Extreme Learning Machine (ELM), proposed by Huang et al., has been shown a promising learning algorithm for single-hidden layer feedforward neural networks (SLFNs). Nevertheless, because of the random choice of input weights and biases, the ELM algorithm sometimes makes the hidden layer output matrix H of SLFN not full column rank, which lowers the effectiveness of ELM. This paper discusses the effectiveness of ELM and proposes an improved algorithm called EELM that makes a proper selection of the input weights and bias before calculating the output weights, which ensures the full column rank of H in theory. This improves to some extend the learning rate (testing accuracy, prediction accuracy, learning time) and the robustness property of the networks. The experimental results based on both the benchmark function approximation and real-world problems including classification and regression applications show the good performances of EELM.
Aristolactam I (ALI) is an active component derived from some Traditional Chinese medicines (TCMs), and also the important metabolite of aristolochic acid. Long-term administration of medicine-containing ALI was reported to be related to aristolochic acid nephropathy (AAN), which was attributed to ALI-induced nephrotoxicity. However, the toxic mechanism of action involved is still unclear. Recently, pathogenic ferroptosis mediated lipid peroxidation was demonstrated to cause kidney injury. Therefore, this study explored the role of ferroptosis induced by mitochondrial iron overload in ALI-induced nephrotoxicity, aiming to identify the possible toxic mechanism of ALI-induced chronic nephropathy. Our results showed that ALI inhibited HK-2 cell activity in a dose-dependent manner and significantly suppressed glutathione (GSH) levels, accompanying by significant increases in intracellular 4-hydroxynonenal (4-HNE) and intracellular iron ions. Moreover, the ALI-mediated cytotoxicity could be reversed by deferoxamine mesylate (DFO). Compared with other inhibitors, Ferrostatin-1 (Fer-1), a ferroptosis inhibitor, obviously alleviated ALI-induced cytotoxicity. Furthermore, we have shown that ALI could remarkably increase the levels of superoxide anion and ferrous ions in mitochondria, and induce mitochondrial damage and condensed mitochondrial membrane density, the morphological characteristics of ferroptosis, all of which could be reversed by DFO. Interestingly, ALI dose-dependently inhibited these protein contents of nuclear factor erythroid 2-related factor 2 (Nrf2), heme oxygenase-1 (HO-1), and glutathione peroxidase 4 (GPX4), which could be partly rescued by Tin-protoporphyrin IX (SnPP) and mitoTEMPO co-treatment. In conclusion, our results demonstrated that mitochondrial iron overload-mediated antioxidant system inhibition would assist ALI-induced ferroptosis in renal tubular epithelial cells, and Nrf2-HO-1/GPX4 antioxidative system could be an important intervention target to prevent medicine containing ALI-induced nephropathy.
Spherical needlets are highly localized radial polynomials on the sphere S d ⊂ R d+1 , d ≥ 2, with centers at the nodes of a suitable cubature rule. The original semidiscrete spherical needlet approximation of Narcowich, Petrushev and Ward is not computable, in that the needlet coefficients depend on inner product integrals. In this work we approximate these integrals by a second quadrature rule with an appropriate degree of precision, to construct a fully discrete needlet approximation. We prove that the resulting approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace series partial sum with inner products replaced by appropriate cubature sums. It follows that the L p -error of discrete needlet approximation of order J for 1 ≤ p ≤ ∞ and s > d/p has for a function f in the Sobolev space W s p (S d ) the optimal rate of convergence in the sense of optimal recovery, namely O 2 −Js . Moreover, this is achieved with a filter function that is of smoothness, in contrast to the usually assumed C ∞ . A numerical experiment for a class of functions in known Sobolev smoothness classes gives L 2 errors for the fully discrete needlet approximation that are almost identical to those for the original semidiscrete needlet approximation. Another experiment uses needlets over the whole sphere for the lower levels together with high-level needlets with centers restricted to a local region. The resulting errors are reduced in the local region away from the boundary, indicating that local refinement in special regions is a promising strategy.Proof. By Theorem 3.10, the approximation by the semidiscrete needlets V need J (f ) is equivalent to that by filtered approximation V 2 J−1 ,H (f ). Then the definition (8) of V need J (f ) and (29) of Theorem 3.6 together with Theorem 3.10 giveTheorem 3.11 and Lemma 3.5 imply a rate of convergence of the approximation error of V need J (f ) for f in a Sobolev space, as follows.7 2 (S 2 ). The function f k has limited smoothness at the centers z i and at the boundary of each cap with center z i . These features make f k relatively difficult to approximate in these regions, especially for small k.L 2 approximation error. We show the L 2 errors when using V need J and by V need J,N . For V need J,N (f ) we compute its L 2 error by discretizing the squared L 2 -norm by a quadrature rule. We cannot compute
Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L 2 pMq in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S 2 as well as numerical examples are given.Keywords: tight framelets, affine system, compact Riemannian manifold, quadrature rule, filter bank, FFT, fast spherical harmonic transform, Laplace-Beltrami operator, unitary extension principle 2010 MSC: 42C15, 42C40, 42B05, 41A55, 57N99, 58C35, 94A12, 94C15, 93C55, 93C95 Introduction and motivationIn the era of information technologies, the rapid development of modern high-tech devices, for example, a super computer, PC, smart phone, wearable and VR/AR device, is driven internally by Moore's Law [55] which contributes to the exponential growth of the computational power, while externally stimulated by the tremendous need of both the public and individual parties in processing massive data from finance, economy, geology, bio-information, cosmology, medical sciences and so on. It has been noticed that Moore's Law is slowing down due to the constrains of the physical law [19] but the volume of data is dramatically increasing. Dealing with Big Data is becoming a crucial part of an individual person, party, government and country.Real-world data often inherit high-dimensionality such as data from a surveillance system, seismology, climatology. High-dimensional data are typically concentrated on a low-dimensional manifold [60,67], for instance, the sphere in remote sensing and CMB data [6], more complex surfaces in brain imaging [68], and discrete graph data from social and traffic networks [61]. Analysis and learning tools on manifolds hence play an increasingly important role in machine learning and statistics.The key to successful manifold learning lies in that data on a manifold may exhibit high complexity on one hand while they are highly sparse at a certain domain via an appropriate multiscale representation system on the other hand. Sparsity within such representations, stemming from computational harmonic analysis, enables efficient analysis and processing of high-dimensional and massive data.Multiresolution analysis in general are designed for data in the Euclidean space R d , d ě 1, for example, a signal in R, an image in R 2 and a video in R 3 . Multiscale representation systems in R d including wavelets, framelets, curvelets, shearlets, etc., which are capable of capturing the sparsity of data, have been well-developed and widely used, see e.g. [7,11,14,17,21,49,50]. The core of the...
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