wanyu/IteraTeR_v2 huggingface dataset:
Original:  We also show that the often used Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case , however, it usually does not converge to the optimal solution -- neither with regard to the Frobenius norm nor the spectral norm.
Revised:  We also show that the often used Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case . However, it usually does not converge to the optimal solution -- neither with regard to the Frobenius norm nor the spectral norm.
Edit intent: fluency
Original:  We also show that the often used Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case , however, it usually does not converge to the optimal solution -- neither with regard to the Frobenius norm nor the spectral norm.
Revised:  We also show that the often used Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case , however, it usually does not converge to the optimal solution , neither with regard to the Frobenius norm nor the spectral norm.
Edit intent: fluency
Original:  We then introduce an adversarial training method that makes DNNs robust to small input noise, and use these evaluations to demonstrate that it greatly improves interpretability.
Revised:  We then introduce an adversarial training method that makes DNNs robust to small input noise, and demonstrate that it greatly improves interpretability.
Edit intent: clarity
Original:  However, many methods for creating these maps fail due to  focusing too much on the input or being extremely sensitive to small input noise .
Revised:  However, methods for creating these maps fail due to focusing too much on the input or being extremely sensitive to small input noise .
Edit intent: clarity
Original:  However, many methods for creating these maps fail due to  focusing too much on the input or being extremely sensitive to small input noise .
Revised:  However, many methods for creating maps often fail due to focusing too much on the input or being extremely sensitive to small input noise .
Edit intent: clarity
Original:  However, many methods for creating these maps fail due to  focusing too much on the input or being extremely sensitive to small input noise .
Revised:  However, many methods for creating these maps fail due to focusing too much on the input and too little on the model .
Edit intent: clarity
Original:  It is also challenging to quantitatively evaluate how well saliency maps correspond to the truly relevant input information  .
Revised:  It is also challenging to evaluate how well saliency maps correspond to the truly relevant input information .
Edit intent: clarity
Original:  The concept is further extended to cover analogous degradation models in a transformed domain, e.g. quantization of the  time-frequency coefficients.
Revised:  The concept is further extended to cover analogous degradation models in a transformed domain, e.g. quantization of the signal's time-frequency coefficients.
Edit intent: clarity
Original:  The problem of restoring an audio signal from degraded observations in two different domains is formulated as an inverse problem, and several algorithmic solutions are developed.
Revised:  The task of reconstructing an audio signal from degraded observations in two different domains is formulated as an inverse problem, and several algorithmic solutions are developed.
Edit intent: clarity
Original:  The viability of the presented concept is demonstrated on an example where audio restoration from partial and quantized observations of both the time-domain signal and its time-frequency coefficients is carried on .
Revised:  The viability of the presented concept is demonstrated on an example where audio reconstruction from partial and quantized observations of both the time-domain signal and its time-frequency coefficients is carried on .
Edit intent: clarity
Original:  The viability of the presented concept is demonstrated on an example where audio restoration from partial and quantized observations of both the time-domain signal and its time-frequency coefficients is carried on .
Revised:  The viability of the presented concept is demonstrated on an example where audio restoration from partial and quantized observations of both the time-domain signal and its time-frequency coefficients is carried out .
Edit intent: fluency
Original:  An attempt to prove this result was presented by Fleischner, Mujuni, Paulusma, and Szeider Theor.
Revised:  An attempt to prove this result was presented in Fleischner, Mujuni, Paulusma, and Szeider Theor.
Edit intent: fluency
Original:  Comput. Sci. 2009], but we realized that there was an apparently non-fixable flaw in their proof  .
Revised:  Comput. Sci. 2009], but there was a flaw in their proof .
Edit intent: clarity
Original:  This would also answer a question posed by Kratochv\'il, Tuza, and Voigt Proc. of WG 2002].
Revised:  This would also answer a question posed in Kratochv\'il, Tuza, and Voigt Proc. of WG 2002].
Edit intent: fluency
Original:  This would also answer a question posed by Kratochv\'il, Tuza, and Voigt Proc. of WG 2002].
Revised:  This would also answer a question posed by Kratochv\'il, Tuza, and Voigt , 2002].
Edit intent: clarity
Original:  Also, the game-theoretic concepts of no-absentmindedness, perfect-information, and (pure-strategy) Nash-equilibrium are shown to be isomorphically invariant .
Revised:  Also, the game-theoretic concepts of no-absentmindedness, perfect-information, and (pure-strategy) Nash-equilibrium .
