wanyu/IteraTeR_v2 huggingface dataset:
Original:  The mean isoperimetry problem for weighted graphs {\it   is a generalization of the  sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V for which the average normalized cut value of the parts is as small as possible.
Revised:  The mean isoperimetry problem for weighted graphs {\it is a generalization of the sparsest cut problem in which, given a parameter k alongside a weighted graph G=(V, E), the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts is as small as possible.
Edit intent: clarity
Original:  In this paper, we present an O(\log^{1.5n \log\log n) approximation algorithm for the }\textsc{mean isoperimetry problem .
Revised:  In this paper, we present an O(\log^{1.5n \log\log n) approximation algorithm for the }\textsc{ .
Edit intent: clarity
Original:  Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- i.e. ' blow-ups'.
Revised:  Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- that is ` blow-ups'.
Edit intent: coherence
Original:  Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- that is `blow-ups'.
Revised:  Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions , which we refer to as `blow-ups'.
Edit intent: clarity
Original: We extend a model of positive feedback and contagion in large mean-field systems  by introducing a common source of noise driven by Brownian motion.
Revised: We extend a model of positive feedback and contagion in large mean-field systems , by introducing a common source of noise driven by Brownian motion.
Edit intent: fluency
Original:  Although the dynamics in the model are continuous, the  feedback effect can lead to jump discontinuities in the solutions, which we refer to as `blow-ups' .
Revised:  Although the driving dynamics are continuous, the feedback effect can lead to jump discontinuities in the solutions, which we refer to as `blow-ups' .
Edit intent: clarity
Original:  As a consequence, we obtain similar bounds on the number of \alpha-shortest vectors in "totally unimodular '' lattices and on the number of \alpha-minimum weight codewords in "regular '' codes.
Revised:  As a consequence, we obtain similar bounds on the number of \alpha-shortest vectors in "totally unimodular " lattices and on the number of \alpha-minimum weight codewords in "regular '' codes.
Edit intent: fluency
Original:  As a consequence, we obtain similar bounds on the number of \alpha-shortest vectors in "totally unimodular '' lattices and on the number of \alpha-minimum weight codewords in "regular '' codes.
Revised:  As a consequence, we obtain similar bounds on the number of \alpha-shortest vectors in "totally unimodular '' lattices and on the number of \alpha-minimum weight codewords in "regular " codes.
Edit intent: fluency
Original:  This zigzag Morse filtration generalizes the %DIFDELCMD < {\em %%% filtered Morse complex of Mischaikow and Nanda ~\mbox{ MischaikowN13 , defined for standard persistence.
Revised:  This zigzag Morse filtration generalizes the %DIFDELCMD < {\em %%% filtered Morse complex of Mischaikow and Nanda , defined for standard persistence.
Edit intent: clarity
Original:  This approach allows us to demonstrate and theoretically explain the emergence of five distinct phenomena reminiscent of cognitive functions:
Revised:  This approach allows us to demonstrate and theoretically explain the emergence of 5 distinct phenomena reminiscent of cognitive functions:
Edit intent: fluency
Original: Machine learning is often used to produce decision-making rules that classify or evaluate individuals.
Revised: Algorithms are often used to produce decision-making rules that classify or evaluate individuals.
Edit intent: clarity
Original:  The overall approach takes advantage of both physicochemical modeling and data-based approaches and allows to construct optimal parameterized reduced dynamics in the number of species, reactions and parameters, while controlling the information loss due to the reduction.
Revised:  This approach takes advantage of both physicochemical modeling and data-based approaches and allows to construct optimal parameterized reduced dynamics in the number of species, reactions and parameters, while controlling the information loss due to the reduction.
Edit intent: coherence
Original:  We demonstrate the effectiveness of our model reduction method on several complex, high-dimensional biochemical reaction networks from the recent literature .
Revised:  We demonstrate the effectiveness of our model reduction method on several complex, high-dimensional chemical reaction networks arising in biochemistry .
Edit intent: clarity
Original: This paperproposes a navigation algorithm ori- ented to multi-agent dynamic environment.
Revised: In this paper, we propose a navigation algorithm ori- ented to multi-agent dynamic environment.
Edit intent: fluency
Original: This paperproposes a navigation algorithm ori- ented to multi-agent dynamic environment.
Revised: This paperproposes a navigation algorithm oriented to multi-agent dynamic environment.
Edit intent: fluency
Original:  The algorithm is expressed as a hierarchical framework which contains a Hidden Markov Model (HMM) and  Deep Reinforcement Learning (DRL)  . For simplification, we term our method Hierarchical Navigation Reinforcement Network (HNRN).
Revised:  The algorithm is expressed as a hierarchical framework that contains a Hidden Markov Model (HMM) and Deep Reinforcement Learning (DRL) . For simplification, we term our method Hierarchical Navigation Reinforcement Network (HNRN).
