wanyu/IteraTeR_full_sent huggingface dataset:
Original:  More broadly, our work sheds light on how other two-sided platforms can control the double-edged impacts that commitment levers have on growth and engagement  .
Revised:  More broadly, our work sheds light on how other two-sided platforms can control the double-edged impacts that commitment levers have on growth and engagement . A one-size-fits-all solution may not be effective, as the optimal design crucially depends on the characteristics of the volunteer population .
Edit intent: meaning-changed
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the starting nodes of random walkers where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: clarity
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes of the graph . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: clarity
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and the standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: fluency
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of weighted node degrees , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: clarity
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. , and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: coherence
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} The characteristics (a) is useful to understand the effect of the graph structure on the first meeting time . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: meaning-changed
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the revealed effect of graph structures , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: meaning-changed
Original:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes  . d_{\rm avg} and d_{\rm std} are the mean and  standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures}  . According to the analysis result , the variance of coefficient for weighted degrees, d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Revised:  and (b)the expected first meeting time is independent of the beginning nodes of multiple random walks where n is the number of nodes . d_{\rm avg} and d_{\rm std} are the mean and standard deviation of the weighted degree of each node , respectively. n and d_{\rm avg, and d_{\rm std} are related to the statistics of graph structures} . According to the analysis result , the variance of coefficient d_{\rm std}/d_{\rm avg}(degree heterogeneity)
Edit intent: coherence
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing the partition function \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb the Boolean cube \{-1, 1\}^n.
Edit intent: meaning-changed
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing \sum_x e^{f(x)}, where f a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb the Boolean cube \{-1, 1\}^n.
Edit intent: clarity
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% %DIFDELCMD < \right) %%% ranges over{\Bbb the Boolean cube \{-1, 1\}^n.
Edit intent: meaning-changed
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% : \{-1, 1\}^n \longrightarrow{\Bbb the Boolean cube \{-1, 1\}^n.
Edit intent: meaning-changed
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb R the Boolean cube \{-1, 1\}^n.
Edit intent: fluency
Original: We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n.
Revised: We consider the problem of computing \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb is a quadratic or cubic polynomial on the Boolean cube \{-1, 1\}^n.
Edit intent: meaning-changed
Original:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
Revised:  In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq } 1-\delta for all i.
Edit intent: clarity
Original:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
Revised:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 in quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq } 1-\delta for all i.
Edit intent: fluency
Original:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
Revised:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time | \leq } 1-\delta for all i.
Edit intent: meaning-changed
Original:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
Revised:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq } if the Lipschitz constant of the non-linear part of f with respect to the \ell^1 metric on the Boolean cube does not exceed 1-\delta for all i.
Edit intent: meaning-changed
Original:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
Revised:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq } 1-\delta , for any \delta >0, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition.
Edit intent: meaning-changed
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f} interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: clarity
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f} interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: clarity
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% | \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f} interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: fluency
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% | \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f} interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: clarity
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% complex-valued polynomials\tilde{f} interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: meaning-changed
Original:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Revised:  We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f} in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model.
Edit intent: meaning-changed
Original: We construct deep ReLU neural networks to approximate functions in dilated shift-invariant spaces generated by a continuous function with compact support and study the approximation rates with respect to the number of neurons .
Revised: We construct deep ReLU neural networks to approximate functions in dilated shift-invariant spaces , which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the number of neurons .
Edit intent: meaning-changed
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 1 ^d \le \le ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithm factor.
Edit intent: meaning-changed
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 0, 1 \le \le ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithm factor.
Edit intent: clarity
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 0, 1 ^d \le p\le ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithm factor.
Edit intent: meaning-changed
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le \infty ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithm factor.
Edit intent: meaning-changed
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le ) approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithm factor.
Edit intent: meaning-changed
Original:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
Revised:  We also give lower bounds of the L^p ( 0, 1 ^d \le \le ) approximation error for Sobolev spaces, which show that our construction is asymptotically optimal up to a logarithmic factor.
Edit intent: fluency