wanyu/IteraTeR_v2 huggingface dataset:
Original:  This quantitative convergence analysis is closely connected to the  J-method's unique feature of sensitivity with respect to spectral shifts.
Revised:  This quantitative convergence analysis is closely connected to the ~ J-method's unique feature of sensitivity with respect to spectral shifts.
Edit intent: fluency
Original:  Contrary to classical gradient flows  this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems.
Revised:  Contrary to classical gradient flows , this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems.
Edit intent: fluency
Original:  This methodology extends to the fully non-linear case the approach recently proposed in HPW19 for semi-linear PDEs.
Revised:  This methodology extends to the fully nonlinear case the approach recently proposed in HPW19 for semi-linear PDEs.
Edit intent: fluency
Original:  The TDNN aims at detecting road friction coefficient under lateral force excitations avoiding the use of standard mathematical tire models, which may provide more efficient methods and more robust results.
Revised:  The TDNN aims at detecting road friction coefficient under lateral force excitations avoiding the use of standard mathematical tire models, which may provide a more efficient method with robust results.
Edit intent: clarity
Original:  Our approach consists in representing the low-rank part of the solution as the product MM^{T}, where M is a rectangular matrix of appropriate size, parameterized by the coefficients of a deep neural network.
Revised:  Our approach consists in representing the low-rank part of the solution as the product MM^{T}, where M is a rectangular matrix of appropriate size, parametrized by the coefficients of a deep neural network.
Edit intent: fluency
Original:  We deduce its convergence speed to a local optimum from the Lipschitz smoothness of our loss function.
Revised:  We deduce its convergence rate to a local optimum from the Lipschitz smoothness of our loss function.
Edit intent: clarity
Original:  We show that the rate of convergence grows polynomially in the dimensions of the input, output, and  each of the hidden layers and hence conclude that our algorithm does not suffer from the curse of dimensionality .
Revised:  We show that the rate of convergence grows polynomially in the dimensions of the input, output, and each of the hidden layers .
Edit intent: coherence
Original:  As a discretization space for the partial differential equation a globally C^2-smooth isogeometric spline space is used, whose construction is simple and works uniformly for all possible multi-patch configurations.
Revised:  As a discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations.
Edit intent: clarity
Original:  The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degrees .
Revised:  The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degree .
Edit intent: fluency
Original:  Again, these points possess better convergence properties than Greville abscissae in case of odd spline degrees .
Revised:  Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree .
Edit intent: fluency
Original:  Unfortunately, a closely related issue to the approximate k-NN graph construction has been long overlooked.
Revised:  Unfortunately, a closely related issue to the graph construction has been long overlooked.
Edit intent: clarity
Original:  One one hand, we address the problem of merging two existing graphs into one by the proposed peer merge .
Revised:  One one hand, we address the problem of merging two existing graphs into one .
Edit intent: clarity
Original:  On the other hand, the problem of merging a raw set into a built k-NN graph is also addressed by the proposed joint merge.
Revised:  On the other hand, the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Edit intent: clarity
Original:  It enables the approximate k-NN graph to be built incrementally.
Revised:  It allows the approximate k-NN graph to be built incrementally.
Edit intent: fluency
Original:  Thus it supports approximate k-NN graph construction for an open set.
Revised:  It therefore supports approximate k-NN graph construction for an open set.
Edit intent: clarity
Original:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types and various data dimensions,  under different distance measures.
Revised:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types and data dimensions, under different distance measures.
Edit intent: clarity
Original:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types and various data dimensions,  under different distance measures.
Revised:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types and various data dimensions, and under different distance measures.
Edit intent: coherence
Original: K-nearest neighbor graph is the fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition and machine learning, etc.
Revised: k-nearest neighbor graph is the fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition and machine learning, etc.
Edit intent: fluency
Original:  One one hand, peer merge is proposed to address the problem of merging two approximate k-NN graphs into one.
Revised:  On the first hand, peer merge is proposed to address the problem of merging two approximate k-NN graphs into one.
Edit intent: fluency
Original:  On the other hand , the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Revised:  In addition , the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Edit intent: coherence
Original: k-nearest neighbor graph is the fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition  and machine learning, etc.
Revised: k-nearest neighbor graph is a fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition and machine learning, etc.
Edit intent: fluency
Original: k-nearest neighbor graph is the fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition  and machine learning, etc.
Revised: k-nearest neighbor graph is the fundamental data structure in many disciplines such as information retrieval, data-mining, pattern recognition , and machine learning, etc.
Edit intent: fluency
Original:  Namely, few literature covers about how to merge two existing k-NN graphs.
Revised:  Few works in the literature cover how to merge two existing k-NN graphs.
Edit intent: clarity
Original:  In addition , the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Revised:  Moreover , the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Edit intent: coherence
Original:  In addition , the problem of merging a raw set into a built k-NN graph is also addressed by joint merge.
Revised:  In addition , the problem of merging a raw set into a built k-NN graph is addressed by the joint merge.
Edit intent: clarity
Original:  Moreover, deriving from  joint merge, an hierarchical approximate k-NN graph construction approach is presented.
