Robust vertex enumeration for convex hulls in high dimensions
The problem of computing the vertices of the convex hull of a given input set S = { v i ∈ R m : i = 1 , ⋯ , n } is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Tria...
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| Published in: | Annals of operations research Vol. 295; no. 1; pp. 37 - 73 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01.12.2020
Springer Springer Nature B.V |
| Subjects: | |
| ISSN: | 0254-5330, 1572-9338 |
| Online Access: | Get full text |
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| Abstract | The problem of computing the vertices of the convex hull of a given input set
S
=
{
v
i
∈
R
m
:
i
=
1
,
⋯
,
n
}
is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present
All Vertex Triangle Algorithm
(AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset
S
¯
of all
K
vertices of the convex hull of
S
so that the convex hull of the approximate subset of vertices is as close to
conv
(
S
) as desired. On the other hand, assuming a known lower bound
γ
on the ratio
Γ
∗
/
R
, where
Γ
∗
the minimum of the distances from each vertex to the convex hull of the remaining vertices and
R
the diameter of
S
, AVTA can recover all of
S
¯
. Furthermore, assuming that instead of
S
the input is an
ε
-perturbation of
S
,
S
¯
ε
, where
‖
v
i
-
v
i
ε
‖
≤
ε
R
, AVTA can compute approximation to
c
o
n
v
(
S
¯
ε
)
, to any prescribed accuracy. Also, given a lower bound to the ratio
Σ
∗
/
R
, where
Σ
∗
is the minimum of the distances from each vertex to the convex hull of the remaining point of
S
, AVTA can recover all of
S
¯
ε
. We show
Σ
∗
≥
ρ
∗
Γ
∗
/
R
, where
ρ
∗
is the minimum distance between distinct pair of points in
S
and prove the following main results:
Given any
t
∈
(
0
,
1
)
, AVTA computes a subset
S
¯
t
of
S
¯
of cardinality
K
(
t
)
in
O
(
n
K
(
t
)
(
m
+
t
-
2
)
)
operations so that for any
p
∈
c
o
n
v
(
S
)
its Euclidean distance to
c
o
n
v
(
S
¯
t
)
is at most
tR
.
Given
γ
≤
γ
∗
=
Γ
∗
/
R
, AVTA computes
S
¯
in
O
(
n
K
(
m
+
γ
-
2
)
)
operations.
If
K
is known, the complexity of AVTA is
O
(
n
K
(
m
+
γ
∗
-
2
)
log
(
γ
∗
-
1
)
)
.
Assuming instead of
S
, its
ε
-perturbation,
S
ε
is given, we prove
Given any
t
∈
(
0
,
1
)
, AVTA computes a subset
S
¯
ε
t
⊂
S
¯
ε
of cardinality
K
ε
(
t
)
in
O
(
n
K
ε
(
t
)
(
m
+
t
-
2
)
)
operations so that for any
p
∈
c
o
n
v
(
S
)
its distance to
c
o
n
v
(
S
¯
ε
t
)
is at most
(
t
+
ε
)
R
.
Given
σ
∈
[
4
ε
,
σ
∗
=
Γ
∗
/
R
]
, AVTA computes
S
¯
ε
in
O
(
n
K
ε
(
m
+
σ
-
2
)
)
operations, where
K
≤
K
ε
≤
n
.
If
γ
≤
γ
∗
=
Γ
∗
/
R
is known satisfying
4
ε
≤
γ
ρ
∗
/
R
,
AVTA
computes
S
¯
ε
in
O
(
n
K
ε
(
m
+
(
γ
ρ
∗
)
-
2
)
)
operations.
Given
σ
∈
[
4
ε
,
σ
∗
]
, if
K
is known, AVTA computes
S
¯
ε
in
O
(
n
K
(
m
+
σ
∗
-
2
)
log
(
σ
∗
-
1
)
)
operations.
