wanyu/IteraTeR_full_sent huggingface dataset:
Original:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi}  1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Revised:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% time algorithm that outputs a list containing a\hat{\Pi} \hat{\Pi} 1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Edit intent: meaning-changed
Original:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi}  1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Revised:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} satisfying \|\hat{\Pi} 1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Edit intent: meaning-changed
Original:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi}  1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Revised:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi} -\Pi_*\|_F \leq O( 1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Edit intent: meaning-changed
Original:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi}  1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Revised:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi} 1/ samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Edit intent: clarity
Original:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi}  1/ \alpha^2) samples and runs in time n^{\log (r \kappa) \tilde{O(1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Revised:  The algorithm needs n=d ^{ \log ( r \kappa ) %DIFDELCMD < \tilde{O}%%% (\hat{\Pi} \hat{\Pi} 1/ \alpha^2) (1/\alpha^4)} time where \kappa is the ratio of the largest to smallest non-zero eigenvalues of \Sigma_*.
Edit intent: clarity
Original:  Our algorithm builds on the recently developed framework for list-decodable learning via the sum-of-squares (SoS)method }%DIFDELCMD < [%%% KKK'19, RY'20 with some key technical and conceptual advancements.
Revised:  Our algorithm builds on the recently developed framework for list-decodable learning via the sum-of-squares (SoS)method }%DIFDELCMD < [%%% with some key technical and conceptual advancements.
Edit intent: coherence
Original:  Our algorithm builds on the recently developed framework for list-decodable learning via the sum-of-squares (SoS)method }%DIFDELCMD < [%%% KKK'19, RY'20 with some key technical and conceptual advancements. Our key conceptual contribution involves showing a (SoS "certified") lower bound on the eigenvalues of covariances of arbitrary small subsamples of an i.i.d. sample of a certifiably anti-concentrated distribution. One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  Our algorithm builds on the recently developed framework for list-decodable learning via the sum-of-squares (SoS)method }%DIFDELCMD < [%%% KKK'19, RY'20 %DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{ Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% . In a concurrent and independent work , to have }\emph{ Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: coherence
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 \alpha). Our result only needs \mathcal{D to have }\emph{ Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{certifiably hypercontractive Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{ degree 2 polynomials - a condition satisfied by a much broader family of distributions in contrast to certifiable anticoncentration. As a result, in addition to Gaussians, our algorithm applies to uniform distribution on the hypercube and q-ary cubes and arbitrary product distributions with subgaussian marginals. Prior work ( Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{ Raghavendra and Yau RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{ Raghavendra and Yau proved related results for list-decodable subspace recovery , 2020) had identified such distributions as potential hard examples as such distributions do not exhibit strong enough anti-concentration. When \mathcal{D satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Edit intent: meaning-changed
Original:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
Revised:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in%DIFDELCMD < [%%% KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{ Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi}-\Pi_*\|_F \leq \eta for any arbitrary \eta > 0 in d^{O(poly(1/\alpha) + \log (1/\eta)) .
Edit intent: meaning-changed
Original:   This is due to the optimality of coding and learning schemes, which have yet to be unveiled. The formulation of biologically inspired networks capable to perform complex computations can mediate a synergetic interaction between Machine Learning and Neuroscience bringing to mutual benefits and helping to improve our understanding of biological and artificial intelligence.
Revised:  This is due to the optimality of coding and learning schemes, which have yet to be clearly understood. This naturally challenges the formulation of biologically inspired networks capable to perform complex computations can mediate a synergetic interaction between Machine Learning and Neuroscience bringing to mutual benefits and helping to improve our understanding of biological and artificial intelligence.
Edit intent: meaning-changed
Original:  The formulation of biologically inspired networks capable to perform complex computations can mediate a synergetic interaction between Machine Learning and Neuroscience bringing to mutual benefits and helping to improve our understanding of biological and artificial intelligence. Even though several models have been proposed , it remains a challenging task to design RSNNs which use biologically plausible mechanisms .
Revised:  The formulation of biologically inspired networks capable to perform complex computations can mediate a synergetic interaction between Machine Learning and Neuroscience bringing to mutual benefits and helping to improve our understanding of biological intelligence and the efficiency of artificial ones. Several spiking network models have been proposed , it remains a challenging task to design RSNNs which use biologically plausible mechanisms .
Edit intent: meaning-changed
Original:  Even though several models have been proposed , it remains a challenging task to design RSNNs which use biologically plausible mechanisms . We propose a general probabilistic framework which relies on the principle of maximizing the likelihood for the network to solve the task .
Revised:  Even though several models have been proposed , it remains a challenging task to design RSNNs which use biologically plausible mechanisms capable of solving complex temporal tasks. We use a general probabilistic framework which relies on the principle of maximizing the likelihood for the network to solve the task .