Edit intent: fluency
Original:  The algorithm is expressed as a hierarchical framework which contains a Hidden Markov Model (HMM) and  Deep Reinforcement Learning (DRL)  . For simplification, we term our method Hierarchical Navigation Reinforcement Network (HNRN).
Revised:  The algorithm is expressed as a hierarchical framework which contains a Hidden Markov Model (HMM) and a Deep Reinforcement Learning (DRL) . For simplification, we term our method Hierarchical Navigation Reinforcement Network (HNRN).
Edit intent: fluency
Original:  In high-level architecture, we train an HMM to evaluate agents environment in order to obtain a score.
Revised:  In high- level architecture, we train an HMM to evaluate agents environment in order to obtain a score.
Edit intent: fluency
Original:  In high-level architecture, we train an HMM to evaluate agents environment in order to obtain a score.
Revised:  In high-level architecture, we train an HMM to evaluate the agent's perception to obtain a score.
Edit intent: clarity
Original:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is  collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Revised:  While in low-level architecture, two sub-systems are introduced, one is a differential target- driven system, which aims at heading to the target , the other is collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Edit intent: fluency
Original:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is  collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Revised:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target ; the other is collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Edit intent: fluency
Original:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is  collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Revised:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is a collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Edit intent: fluency
Original:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is  collision avoidance DRL system, which is used for avoiding obstacles in the dynamic environment .
Revised:  While in low-level architecture, two sub-systems are introduced, one is a differential target-driven system, which aims at heading to the target , the other is collision avoidance DRL system, which is used for avoiding dynamic obstacles .
Edit intent: clarity
Original:  The advantage of this hierarchical system is to decouple the target-driven and collision avoidance tasks, leading to a faster and easier model to be trained.
Revised:  The advantage of this hierarchical structure is decoupling the target-driven and collision avoidance tasks, leading to a faster and easier model to be trained.
Edit intent: clarity
Original:  The advantage of this hierarchical system is to decouple the target-driven and collision avoidance tasks, leading to a faster and easier model to be trained.
Revised:  The advantage of this hierarchical system is to decouple the target-driven and collision avoidance tasks, leading to a faster and more stable model to be trained.
Edit intent: clarity
Original:  As the experiments manifest, our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Revised:  The experiments indicate that our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Edit intent: clarity
Original:  As the experiments manifest, our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Revised:  As the experiments manifest, our algorithm has higher learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Edit intent: clarity
Original:  As the experiments manifest, our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Revised:  As the experiments manifest, our algorithm has faster learning efficiency and rate of success than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Edit intent: clarity
Original:  As the experiments manifest, our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms and hybrid DRL method.
Revised:  As the experiments manifest, our algorithm has faster learning efficiency and a higher success rate than traditional Velocity Obstacle (VO) algorithms or hybrid DRL method.
Edit intent: fluency
Original:  The system is available at URL 
Revised:  The system is available at URL .
Edit intent: fluency
Original: We present the second part of a stabilized cut discontinuous Galerkin ( cutDG) framework initiated in 1%DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Revised: We develop novel stabilized cut discontinuous Galerkin ( cutDG) framework initiated in 1%DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Edit intent: clarity
Original: We present the second part of a stabilized cut discontinuous Galerkin ( cutDG) framework initiated in 1%DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Revised: We present the second part of a stabilized cut discontinuous Galerkin ( 1%DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Edit intent: clarity
Original: We present the second part of a stabilized cut discontinuous Galerkin ( cutDG) framework initiated in 1%DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Revised: We present the second part of a stabilized cut discontinuous Galerkin ( cutDG) framework initiated in %DIFDELCMD < ] %%% and develop a novel cutDG method for scalar hyperbolic problems.
Edit intent: fluency
Original: The discontinuous DNA distribution in angiosperms is deduced from the eigen-value equation of the genome evolution operator H.
Revised: The nuclear DNA amount in angiosperms is deduced from the eigen-value equation of the genome evolution operator H.
Edit intent: clarity
Original: The discontinuous DNA distribution in angiosperms is deduced from the eigen-value equation of the genome evolution operator H.
Revised: The discontinuous DNA distribution in angiosperms is studied from the eigen-value equation of the genome evolution operator H.
Edit intent: clarity
Original:  Their analysis is based on the Contextuality-by-Default (CbD) theory , one that allows one to separate true contextuality from direct influences in the overall context-dependence.
Revised:  Their analysis is based on the Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence.
Edit intent: clarity
Original:  In this article  we propose an exponentially accurate nonconforming spectral element method for these problems based on [7, 18].
Revised:  In this article , we propose an exponentially accurate nonconforming spectral element method for these problems based on [7, 18].
