wanyu/IteraTeR_v2 huggingface dataset:
Original:  In case of high-frequency phase variations ESPIRiT yields additional set of maps which capture the high-frequency phase component.
Revised:  In case of high-frequency phase variations ESPIRiT yields additional set of maps that indicates the existence of a high-frequency phase component.
Edit intent: clarity
Original:  Conclusion: The extended VCC-ESPIRiT is a new promising framework for estimation of phase information and phase-constrained imaging.
Revised:  Conclusion: The extended VCC-ESPIRiT is a useful tool for phase-constrained imaging.
Edit intent: clarity
Original: Scholars have engaged in a longstanding debate about how best to measure residential segregation , and scores of indexes have been developed in response .
Revised: Residential segregation is defined in a variety of ways to address a common concern: to what extent do social groups reside in separate or distinct places. The spatial pattern of segregation varies widely across cities, and distinct spatial patterns can be generated by different mechanisms and have different consequences for residents and their communities .
Edit intent: clarity
Original:  I demonstrate the contribution of my approach using a series of simulated cities .
Revised:  I demonstrate the contribution of my method using a series of simulated cities .
Edit intent: clarity
Original:  However, the methods commonly employed to measure segregation ignore how a city is spatially organized.
Revised:  However, the methods used to measure segregation ignore how a city is spatially organized.
Edit intent: clarity
Original: Given a d-dimensional weighted flag variety w\Sigma(\mu,u)
Revised: Given a weighted flag variety w\Sigma(\mu,u)
Edit intent: clarity
Original:  corresponding to chosen fixed parameters \mu and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections of w\Sigma(\mu,u) such that d > n%DIFDELCMD < \ge %%% 2. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted G_2 variety.
Revised:  corresponding to chosen fixed parameters \mu and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections in w\Sigma(\mu,u) such that d > n%DIFDELCMD < \ge %%% 2. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted G_2 variety.
Edit intent: fluency
Original:  corresponding to chosen fixed parameters \mu and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections of w\Sigma(\mu,u) such that d > n%DIFDELCMD < \ge %%% 2. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted G_2 variety.
Revised:  corresponding to chosen fixed parameters \mu and u, we present an algorithm to compute lists of all possible projectively Gorenstein n-folds, having canonical weight k and isolated orbifold points, appearing as weighted complete intersections of w\Sigma(\mu,u) %DIFDELCMD < \ge %%% 2. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted G_2 variety.
Edit intent: clarity
Original:  The new observation (by Szybowski), that " no PC submatrix may have a worse inconsistency indicator than the given PC matrix " is an essential simplification of the axiomatization.
Revised:  It is based on the new findings that " no PC submatrix may have a worse inconsistency indicator than the given PC matrix " is an essential simplification of the axiomatization.
Edit intent: clarity
Original:  The new observation (by Szybowski), that " no PC submatrix may have a worse inconsistency indicator than the given PC matrix " is an essential simplification of the axiomatization.
Revised:  The new observation (by Szybowski), that " PC submatrix cannot have a worse inconsistency indicator than the given PC matrix " is an essential simplification of the axiomatization.
Edit intent: clarity
Original:  The new observation (by Szybowski), that " no PC submatrix may have a worse inconsistency indicator than the given PC matrix " is an essential simplification of the axiomatization.
Revised:  The new observation (by Szybowski), that " no PC submatrix may have a worse inconsistency indicator than the PC matrix containing it " is an essential simplification of the axiomatization.
Edit intent: clarity
Original:  We show that bursty influx exponentially decreases the mean escape time compared to the "usual  " , single-step influx case 0->A .
Revised:  We show that bursty influx exponentially decreases the mean escape time compared to the "usual case " , single-step influx case 0->A .
Edit intent: fluency
Original:  We show that bursty influx exponentially decreases the mean escape time compared to the "usual  " , single-step influx case 0->A .
Revised:  We show that bursty influx exponentially decreases the mean escape time compared to the "usual " of single-step influx case 0->A .
Edit intent: clarity
Original: A matrix X \in \mathbb{R^{n \times n} is positive semi-definite (PSD) if and only if it can be written as the product UU^\top, for some matrix U.
Revised: ^{n \times n} is positive semi-definite (PSD) if and only if it can be written as the product UU^\top, for some matrix U.
