wanyu/IteraTeR_v2 huggingface dataset:
Original:  Thus this investigation reveals that the foundational problems of Geometry (classification and GRH ) and Complexity Theory (lower bounds and derandomization) share a common root difficulty that lies at the junction of these two fields.
Revised:  Thus this investigation reveals that the fundamental problems of Geometry (classification and GRH ) and Complexity Theory (lower bounds and derandomization) share a common root difficulty that lies at the junction of these two fields.
Edit intent: fluency
Original:  Thus this investigation reveals that the foundational problems of Geometry (classification and GRH ) and Complexity Theory (lower bounds and derandomization) share a common root difficulty that lies at the junction of these two fields.
Revised:  Thus this investigation reveals that the foundational problems of Geometry (classification ) and Complexity Theory (lower bounds and derandomization) share a common root difficulty that lies at the junction of these two fields.
Edit intent: clarity
Original:  We refer to it as the GCT chasm{\em  .
Revised:  We call it the{\em .
Edit intent: clarity
Original:  in a strong form for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices can be brought down from EXPSPACE ( where it was earlier ) to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Revised:  that lies at the heart of the wild problem of classifying tuples of matrices can be brought down from EXPSPACE ( where it was earlier ) to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Edit intent: clarity
Original:  in a strong form for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices can be brought down from EXPSPACE ( where it was earlier ) to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Revised:  in a strong form for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices can be brought down from EXPSPACE , where it was earlier ) to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Edit intent: fluency
Original:  in a strong form for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices can be brought down from EXPSPACE ( where it was earlier ) to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Revised:  in a strong form for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices can be brought down from EXPSPACE ( where it was earlier , to PSPACE unconditionally, to PH assuming the Generalized Riemann Hypothesis (GRH), and to P assuming the black-box derandomization hypothesis for symbolic trace identity testing (STIT) or equivalently symbolic determinantidentity testing (SDIT).
Edit intent: fluency
Original:  Furthermore, it is shown that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from EXPSPACE, where it is currently, to P assuming a strengthened form of the black-box derandomization hypothesis  for polynomial identity testing (PIT), and to quasi-P assuming that some exponential-time-computable multilinear integral polynomial cannot be approximated infinitesimally closely by arithmetic circuits of sub-exponential size.
Revised:  Furthermore, it is shown that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from EXPSPACE, where it is currently, to P assuming a strengthened form of the black-box derandomization hypothesis (BDH) for polynomial identity testing (PIT), and to quasi-P assuming that some exponential-time-computable multilinear integral polynomial cannot be approximated infinitesimally closely by arithmetic circuits of sub-exponential size.
Edit intent: clarity
Original:  Furthermore, it is shown that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from EXPSPACE, where it is currently, to P assuming a strengthened form of the black-box derandomization hypothesis  for polynomial identity testing (PIT), and to quasi-P assuming that some exponential-time-computable multilinear integral polynomial cannot be approximated infinitesimally closely by arithmetic circuits of sub-exponential size.
Revised:  Furthermore, it is shown that the problem of derandomizing Noether's Normalization Lemma for any explicit variety can be brought down from EXPSPACE, where it is currently, to P assuming a strengthened form of the black-box derandomization hypothesis for polynomial identity testing (PIT), and to quasi-P assuming that some exponential-time-computable multilinear polynomial cannot be approximated infinitesimally closely by arithmetic circuits of sub-exponential size.
Edit intent: clarity
Original:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Revised:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Edit intent: clarity
Original:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Revised:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry (classification) and Complexity Theory share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Edit intent: coherence
Original:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the formidable EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Revised:  The converse also holds for a strict form of NNL. These and related results in this paper reveal that the fundamental problems of Geometry (classification) and Complexity Theory (lower bounds and derandomization) share a common root difficulty, namely, the problem of overcoming the EXPSPACE vs. P gap in the complexity of NNL for explicit varieties.
Edit intent: clarity
Original:  We call this gap the GCT chasm.
Revised:  This gap is called the GCT chasm.
Edit intent: coherence
Original:  On the positive side, we show that NNL for the ring of invariants for any finite dimensional rational representation of the special linear group of fixed dimension can be brought down from EXPSPACE to quasi-P unconditionally  .
Revised:  On the positive side, it is shown that NNL for the ring of invariants for any finite dimensional rational representation of the special linear group of fixed dimension can be brought down from EXPSPACE to quasi-P unconditionally .
Edit intent: clarity
Original:  On the positive side, we show that NNL for the ring of invariants for any finite dimensional rational representation of the special linear group of fixed dimension can be brought down from EXPSPACE to quasi-P unconditionally  .
Revised:  On the positive side, we show that NNL for the ring of invariants for any finite dimensional representation of the special linear group of fixed dimension can be brought down from EXPSPACE to quasi-P unconditionally .