We also consider the application of AVTA in the recovery of vertices through the projection of
S
or
S
ε
under a Johnson–Lindenstrauss randomized linear projection
L
:
R
m
→
R
m
′
. Denoting
U
=
L
(
S
)
and
U
ε
=
L
(
S
ε
)
, by relating the robustness parameters of
conv
(
U
) and
c
o
n
v
(
U
ε
)
to those of
conv
(
S
) and
c
o
n
v
(
S
ε
)
, we derive analogous complexity bounds for probabilistic computation of the vertex set of
conv
(
U
) or those of
c
o
n
v
(
U
ε
)
, or an approximation to them. Finally, we apply AVTA to design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models, our new algorithm leads to significantly better reconstruction of the topic-word matrix than state of the art approaches of Arora et al. (International conference on machine learning, pp 280–288, 2013) and Bansal et al. (Advances in neural information processing systems, pp 1997–2005, 2014). Additionally, we provide a robust analysis of AVTA and empirically demonstrate that it can handle larger amounts of noise than existing methods. For non-negative matrix factorization we show that AVTA is competitive with existing methods that are specialized for this task in Arora et al. (Proceedings of the forty-fourth annual ACM symposium on theory of computing, ACM, pp 145–162, 2012a). We also contrast AVTA with Blum et al. (Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, Society for Industrial and Applied Mathematics, pp 548–557, 2016)
Greedy Clustering
coreset algorithm for computing approximation to the set of vertices and argue that not only there are regimes where AVTA outperforms that algorithm but it can also be used as a pre-processing step to their algorithm. Thus the two algorithms in fact complement each other. |
|---|---|
| AbstractList | The problem of computing the vertices of the convex hull of a given input set [Formula omitted] is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Triangle Algorithm (AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset [Formula omitted] of all K vertices of the convex hull of S so that the convex hull of the approximate subset of vertices is as close to conv(S) as desired. On the other hand, assuming a known lower bound [Formula omitted] on the ratio [Formula omitted], where [Formula omitted] the minimum of the distances from each vertex to the convex hull of the remaining vertices and R the diameter of S, AVTA can recover all of [Formula omitted]. Furthermore, assuming that instead of S the input is an [Formula omitted]-perturbation of S, [Formula omitted], where [Formula omitted], AVTA can compute approximation to [Formula omitted], to any prescribed accuracy. Also, given a lower bound to the ratio [Formula omitted], where [Formula omitted] is the minimum of the distances from each vertex to the convex hull of the remaining point of S, AVTA can recover all of [Formula omitted]. We show [Formula omitted], where [Formula omitted] is the minimum distance between distinct pair of points in S and prove the following main results:
Given any [Formula omitted], AVTA computes a subset [Formula omitted] of [Formula omitted] of cardinality [Formula omitted] in [Formula omitted] operations so that for any [Formula omitted] its Euclidean distance to [Formula omitted] is at most tR. The problem of computing the vertices of the convex hull of a given input set S={vi∈Rm:i=1,⋯,n} is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Triangle Algorithm (AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset S¯ of all K vertices of the convex hull of S so that the convex hull of the approximate subset of vertices is as close to conv(S) as desired. On the other hand, assuming a known lower bound γ on the ratio Γ∗/R, where Γ∗ the minimum of the distances from each vertex to the convex hull of the remaining vertices and R the diameter of S, AVTA can recover all of S¯. Furthermore, assuming that instead of S the input is an ε-perturbation of S, S¯ε, where ‖vi-viε‖≤εR, AVTA can compute approximation to conv(S¯ε), to any prescribed accuracy. Also, given a lower bound to the ratio Σ∗/R, where Σ∗ is the minimum of the distances from each vertex to the convex hull of the remaining point of S, AVTA can recover all of S¯ε. We show Σ∗≥ρ∗Γ∗/R, where ρ∗ is the minimum distance between distinct pair of points in S and prove the following main results: Given any t∈(0,1), AVTA computes a subset S¯t of S¯ of cardinality K(t) in O(nK(t)(m+t-2)) operations so that for any p∈conv(S) its Euclidean distance to conv(S¯t) is at most tR.Given γ≤γ∗=Γ∗/R, AVTA computes S¯ in O(nK(m+γ-2)) operations.If K is known, the complexity of AVTA is O(nK(m+γ∗-2)log(γ∗-1)).Assuming instead of S, its ε-perturbation, Sε is given, we prove Given any