Edit intent: clarity
Original: This paper proposes a quantum approximate optimization algorithm (QAOA) method for multi-scale wireless scheduling problems.
Revised: This paper proposes a quantum approximate optimization algorithm (QAOA) method for wireless scheduling problems.
Edit intent: clarity
Original:  First of all, this paper formulates a multi-scale scheduling problem using maximum weight independent set (MWIS) formulation .
Revised:  First of all, this paper formulates a multi-scale scheduling problem using maximum weight independent set (MWIS) .
Edit intent: clarity
Original:  Finally, for  linear recurrent networks and independent inputs it is proved that the memory capacity is given by the rank of the associated controllability matrix  .
Revised:  Finally, for linear recurrent networks with independent inputs it is proved that the memory capacity is given by the rank of the associated controllability matrix .
Edit intent: fluency
Original:  In this paper, we present a unified theory for tensor ranks such that they are natural extension of matrix ranks.
Revised:  In this paper, we explore this property for tensors. We first present a theory for tensor ranks such that they are natural extension of matrix ranks.
Edit intent: clarity
Original:  The CP rank, and the max-Tucker rank are not the smallest tensor rank function .
Revised:  We show that the smallest tensor rank function .
Edit intent: clarity
Original:  We define the closure of a strongly proper tensor rank function , and show that it is also a strongly proper tensor rank function .
Revised:  We define the closure of a strongly proper tensor rank function is a strongly proper tensor rank function .
Edit intent: clarity
Original: One fundamental property in matrix theory is that the rank of a matrix is always equal to the maximum value of all of its full rank submatrices.
Revised: A matrix always has a full-rank submatrix such that the rank of a matrix is always equal to the maximum value of all of its full rank submatrices.
Edit intent: clarity
Original: One fundamental property in matrix theory is that the rank of a matrix is always equal to the maximum value of all of its full rank submatrices.
Revised: One fundamental property in matrix theory is that the rank of this matrix is equal to the maximum value of all of its full rank submatrices.
Edit intent: clarity
Original:  We present some axioms for tensor rank functions.
Revised:  We propose some axioms for tensor rank functions.
Edit intent: clarity
Original:  We  show that the smallest tensor rank function is strongly proper and has the sub-full-rank property .
Revised:  We show that the smallest tensor rank function always has the sub-full-rank property .
Edit intent: clarity
Original:  We also show that the closure of a strongly proper tensor rank function is a strongly proper tensor rank function with the sub-full-rank property.
Revised:  We also show that the closure of a proper tensor rank function is a strongly proper tensor rank function with the sub-full-rank property.
Edit intent: fluency
Original:  We also show that the closure of a strongly proper tensor rank function is a strongly proper tensor rank function with the sub-full-rank property.
Revised:  We also show that the closure of a strongly proper tensor rank function is a proper tensor rank function with the sub-full-rank property.
Edit intent: fluency
Original:  These two properties form the corner stones of matrix theory.
Revised:  These two properties are two corner stones of matrix theory.
Edit intent: clarity
Original:  We may call that submatrix a base submatrix of this matrix, and the two properties the sub-full-rank property and the base submatrix property, respectively.
Revised:  We call that submatrix a base submatrix of this matrix, and the two properties the sub-full-rank property and the base submatrix property, respectively.
Edit intent: coherence
Original: A matrix always has a full-rank submatrix such that the rank of this matrix is equal to the rank of that submatrix.
Revised: A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix.
Edit intent: fluency
Original:  We call that submatrix a base submatrix of this matrix, and the two properties the sub-full-rank property and the base submatrix property, respectively.
Revised:  We call that submatrix a base submatrix of this matrix, and the sub-full-rank property and the base submatrix property, respectively.
Edit intent: clarity
Original:  We introduce the concepts of full-rank tensorsand base subtensors , and show that the max-Tucker tensor rank function always has the sub-full-rank property and the base subtensor property.
Revised:  We introduce the concept of full rank tensors , and show that the max-Tucker tensor rank function always has the sub-full-rank property and the base subtensor property.
Edit intent: fluency
Original:  We then define the closure for a proper tensor rank function, and show that the closure of a proper tensor rank function is a proper tensor rank function with the sub-full-rank property.
Revised:  We then define the closure for a proper tensor rank function, and show that the closure of any proper tensor rank function is a proper tensor rank function with the sub-full-rank property.
Edit intent: clarity
Original:  We call this property the sub-full-rank property.
