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Manfred K. Warmuth - One of the best experts on this subject based on the ideXlab platform.
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EuroCOLT - Sample Compression, Learnability, and the Vapnik-Chervonenkis Dimension
Lecture Notes in Computer Science, 1997Co-Authors: Manfred K. WarmuthAbstract:Within the framework of pac-learning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of sizek for a concept class\(C \subseteq 2^X \) consists of a compression Function and a Reconstruction Function. The compression Function receives a finite sample set consistent with some concept inC and chooses a subset ofk examples as the compression set. The Reconstruction Function forms a hypothesis onX from a compression set ofk examples. For any sample set of a concept inC the compression set produced by the compression Function must lead to a hypothesis consistent with the whole original sample set when it is fed to the Reconstruction Function. We demonstrate that the existence of a sample compression scheme of fixed-size for a classC is sufficient to ensure that the classC is pac-learnable.
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Sample compression, learnability, and the Vapnik-Chervonenkis dimension
Machine Learning, 1995Co-Authors: Sally Floyd, Manfred K. WarmuthAbstract:Within the framework of pac-learning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class $$C \subseteq 2^X $$ consists of a compression Function and a Reconstruction Function. The compression Function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The Reconstruction Function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression Function must lead to a hypothesis consistent with the whole original sample set when it is fed to the Reconstruction Function. We demonstrate that the existence of a sample compression scheme of fixed-size for a class C is sufficient to ensure that the class C is pac-learnable. Previous work has shown that a class is pac-learnable if and only if the Vapnik-Chervonenkis (VC) dimension of the class is finite. In the second half of this paper we explore the relationship between sample compression schemes and the VC dimension. We define maximum and maximal classes of VC dimension d . For every maximum class of VC dimension d , there is a sample compression scheme of size d , and for sufficiently-large maximum classes there is no sample compression scheme of size less than d . We discuss briefly classes of VC dimension d that are maximal but not maximum. It is an open question whether every class of VC dimension d has a sample compression scheme of size O( d ).
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Sample Compression, Learnability, and the Vapnik-Chervonenkis Dimension
Machine Learning, 1995Co-Authors: Sally Floyd, Manfred K. WarmuthAbstract:Within the framework of pac-learning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ⊆ 2X consists of a compression Function and a Reconstruction Function. The compression Function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The Reconstruction Function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression Function must lead to a hypothesis consistent with the whole original sample set when it is fed to the Reconstruction Function. We demonstrate that the existence of a sample compression scheme of fixed-size for a class C is sufficient to ensure that the class C is pac-learnable. Previous work has shown that a class is pac-learnable if and only if the Vapnik-Chervonenkis (VC) dimension of the class is finite. In the second half of this paper we explore the relationship between sample compression schemes and the VC dimension. We define maximum and maximal classes of VC dimension d. For every maximum class of VC dimension d, there is a sample compression scheme of size d, and for sufficiently-large maximum classes there is no sample compression scheme of size less than d. We discuss briefly classes of VC dimension d that are maximal but not maximum. It is an open question whether every class of VC dimension d has a sample compression scheme of size O(d).
Sally Floyd - One of the best experts on this subject based on the ideXlab platform.
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Sample compression, learnability, and the Vapnik-Chervonenkis dimension
Machine Learning, 1995Co-Authors: Sally Floyd, Manfred K. WarmuthAbstract:Within the framework of pac-learning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class $$C \subseteq 2^X $$ consists of a compression Function and a Reconstruction Function. The compression Function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The Reconstruction Function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression Function must lead to a hypothesis consistent with the whole original sample set when it is fed to the Reconstruction Function. We demonstrate that the existence of a sample compression scheme of fixed-size for a class C is sufficient to ensure that the class C is pac-learnable. Previous work has shown that a class is pac-learnable if and only if the Vapnik-Chervonenkis (VC) dimension of the class is finite. In the second half of this paper we explore the relationship between sample compression schemes and the VC dimension. We define maximum and maximal classes of VC dimension d . For every maximum class of VC dimension d , there is a sample compression scheme of size d , and for sufficiently-large maximum classes there is no sample compression scheme of size less than d . We discuss briefly classes of VC dimension d that are maximal but not maximum. It is an open question whether every class of VC dimension d has a sample compression scheme of size O( d ).
