extremity and Brain sites had the highest accuracy (54%). Higher sensitivity than specificity indicated that the model was more able to predict true positives. For NN model, testing set sensitivity, specificity and accuracy ranged from 23-62%, 66-90% and 64-86%, respectively. Higher specificity than sensitivity was observed. The Brain site had highest accuracy (86%). Accuracy of NN was higher than LASSO for all treatment sites (68% vs. 50%), which is particularly true for the brain (86% vs. 54%) and the pelvis/prostate (77% vs. 35%). This result indicates that the linear hyperplane of LASSO was insufficient to classify the underlying complex clinical dataset accurately. Including more input features is expected to improve the accuracy for both models. <h3>Conclusion</h3> Our results provide proof-of-concept that AI can be used as a screening tool to aid the peer review process. It can help to predict treatment interruptions, which include replanning or treatment cessation. Early identification of patients at risk of radiotherapy interruptions using this method may potentially translate into higher treatment completion rates. The study is being continued at our institution to further explore the capabilities of AI in the peer review process.

An Optimal Allocation Method of Distributed PV and Energy Storage Considering Moderate Curtailment Measure

Increasing distributed generations (DGs) are integrated into the distribution network. The risk of not satisfying operation constraints caused by the uncertainty of renewable energy output is increasing. The energy storage (ES) could stabilize the fluctuation of renewable energy generation output. Therefore, it can promote the consumption of renewable energy. A distributed photovoltaic (PV) and ES optimal allocation method based on the security region is proposed. Firstly, a bi-level optimal allocation model of PV and ES is established. The outer layer is a nonlinear optimization model, taking the maximum power supply benefit as the objective function. The inner layer is a day-ahead economic dispatching model. Then, a quick model solving method based on the steady-state security region is proposed. An initial allocation scheme of PV and ES is determined with the redundancy capacity. In addition, the linear hyperplane coefficient of the security region is used to convert the nonlinear day-ahead economic dispatching model into a linear one. Finally, the proposed method is used to analyze the improved IEEE 33-node system. It is found that a moderate curtailment measure of distributed PV peak output and the allocation of energy storage have a significant effect on the power supply benefit of the distribution system. The optimal quota capacity of DG exceeds the sum of the maximum load and the branch capacity. In addition, the optimal allocation scheme is closely related to the cost and technical parameters of distributed PV and ES. Dynamic allocation schemes should be formulated for distribution network.

Global Translation of Classification Models

The widespread and growing usage of machine learning models, particularly for critical areas such as law, predicate the need for global interpretability. Models that cannot be audited are vulnerable to biases inherited from the datasets that were used to develop them. Moreover, locally interpretable models are vulnerable to adversarial attacks. To address this issue, the present paper proposes a new methodology that can translate any existing machine learning model into a globally interpretable one. MTRE-PAN is a hybrid SVM-decision tree architecture that leverages the interpretability of linear hyperplanes by creating a set of polygons that delimit the decision boundaries of the target model. Moreover, the present paper introduces two new metrics: certain and boundary model parities. These metrics can be used to accurately evaluate the performance of the interpretable model near the decision boundaries. These metrics are used to compare MTRE-PAN to a previously proposed interpretable architecture called TRE-PAN. As in the case of TRE-PAN, MTRE-PAN aims at providing global interpretability. The comparisons are performed over target models developed using three benchmark datasets: Abalone, Census and Diabetes data. The results show that MTRE-PAN generates interpretable models that have a lower number of leaves and a higher agreement with the target models, especially around the most important regions in the feature space, namely the decision boundaries.

Trims and extensions of quadratic APN functions

In this work, we study functions that can be obtained by restricting a vectorial Boolean function F :mathbb {F}_{2}^n rightarrow mathbb {F}_{2}^n to an affine hyperplane of dimension n-1 and then projecting the output to an n-1-dimensional space. We show that a multiset of 2 cdot (2^n-1)^2 EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on mathbb {F}_{2}^n. Further, for all of the known quadratic APN functions in dimension n < 10, we determine the restrictions that are also APN. Moreover, we construct 6368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function F :mathbb {F}_{2}^n rightarrow mathbb {F}_{2}^n with linearity of 2^{n-1} by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity 2^7 up to EA-equivalence.

A metric for quantifying nonlinearity in k-dimensional complex-valued functions

Modeling and simulation is a proven cost-efficient means for studying the behavioral dynamics of modern systems of systems. Our research is focused on evaluating the ability of neural networks to approximate multivariate, nonlinear, complex-valued functions. In order to evaluate the accuracy and performance of neural network approximations as a function of nonlinearity (NL), it is required to quantify the amount of NL present in the complex-valued function. In this paper, we introduce a metric for quantifying NL in multi-dimensional complex-valued functions. The metric is an extension of a real-valued NL metric into the k-dimensional complex domain. The metric is flexible as it uses discrete input–output data pairs instead of requiring closed-form continuous representations for calculating the NL of a function. The metric is calculated by generating a best-fit, least-squares solution (LSS) linear k-dimensional hyperplane for the function; calculating the L2 norm of the difference between the hyperplane and the function being evaluated; and scaling the result to yield a value between zero and one. The metric is easy to understand, generalizable to multiple dimensions, and has the added benefit that it does not require a closed-form continuous representation of the function being evaluated.