Edit intent: coherence
Original:  There are two proper tensor rank functions associated with the Tucker decomposition , the max-Tucker rank and the submax-Tucker rank  .
Revised:  There are two proper tensor rank functions , the max-Tucker rank and the submax-Tucker rank .
Edit intent: clarity
Original:  We introduce the concept of full rank tensors, and show that the max-Tucker tensor rank function and the smallest tensor rank function have the sub-full-rank property.
Revised:  We introduce the concept of full rank tensors, and show that the max-Tucker tensor rank function and the smallest tensor rank function have this property.
Edit intent: clarity
Original:  We define the closure for a proper tensor rank function, and show that the closure of any proper tensor rank function is a proper tensor rank function with the sub-full-rank property.
Revised:  We define the closure for a proper tensor rank function, and show that it is still a proper tensor rank function with the sub-full-rank property.
Edit intent: clarity
Original:  The dynamics consists of a Susceptible-Infected (SI) model based on the logarithmic Caputo fractional derivative.
Revised:  The dynamics consists of a Susceptible-Infected (SI) model based on the Caputo fractional-order derivative.
Edit intent: clarity
Original:  We find an approximate analytical solution for this model which represents an upper bound for the fractional SI dynamics on a network.
Revised:  We find an upper bound to the analytical solution of this model which represents an upper bound for the fractional SI dynamics on a network.
Edit intent: clarity
Original:  We find an approximate analytical solution for this model which represents an upper bound for the fractional SI dynamics on a network.
Revised:  We find an approximate analytical solution for this model which represents the worse-case scenario on the propagation of perturbations across a protein residue network.
Edit intent: clarity
Original:  This upper bound is expressed in terms of the Mittag-Leffler function of the adjacency matrix of the network of inter-amino acids interactions.
Revised:  This upper bound is expressed in terms of Mittag-Leffler function of the adjacency matrix of the network of inter-amino acids interactions.
Edit intent: fluency
Original:  This upper bound is expressed in terms of the Mittag-Leffler function of the adjacency matrix of the network of inter-amino acids interactions.
Revised:  This upper bound is expressed in terms of the Mittag-Leffler functions of the adjacency matrix of the network of inter-amino acids interactions.
Edit intent: fluency
Original: We investigate the use of Genetic Programming (GP) as a convolutional predictor for supervised learning tasks in signal processing, focusing on the use case of predicting missing pixels in images.
Revised: We investigate the use of Genetic Programming (GP) as a convolutional predictor for missing pixels in images.
Edit intent: clarity
Original:  The training  is performed by sweeping a small sliding window on the available pixels : all pixels in the window except for the central one are fed in input to a GP tree whose output is taken as the predicted value for the central pixel.
Revised:  The training phase is performed by sweeping a small sliding window on the available pixels : all pixels in the window except for the central one are fed in input to a GP tree whose output is taken as the predicted value for the central pixel.
Edit intent: clarity
Original:  The training  is performed by sweeping a small sliding window on the available pixels : all pixels in the window except for the central one are fed in input to a GP tree whose output is taken as the predicted value for the central pixel.
Revised:  The training is performed by sweeping a small sliding window on the border represent the inputs of a GP tree whose output is taken as the predicted value for the central pixel.
Edit intent: clarity
Original:  The training  is performed by sweeping a small sliding window on the available pixels : all pixels in the window except for the central one are fed in input to a GP tree whose output is taken as the predicted value for the central pixel.
Revised:  The training is performed by sweeping a small sliding window on the available pixels : all pixels in the window except for the central one are fed in input to a GP tree . The output of the tree is taken as the predicted value for the central pixel.
Edit intent: clarity
Original:   The best GP tree in the population scoring the lowest prediction error over all available pixels in the population is then tested on the actual missing pixels of the degraded image .
Revised:  The best GP tree scoring the lowest prediction error over all available pixels in the population is then tested on the actual missing pixels of the degraded image .
Edit intent: fluency
Original:   The best GP tree in the population scoring the lowest prediction error over all available pixels in the population is then tested on the actual missing pixels of the degraded image .
Revised:  The best GP tree in the population scoring the lowest prediction error over the training set is then used to predict the pixels in the test set .
Edit intent: clarity
Original:  We experimentally assess this approach by training over four target images , removing up to 20%DIFDELCMD < \\%%% %DIF < of the pixels for the testing phase.
Revised:  We experimentally assess %DIFDELCMD < \\%%% %DIF < of the pixels for the testing phase.
Edit intent: clarity
Original:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better than the Von Neumann neighborhood , although the latter allows for a larger training set size .
