wanyu/IteraTeR_v2 huggingface dataset:
Original:  We prove worst-case recovery guarantees and improve on the recent new upper bounds for sampling numbers in Krieg, M. Ullrich by explicitly controlling all the involved constants with respect to the underlying spatial dimension d . This leads to new preasymptotic recovery bounds with high probability for the error of Hyperbolic Fourier Regression for multivariate functions .
Revised:  We prove worst-case recovery guarantees and improve on the recent new upper bounds for sampling numbers in Krieg, M. Ullrich by explicitly controlling all the involved constants with respect to the underlying spatial dimension d . This leads to new preasymptotic recovery bounds with high probability for the error of Hyperbolic Fourier Regression on multivariate data .
Edit intent: clarity
Original:  In addition, we analyze the algorithm Hyperbolic Wavelet Regression also based on least-squares to recover non-periodic functions from random samples.
Revised:  In addition, we further investigate its counterpart Hyperbolic Wavelet Regression also based on least-squares to recover non-periodic functions from random samples.
Edit intent: clarity
Original:  As a further application we reconsider the analysis of a cubature method based on plain random points with optimal weights introduced by Oettershagen.
Revised:  Finally, we reconsider the analysis of a cubature method based on plain random points with optimal weights introduced by Oettershagen.
Edit intent: clarity
Original:  Agents learn about a binary state and take turns choosing actions based on own signals and observations of network neighbors' behavior.
Revised:  Agents learn about a binary state and take turns choosing actions based on own signals and network neighbors' behavior.
Edit intent: clarity
Original:  When this observation structure is sufficiently connected and symmetric, the additional information aggregated by each generation is asymptotically equivalent to fewer than two independent signals, even when generations are arbitrarily large.
Revised:  When this observation structure is sufficiently symmetric, the additional information aggregated by each generation is asymptotically equivalent to fewer than two independent signals, even when generations are arbitrarily large.
Edit intent: clarity
Original: We consider a sequential social-learning environment with rational agents and Gaussian private signals , focusing on how the observation network affects the speed of learning.
Revised: In social-learning environment with rational agents and Gaussian private signals , focusing on how the observation network affects the speed of learning.
Edit intent: clarity
Original:  We then consider a network structure where agents move in generations and observe some members of the previous generation.
Revised:  We then consider a network structure where agents move in generations and observe the previous generation.
Edit intent: clarity
Original:  When this observation structure is sufficiently symmetric, the additional information aggregated by each generation is asymptotically equivalent to fewer than two independent signals , even when generations are arbitrarily large .
Revised:  Generations after the first contribute very little additional information , even when generations are arbitrarily large .
Edit intent: clarity
Original: In social-learning settings where individuals receive private signals and observe network neighbors' actions, the network structure often obstructs information aggregation .
Revised: When individuals in a social network learn about an unknown state from private signals and observe network neighbors' actions, the network structure often obstructs information aggregation .
Edit intent: clarity
Original: In social-learning settings where individuals receive private signals and observe network neighbors' actions, the network structure often obstructs information aggregation .
Revised: In social-learning settings where individuals receive private signals and neighbors' actions, the network structure often obstructs information aggregation .
Edit intent: clarity
Original: In social-learning settings where individuals receive private signals and observe network neighbors' actions, the network structure often obstructs information aggregation .
Revised: In social-learning settings where individuals receive private signals and observe network neighbors' actions, the network structure often causes information loss .
Edit intent: clarity
Original:  We consider sequential social learning with rational agents and Gaussian signals  and ask how the  efficiency of signal aggregation changes with the network . Rational actions in our model  admit a signal-counting interpretation of accuracy ,
Revised:  We consider rational agents and Gaussian signals and ask how the efficiency of signal aggregation changes with the network . Rational actions in our model admit a signal-counting interpretation of accuracy ,
Edit intent: clarity
Original:  We consider sequential social learning with rational agents and Gaussian signals  and ask how the  efficiency of signal aggregation changes with the network . Rational actions in our model  admit a signal-counting interpretation of accuracy ,
Revised:  We consider sequential social learning with rational agents and Gaussian signals and ask how the efficiency of signal aggregation . Rational actions in our model admit a signal-counting interpretation of accuracy ,
Edit intent: clarity
Original:  An important observation in this regard is that agents in the real world often only directly affect a limited set of neighboring agents.