Patch-based Medical Image Segmentation using Matrix Product State Tensor Networks

Tensor networks are efficient factorisations of high dimensional tensors into network of lower order tensors. They have been most commonly used to model entanglement in quantum many-body systems and more recently are witnessing increased applications in supervised machine learning. In this work, we formulate image segmentation in a supervised setting with tensor networks. The key idea is to first lift the pixels in image patches to exponentially high dimensional feature spaces and using a linear decision hyper-plane to classify the input pixels into foreground and background classes. The high dimensional linear model itself is approximated using the matrix product state (MPS) tensor network. The MPS is weight-shared between the non-overlapping image patches resulting in our &lt;em&gt;strided tensor network&lt;/em&gt; model. The performance of the proposed model is evaluated on three three 2D- and one 3D- biomedical imaging datasets. The performance of the proposed tensor network segmentation model is compared with relevant baseline methods. In the 2D experiments, the tensor network model yeilds competitive performance compared to the baseline methods while being more resource efficient.

The Varchenko matrix for topoplane arrangements

A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements.

On the isometric conjecture of Banach

Let $V$ be a Banach space where for fixed $n$, $1<n<\dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for even $n$ and all $V$. In this paper we give a positive answer for real $V$ and odd $n$ of the form $n=4k+1$, with the possible exception of $n=133.$ Our proof relies on a new characterization of ellipsoids in ${\mathbb{R}}^n$, $n\geq 5$, as the only symmetric convex bodies all of whose linear hyperplane sections are linearly equivalent affine bodies of revolution.

An algebraic representation of Steiner triple systems of order 13

In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V=GF(5)13, with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V, which can also be characterized as the intersections of B with thirteen linear hyperplanes of V.

Universal Active Learning via Conditional Mutual Information Minimization

Modern machine learning systems require massive amounts of labeled training data in order to achieve high accuracy rates which is very expensive in terms of time and cost. Active learning is an approach which uses feedback to only label the most informative data points and significantly reduce the labeling effort. Many heuristics for selecting data points have been developed in recent years which are usually tailored to a specific task and a general unified framework is lacking. In this work, a new information theoretic criterion is proposed based on a minimax log-loss regret formulation of the active learning problem. First, a Redundancy Capacity theorem for active learning is derived along with an optimal learner. This leads to a new active learning criterion which naturally induces an exploration - exploitation trade-off in feature selection and generalizes previously proposed heuristic criteria. The new criterion is compared analytically and via empirical simulation to other commonly used information theoretic active learning criteria. Next, the linear hyper-plane hypotheses class with possibly asymmetric label noise is considered. The achievable performance for the proposed criterion is analyzed using a new low complexity greedy algorithm based on the Posterior Matching scheme for communication with feedback. It is shown that for general label noise and bounded feature distribution, the new information theoretic criterion decays exponentially fast to zero.

A bound for crystallographic arrangements

A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by Heckenberger and the author. However, this classification is based on two computer proofs checking millions of cases. In the present paper, we prove without using a computer that, up to equivalence, there are only finitely many irreducible crystallographic arrangements in each rank greater than two.

Decision regions and decision boundaries of generalized K mean algorithm based on various norm criteria

This paper derives the decision regions and the decision boundaries of the generalized K mean algorithms based on the L1 norm criterion, the L2 norm criterion and the L∞ norm criterion. The decision boundaries of these three generalized K mean algorithms are all linear hyperplanes. However, the total numbers of the decision boundaries of the generalized K mean algorithms based on both the L1 norm criterion and the L∞ norm criterion are more than that based on the L2 norm criterion. On the other hand, the decision regions of the generalized K mean algorithm based on the L2 norm criterion are convex while that based on both the L1 norm criterion and the L∞ norm criterion are in general nonconvex. The computer numerical simulations on a toy example demonstrate the above phenomena. Besides, two examples are illustrated. The first example on the patent image retrieval system shows that the recognition accuracies of using the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 58.25%, 61% and 58.75%, respectively. The second example on the electromyogram based Parkinson’s disease detection system shows that the recognition accuracies of using the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 60%, 60% and 67%, respectively, if the signals are classified directly in the time domain. On the other hand, the recognition accuracies of the generalized K mean algorithms based on the L2 norm criterion, the L1 norm criterion and the L∞ norm criterion are 60%, 87% and 60%, respectively, if the signals are classified directly in the discrete cosine transform domain. The improvements are due to the nonconvexity of the decision regions of the generalized K mean algorithm based on the L1 norm criterion and the L∞ norm criterion.

A new approach to weak convergence of random cones and polytopes

AbstractA new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated bynindependent and uniformly distributed random linear hyperplanes in$\mathbb {R}^{d+1}$, weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in$\mathbb {R}^d$, as$n\to \infty $.

A data-driven reduced-order model of nonlinear processes based on diffusion maps and artificial neural networks

A framework, is presented, for deriving accurate predictive models of nonlinear processes based on data, either experimental or computational or a combination thereof. The derived model is low-dimensional, hence appropriate to be used as an efficient Digital Twin of the process at hand. Diffusion Maps, a nonlinear dimensionality reduction algorithm is implemented in order to identify an economical description of the data. This can be thought of as a nonlinear counterpart of Principal Component Analysis (PCA), that instead of passing linear hyperplanes through the data, identifies a parametrization of the low-dimensional manifold on which the data are sampled from. As a result, in nonlinear processes, such as the tubular reactor model used here as a case study, where multiple states may be observed, experimentally or in simulations, for the same set of parameter values, the proposed approach is able to identify a low-dimensional, yet accurate representation of the data. In order to make out-of-sample predictions, appropriately trained artificial neural networks are implemented. Both the training and application of the neural networks requires limited computational effort, because of the dimensionality reduction step that precedes.