wanyu/IteraTeR_v2 huggingface dataset:
Original:  Although several candidate lower bounds were suggested in the past, such as WB-1 and WB-2 bounds by Wilber, and Independent Rectangles bound by Demaine et al., the only currently known non-trivial approximation algorithm achieves an O(\log \log n) approximation factor by comparing OPT with a weak variant of WB-1, that uses afixed partitioning of the keys.
Revised:  Although several candidate lower bounds were suggested in the past, such as WB-1 and WB-2 bounds by Wilber, and Independent Rectangles bound by Demaine et al., the only currently known non-trivial approximation algorithm achieves an O(\log \log n) approximation factor by comparing OPT with a weak variant of WB-1, that uses a partitioning of the keys.
Edit intent: fluency
Original:  To overcome this obstacle, it is natural to consider a stronger variant of WB-1 , that maximizes the bound overall partitionings of the keys.
Revised:  To overcome this obstacle, it is natural to consider a stronger variant of WB-1 all partitionings of the keys.
Edit intent: clarity
Original:  To overcome this obstacle, it is natural to consider a stronger variant of WB-1 , that maximizes the bound overall partitionings of the keys.
Revised:  To overcome this obstacle, it is natural to consider a stronger variant of WB-1 , that maximizes the bound over partitionings of the keys.
Edit intent: fluency
Original:  We also provide algorithmic results: for any parameter D, we present a simple O(D)-approximation algorithm with running time \exp\left(O\left (n^{1/2 ^D }\log n\right )\right ).
Revised:  We also provide algorithmic results: for any parameter D, we present a simple O(D)-approximation in time \exp\left(O\left (n^{1/2 ^D }\log n\right )\right ).
Edit intent: clarity
Original:  In doing so, we provide a theoretical analysis of the feasibility and profitability of a block stealing attack that we call selfish endorsing , a concrete instance of an attack previously only theoretically considered.
Revised:  In doing so, we provide a theoretical analysis of the feasibility and profitability of a block stealing attack that we call selfish endorsing , a concrete instance of an attack previously only theoretically considered.
Edit intent: fluency
Original:  Regression counterfactual regret minimization (RCFR) is a flexible and simple algorithm for approximately solving imperfect information games with policies parameterized by a normalized rectified linear unit (ReLU)  .
Revised:  Regression counterfactual regret minimization (RCFR) is a simple algorithm for approximately solving imperfect information games with policies parameterized by a normalized rectified linear unit (ReLU) .
Edit intent: clarity
Original:  Regression counterfactual regret minimization (RCFR) is a flexible and simple algorithm for approximately solving imperfect information games with policies parameterized by a normalized rectified linear unit (ReLU)  .
Revised:  Regression counterfactual regret minimization (RCFR) is a flexible and simple algorithm for approximately solving imperfect information games with normalized rectified linear unit (ReLU) .
Edit intent: clarity
Original:  In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and has a regret bound with a better dependence on the number of actions in the tabular case .
Revised:  In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and yields a regret bound with a better dependence on the number of actions in the tabular case .
Edit intent: clarity
Original:  In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and has a regret bound with a better dependence on the number of actions in the tabular case .
Revised:  In contrast, the more conventional softmax parameterization is standard in the field of reinforcement learning and has a regret bound with a better dependence on the number of actions .
Edit intent: coherence
Original:  These bounds recover a tighter bound for RCFR and provides a theoretical justification for RCFR implementations with alternative policy parameterizations (f-RCFR), including softmax.
Revised:  These bounds recover a tighter bound for RCFR and provide a theoretical justification for RCFR implementations with alternative policy parameterizations (f-RCFR), including softmax.
Edit intent: fluency
Original:  We provide exploitability bounds for f-RCFR with the polynomial and exponential link functions in zero-sum imperfect information games , and examine empirically how the link function interacts with the severity of the approximation to determine exploitability performance in practice.
Revised:  We provide exploitability bounds for f-RCFR with the polynomial and exponential link functions in zero-sum imperfect information games and examine empirically how the link function interacts with the severity of the approximation to determine exploitability performance in practice.