Edit intent: clarity
Original:  We prove that in any Sturmian word the  ratio between the maximal exponent of an abelian repetition of period m and m converges to a number greater than or equal to 5, and the equality holds for the Fibonacci infinite word.
Revised:  We prove that in any Sturmian word the ratio between the maximal exponent of an abelian repetition of period m and m is a number \geq 5, and the equality holds for the Fibonacci infinite word.
Edit intent: clarity
Original: The regularized random forest (RRF) has recently been proposed for feature selection by building only one ensemble.
Revised: The regularized random forest (RRF) was recently proposed for feature selection by building only one ensemble.
Edit intent: clarity
Original:  However, in RRF the features are evaluated by a part of the training data at each tree node , and thus the feature selection process may not be stable .
Revised:  In RRF the features are evaluated by a part of the training data at each tree node , and thus the feature selection process may not be stable .
Edit intent: coherence
Original:  However, in RRF the features are evaluated by a part of the training data at each tree node , and thus the feature selection process may not be stable .
Revised:  However, in RRF the features are evaluated on a part of the training data at each tree node , and thus the feature selection process may not be stable .
Edit intent: clarity
Original:  Here an enhanced RRF, referred to as  guided RRF (GRRF), is proposed.
Revised:  Here an enhanced RRF, referred to as the guided RRF (GRRF), is proposed.
Edit intent: fluency
Original:  In GRRF, the importance scores from an ordinary random forest (RF) are used to guide the feature selection process in RRF. Experimental studies show that GRRF , in general, is able to select compact feature sub sets and is better than RRF, varSelRF and LASSO logistic regression in terms of the accuracy of RF and a decision tree method C4.5 . Both RRF and GRRF were implemented in the "RRF"  package available at CRAN URL the official R package archive.
Revised:  In GRRF, the importance scores from an ordinary random forest (RF) are used to guide the feature selection process in RRF. Experimental studies show that GRRF , in general, is able to select compact feature sub sets and is better than RRF, varSelRF and LASSO logistic regression in terms of the accuracy of RF and a decision tree method C4.5 . Both RRF and GRRF were implemented in the "RRF" R package available at CRAN URL the official R package archive.
Edit intent: coherence
Original:  In GRRF, the importance scores from an ordinary random forest (RF) are used to guide the feature selection process in RRF. Experimental studies show that GRRF , in general, is able to select compact feature sub sets and is better than RRF, varSelRF and LASSO logistic regression in terms of the accuracy of RF and a decision tree method C4.5 . Both RRF and GRRF were implemented in the "RRF"  package available at CRAN URL the official R package archive.
Revised:  In GRRF, the importance scores from an ordinary random forest (RF) are used to guide the feature selection process in RRF. Experimental studies show that GRRF , in general, is able to select compact feature sub sets and is better than RRF, varSelRF and LASSO logistic regression in terms of the accuracy of RF and a decision tree method C4.5 . Both RRF and GRRF were implemented in the "RRF" package available at CRAN , the official R package archive.
Edit intent: fluency
Original:  The application of this algorithm to six different published conductance based models always finds the transcritical bifurcation in a physiological parameter range , which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons .
Revised:  Our analysis of six different published conductance based models always finds the transcritical bifurcation in a physiological parameter range , which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons .
Edit intent: clarity
Original:  The application of this algorithm to six different published conductance based models always finds the transcritical bifurcation in a physiological parameter range , which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons .
Revised:  The application of this algorithm to six different published conductance based models always finds the transcritical bifurcation and the associated switch in excitability , which suggests that the mathematical predictions have a physiological relevance and that a same regulatory mechanism is potentially involved in the excitability and signaling of many neurons .
Edit intent: clarity
Original:  In this paper, we  analyze and compare two types of such models: global models and local models .
Revised:  In this paper, we review, analyze and compare two types of such models: global models and local models .
Edit intent: clarity
Original:  In this paper, we  analyze and compare two types of such models: global models and local models .
Revised:  In this paper, we analyze and compare different statistical models: from those that analyze the variation in geometry globally to those that analyze the variation in geometry locally .
Edit intent: clarity
Original:  We first review how both types of models have been used in the literature, then proceed to define the models and analyze them theoretically, in terms of both their statistical and computational aspects.
Revised:  We first review how different types of models have been used in the literature, then proceed to define the models and analyze them theoretically, in terms of both their statistical and computational aspects.
Edit intent: clarity
Original:  We then perform extensive experimental comparison to assess which type of model is better for a given task .
Revised:  We then perform extensive experimental comparison to assess which type of model is better for a few applications .