Edit intent: clarity
Original:  specifically, we consider }  the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Revised:  specifically, we consider } In this paper, we study the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Edit intent: clarity
Original:  specifically, we consider }  the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Revised:  specifically, we consider } the minimization of a convex function f (X) over the space of n\times n positive semidefinite matrices X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Edit intent: clarity
Original:  specifically, we consider }  the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Revised:  specifically, we consider } the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), with U being an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Edit intent: coherence
Original:  specifically, we consider }  the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix for some r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Revised:  specifically, we consider } the minimization of a convex function f over the positive semi-definite cone X\succeq 0, but via gradient descent on f(UU^\top), which is a non-convex function of U . We focus on the (empirically quite popular) approach where, for computational or statistical reasons, U is set to be an n\times r matrix and r\leq n , and correspondingly f satisfies restricted strong convexity (setting r=n recovers the exact case with global strong convexity) .
Edit intent: clarity
Original:  We develop a special choice of step size, and show that updating U via gradient descent with this choice results in linear convergence to the top-r components of the optimum of f, provided we start from a point which has constant relative distance to the optimum.
Revised:  We develop a special choice of step size, and with this choice results in linear convergence to the top-r components of the optimum of f, provided we start from a point which has constant relative distance to the optimum.
Edit intent: clarity
Original: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Revised: We study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Edit intent: clarity
Original: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Revised: In this paper, we study the minimization of a convex function f(X) over the set of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Edit intent: clarity
Original: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Revised: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semi-definite matrices , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Edit intent: clarity
Original: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Revised: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Edit intent: clarity
Original: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) where g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Revised: In this paper, we study the minimization of a convex function f(X) over the space of n\times n positive semidefinite matricesX\succeq 0 , but when the problem is recast as the non-convex problem \min_U g(U) := f(UU^\top), with U being an n\times r matrix^{n \times r} } and r \leq n.
Edit intent: coherence
Original:  We study the performance of gradient descent on g -- which we refer to as Factored Gradient Descent (FGD) -- under standard assumptions on the original function f.
Revised:  We study the performance of gradient descent on g---which we refer to as Factored Gradient Descent (FGD) -- under standard assumptions on the original function f.
Edit intent: fluency
Original:  We study the performance of gradient descent on g -- which we refer to as Factored Gradient Descent (FGD) -- under standard assumptions on the original function f.
Revised:  We study the performance of gradient descent on g -- which we refer to as Factored Gradient Descent (FGD) ---under standard assumptions on the original function f.
Edit intent: fluency
Original:  We provide a rule for selecting the step size , and with this choice  show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f -- the error after k steps  is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Revised:  We provide a rule for selecting the step size and, with this choice show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f -- the error after k steps is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Edit intent: fluency
Original:  We provide a rule for selecting the step size , and with this choice  show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f -- the error after k steps  is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Revised:  We provide a rule for selecting the step size , and with this choice , show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f -- the error after k steps is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Edit intent: fluency
Original:  We provide a rule for selecting the step size , and with this choice  show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f -- the error after k steps  is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Revised:  We provide a rule for selecting the step size , and with this choice show that the local convergence rate of FGD mirrors that of standard gradient descent on the original f : i.e., after k steps is O(1/k) for smooth f, and exponentially small in k when f is (restricted) strongly convex.
Edit intent: clarity
Original:  to the best of our knowledge, ours is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions.
Revised:  to the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions.
Edit intent: clarity
Original:  We formalise the concept of a measure of partial balance, compare several known measures on real-world and synthetic datasets , as well as investigating their axiomatic properties.
Revised:  We formalize the concept of a measure of partial balance, compare several known measures on real-world and synthetic datasets , as well as investigating their axiomatic properties.
Edit intent: fluency
Original:  We formalise the concept of a measure of partial balance, compare several known measures on real-world and synthetic datasets , as well as investigating their axiomatic properties.
Revised:  We formalise the concept of a measure of partial balance, compare several known measures on real-world and synthetic datasets and investigate their axiomatic properties.
Edit intent: coherence
Original:  We find that under all our measures, real-world networks are more balanced than random networks .
Revised:  We find that under all our measures, real-world networks are more balanced than what is expected by chance .
Edit intent: clarity
Original:  We formalize the concept of a measure of partial balance, compare several known measures on real-world and synthetic datasets and investigate their axiomatic properties.
Revised:  We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on real-world and synthetic datasets and investigate their axiomatic properties.
Edit intent: clarity
Original:  The synthetic data involves both Erd\H{os-R\'enyi and Barab\'asi-Albert }  graphs.