Edit intent: meaning-changed
Original:  This principle permits to analytically work out an explicit and completely local plasticity rulesupporting the efficient solution of several tasks. We show that the learning algorithm can be achieved in a very few iterations, and that the online approximation of the likelihood maximization is extremely beneficial to fast learning .
Revised:  This principle permits to analytically work out an explicit and biologically plausible plasticity rule. Here we propose a novel target-based learning scheme in which such a rule can be used to efficiently train a RSNN to solve several temporal tasks such as learning multidimensional trajectory and an implementation of the temporal XOR. We finally show that an online approximation of the likelihood maximization is extremely beneficial to fast learning .
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for a family of convex polygons in the plane in O((p-q+1)n^{4/3}\log^{ 2 + \epsilon (n ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane with a total of n vertices in O((p-q+1)n^{4/3}\log^{ 2 + \epsilon (n ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 8 + \epsilon (n ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 2 n(\log\log n)^{1/3 + \epsilon (n ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 2 + (n ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 2 + \epsilon np^2 ) + p^2\log(p) ) expected time.
Edit intent: meaning-changed
Original:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane  in O((p-q+1)n^{4/3}\log^{ 2  + \epsilon (n ) + p^2\log(p) ) expected time.
Revised:  We show how such a stabbing set can be computed for n convex polygons of constant size in the plane in O((p-q+1)n^{4/3}\log^{ 2 + \epsilon (n ) expected time.
Edit intent: clarity
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon } n(\log\log n)^{1/6} } + p^4 ) expected time.
Edit intent: clarity
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For convex polyhedra in R^3, we get an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon } n(\log\log n)^{1/6} } + p^4 ) expected time.
Edit intent: clarity
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 5 +\epsilon } n(\log\log n)^{1/6} } + p^4 ) expected time.
Edit intent: fluency
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 /2 } n(\log\log n)^{1/6} } + p^4 ) expected time.
Edit intent: meaning-changed
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon } \log^{10 n(\log\log n)^{1/6} } + p^4 ) expected time.
Edit intent: meaning-changed
Original:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon }  n(\log\log n)^{1/6} } + p^4 ) expected time.
Revised:  For convex polyhedra in R^3, the method yields an algorithm running in O((p-q+1)n^{ 13/ 5 +\epsilon } n(\log\log n)^{1/6} } + np^3 ) expected time.
Edit intent: meaning-changed
Original:  We also  show that analogous results of the Hadwiger and Debrunner (p,q)-theorem hold in other settings, such as convex sets in \mathbb{R}^d\times\mathbb{Z}^k or abstract convex geometries.
Revised:  We also investigate other conditions on convex polygons for which our algorithm can find a fixed number of points stabbing them. Finally, we show that analogous results of the Hadwiger and Debrunner (p,q)-theorem hold in other settings, such as convex sets in \mathbb{R}^d\times\mathbb{Z}^k or abstract convex geometries.
Edit intent: meaning-changed
Original:  We study the maximal independent set (MIS) and maximum independent set ( MAX-IS ) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs.
Revised: Map labeling is a classical problem in cartography and geographic information systems that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. We study the maximal independent set (MIS) and maximum independent set ( MAX-IS ) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs.
Edit intent: meaning-changed
Original:  We study the maximal independent set (MIS) and maximum independent set ( MAX-IS ) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs.
Revised:  We study the maximal independent set (MIS) and maximum independent set ( Max-IS ) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs.
Edit intent: clarity
Original:  We study the maximal independent set (MIS) and maximum independent set ( MAX-IS ) problems on dynamic sets of O(n) axis-parallel rectangles, which can be modeled as dynamic rectangle intersection graphs. We consider the fully dynamic vertex update (insertion/deletion ) modelfor two types of rectangles: (i) uniform height and width and (ii) uniform height and arbitrary width .
Revised:  We study the maximal independent set (MIS) and maximum independent set ( MAX-IS ) problems on fully dynamic (insertion/deletion ) modelfor two types of rectangles: (i) uniform height and width and (ii) uniform height and arbitrary width .
Edit intent: clarity
Original:  We consider the fully dynamic vertex update (insertion/deletion ) modelfor two types of rectangles: (i) uniform height and width and (ii) uniform height and arbitrary width . These types of dynamic vertex update problems arise, e.g., in interactive map labeling.
Revised:  We consider the fully dynamic vertex update (insertion/deletion ) modelfor two types of rectangles: (i) uniform height and width and (ii) uniform height and arbitrary width ; both settings can be modeled as rectangle intersection graphs.
Edit intent: meaning-changed
Original:  We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate MAX-IS ) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time.
Revised:  We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate Max-IS ) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time.
Edit intent: fluency
Original:  This breaks the natural barrier of O (\Delta) update time (where \Delta is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018).
Revised:  This breaks the natural barrier of \Omega (\Delta) update time (where \Delta is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018).
Edit intent: meaning-changed