Edit intent: fluency
Original:  Although a ReLU parameterized policy is typically the best choice, a softmax parameterization can perform as well or better in settings that require aggressive approximation .
Revised:  Although a ReLU parameterized policy is typically the best choice, a softmax parameterization can perform better when the approximation error is large .
Edit intent: clarity
Original: Given a sample of an abstract manifold immerged in some Euclidean space, we describe a way to recover the singular homology of the original manifold.
Revised: Given a sample of an abstract manifold immersed in some Euclidean space, we describe a way to recover the singular homology of the original manifold.
Edit intent: fluency
Original:  It consists in estimating its tangent bundle-seen as subset of another Euclidean space-in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology.
Revised:  It consists in estimating its tangent bundle---seen as subset of another Euclidean space-in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology.
Edit intent: fluency
Original:  It consists in estimating its tangent bundle-seen as subset of another Euclidean space-in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology.
Revised:  It consists in estimating its tangent bundle-seen as subset of another Euclidean space---in a measure theoretic point of view, and in applying measure-based filtrations for persistent homology.
Edit intent: fluency
Original: For the approximation of multivariate non-periodic functions h on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function with a transformation \psi yields a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Revised: We combine a periodization strategy for weighted L_{2 multivariate non-periodic functions h on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function with a transformation \psi yields a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Edit intent: clarity
Original: For the approximation of multivariate non-periodic functions h on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function with a transformation \psi yields a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Revised: For the approximation of multivariate non-periodic functions on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function with a transformation \psi yields a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Edit intent: clarity
Original: For the approximation of multivariate non-periodic functions h on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function with a transformation \psi yields a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Revised: For the approximation of multivariate non-periodic functions h on the high-dimensional cube \left[-1{2},1{2}\right]^{d} we combine a periodization strategy for weighted L_{2 d-variate torus-to-cube transformations {\psi:\left[-1{2},1{2}\right]^{d}\to\left[-1{2},1{2}\right]^{d}} and on the non-negative weight function \omega such that the composition of a possibly non-periodic function is transformed into a smooth function in the Sobolev space \mathcal H^{m}(T^{d}) .
Edit intent: clarity
Original:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting.
Revised:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting.
Edit intent: clarity
Original:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting.
Revised:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting.
Edit intent: clarity
Original:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus to the non-periodic setting.
Revised:  In this framework we adapt certain L_{\infty}(T^{d})- and L_{2}(T^{d})-approximation error estimates for single rank-1 lattice approximation methods as well as algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting.
Edit intent: clarity
Original:  Various numerical tests in up to dimension d=5 confirm the obtained theoretical results for the transformed approximation methods .
Revised:  We illustrate the theoretical findings by means of numerical tests in up to dimension d=5 confirm the obtained theoretical results for the transformed approximation methods .
Edit intent: clarity
Original:  Various numerical tests in up to dimension d=5 confirm the obtained theoretical results for the transformed approximation methods .
Revised:  Various numerical tests in up to d=5 confirm the obtained theoretical results for the transformed approximation methods .
Edit intent: clarity
Original:  Various numerical tests in up to dimension d=5 confirm the obtained theoretical results for the transformed approximation methods .
Revised:  Various numerical tests in up to dimension d=5 dimensions .
Edit intent: clarity
Original: Let G be a graph and D_s and D _{\textsf{t be two dominating sets of G of size k.
Revised: Let G be a graph and D_s and D _t be two dominating sets of G of size k.
Edit intent: fluency
Original:  We first prove that this problem is PSPACE-complete, even when restricted to split, bipartite or bounded tree-width graphs.
Revised:  We first prove that this problem is PSPACE-complete, even when restricted to split, bipartite or bounded treewidth graphs.
Edit intent: fluency
Original:  A  potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice (2019)) . This paper extends this framework to the random (geometric) stopping version.
Revised:  A new potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice (2019)) . This paper extends this framework to the random (geometric) stopping version.
Edit intent: clarity
Original:  A  potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice (2019)) . This paper extends this framework to the random (geometric) stopping version.