Revised:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better , although the Von Neumann neighborhood , although the latter allows for a larger training set size .
Edit intent: coherence
Original:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better than the Von Neumann neighborhood , although the latter allows for a larger training set size .
Revised:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better than the Von Neumann neighborhood allows for a larger training set size .
Edit intent: coherence
Original:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better than the Von Neumann neighborhood , although the latter allows for a larger training set size .
Revised:  The results indicate that our method can learn to some extent the distribution of missing pixels in an image and that GP with Moore neighborhood works better than the Von Neumann neighborhood , although the latter allows for a larger training set .
Edit intent: clarity
Original:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%%  , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Revised:  %DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%% , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Edit intent: coherence
Original:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%%  , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Revised:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% %DIFDELCMD < \right)%%% , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Edit intent: coherence
Original:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%%  , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Revised:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%% , where N is the order of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Edit intent: clarity
Original:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%%  , where N is the degree of the transform , the computational complexity of our method is O\left(N ^2 \log^3(N) \right) .
Revised:  While existing methods have total running times (including all precomputations) which grow at least as fast as O%DIFDELCMD < \left(%%% N^3%DIFDELCMD < \right)%%% , where N is the degree of the transform . This is nearly asymptotically optimal. Moreover, unlike existing algorithms which are asymptotically optimal or nearly so, the constant in the running time of our algorithm is small enough to make it competitive with state-of-the-art O\left(N ^2 \log^3(N) \right) .
Edit intent: clarity
Original:  I address that potential problem by fitting a generalized additive model to past weekly provisional counts of all registered deaths in England and Wales.
Revised:  I address that critique by fitting a generalized additive model to past weekly provisional counts of all registered deaths in England and Wales.
Edit intent: clarity
Original:  The Department of Health and Social Care's counts of in-hospital COVID-19-associated deaths over the same period are even less, indicating that they are more severe undercounts .
Revised:  The Department of Health and Social Care's counts of in-hospital COVID-19-associated deaths over the same period are even lower, signalling more severe undercounting .
Edit intent: clarity
Original:  If underreporting rates have held steady, about 32,000 COVID-19 deaths might already have been registered but not publicly reported .
Revised:  If underreporting rates have held steady, about 32,000 COVID-19 deaths might already have been registered but not yet publicly reported in full .
Edit intent: clarity
Original:  I address that critique by fitting a generalized additive model to past weekly provisional counts of all registered deaths in England and Wales.
Revised:  I address that critique by fitting a generalized additive model to past weekly counts of all registered deaths in England and Wales.
Edit intent: clarity
Original:  The Department of Health and Social Care's counts of in-hospital COVID-19-associated deaths over the same period are even lower, signalling more severe undercounting .
Revised:  The Department of Health and Social Care's counts of in-hospital COVID-19-associated deaths , are appreciably less, implying that their counting methods have underestimated rather than overestimated the pandemic's true death toll .
Edit intent: clarity
Original:  If underreporting rates have held steady, about 41,300 COVID-19 deaths might already have been registered  but not yet publicly reported in full.
Revised:  If underreporting rates have held steady, about 41,300 COVID-19 deaths might have been registered but not yet publicly reported in full.
Edit intent: clarity
Original:  I address that critique by fitting a generalized additive model to past weekly counts of all registered deaths in England and Wales  .
Revised:  I address that critique by fitting a generalized additive model to weekly counts of all registered deaths in England and Wales .
Edit intent: clarity
Original:  I address that critique by fitting a generalized additive model to weekly counts of all registered deaths in England and Wales during the 2010s.
Revised:  I address that critique by fitting a generalized additive model to weekly counts of all deaths registered in England and Wales during the 2010s.
Edit intent: fluency
Original:  If underreporting rates have held steady during May, about 56,600 direct and indirect COVID-19 deaths might have been registered through 25 May but not yet publicly reported in full.
Revised:  If underreporting rates have held steady during May, about 56,600 direct and indirect COVID-19 deaths might have been registered through the end of May but not yet publicly reported in full.
Edit intent: clarity
Original:  We, thus, make a distinction between items that are good for everyone (i.e. goods), bad for everyone (i.e. bads) , and good for some agents and bad for other agents (i.e. mixed ).
Revised:  We, thus, make a distinction between items that are good for some agents and bad for other agents (i.e. goods), bad for everyone (i.e. bads) , and good for some agents and bad for other agents (i.e. mixed ).
Edit intent: fluency