Edit intent: clarity
Original: Document ranking experiments should be repeatable : running the same ranking model over the same collection with the same queries should yield exactly the same output .
Revised: Document ranking experiments should be repeatable .
Edit intent: clarity
Original:  However, the presence of different documents with the same score may yield non-deterministic rankings, making repeatability not as trivial as one might imagine.
Revised:  However, the interaction between multi-threaded indexing and score ties during retrieval may yield non-deterministic rankings, making repeatability not as trivial as one might imagine.
Edit intent: clarity
Original:  In the context of our work using the open-source Lucene search engine, score ties are broken by internal document ids, which are assigned at index time.
Revised:  In the context of the open-source Lucene search engine, score ties are broken by internal document ids, which are assigned at index time.
Edit intent: clarity
Original:  This approach, however, comes with measurable efficiency costs due to the necessity of consulting external identifiers during the inner loop of query evaluation.
Revised:  This approach, however, comes with measurable efficiency costs due to the necessity of consulting external identifiers during query evaluation.
Edit intent: clarity
Original: Given a directed graph, the vertex connectivity from u to v is the maximum number of internally vertex-disjoint paths from u to v. We design faster algorithms that, given as input a directed graph G with unit node capacities and a threshold k, report for all vertex pairs (s,t) the size of a minimum st-vertex cut (or maximum st-flow or vertex connectivity) if it is <k, or report that it is%DIFDELCMD < \ge %%% k otherwise.
Revised: The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/ vertex connectivity) if it is <k, or report that it is%DIFDELCMD < \ge %%% k otherwise.
Edit intent: clarity
Original: Given a directed graph, the vertex connectivity from u to v is the maximum number of internally vertex-disjoint paths from u to v. We design faster algorithms that, given as input a directed graph G with unit node capacities and a threshold k, report for all vertex pairs (s,t) the size of a minimum st-vertex cut (or maximum st-flow or vertex connectivity) if it is <k, or report that it is%DIFDELCMD < \ge %%% k otherwise.
Revised: Given a directed graph, the vertex connectivity from u to v is the maximum number of internally vertex-disjoint paths from u to v. We design faster algorithms that, given as input a directed graph G with unit node capacities and a threshold k, report for all vertex pairs (s,t) the size of a minimum st-vertex cut (or maximum st-flow or vertex connectivity) %DIFDELCMD < \ge %%% k otherwise.
Edit intent: coherence
Original:  This result stems from an application of the network coding method by Cheung, Lau, and Leung SICOMP 2013%DIFDELCMD < ] %%% to vertex-capacitated digraphs.
Revised:  This result stems from an application of the network coding method by Cheung, Lau, and Leung %DIFDELCMD < ] %%% to vertex-capacitated digraphs.
Edit intent: coherence
Original:  The second algorithm is faster on dense DAGs and runs in time O((k\log n)^{4^k+o(k)} n^{\omega}).
Revised:  The second algorithm can be faster on dense DAGs and runs in time O((k\log n)^{4^k+o(k)} n^{\omega}).
Edit intent: clarity
Original:  Notice that a solution even to kAPMVC, for any k%DIFDELCMD < \ge %%% 1, implies a solution to triangle finding and to transitive closure: thus, our bounds for k=o(\log n) and for k=o(\log\log n), are tight up to subpolynomial factors in n, where the former applies to combinatorial algorithms Abboud and Williams, FOCS 2014%DIFDELCMD < ]%%% . Our results rule out that kAPMVC can be solved as efficiently as a transitive closure computation for all k.
Revised:  %DIFDELCMD < \ge %%% 1, implies a solution to triangle finding and to transitive closure: thus, our bounds for k=o(\log n) and for k=o(\log\log n), are tight up to subpolynomial factors in n, where the former applies to combinatorial algorithms Abboud and Williams, FOCS 2014%DIFDELCMD < ]%%% . Our results rule out that kAPMVC can be solved as efficiently as a transitive closure computation for all k.
Edit intent: clarity
Original:  Notice that a solution even to kAPMVC, for any k%DIFDELCMD < \ge %%% 1, implies a solution to triangle finding and to transitive closure: thus, our bounds for k=o(\log n) and for k=o(\log\log n), are tight up to subpolynomial factors in n, where the former applies to combinatorial algorithms Abboud and Williams, FOCS 2014%DIFDELCMD < ]%%% . Our results rule out that kAPMVC can be solved as efficiently as a transitive closure computation for all k.
Revised:  Notice that a solution even to kAPMVC, for any k%DIFDELCMD < \ge %%% Abboud and Williams, FOCS 2014%DIFDELCMD < ]%%% . Our results rule out that kAPMVC can be solved as efficiently as a transitive closure computation for all k.