Revised:  Moreover, deriving from the joint merge, an hierarchical approximate k-NN graph construction approach is presented.
Edit intent: fluency
Original:  Moreover, deriving from  joint merge, an hierarchical approximate k-NN graph construction approach is presented.
Revised:  Moreover, deriving from joint merge, a hierarchical approximate k-NN graph construction approach is presented.
Edit intent: fluency
Original:  In this paper, we address the  k-NN graph merge issue of two different scenarios.
Revised:  In this paper, we address the issue of k-NN graph merge issue of two different scenarios.
Edit intent: fluency
Original:  In this paper, we address the  k-NN graph merge issue of two different scenarios.
Revised:  In this paper, we address the k-NN graph merging in two different scenarios.
Edit intent: clarity
Original:  On the first hand, the symmetric merge is proposed to address the problem of merging two approximate k-NN graphs into one.
Revised:  In the first scenario, a symmetric merge algorithm is proposed to address the problem of merging two approximate k-NN graphs into one.
Edit intent: clarity
Original:  On the first hand, the symmetric merge is proposed to address the problem of merging two approximate k-NN graphs into one.
Revised:  On the first hand, the symmetric merge is proposed to combine two approximate k-NN graphs into one.
Edit intent: clarity
Original:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types and data dimensions, and under different distance measures.
Revised:  With the support of produced graph hierarchy, superior performance is observed on the large-scale NN search task across various data types , dimensionality and distance measures.
Edit intent: clarity
Original:  The algorithm facilitates large-scale processing by  efficient merging of k-NN graphs that are produced in parallel.
Revised:  The algorithm facilitates large-scale processing by the efficient merging of k-NN graphs that are produced in parallel.
Edit intent: fluency
Original:  The algorithm enables  incremental construction of a hierarchical approximate k-NN graph.
Revised:  The algorithm enables the incremental construction of a hierarchical approximate k-NN graph.
Edit intent: fluency
Original:  Superior performance is attained when leveraging the hierarchy for NN search of various data types, dimensionality  and distance measures.
Revised:  Superior performance is attained when leveraging the hierarchy for NN search of various data types, dimensionality , and distance measures.
Edit intent: fluency
Original:  Our approach results in significantly faster solutions on a subset of available SMT benchmarks .
Revised:  Our approach can result in significantly faster solutions on a subset of available SMT benchmarks .
Edit intent: clarity
Original: Proximal based methods are well-suited to nonsmooth optimization problems with important applications in signal processing, control theory, statistics and machine learning.
Revised: Proximal based algorithms are well-suited to nonsmooth optimization problems with important applications in signal processing, control theory, statistics and machine learning.
Edit intent: clarity
Original:  There are essentially four basic types of proximal algorithms  currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford  and the very recent Davis-Yin three-operator splitting.
Revised:  There are essentially four basic types of proximal algorithms currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford , and the very recent Davis-Yin three-operator splitting.
Edit intent: fluency
Original:  Overall, our results strengthen the connections between optimization and continuous dynamical systems and offers a more unified perspective on accelerated methods.
Revised:  Overall, our results strengthen the connections between optimization and continuous dynamical systems and offer a more unified perspective on accelerated methods.
Edit intent: fluency
Original: Proximal based algorithms are well-suited to nonsmooth optimization problems with important applications in signal processing, control theory, statistics and machine learning .
Revised: Proximal algorithms are well-suited to nonsmooth optimization problems with important applications in signal processing, control theory, statistics and machine learning .
Edit intent: coherence
Original: Proximal based algorithms are well-suited to nonsmooth optimization problems with important applications in signal processing, control theory, statistics and machine learning .
Revised: Proximal based algorithms are well-suited for nonsmooth and constrained large-scale optimization problems and therefore suitable for applications in many areas of science .
Edit intent: clarity
Original:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Revised:  There are essentially four proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Edit intent: clarity
Original:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Revised:  There are essentially four basic types of proximal algorithms based on fixed-point iterations currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Edit intent: fluency
Original:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Revised:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the Davis-Yin three-operator splitting.
Edit intent: clarity
Original:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three-operator splitting.
Revised:  There are essentially four basic types of proximal algorithms based on fixed-point iteration currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the very recent Davis-Yin three operator splitting.
Edit intent: fluency
Original:  In this paper , we show that all these different methods can be derived from the gradient flow by using splitting methods for ordinary differential equations.
Revised:  In this paper we show that all these different methods can be derived from the gradient flow by using splitting methods for ordinary differential equations.
Edit intent: fluency
Original:  In this paper , we show that all these different methods can be derived from the gradient flow by using splitting methods for ordinary differential equations.
Revised:  In this paper , we show that all of these algorithms can be derived from the gradient flow by using splitting methods for ordinary differential equations.
Edit intent: clarity
Original:  In this paper , we show that all these different methods can be derived from the gradient flow by using splitting methods for ordinary differential equations.
Revised:  In this paper , we show that all these different methods can be derived from the gradient flow by using splitting methods for differential equations.