Revised:  A potential-based framework for the fixed horizon version of this problem has been recently developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice (2019)) . This paper extends this framework to the random (geometric) stopping version.
Edit intent: clarity
Original:  A  potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory (Kobzar, Kohn and Wang, New Potential-Based Bounds for Prediction with Expert Advice (2019)) . This paper extends this framework to the random (geometric) stopping version.
Revised:  A potential-based framework for the fixed horizon version of this problem was previously developed using verification arguments from optimal control theory . This paper extends this framework to the random (geometric) stopping version.
Edit intent: coherence
Original:  Taking advantage of these ideas , we construct potentials for the geometric version of prediction with expert advice from potentials used for the fixed horizon version  .
Revised:  Taking advantage of these ideas , we construct potentials for the geometric version from potentials used for the fixed horizon version .
Edit intent: clarity
Original:  Taking advantage of these ideas , we construct potentials for the geometric version of prediction with expert advice from potentials used for the fixed horizon version  .
Revised:  Taking advantage of these ideas , we construct potentials for the geometric version of prediction with expert advice from potentials used for the fixed horizon version of the problem .
Edit intent: clarity
Original: We study a natural variant of scheduling that we call \emph{partial scheduling}: In this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs  have to be processed.
Revised: We study a natural variant of scheduling that we call \emph{partial scheduling}: In this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs , have to be processed.
Edit intent: fluency
Original:  We study the Fixed Parameter Tractability of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints.
Revised:  Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints.
Edit intent: clarity
Original:  That is, we investigate whether algorithms with runtimes of the type \mathcal{O f(k) ) exist , where the O^*(\cdot) notation omits factors polynomial in the input size.
Revised:  That is, we investigate whether algorithms with runtimes of the type f(k) ) exist , where the O^*(\cdot) notation omits factors polynomial in the input size.
Edit intent: clarity
Original:  We obtain atrichotomy by categorizing each variant to be either in P, NP-complete and Fixed Parameter Tractable by k, or W 1 -hard  by k.
Revised:  We obtain a by categorizing each variant to be either in P, NP-complete and Fixed Parameter Tractable by k, or W 1 -hard by k.
Edit intent: clarity
Original:  We obtain atrichotomy by categorizing each variant to be either in P, NP-complete and Fixed Parameter Tractable by k, or W 1 -hard  by k.
Revised:  We obtain atrichotomy by categorizing each variant to be either in P, NP-complete and Fixed Parameter Tractable by k, or 1 -hard by k.
Edit intent: fluency
Original:   As one of our main technical contributions, we give an O ^* (8^k  ) time algorithm to solve instances of k-scheduling problems minimizing the makespan with unit job lengths , precedence constraints and release dates  .
Revised:  As one of our main technical contributions, we give an O (8^k ) time algorithm to solve instances of k-scheduling problems minimizing the makespan with unit job lengths , precedence constraints and release dates .
Edit intent: coherence
Original:   As one of our main technical contributions, we give an O ^* (8^k  ) time algorithm to solve instances of k-scheduling problems minimizing the makespan with unit job lengths , precedence constraints and release dates  .
Revised:  As one of our main technical contributions, we give an O ^* (8^k ) time algorithm to solve instances of k-scheduling problems minimizing the makespan with unit length jobs , precedence constraints and release dates .
Edit intent: fluency
Original:  For occupancy testing , NASA circumvents the complexity of meshes and mitigates the issue of water-tightness.
Revised:  For occupancy testing , NASA circumvents the complexity of meshes and the issue of water-tightness.
Edit intent: clarity
Original:  If the dimensionality of the target distribution is high, the number of accepted proposals obtained with a given computational effort may be multiplied by a factor of three or more by switching to alternative integrators very similar to leapfrog.
Revised:  When the dimensionality of the target distribution is high, the number of accepted proposals obtained with a given computational effort may be multiplied by a factor of three or more by switching to alternative integrators very similar to leapfrog.
Edit intent: fluency
Original:  In addition, we provide several theoretical results on the acceptance rate of HMC . We prove that, at stationarity, one out of two accepted proposals comes from an integration leg where energy is lost.