Revised:  An important observation in this regard is that agents in the real world often only directly affect a limited set of neighbouring agents.
Edit intent: fluency
Original:  As opposed to in the planning setting, for learning methods it is challenging to establish theoretical guarantees.
Edit intent: coherence
Original:  Finally , we empirically show that MATS outperforms the state-of-the-art algorithm, MAUCE, on two synthetic benchmarks, a realistic wind farm control task , and a novel benchmark with Poisson distributions .
Revised:  Additionally , we empirically show that MATS outperforms the state-of-the-art algorithm, MAUCE, on two synthetic benchmarks, a realistic wind farm control task , and a novel benchmark with Poisson distributions .
Edit intent: coherence
Original:  In this work, we address this issue by introducing HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping  . HILUCSI differs from the state-of-the-art multilevel ILU in three aspects.
Revised:  In this work, we introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping . HILUCSI differs from the state-of-the-art multilevel ILU in three aspects.
Edit intent: coherence
Original:  In this work, we address this issue by introducing HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping  . HILUCSI differs from the state-of-the-art multilevel ILU in three aspects.
Revised:  In this work, we address this issue by introducing HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping . HILUCSI differs from the state-of-the-art ILU techniques in two main aspects.
Edit intent: clarity
Original:  Third, to improve the effectiveness for large-scale problems from PDEs, HILUCSI introduces a scalability-oriented dropping in conjunction with a modified inverse-based dropping  .
Revised:  Third, to improve the effectiveness for large-scale problems from PDEs, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping .
Edit intent: clarity
Original:  Third, to improve the effectiveness for large-scale problems from PDEs, HILUCSI introduces a scalability-oriented dropping in conjunction with a modified inverse-based dropping  .
Revised:  Third, to improve the effectiveness for large-scale problems from PDEs, HILUCSI introduces a scalability-oriented dropping in conjunction with a modified inverse-based dropping to improve the efficiency for large-scale problems from PDEs .
Edit intent: clarity
Original:  We demonstrate the robustness and efficiency of HILUCSI for benchmark problems from a wide range of applications against ILUPACK, the supernodal ILUTP in SuperLU, and multithreaded direct solvers in PARDISO and MUMPS.
Revised:  We demonstrate the effectiveness of HILUCSI for benchmark problems from a wide range of applications against ILUPACK, the supernodal ILUTP in SuperLU, and multithreaded direct solvers in PARDISO and MUMPS.
Edit intent: clarity
Original:  Its multilevel variants, such as those in ILUPACK and ARMS, have been shown to be more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques.
Revised:  Its multilevel variants, such as those in ILUPACK and ARMS, are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques.
Edit intent: clarity
Original:  Its multilevel variants, such as those in ILUPACKand ARMS , are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques.
Revised:  Its multilevel variants, such as ILUPACK , are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques.
Edit intent: clarity
Original:  However, multilevel ILU still lacked robustness and efficiency for some large-scale saddle-point problems, which often arise from systems of partial differential equations (PDEs).
Revised:  However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large-scale saddle-point problems, which often arise from systems of partial differential equations (PDEs).
Edit intent: clarity
Original:  In this work, we introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping , which is specifically designed to take advantage of some special features of such systems.
Revised:  We introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-oriented and Inverse-based dropping , which is specifically designed to take advantage of some special features of such systems.
Edit intent: clarity
Original:  First, HILUCSI leverages the near or partial symmetry of the underlying problems and the inherent block structures of multilevel ILU to improve robustness and to simplify the treatment of\times1 \times2  indefinite systems.
Revised:  First, HILUCSI leverages the near or partial symmetry of the underlying problems and the inherent block structures of multilevel ILU to improve robustness and to simplify the treatment of\times1 and 2\times2 indefinite systems.
Edit intent: clarity
Original:  Second, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping to improve the efficiency for large-scale problems from PDEs .
Revised:  Second, HILUCSI introduces a scalability-oriented dropping in conjunction with a variant of inverse-based dropping .
Edit intent: coherence
Original:  We demonstrate the effectiveness of HILUCSI for a number of benchmark problems, including those from  mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations.
Revised:  We demonstrate the effectiveness of HILUCSI for dozens of benchmark problems, including those from mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations.
Edit intent: clarity
Original:  We demonstrate the effectiveness of HILUCSI for a number of benchmark problems, including those from  mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations.
