wanyu/IteraTeR_full_sent huggingface dataset:
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% ^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } walk and discrete variants. ( 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ) For n points in \mathbb{R ] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: clarity
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } SODA, 2008] ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ], and Bamberger and Krahmer ] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] arXiv, 2017] ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ]. Our algorithm is memory-optimal, and outperforms existing algorithms in regimes when n is sufficiently large and the distortion parameter is sufficiently small. In particular, this confirms a conjecture of Ailon and Chazelle ] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] STOC, 2006] in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in a stronger form. (2) The same construction gives a simple transform with optimal Restricted Isometry Property (RIP) which can be applied in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, }  1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed }  ] ] ]  in time O(d\log{d} + \min\{d\log{n, \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Revised:  given a set X of n points in \mathbb{R^{d} and an error parameter \epsilon, this is a linear transformation \Psi: R^{d} }%DIFDELCMD < \to %%% \mathbb{R^{O(\epsilon^{-2}n)} such that \|\Psi x\|_{2} \in (1- \epsilon, } 1 +\epsilon)\cdot \|x\|_{2 for all x\in X, and such that for each x\in X, \Psi x can be computed } ] ] ] in time O(d\log{d} , \epsilon^{-2}\log^{3}{n}\log^{3}(\epsilon^{-1}\log{n}) \}) with only a constant amount of memory overhead.
Edit intent: meaning-changed
Original:  In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle}  ] ] .
Revised:  In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle} ) for essentially the same range of sparsity as in the best-known such transform due to Ailon and Rauhut ] ] .
Edit intent: meaning-changed
Original:  In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle}  ] ] .
Revised:  In some parameter regimes, our algorithm is best known, and essentially confirms a conjecture of Ailon and Chazelle} ]. (3) We show that by fixing the angle in the Kac walk to be \pi/4 throughout, one obtains optimal JL and RIP transforms with almost the same running time, thereby confirming -- up to a \log\log{d ] .
Edit intent: meaning-changed
Original:  We find that our model provides a surprisingly good fit to the currently-available data . It predicts that the disease will run its full course over six to seven months , starting from the date on which the world URLanisation provided the first ' Situation report' (21 January 2020 - day 1).
Revised:  We find that our model provides a surprisingly good fit to the currently-available data , which exhibits a cross-over from exponential to power-law growth, as lock-down measures begin to take effect. Before these measures, our model predicts exponential growth from day 30 to 69 , starting from the date on which the world URLanisation provided the first ' Situation report' (21 January 2020 - day 1).
Edit intent: meaning-changed
Original:  It predicts that the disease will run its full course over six to seven months , starting from the date on which the world URLanisation provided the first ' Situation report' (21 January 2020 - day 1).
Revised:  It predicts that the disease will run its full course over six to seven months , starting from the date on which the world URLanisation provided the first ` Situation report' (21 January 2020 - day 1).
Edit intent: fluency
Original:  Unless drastic new interventions become possible , the global number of new cases is predicted to peak on day 129 (about the middle of May 2020), with an estimated 65 million new cases per day.
Revised:  Unless drastic new interventions become possible , the global number of new cases is predicted to peak on day 133 (about the middle of May 2020), with an estimated 65 million new cases per day.
Edit intent: meaning-changed
Original:  Unless drastic new interventions become possible , the global number of new cases is predicted to peak on day 129 (about the middle of May 2020), with an estimated 65 million new cases per day.
Revised:  Unless drastic new interventions become possible , the global number of new cases is predicted to peak on day 129 (about the middle of May 2020), with an estimated 60 million new cases per day.
Edit intent: meaning-changed
Original:  Since our simple model fits the available global data so well, we suggest that the measures being taken so far to contain the pandemic may be ineffective in curbing the global  spread of the virus .
Revised:  If current lock-down measures can be maintained, our model predicts power law growth from day 69 onward. Such growth is comparatively slow and would have to continue for several decades before a sufficient number of people (at least 23\% of the global spread of the virus .
Edit intent: meaning-changed
Original:  Since our simple model fits the available global data so well, we suggest that the measures being taken so far to contain the pandemic may be ineffective in curbing the global  spread of the virus .
Revised:  Since our simple model fits the available global data so well, we suggest that the measures being taken so far to contain the pandemic may be ineffective in curbing the global population) have developed immunity to the disease through being infected. Lock-down measures appear to be very effective in postponing the unimaginably large peak in the daily number of new cases that would occur in the absence of any interventions. However, should these measure be relaxed, the spread of the virus .
Edit intent: meaning-changed
Original:  Since our simple model fits the available global data so well, we suggest that the measures being taken so far to contain the pandemic may be ineffective in curbing the global  spread of the virus .
Revised:  Since our simple model fits the available global data so well, we suggest that the measures being taken so far to contain the pandemic may be ineffective in curbing the global spread of the disease will most likely revert back to its original exponential growth pattern .
Edit intent: meaning-changed
Original:  A bistellar flip on a partial triangulation either flips an edge  , removes a non-extreme point of degree 3, or adds a point in P \setminus P' as vertex of degree 3.
Revised:  A bistellar flip on a partial triangulation flips an edge , removes a non-extreme point of degree 3, or adds a point in P \setminus P' as vertex of degree 3.
Edit intent: clarity
Original:  A bistellar flip on a partial triangulation either flips an edge  , removes a non-extreme point of degree 3, or adds a point in P \setminus P' as vertex of degree 3.
Revised:  A bistellar flip on a partial triangulation either flips an edge (an edge flip) , removes a non-extreme point of degree 3, or adds a point in P \setminus P' as vertex of degree 3.
Edit intent: meaning-changed
Original:  A bistellar flip on a partial triangulation either flips an edge  , removes a non-extreme point of degree 3, or adds a point in P \setminus P' as vertex of degree 3.
Revised:  A bistellar flip on a partial triangulation either flips an edge , removes a non-extreme point of degree 3, or adds a point in P \ P' as vertex of degree 3.
Edit intent: meaning-changed
Original:  For sets P of n points in  general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Revised:  For sets of n points in general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Edit intent: clarity
Original:  For sets P of n points in  general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Revised:  For sets P of n points in the plane in general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Edit intent: meaning-changed
Original:  For sets P of n points in  general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Revised:  For sets P of n points in general position, we show that the edge flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Edit intent: meaning-changed
Original:  For sets P of n points in  general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Revised:  For sets P of n points in general position, we show that the bistellar flip graph is ( n/2-2 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos).
Edit intent: meaning-changed
Original:  For sets P of n points in  general position, we show that the bistellar flip graph is ( n-3 )-connected, answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos). This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  For sets P of n points in general position, we show that the bistellar flip graph is ( n-3 )-connected, and the bistellar flip graph is (n-3)-connected; both results are tight. The latter bound matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: meaning-changed
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for the subfamily of regular triangulations, ie. partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: meaning-changed
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: meaning-changed
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull . Here (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: coherence
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the late 80s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: meaning-changed
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope due to Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Edit intent: clarity
Original:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov , Zelevinsky ) and Balinski's Theorem.
Revised:  This matches the situation for regular triangulations ( partial triangulations obtained by lifting the points and projecting the lower convex hull ), where (n-3)-connectivity has been known since the 1980s via the secondary polytope ( Gelfand, Kapranov Zelevinsky and Balinski's Theorem.
Edit intent: fluency