Edit intent: clarity
Original:  We employ a mean-field model which allows for the construction of a population-specific socially informed model for predicting links from both network and social properties in large social networks.
Revised:  We employ a mean-field model which allows for the construction of a population-specific socially informed model for predicting links from both network and social properties in large social networks.
Edit intent: fluency
Original:  From the expression of Newton-Raphson's iteration we then derive a recurrence relation and an actually explicit formula for the modular inverse, generalizing to any prime power modulus.
Revised:  From the expression of Newton-Raphson's iteration we then derive an actually explicit formula for the modular inverse, generalizing to any prime power modulus.
Edit intent: clarity
Original:  We first give another proof of the formulas of Arazi and Qi for the modular inverse modulo 2^m, derived from Newton-Raphson iteration over p-adic fields, namely Hensel's lifting.
Revised:  Newton-Raphson iteration over p-adic fields, namely Hensel's lifting.
Edit intent: coherence
Original:  Overall  a hybrid combination of the two latter algorithms yield a constant factor improvement  also for large exponents.
Revised:  Overall we thus propose a hybrid combination of the two latter algorithms yield a constant factor improvement also for large exponents.
Edit intent: coherence
Original:  Overall  a hybrid combination of the two latter algorithms yield a constant factor improvement  also for large exponents.
Revised:  Overall a hybrid combination of the two latter algorithms yield a constant factor improvement , also for large exponents.
Edit intent: fluency
Original: We give the first linear kernels for %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%%  Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.
Revised: We give the first linear kernels for %DIFDELCMD < {\sc %%% and%DIFDELCMD < {\sc %%% Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.
Edit intent: fluency
Original: We give the first linear kernels for %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%%  Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.
Revised: We give the first linear kernels for %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.
Edit intent: fluency
Original:  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G' on %DIFDELCMD < \cO%%%  (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Revised:  In other words, we give polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer k, output an H-topological-minor free graph G' on %DIFDELCMD < \cO%%% (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Edit intent: fluency
Original:  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G' on %DIFDELCMD < \cO%%%  (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Revised:  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor-free graph G' on %DIFDELCMD < \cO%%% (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Edit intent: fluency
Original:  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G' on %DIFDELCMD < \cO%%%  (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Revised:  In other words, we give polynomial time algorithms that, for a given H-topological-minor free graph G and a positive integer k, output an H-topological-minor free graph G' on %DIFDELCMD < \cO%%% O (k) vertices such that G has a (connected) dominating set of size k if and only if G' has.
Edit intent: clarity
Original:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%%  Connected Dominating Set problems admit linear kernels.
Revised:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% and%DIFDELCMD < {\sc %%% Connected Dominating Set problems admit linear kernels.
Edit intent: fluency
Original:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%%  Connected Dominating Set problems admit linear kernels.
Revised:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% Dominating Set %DIFDELCMD < {\sc %%% Connected Dominating Set problems admit linear kernels.
Edit intent: clarity
Original:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%%  Connected Dominating Set problems admit linear kernels.
Revised:  Our results extend the known classes of graphs on which %DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% Dominating Set and Connected Dominating Set problems admit linear kernels.
Edit intent: fluency
Original:  The kernelization algorithm is based on a non-trivial combination of the following ingredients \bullet The structural theorem of Grohe and Marx [STOC 2012] for graphs excluding a fixed graph H as a topological subgraph;
Revised:  The kernelization algorithm is based on a non-trivial combination of the following ingredients The structural theorem of Grohe and Marx [STOC 2012] for graphs excluding a fixed graph H as a topological subgraph;
Edit intent: coherence
Original:  \bullet A novel notion of protrusions, different that the one defined in FOCS 2009%DIFDELCMD < ]%%% ;
Revised:  FOCS 2009%DIFDELCMD < ]%%% ;
Edit intent: coherence
Original:  \bullet A novel notion of protrusions, different that the one defined in FOCS 2009%DIFDELCMD < ]%%% ;
Revised:  \bullet A novel notion of protrusions, different that the one defined in %DIFDELCMD < ]%%% ;
Edit intent: clarity
Original:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% Connected Dominating Set from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Revised:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% and%DIFDELCMD < {\sc %%% Connected Dominating Set from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Edit intent: fluency
Original:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% Connected Dominating Set from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Revised:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% Dominating Set %DIFDELCMD < {\sc %%% Connected Dominating Set from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Edit intent: fluency
Original:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% Connected Dominating Set from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Revised:  \bullet Reinterpretations of reduction techniques developed for kernelization algorithms for%DIFDELCMD < {\sc %%% Dominating Set and%DIFDELCMD < {\sc %%% from%DIFDELCMD < [%%% SODA 2012 . A protrusion is a subgraph of constant treewidth separated from the remaining vertices by a constant number of vertices.
Edit intent: clarity