On the Rates of Convergence From Surrogate Risk Minimizers to the Bayes Optimal Classifier

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

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Distributed learning with indefinite kernels

Analysis and Applications ◽

10.1142/s021953051850032x ◽

2019 ◽

Vol 17 (06) ◽

pp. 947-975 ◽

Cited By ~ 2

Author(s):

Lei Shi

Keyword(s):

Large Scale ◽

Substantial Reduction ◽

Computation Time ◽

Distributed Learning ◽

Rates Of Convergence ◽

Regression Problem ◽

Data Set ◽

Regularization Scheme ◽

Original Algorithm ◽

Indefinite Kernel

We investigate the distributed learning with coefficient-based regularization scheme under the framework of kernel regression methods. Compared with the classical kernel ridge regression (KRR), the algorithm under consideration does not require the kernel function to be positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods. The distributed learning approach partitions a massive data set into several disjoint data subsets, and then produces a global estimator by taking an average of the local estimator on each data subset. Easy exercisable partitions and performing algorithm on each subset in parallel lead to a substantial reduction in computation time versus the standard approach of performing the original algorithm on the entire samples. We establish the first mini-max optimal rates of convergence for distributed coefficient-based regularization scheme with indefinite kernels. We thus demonstrate that compared with distributed KRR, the concerned algorithm is more flexible and effective in regression problem for large-scale data sets.

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$$L_2$$-norm sampling discretization and recovery of functions from RKHS with finite trace

Sampling Theory, Signal Processing, and Data Analysis ◽

10.1007/s43670-021-00013-3 ◽

2021 ◽

Vol 19 (2) ◽

Author(s):

Moritz Moeller ◽

Tino Ullrich

Keyword(s):

Random Function ◽

Additional Assumption ◽

Main Tool ◽

Singular Values ◽

Rates Of Convergence ◽

Concentration Inequality ◽

Worst Case ◽

Mercer Kernels ◽

Finite Trace ◽

Complex Valued

AbstractIn this paper we study $$L_2$$ L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $$D \subset \mathbb {R}^d$$ D ⊂ R d based on random function samples. We only assume the finite trace of the kernel (Hilbert–Schmidt embedding into $$L_2$$ L 2 ) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in n, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.

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Classification of standing and sitting phases based on in-socket piezoelectric sensors in a transfemoral amputee

Biomedical Engineering / Biomedizinische Technik ◽

10.1515/bmt-2018-0249 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Tawfik Yahya ◽

Nur Azah Hamzaid ◽

Sadeeq Ali ◽

Farahiyah Jasni ◽

Hanie Nadia Shasmin

Keyword(s):

Time Domain ◽

Classification Accuracy ◽

Sensory System ◽

Metabolic Energy ◽

Support Vector ◽

Data Set ◽

Transfemoral Amputee ◽

Sit To Stand ◽

Optimal Classifier ◽

The Impact

AbstractA transfemoral prosthesis is required to assist amputees to perform the activity of daily living (ADL). The passive prosthesis has some drawbacks such as utilization of high metabolic energy. In contrast, the active prosthesis consumes less metabolic energy and offers better performance. However, the recent active prosthesis uses surface electromyography as its sensory system which has weak signals with microvolt-level intensity and requires a lot of computation to extract features. This paper focuses on recognizing different phases of sitting and standing of a transfemoral amputee using in-socket piezoelectric-based sensors. 15 piezoelectric film sensors were embedded in the inner socket wall adjacent to the most active regions of the agonist and antagonist knee extensor and flexor muscles, i. e. region with the highest level of muscle contractions of the quadriceps and hamstring. A male transfemoral amputee wore the instrumented socket and was instructed to perform several sitting and standing phases using an armless chair. Data was collected from the 15 embedded sensors and went through signal conditioning circuits. The overlapping analysis window technique was used to segment the data using different window lengths. Fifteen time-domain and frequency-domain features were extracted and new feature sets were obtained based on the feature performance. Eight of the common pattern recognition multiclass classifiers were evaluated and compared. Regression analysis was used to investigate the impact of the number of features and the window lengths on the classifiers’ accuracies, and Analysis of Variance (ANOVA) was used to test significant differences in the classifiers’ performances. The classification accuracy was calculated using k-fold cross-validation method, and 20% of the data set was held out for testing the optimal classifier. The results showed that the feature set (FS-5) consisting of the root mean square (RMS) and the number of peaks (NP) achieved the highest classification accuracy in five classifiers. Support vector machine (SVM) with cubic kernel proved to be the optimal classifier, and it achieved a classification accuracy of 98.33 % using the test data set. Obtaining high classification accuracy using only two time-domain features would significantly reduce the processing time of controlling a prosthesis and eliminate substantial delay. The proposed in-socket sensors used to detect sit-to-stand and stand-to-sit movements could be further integrated with an active knee joint actuation system to produce powered assistance during energy-demanding activities such as sit-to-stand and stair climbing. In future, the system could also be used to accurately predict the intended movement based on their residual limb’s muscle and mechanical behaviour as detected by the in-socket sensory system.

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A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

IMA Journal of Numerical Analysis ◽

10.1093/imanum/draa035 ◽

2020 ◽

Author(s):

Sergio Caucao ◽

Ivan Yotov

Keyword(s):

Banach Space ◽

Time Discretization ◽

Monotone Operators ◽

Rates Of Convergence ◽

Mixed Formulation ◽

Model Parameters ◽

Weak Formulation ◽

Finite Element Approximations ◽

Backward Euler ◽

Discrete Finite Element

Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

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Multivariate linear time series models

Advances in Applied Probability ◽

10.2307/1427286 ◽

1984 ◽

Vol 16 (3) ◽

pp. 492-561 ◽

Cited By ~ 107

Author(s):

E. J. Hannan ◽

L. Kavalieris

Keyword(s):

Limit Theorems ◽

Analysis Of Algorithms ◽

Linear Time ◽

Asymptotic Properties ◽

Real Data ◽

Rates Of Convergence ◽

Structure Theory ◽

Time Series Models ◽

Transfer Model ◽

Rational Transfer

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

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