Edit intent: coherence
Original: In this paper , we consider the relation between the spectrum and the number of short cycles in large graphs.
Revised: In this paper we consider the relation between the spectrum and the number of short cycles in large graphs.
Edit intent: fluency
Original:  This is deduced by way of an analogous spectral rigidity theorem proved for certain infinite sophic graphs .
Revised:  This is deduced by way of an analogous theorem proved for certain infinite sophic graphs .
Edit intent: clarity
Original: A number of recent works have studied algorithms for entrywise \ell_p-low rank approximation, namely  algorithms which given an n \times d matrix A (with n \geq d), output a rank-k matrix B minimizing \|A-B\|_p^p=i,j|A_{i,j} - B _{i,j}|^p .
Revised: A number of recent works have studied algorithms for entrywise \ell_p-low rank approximation, namely , algorithms which given an n \times d matrix A (with n \geq d), output a rank-k matrix B minimizing \|A-B\|_p^p=i,j|A_{i,j} - B _{i,j}|^p .
Edit intent: fluency
Original: A number of recent works have studied algorithms for entrywise \ell_p-low rank approximation, namely  algorithms which given an n \times d matrix A (with n \geq d), output a rank-k matrix B minimizing \|A-B\|_p^p=i,j|A_{i,j} - B _{i,j}|^p .
Revised: A number of recent works have studied algorithms for entrywise \ell_p-low rank approximation, namely algorithms which given an n \times d matrix A (with n \geq d), output a rank-k matrix B minimizing \|A-B\|_p^p=i,j|A_{i,j} -B _{i,j}|^p .
Edit intent: fluency
Original:  We show the following:\neq B_{i,j}] for p=0. } On the algorithmic side, for p \in (0,2), we give the first n^{\text{poly(k/\epsilon)} time } (1+\epsilon)-approximation algorithm .
Revised:  We show the following:\neq B_{i,j}] for p=0. } On the algorithmic side, for p \in (0,2), we give the first (k/\epsilon)} time } (1+\epsilon)-approximation algorithm .
Edit intent: clarity
Original:  For p = 0, there are various problem formulations, a common one being the binary setting for which A\in\{ 0, 1\}^{n\times d and B = U \cdot V, where U\in\{0,1\}^{n \times k} and V\in\{0,1\}^{k \times d}.
Revised:  For p = 0, 1\}^{n\times d and B = U \cdot V, where U\in\{0,1\}^{n \times k} and V\in\{0,1\}^{k \times d}.
Edit intent: coherence
Original:  For p = 0, there are various problem formulations, a common one being the binary setting for which A\in\{ 0, 1\}^{n\times d and B = U \cdot V, where U\in\{0,1\}^{n \times k} and V\in\{0,1\}^{k \times d}.
Revised:  For p = 0, there are various problem formulations, a common one being the binary setting for which A\in\{ 0, and B = U \cdot V, where U\in\{0,1\}^{n \times k} and V\in\{0,1\}^{k \times d}.
Edit intent: fluency
Original:  We }  give the first PTAS for what we call the Generalized Binary \ell_0-Rank-k Approximation problem, for which these variants are special cases .
Revised:  We } we give the first PTAS for what we call the Generalized Binary \ell_0-Rank-k Approximation problem, for which these variants are special cases .
Edit intent: fluency
Original:  Our algorithm runs in time (1/\epsilon)^{2^{O(k)}/\epsilon^{2}} \cdot nd \cdot \log ^{ 2^k%DIFDELCMD < \MBLOCKRIGHTBRACE %%% d. For the specific case of finite fields of constant size, we obtain an alternate algorithm with time n \cdot d^{poly(k/\epsilon)
Revised:  Our algorithm runs in time (1/\epsilon)^{2^{O(k)}/\epsilon^{2}} \cdot nd \cdot \log ^{ %DIFDELCMD < \MBLOCKRIGHTBRACE %%% d. For the specific case of finite fields of constant size, we obtain an alternate algorithm with time n \cdot d^{poly(k/\epsilon)
Edit intent: clarity
Original:   On the hardness front , for p \in (1,2), we show under the Small Set Expansion Hypothesis and  Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2^{k^{\delta} for a constant \delta }  > 0 , showing an exponential dependence on k is necessary.
Revised:  . On the hardness front , for p \in (1,2), we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2^{k^{\delta} for a constant \delta } > 0 , showing an exponential dependence on k is necessary.
Edit intent: fluency
Original:   On the hardness front , for p \in (1,2), we show under the Small Set Expansion Hypothesis and  Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2^{k^{\delta} for a constant \delta }  > 0 , showing an exponential dependence on k is necessary.
Revised:  On the hardness of approximation side , for p \in (1,2), we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2^{k^{\delta} for a constant \delta } > 0 , showing an exponential dependence on k is necessary.
Edit intent: clarity