Revised:  The synthetic data involves s-R\'enyi and Barab\'asi-Albert } graphs.
Edit intent: clarity
Original:  Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas , such as international relations .
Revised:  Since then the idea that signed networks tend over time to become more balanced has been widely used in several application areas .
Edit intent: clarity
Original:  We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on real-world and synthetic datasets and investigate their axiomatic properties.
Revised:  We formalize the concept of a measure of partial balance, discuss various measures, compare the measures on synthetic datasets, and investigate their axiomatic properties.
Edit intent: clarity
Original:  We use both well-known datasets from the sociology literature, such as Read's New Guinean tribes, and much more recent ones involving senate bill co-sponsorship.
Edit intent: coherence
Original:  We show that some measures behave better than others in terms of axioms , computational tractability and ability to differentiate between graphs.
Revised:  We show that some measures behave better than others in terms of axioms and ability to differentiate between graphs.
Edit intent: clarity
Original: Is the enemy of an enemy necessarily a friend , or a friend of a friend a friend ? If not, to what extent does this tend to hold?
Revised: Is the enemy of an enemy necessarily a friend ? If not, to what extent does this tend to hold?
Edit intent: coherence
Original:  However  investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice.
Revised:  However , investigation of this hypothesis has been complicated by the lack of a standard measure of partial balance, since complete balance is almost never achieved in practice.
Edit intent: fluency
Original:  The synthetic data involves Erdo s-R\'enyi and specially structured random graphs.
Revised:  The synthetic data involves Erdo s-R\'{e and specially structured random graphs.
Edit intent: fluency
Original:  The synthetic data involves Erdo s-R\'{e and specially structured random graphs.
Revised:  The synthetic data involves Erdo s-R\'enyi and specially structured random graphs.
Edit intent: fluency
Original:  This new perspective not only allows to rationalize the normal cellular functions, but also helps to understand the emergence of pathological states, in which the epigenetic landscape becomes disfunctional .
Revised:  This new perspective not only allows to rationalize the normal cellular functions, but also helps to understand the emergence of pathological states, in which the epigenetic landscape becomes dysfunctional .
Edit intent: fluency
Original:  These include not only the change in the molecular properties associated with the chemical modifications of DNA and histone proteins - such as methylation and acetylation - but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Revised:  These include not only the changes in the molecular properties associated with the chemical modifications of DNA and histone proteins - such as methylation and acetylation - but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Edit intent: fluency
Original:  These include not only the change in the molecular properties associated with the chemical modifications of DNA and histone proteins - such as methylation and acetylation - but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Revised:  These include not only the change in the molecular properties associated with the chemical modifications of DNA and histone proteins , such as methylation and acetylation - but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Edit intent: fluency
Original:  These include not only the change in the molecular properties associated with the chemical modifications of DNA and histone proteins - such as methylation and acetylation - but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Revised:  These include not only the change in the molecular properties associated with the chemical modifications of DNA and histone proteins - such as methylation and acetylation , but also less conventional ones, such as the physics that governs the three-dimensional organization of the genome in cell nuclei.
Edit intent: fluency
Original: We propose a method to decompose a multivariate dynamical system into weakly-coupled modules based on the idea that module boundaries constrain the spread of perturbations.
Revised: We propose a method to decompose dynamical systems based on the idea that module boundaries constrain the spread of perturbations.
Edit intent: clarity
Original: We propose a method to decompose a multivariate dynamical system into weakly-coupled modules based on the idea that module boundaries constrain the spread of perturbations.
Revised: We propose a method to decompose a multivariate dynamical system into weakly-coupled modules based on the idea that modules constrain the spread of perturbations.
Edit intent: clarity
Original:  Using a novel quality function called 'perturbation modularity', we find system coarse-grainings that optimally separate the dynamics of perturbation spreading into fast intra-modular and slow inter-modular components.
Revised:  We find partitions of system variables that maximize 'perturbation modularity', we find system coarse-grainings that optimally separate the dynamics of perturbation spreading into fast intra-modular and slow inter-modular components.
Edit intent: clarity
Original:  Due to this, we are able to capture the variation of modular organization across states , timescales, and in response to different perturbations, aspects of modularity which are all relevant to real-world dynamical systems.
Revised:  Due to this, we are able to capture the variation of modular organization across different system states, time scales , timescales, and in response to different perturbations, aspects of modularity which are all relevant to real-world dynamical systems.
Edit intent: fluency