Revised:  We demonstrate the effectiveness of HILUCSI for a number of benchmark problems, including those from the mixed formulation of the Poisson equation, Stokes equations, and Navier-Stokes equations.
Edit intent: fluency
Original:  ~\mbox{ \cite{CaiDW12a, CaiDW12b,CaiDW13a,CaiDW13b show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (\varepsilon-BIC).
Revised:  show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (\varepsilon-BIC).
Edit intent: coherence
Original:  Other \varepsilon-BIC to BIC transformations  exist in the literature ~\mbox{ \cite{DaskalakisW12, RubinsteinW15, CaiZ17 but all require exponential time to run  .
Revised:  Other \varepsilon-BIC to BIC transformations exist in the literature but all require exponential time to run .
Edit intent: clarity
Original:  ~\mbox{ \cite{CaiDW12a, CaiDW12b,CaiDW13a,CaiDW13b to generate an exactly BIC mechanism.
Revised:  to generate an exactly BIC mechanism.
Edit intent: coherence
Original:  We provide a new polynomial time transformation that converts any \varepsilon-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss.
Revised:  We provide a new polynomial time transformations that convert any \varepsilon-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss.
Edit intent: fluency
Original:  Our  transformation applies to a very general mechanism design setting  and only requires sample access to the agents' type distributions and query access to the original \varepsilon-BIC mechanism.
Revised:  Our transformation applies to any mechanism design setting and only requires sample access to the agents' type distributions and query access to the original \varepsilon-BIC mechanism.
Edit intent: clarity
Original:  Other \varepsilon-BIC to BIC transformations for revenue exist in the literature but all require exponential time to run in the general setting we consider.
Revised:  Other \varepsilon-BIC to BIC transformations for revenue exist in the literature but all require exponential time to run in both of the settings we consider.
Edit intent: clarity
Original:  As an application of our transformation , we improve the reduction by Cai et al. to generate an exactly BIC mechanism .
Revised:  As an application of our transformations , we improve the reduction by Cai et al. to generate an exactly BIC mechanism .
Edit intent: fluency
Original: Modeling the relationship between chemical structure and molecular activity is a key task in drug development and precision medicine.
Revised: Modeling the relationship between chemical structure and molecular activity is a key goal in drug development and precision medicine.
Edit intent: clarity
Original:  We also  separate the power of Sherali-Adams versus Lovasz-Schrijver hierarchies for approximating Max-Cut, since it is known that (1{2}+\varepsilon) approximation of Max Cut requires \Omega_\varepsilon (n) rounds in the Lovasz-Schrijver hierarchy.
Revised:  We also separate the power of Sherali-Adams versus Lov\'asz-Schrijver hierarchies for approximating Max-Cut, since it is known that (1{2}+\varepsilon) approximation of Max Cut requires \Omega_\varepsilon (n) rounds in the Lovasz-Schrijver hierarchy.
Edit intent: fluency
Original:  We also  separate the power of Sherali-Adams versus Lovasz-Schrijver hierarchies for approximating Max-Cut, since it is known that (1{2}+\varepsilon) approximation of Max Cut requires \Omega_\varepsilon (n) rounds in the Lovasz-Schrijver hierarchy.
Revised:  We also separate the power of Sherali-Adams versus Lovasz-Schrijver hierarchies for approximating Max-Cut, since it is known that (1{2}+\varepsilon) approximation of Max Cut requires \Omega_\varepsilon (n) rounds in the Lov\'asz-Schrijver hierarchy.
Edit intent: fluency
Original:  We also provide a subexponential time approximation for Khot's Unique Games problem: we show that for every \varepsilon > 0 the degree-n ^\varepsilon \log q) Sherali-Adams linear program distinguishes instances of Unique Games of value \geq 1-\varepsilon' from instances of value \leq \varepsilon', for some \varepsilon'( \varepsilon) >0, where q is the alphabet size.
Revised:  We also provide a subexponential time approximation for Khot's Unique Games problem: we show that for every \varepsilon > 0 the degree-(n ^\varepsilon \log q) Sherali-Adams linear program distinguishes instances of Unique Games of value \geq 1-\varepsilon' from instances of value \leq \varepsilon', for some \varepsilon'( \varepsilon) >0, where q is the alphabet size.