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Sample Compression, Learnability, and the Vapnik-Chervonenkis Dimension
Machine Learning, 1995Co-Authors: Sally Floyd, Manfred K. WarmuthAbstract:Within the framework of pac-learning, we explore the learnability of concepts from samples using the paradigm of sample compression schemes. A sample compression scheme of size k for a concept class C ⊆ 2X consists of a compression Function and a Reconstruction Function. The compression Function receives a finite sample set consistent with some concept in C and chooses a subset of k examples as the compression set. The Reconstruction Function forms a hypothesis on X from a compression set of k examples. For any sample set of a concept in C the compression set produced by the compression Function must lead to a hypothesis consistent with the whole original sample set when it is fed to the Reconstruction Function. We demonstrate that the existence of a sample compression scheme of fixed-size for a class C is sufficient to ensure that the class C is pac-learnable. Previous work has shown that a class is pac-learnable if and only if the Vapnik-Chervonenkis (VC) dimension of the class is finite. In the second half of this paper we explore the relationship between sample compression schemes and the VC dimension. We define maximum and maximal classes of VC dimension d. For every maximum class of VC dimension d, there is a sample compression scheme of size d, and for sufficiently-large maximum classes there is no sample compression scheme of size less than d. We discuss briefly classes of VC dimension d that are maximal but not maximum. It is an open question whether every class of VC dimension d has a sample compression scheme of size O(d).
Feng Xiao - One of the best experts on this subject based on the ideXlab platform.
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Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm
Computers & Fluids, 2020Co-Authors: Xi Deng, Bin Xie, Yuya Shimizu, Feng XiaoAbstract:Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called PnTm - BVD (polynomial of n-degree and THINC Function of m-level Reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness as the Reconstruction candidates. The final Reconstruction Function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new Reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n + 1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new Reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.
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Implicit large eddy simulation of compressible turbulence flow with PnTm − BVD scheme
Applied Mathematical Modelling, 2020Co-Authors: Xi Deng, Feng Xiao, Zhen-hua Jiang, Chao YanAbstract:Implicit large eddy simulation (ILES) of compressible turbulence with shock capturing schemes requires wide investigations and numerical experiments. In this study, a newly proposed PnTm - BVD (polynomial of n-degree and THINC Function of m-level Reconstruction based on BVD algorithm) shock capturing scheme is introduced to simulate compressible turbulence flow with ILES. The new scheme is designed by employing high-order linear-weight polynomials and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness as the Reconstruction candidates. The final Reconstruction Function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. Numerical tests involving shock waves and broadband turbulence are conducted in comparison with WENO (Weighted Essentially Non-oscillatory) schemes which are widely used in ILES. The results demonstrate performing ILES with PnTm- BVD scheme is able to obtain higher resolution and more faithful results than WENO does. Importantly, the superiority of PnTm-BVD becomes more notable in high wave-number region. Thus this paper provides and verifies a new scheme which is promising in providing high-resolution results for real-case ILES of compressible turbulence flow.
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A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm
Journal of Computational Physics, 2019Co-Authors: Xi Deng, Yuya Shimizu, Feng XiaoAbstract:Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC Function of 2-level Reconstruction based on BVD algorithm), is formulated as a two-stage spatial Reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) Functions with two-level steepness are used as the candidate Reconstruction Functions. The final Reconstruction Function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.
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A fifth-order shock capturing scheme with BVD algorithm.
arXiv: Computational Physics, 2018Co-Authors: Xi Deng, Yuya Shimizu, Feng XiaoAbstract:A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P4-THINC-BVD (4th degree polynomial and THINC Reconstruction based on BVD algorithm), is formulated as a two-stage cascade BVD (Boundary Variation Diminishing) algorithm following the BVD principle that minimizes the jumps of reconstructed values at cell boundaries. In the P4-THINC-BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness are used as the candidate Reconstruction Functions. The final Reconstruction Function is selected from the candidate Functions by a two-stage cascade BVD algorithm so as to effectively control numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P4-THINC-BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it faithfully maintains the free-mode solutions in long term computation. P4-THINC-BVD, as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.