Revised:  In addition, we provide some theoretical results on the acceptance rate of HMC . We prove that, at stationarity, one out of two accepted proposals comes from an integration leg where energy is lost.
Edit intent: clarity
Original: We give numerical evidence that the standard leapfrog algorithm may not be the best integrator to use within the Hamiltonian Monte Carlo (HMC) method and its variants.
Revised: The leapfrog integrator is routinely used within the Hamiltonian Monte Carlo (HMC) method and its variants.
Edit intent: clarity
Original: We give numerical evidence that the standard leapfrog algorithm may not be the best integrator to use within the Hamiltonian Monte Carlo (HMC) method and its variants.
Revised: We give numerical evidence that the standard leapfrog algorithm may not be the best integrator to use within the Hamiltonian Monte Carlo method and its variants.
Edit intent: clarity
Original:   When the dimensionality of the target distribution is high, the number of accepted proposals obtained with a given computational effort may be multiplied by a factor of three or more by switching to alternative integrators very similar to leapfrog.
Revised:  When the dimensionality of the target distribution is high, the number of accepted proposals may be multiplied by a factor of three or more by switching to alternative integrators very similar to leapfrog.
Edit intent: clarity
Original:  We show how our 3 modules consistently improve state-of-the-art methods in all point upsampling metrics .
Revised:  Extensive experiments show how these modules consistently improve state-of-the-art methods in all point upsampling metrics .
Edit intent: clarity
Original:  We show how our 3 modules consistently improve state-of-the-art methods in all point upsampling metrics .
Revised:  We show how our 3 modules consistently improve state-of-the-art upsampling methods .
Edit intent: clarity
Original:  By aggregating data at multiple scales, our new feature extractor is more resilient to density changes along point cloud surfaces .
Revised:  By aggregating data at multiple scales, this feature extractor enables further performance gain in the final upsampled point clouds .
Edit intent: clarity
Original:  We show both qualitatively and quantitatively the  advantages of PU-GCN against the state-of-the-art in terms of fine-grained upsampling quality and point cloud uniformity .
Revised:  We show both qualitatively and quantitatively the advantages of PU-GCN against the state-of-the-art .
Edit intent: clarity
Original:  The website and source code of this work is available at URL and URL respectively.
Revised:  The website and source code of this work are available at URL and URL respectively.
Edit intent: fluency
Original:  We also propose a new multi-scale point feature extractor, called Inception DenseGCN.
Revised:  For feature extraction, we also propose a new multi-scale point feature extractor, called Inception DenseGCN.
Edit intent: coherence
Original:  We combine Inception DenseGCN with one of our upsampling modules (NodeShuffle ) into a new point upsampling pipeline : PU-GCN.
Revised:  We combine Inception DenseGCN with NodeShuffle into a new point upsampling pipeline : PU-GCN.
Edit intent: clarity
Original:  We combine Inception DenseGCN with one of our upsampling modules (NodeShuffle ) into a new point upsampling pipeline : PU-GCN.
Revised:  We combine Inception DenseGCN with one of our upsampling modules (NodeShuffle ) into a new point upsampling pipeline called PU-GCN.
Edit intent: fluency
Original:  We show qualitatively and quantitatively the significant advantages of PU-GCN over the state-of-the-art.
Revised:  PU-GCN over the state-of-the-art.
Edit intent: clarity
Original:  Most significant bit (MSB) of encrypted pixels are modulated to vacate room for embedding data bits.
Revised:  Most significant bit (MSB) of encrypted pixels are integrated to vacate room for embedding data bits.
Edit intent: clarity
Original:  Modulated ones will be more resistant against failure of reconstruction if they modify for embedding data bits.
Revised:  Integrated ones will be more resistant against failure of reconstruction if they modify for embedding data bits.
Edit intent: fluency
Original:  Modulated ones will be more resistant against failure of reconstruction if they modify for embedding data bits.
Revised:  Modulated ones will be more resistant against failure of reconstruction if they are modified for embedding data bits.
Edit intent: fluency