Input space regularization stabilizes pre-images for kernel PCA de-noising

Abrahamsen, Trine Julie; Hansen, Lars Kai

doi:10.1109/mlsp.2009.5306191

Cited by 15 publications

(12 citation statements)

References 13 publications

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“…Since the map Γ in (16) has no inverse, finding Υ * in X from τ * defines an ill-posed problem [70,71,[76][77][78]. In this setting, the determination of Υ * from τ * , requires the use of a P projector on τ 0 (20) in H, which yields a P (τ 0 ) element in H. If τ * lies in (or close to) the span of {Γ (Υ i )}, where Υ i is an i-th training data, Υ i ∈ X , from a training set S X of N training data,…”

Section: Connectivity Of the Objective Function In The Target Statementioning

confidence: 99%

Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers

Gyöngyösi

2020

Sci Rep

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A computational problem fed into a gate-model quantum computer identifies an objective function with a particular computational pathway (objective function connectivity). The solution of the computational problem involves identifying a target objective function value that is the subject to be reached. A bottleneck in a gate-model quantum computer is the requirement of several rounds of quantum state preparations, high-cost run sequences, and multiple rounds of measurements to determine a target (optimal) state of the quantum computer that achieves the target objective function value. Here, we define a method for optimal quantum state determination and computational path evaluation for gate-model quantum computers. We prove a state determination method that finds a target system state for a quantum computer at a given target objective function value. The computational pathway evaluation procedure sets the connectivity of the objective function in the target system state on a fixed hardware architecture of the quantum computer. The proposed solution evolves the target system state without requiring the preparation of intermediate states between the initial and target states of the quantum computer. Our method avoids high-cost system state preparations and expensive running procedures and measurement apparatuses in gate-model quantum computers. The results are convenient for gate-model quantum computations and the near-term quantum devices of the quantum Internet.A computational problem fed into a quantum computer defines an objective function with a particular connectivity (computational pathway) [10]. The solution of this computational problem in the quantum computer involves identifying an objective function with a target value that is subject to be reached. To achieve the target objective function value, the quantum computer must reach a particular system state such that the gate parameters of the unitary operations satisfy the target value. These optimal gate parameter values of the unitary operations of the quantum computer identify the optimal state of the quantum computer. This optimal system state is referred to as the target system state of the quantum computer. Finding the target system state involves multiple measurement rounds and iterations, with high-cost system state preparations 1 , quantum computations, and measurement procedures. Therefore, optimizing the determination procedure of the target system state is essential for gate-model quantum computers.Here, we define a method for state determination and computational path evaluation for gatemodel quantum computers. The aim of state determination is to find a target system state for a quantum computer such that the pre-determined target objective function value is reached. The aim of the computational path evaluation is to find the connectivity of the objective function in the target system state on the fixed hardware architecture [10] of the quantum computer. To resolve these issues, we define a framework that utilizes the theory of kernel methods ...

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Section: Connectivity Of the Objective Function In The Target Statementioning

confidence: 99%

Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers

Gyöngyösi

2020

Sci Rep

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“…We see that in kernel MAF analysis where we find one linear combination only, we need not regularize. Still we may wish to regularize and if so one of several possible alternative versions of the primal formulation is (20) which in the dual version becomes (21) which in turn kernelizes to (22) …”

Section: Regularization and Kernelizationmentioning

confidence: 99%

“…For the kernel MAF/MNF variates to be used in for example denoising we must look into the so-called preimage problem, [20]- [22]. This deals with the complicated problem of mapping back from the feature space defined implicitly by the kernel function to the original variable space.…”

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confidence: 99%

Kernel Maximum Autocorrelation Factor and Minimum Noise Fraction Transformations

Nielsen

2011

IEEE Trans. on Image Process.

124

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Abstract-This paper introduces kernel versions of maximum autocorrelation factor (MAF) analysis and minimum noise fraction (MNF) analysis. The kernel versions are based upon a dual formulation also termed Q-mode analysis in which the data enter into the analysis via inner products in the Gram matrix only. In the kernel version, the inner products of the original data are replaced by inner products between nonlinear mappings into higher dimensional feature space. Via kernel substitution also known as the kernel trick these inner products between the mappings are in turn replaced by a kernel function and all quantities needed in the analysis are expressed in terms of this kernel function. This means that we need not know the nonlinear mappings explicitly.

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“…This helps to reduce the computational complexity of forming the decision boundary while maintaining high classification accuracy. Schölkopf et al 7,8,9,10,11 propose a kernel-based principle component analysis (PCA) to build the pre-images in feature space, which improves the performance of image denoising. Lillholm and Nielsen discuss the relation between an image and its features, 12, 13 define metamery classes of images and the corresponding information content of a canonical least informative representative of the class.…”

Section: Introductionmentioning

confidence: 99%

Towards exaggerated emphysema stereotypes

et al. 2012

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Classification is widely used in the context of medical image analysis and in order to illustrate the mechanism of a classifier, we introduce the notion of an exaggerated image stereotype based on training data and trained classifier. The stereotype of some image class of interest should emphasize/exaggerate the characteristic patterns in an image class and visualize the information the employed classifier relies on. This is useful for gaining insight into the classification and serves for comparison with the biological models of disease.In this work, we build exaggerated image stereotypes by optimizing an objective function which consists of a discriminative term based on the classification accuracy, and a generative term based on the class distributions. A gradient descent method based on iterated conditional modes (ICM) is employed for optimization. We use this idea with Fisher's linear discriminant rule and assume a multivariate normal distribution for samples within a class. The proposed framework is applied to computed tomography (CT) images of lung tissue with emphysema. The synthesized stereotypes illustrate the exaggerated patterns of lung tissue with emphysema, which is underpinned by three different quantitative evaluation methods.

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Input space regularization stabilizes pre-images for kernel PCA de-noising

Cited by 15 publications

References 13 publications

Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers

Quantum State Optimization and Computational Pathway Evaluation for Gate-Model Quantum Computers

Kernel Maximum Autocorrelation Factor and Minimum Noise Fraction Transformations

Towards exaggerated emphysema stereotypes

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