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Toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids: The THINC method with quadratic surface representation and Gaussian quadrature
Journal of Computational Physics, 2017Co-Authors: Bin Xie, Feng XiaoAbstract:Abstract A novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids. Different from conventional volume of fluid (VOF) schemes that require complicated geometric manipulations for interface Reconstructions, the present method can be viewed as a hybrid geometric/algebraic type VOF approach, i.e. the interface is implictly retrieved from an algebraic Function that effectively makes use of the geometrical information, such as normal direction and curvature of the interface. Unlike previous versions of the THINC (tangent of hyperbola interface capturing) method, the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme. The Gaussian quadrature is used to estimate the integration of the cell-wise multi-dimensional hyperbolic tangent Reconstruction Function which is then used to retrieve the interface from the volume fraction value of the target cell. The Gaussian quadrature is also used to compute the numerical fluxes from the Reconstruction Function. Numerical accuracy for Reconstruction and flux computation can be effectively improved by increasing quadrature points. The whole solution procedure follows the finite volume method for advection transport, and is thus simple and easy to use for cells of arbitrary shapes. As verified in the benchmark tests in this paper, the presented scheme, so-called THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme, shows significantly improved geometrical fidelity of interface representation particularly for curved surface. Despite algorithmic simplicity, the solution quality of THINC/QQ is comparable to other existing VOF methods with PLIC geometrical interface Reconstructions, and thus an accurate and efficient VOF scheme of great practical significance for unstructured grids.
Ahmet M. Kondoz - One of the best experts on this subject based on the ideXlab platform.
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Enhanced Reconstruction algorithm for unidirectional distributed video coding
IET Image Processing, 2009Co-Authors: W.a.r.j. Weerakkody, W.a.c. Fernando, Ahmet M. KondozAbstract:Distributed video coding (DVC) is an emerging video coding technology that utilises the distributed source coding principles to build very low cost video encoders, yet with remarkable error resilience. In the most common DVC framework, the Reconstruction Function plays a vital role that has a direct impact on the output video quality. In this study, a novel algorithm is proposed for the Reconstruction Function, particularly focusing on a unidirectional DVC architecture. The proposed technique exploits the variations of the bit error rate of the Wyner-Ziv decoded bit stream and the assumed noise model in the side information stream. The simulation results show that the proposed algorithm yields a significant improvement of the objective and subjective video quality at no additional bit rate cost.
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An enhanced Reconstruction algorithm for unidirectional Distributed Video Coding
2008 IEEE International Symposium on Consumer Electronics, 2008Co-Authors: W.a.r.j. Weerakkody, W.a.c. Fernando, Ahmet M. KondozAbstract:Distributed Video Coding (DVC) is an emerging video coding technology that utilizes the distributed source coding principles to build very low cost video encoders, yet with remarkable error resilience. In the common DVC framework, the Reconstruction Function plays a vital role that has a direct impact on the output video quality. In this paper, a novel algorithm is proposed for the Reconstruction Function, particularly focusing on the unidirectional DVC architecture. The proposed technique exploits the variations of the bit error rate of the Wyner-Ziv decoded bit stream and the side information stream. The simulation results show that the proposed algorithm yields a significant improvement of the objective and subjective video quality at no additional bit rate cost.
Xi Deng - One of the best experts on this subject based on the ideXlab platform.
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Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm
Computers & Fluids, 2020Co-Authors: Xi Deng, Bin Xie, Yuya Shimizu, Feng XiaoAbstract:Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called PnTm - BVD (polynomial of n-degree and THINC Function of m-level Reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness as the Reconstruction candidates. The final Reconstruction Function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new Reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n + 1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new Reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.
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Implicit large eddy simulation of compressible turbulence flow with PnTm − BVD scheme
Applied Mathematical Modelling, 2020Co-Authors: Xi Deng, Feng Xiao, Zhen-hua Jiang, Chao YanAbstract:Implicit large eddy simulation (ILES) of compressible turbulence with shock capturing schemes requires wide investigations and numerical experiments. In this study, a newly proposed PnTm - BVD (polynomial of n-degree and THINC Function of m-level Reconstruction based on BVD algorithm) shock capturing scheme is introduced to simulate compressible turbulence flow with ILES. The new scheme is designed by employing high-order linear-weight polynomials and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness as the Reconstruction candidates. The final Reconstruction Function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. Numerical tests involving shock waves and broadband turbulence are conducted in comparison with WENO (Weighted Essentially Non-oscillatory) schemes which are widely used in ILES. The results demonstrate performing ILES with PnTm- BVD scheme is able to obtain higher resolution and more faithful results than WENO does. Importantly, the superiority of PnTm-BVD becomes more notable in high wave-number region. Thus this paper provides and verifies a new scheme which is promising in providing high-resolution results for real-case ILES of compressible turbulence flow.