Edit intent: clarity
Original:  Furthermore, applying similar discretization scheme to a particular second order differential equation results in accelerated variants of the respective algorithm, which can be of Nesterov or heavy ball type, although we treat both simultaneously .
Revised:  Moreover, employing similar discretization schemes to a particular second order differential equation results in accelerated variants of the respective algorithm, which can be of Nesterov or heavy ball type, although we treat both simultaneously .
Edit intent: fluency
Original:  Many of the optimization algorithms we derive are new . For instance, we propose accelerated variants of Davis-Yin and two extensions of ADMM together with their accelerated variants .
Revised:  Many of the optimization algorithms we derive are new . For instance, we propose accelerated variants of Davis-Yin and two extensions of ADMM .
Edit intent: clarity
Original:  Interestingly, we show that (accelerated) ADMM corresponds to a rebalanced splitting which is a recent technique designed to preserve steady states of the  differential equation.
Revised:  Interestingly, we show that (accelerated) ADMM corresponds to a rebalanced splitting which is a recent technique designed to preserve steady states of the underlying differential equation.
Edit intent: clarity
Original:  Overall, our results strengthen the connections between optimization and continuous dynamical systems and offer a more unified perspective on accelerated methods .
Revised:  Overall, our results strengthen the connections between optimization and continuous dynamical systems , offer a unified perspective on accelerated methods .
Edit intent: clarity
Original:  Overall, our results strengthen the connections between optimization and continuous dynamical systems and offer a more unified perspective on accelerated methods .
Revised:  Overall, our results strengthen the connections between optimization and continuous dynamical systems and offer a more unified perspective on accelerated algorithms, and provide new accelerated algorithms .
Edit intent: clarity
Original: Proximal algorithms are well-suited for nonsmoothand constrained large-scale optimization problems and therefore suitable for applications in many areas of science .
Revised: Proximal algorithms are well-suited to nonsmooth, constrained, large-scale optimization problems and therefore suitable for applications in many areas of science .
Edit intent: fluency
Original:  There are essentially four proximal algorithms based on fixed-point iterations currently known : forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the Davis-Yin three operator splitting.
Revised:  There are essentially four proximal algorithms based on fixed-point iterations currently known : forward-backward splitting, Tseng splitting, Douglas-Rachford, and the Davis-Yin three operator splitting.
Edit intent: clarity
Original:  In this paper  we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely the simple gradient flow  .
Revised:  In this paper , we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely the simple gradient flow .
Edit intent: fluency
Original:  Moreover, employing similar discretization schemes to a particular second-order differential equation , which we refer to as the accelerated gradient flow, results in accelerated variants of each respective proximal algorithm;
Revised:  Moreover, employing similar discretization schemes to a particular second-order differential equation , which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of each respective proximal algorithm;
Edit intent: clarity
Original: Optimization is at the heart of machine learning, statistics , and several applied scientific disciplines.
Revised: Optimization is at the heart of machine learning, statistics and many applied scientific disciplines.
Edit intent: clarity
Original:  There are essentially five proximal algorithms currently known , each proposed in seminal work: forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multipliers, and the more recent Davis-Yin.
Revised:  There are essentially five proximal algorithms currently known : Forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multipliers, and the more recent Davis-Yin.
Edit intent: clarity
Original:  Such methods sit on a higher level of abstraction compared to gradient-based methods, having deep roots in nonlinear functional analysis.
Revised:  These methods sit on a higher level of abstraction compared to gradient-based methods, having deep roots in nonlinear functional analysis.
Edit intent: fluency
Original:  Such methods sit on a higher level of abstraction compared to gradient-based methods, having deep roots in nonlinear functional analysis.
Revised:  Such methods sit on a higher level of abstraction compared to gradient-based ones, with deep roots in nonlinear functional analysis.
Edit intent: clarity
Original:  In this paper, we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely  the simple gradient flow which dates back to Cauchy (1847).
Revised:  We show that all of these algorithms can be derived as different discretizations of a single differential equation, namely the simple gradient flow which dates back to Cauchy (1847).
Edit intent: coherence
Original:  In this paper, we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely  the simple gradient flow which dates back to Cauchy (1847).
Revised:  In this paper, we show that all of these methods are actually different discretizations of a single differential equation, namely the simple gradient flow which dates back to Cauchy (1847).
Edit intent: clarity
Original:  In this paper, we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely  the simple gradient flow which dates back to Cauchy (1847).
Revised:  In this paper, we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely , the simple gradient flow which dates back to Cauchy (1847).
Edit intent: fluency
Original:  We show that by employing similar discretization schemes  to Newton's classical equation of motion with an additional dissipative force, which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms ;
Revised:  We show that similar discretization schemes to Newton's classical equation of motion with an additional dissipative force, which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms ;
Edit intent: clarity
Original:  We show that by employing similar discretization schemes  to Newton's classical equation of motion with an additional dissipative force, which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms ;
Revised:  We show that by employing similar discretization schemes applied to Newton's classical equation of motion with an additional dissipative force, which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms ;
Edit intent: clarity