Edit intent: fluency
Original: Social categorizations are an important aspect of human society, dividing people into "us" and "them  " often along  attributes such as race or even political ideology .
Revised: Social categorizations divide people into "us" and "them " often along attributes such as race or even political ideology .
Edit intent: clarity
Original: Social categorizations are an important aspect of human society, dividing people into "us" and "them  " often along  attributes such as race or even political ideology .
Revised: Social categorizations are an important aspect of human society, dividing people into "us" and "them , " often along attributes such as race or even political ideology .
Edit intent: fluency
Original: Social categorizations are an important aspect of human society, dividing people into "us" and "them  " often along  attributes such as race or even political ideology .
Revised: Social categorizations are an important aspect of human society, dividing people into "us" and "them " often along continuous attributes such as race or even political ideology .
Edit intent: fluency
Original:   We test the model's prediction using US political survey data on how political independents are perceived by registered party members , as well as existing experiments on the perception of racially ambiguous faces, and find support .
Revised:  We test this prediction using U.S. political survey data on how political independents are perceived by registered party members , as well as existing experiments on the perception of racially ambiguous faces, and find support .
Edit intent: clarity
Original:   We test the model's prediction using US political survey data on how political independents are perceived by registered party members , as well as existing experiments on the perception of racially ambiguous faces, and find support .
Revised:  We test the model's prediction using US political survey data on how political independents are perceived by registered party members as well as existing experiments on the perception of racially ambiguous faces, and find support .
Edit intent: fluency
Original:  From this viewpoint, we establish the detailed connections in a technical level between the traditional iterative schemes for constrained linear system and the architecture for the basic block of ResNet.
Revised:  From this viewpoint, we establish the detailed connections in a technical level between the traditional iterative schemes for constrained linear system and the architecture for the basic blocks of ResNet.
Edit intent: fluency
Original:  Under these connections, we propose some natural modifications of ResNet type models which will have less parameters but can keep almost the same accuracy as these  original models.
Revised:  Under these connections, we propose some natural modifications of ResNet type models which will have less parameters but still maintain almost the same accuracy as these original models.
Edit intent: clarity
Original: Projection methods (including collocation methods) are always considered in numerical analysis for differential and integral equations Ax=b .
Revised: Projection methods (including collocation methods) are always considered in numerical solutions of differential and integral equations Ax=b .
Edit intent: clarity
Original:  \notin  In this paper, we project more insight to the numerical behavior of projection methods when b \notin R(A) and propose an non-conditional divergence result of projection method with arbitrary basis into compact operator equation (Theorem 1.1) . Several applications show its power and further discussion on divergence rate is given .
Revised:  \notin In this paper, we consider the numerical behavior of projection methods when b \notin R(A) and propose an non-conditional divergence result of projection method with arbitrary basis into compact operator equation (Theorem 1.1) . Several applications show its power and further discussion on divergence rate is given .
Edit intent: clarity
Original:  \notin  In this paper, we project more insight to the numerical behavior of projection methods when b \notin R(A) and propose an non-conditional divergence result of projection method with arbitrary basis into compact operator equation (Theorem 1.1) . Several applications show its power and further discussion on divergence rate is given .
Revised:  \notin In this paper, we project more insight to the numerical behavior of projection methods when b \notin R(A) and propose an non-conditional divergence result of projection method with arbitrary basis into compact operator equation (Theorem 1.1) . Several applications show its power .
Edit intent: coherence
Original: In the list colouring problem for two matroids, we are given matroids M_1=(S, \mathcal{I {\cal  _1)
Revised: In the list coloring problem for two matroids, we are given matroids M_1=(S, \mathcal{I {\cal _1)
Edit intent: fluency
Original: In the list colouring problem for two matroids, we are given matroids M_1=(S, \mathcal{I {\cal  _1)
Revised: In the list colouring problem for two matroids, we are given matroids M_1=(S, {\cal _1)
Edit intent: clarity
Original:  and M_2=(S, \mathcal{I {\cal  _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Revised:  and M_2=(S, {\cal _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Edit intent: clarity
Original:  and M_2=(S, \mathcal{I {\cal  _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Revised:  and M_2=(S, \mathcal{I {\cal I _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Edit intent: clarity
Original:  and M_2=(S, \mathcal{I {\cal  _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Revised:  and M_2=(S, \mathcal{I {\cal _2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colors for s\in S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer.
Edit intent: fluency