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Discontinuity-resolving shock-capturing schemes on unstructured grids
2020Co-Authors: Cheng Lidong, Xi Deng, Xie Bin, Yi Jiang, Xiao FengAbstract:Solving compressible flows containing discontinuities remains a major challenge for numerical methods especially on unstructured grids. Thus in this work, we make contributions to shock capturing schemes on unstructured grids with aim of resolving discontinuities with low numerical dissipation. Different from conventional shock capturing schemes which only use polynomials as interpolation Functions on unstructured grids, the proposed scheme employs the linear polynomial as well as non-polynomial as Reconstruction candidates. For linear polynomial, the second order MUSCL scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) Function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial Reconstruction candidate. With these Reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which aims to minimize numerical dissipation is designed on unstructured grids to select the final Reconstruction Function. The resulted shock capturing scheme is named as MUSCL-THINC/QQ-BVD. The performance of the proposed scheme is demonstrated through solving compressible single-phase and multi-phase problems where the discontinuity is the typical flow structure. The numerical results show that the proposed scheme is capable of capturing sharp discontinuous profiles without numerical oscillations as well as resolving vortices associated with Kelvin-Helmholtz instabilities along shear layers and material interfaces. In comparison with schemes only replying on high order polynomials, the proposed scheme shows significant improvement of resolution across discontinuities. Thus, this work provides an accurate and robust shock-capturing scheme to resolve discontinuities in compressible flows.Comment: 18 pages, 18 figure
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A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm
Journal of Computational Physics, 2019Co-Authors: Xi Deng, Yuya Shimizu, Feng XiaoAbstract:Abstract A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P 4 T 2 − BVD (polynomial of 4-degree and THINC Function of 2-level Reconstruction based on BVD algorithm), is formulated as a two-stage spatial Reconstruction scheme following the BVD (Boundary Variation Diminishing) principle that minimizes the jumps of the reconstructed values at cell boundaries. In the P 4 T 2 − BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) Functions with two-level steepness are used as the candidate Reconstruction Functions. The final Reconstruction Function is selected through the two-stage BVD algorithm so as to effectively control both numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P 4 T 2 − BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it produces accurate solutions in long term computation. P 4 T 2 − BVD , as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.
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A fifth-order shock capturing scheme with BVD algorithm.
arXiv: Computational Physics, 2018Co-Authors: Xi Deng, Yuya Shimizu, Feng XiaoAbstract:A novel 5th-order shock capturing scheme is presented in this paper. The scheme, so-called P4-THINC-BVD (4th degree polynomial and THINC Reconstruction based on BVD algorithm), is formulated as a two-stage cascade BVD (Boundary Variation Diminishing) algorithm following the BVD principle that minimizes the jumps of reconstructed values at cell boundaries. In the P4-THINC-BVD scheme, polynomial of degree four and THINC (Tangent of Hyperbola for INterface Capturing) Functions with adaptive steepness are used as the candidate Reconstruction Functions. The final Reconstruction Function is selected from the candidate Functions by a two-stage cascade BVD algorithm so as to effectively control numerical oscillation and dissipation. Spectral analysis and numerical verifications show that the P4-THINC-BVD scheme possesses the following desirable properties: 1) it effectively suppresses spurious numerical oscillation in the presence of strong shock or discontinuity; 2) it substantially reduces numerical dissipation errors; 3) it automatically retrieves the underlying linear 5th-order upwind scheme for smooth solution over all wave numbers; 4) it is able to resolve both smooth and discontinuous flow structures of all scales with substantially improved solution quality in comparison to other existing methods; and 5) it faithfully maintains the free-mode solutions in long term computation. P4-THINC-BVD, as well as the underlying idea presented in this paper, provides an innovative and practical approach to design high-fidelity numerical schemes for compressible flows involving strong discontinuities and flow structures of wide range scales.