t∈(0,1), AVTA computes a subset S¯εt⊂S¯ε of cardinality Kε(t) in O(nKε(t)(m+t-2)) operations so that for any p∈conv(S) its distance to conv(S¯εt) is at most (t+ε)R.Given σ∈[4ε,σ∗=Γ∗/R], AVTA computes S¯ε in O(nKε(m+σ-2)) operations, where K≤Kε≤n.If γ≤γ∗=Γ∗/R is known satisfying 4ε≤γρ∗/R, AVTA computes S¯ε in O(nKε(m+(γρ∗)-2)) operations.Given σ∈[4ε,σ∗], if K is known, AVTA computes S¯ε in O(nK(m+σ∗-2)log(σ∗-1)) operations.We also consider the application of AVTA in the recovery of vertices through the projection of S or Sε under a Johnson–Lindenstrauss randomized linear projection L:Rm→Rm′. Denoting U=L(S) and Uε=L(Sε), by relating the robustness parameters of conv(U) and conv(Uε) to those of conv(S) and conv(Sε), we derive analogous complexity bounds for probabilistic computation of the vertex set of conv(U) or those of conv(Uε), or an approximation to them. Finally, we apply AVTA to design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models, our new algorithm leads to significantly better reconstruction of the topic-word matrix than state of the art approaches of Arora et al. (International conference on machine learning, pp 280–288, 2013) and Bansal et al. (Advances in neural information processing systems, pp 1997–2005, 2014). Additionally, we provide a robust analysis of AVTA and empirically demonstrate that it can handle larger amounts of noise than existing methods. For non-negative matrix factorization we show that AVTA is competitive with existing methods that are specialized for this task in Arora et al. (Proceedings of the forty-fourth annual ACM symposium on theory of computing, ACM, pp 145–162, 2012a). We also contrast AVTA with Blum et al. (Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, Society for Industrial and Applied Mathematics, pp 548–557, 2016) Greedy Clustering coreset algorithm for computing approximation to the set of vertices and argue that not only there are regimes where AVTA outperforms that algorithm but it can also be used as a pre-processing step to their algorithm. Thus the two algorithms in fact complement each other. The problem of computing the vertices of the convex hull of a given input set [Formula omitted] is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Triangle Algorithm (AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset [Formula omitted] of all K vertices of the convex hull of S so that the convex hull of the approximate subset of vertices is as close to conv(S) as desired. On the other hand, assuming a known lower bound [Formula omitted] on the ratio [Formula omitted], where [Formula omitted] the minimum of the distances from each vertex to the convex hull of the remaining vertices and R the diameter of S, AVTA can recover all of [Formula omitted]. Furthermore, assuming that instead of S the input is an [Formula omitted]-perturbation of S, [Formula omitted], where [Formula omitted], AVTA can compute approximation to [Formula omitted], to any prescribed accuracy. Also, given a lower bound to the ratio [Formula omitted], where [Formula omitted] is the minimum of the distances from each vertex to the convex hull of the remaining point of S, AVTA can recover all of [Formula omitted]. We show [Formula omitted], where [Formula omitted] is the minimum distance between distinct pair of points in S and prove the following main results: Given any [Formula omitted], AVTA computes a subset [Formula omitted] of [Formula omitted] of cardinality [Formula omitted] in [Formula omitted] operations so that for any [Formula omitted] its Euclidean distance to [Formula omitted] is at most tR. Given [Formula omitted], AVTA computes [Formula omitted] in [Formula omitted] operations. If K is known, the complexity of AVTA is [Formula omitted]. Assuming instead of S, its [Formula omitted]-perturbation, [Formula omitted] is given, we prove Given any [Formula omitted], AVTA computes a subset [Formula omitted] of cardinality [Formula omitted] in [Formula omitted] operations so that for any [Formula omitted] its distance to [Formula omitted] is at most [Formula omitted]. Given [Formula omitted], AVTA computes [Formula omitted] in [Formula omitted] operations, where [Formula omitted]. If [Formula omitted] is known satisfying [Formula omitted], AVTA computes [Formula omitted] in [Formula omitted] operations. Given [Formula omitted], if K is known, AVTA computes [Formula omitted] in [Formula omitted] operations. We also consider the application of AVTA in the recovery of vertices through the projection of S or [Formula omitted] under a Johnson-Lindenstrauss randomized linear projection [Formula omitted]. Denoting [Formula omitted] and [Formula omitted], by relating the robustness parameters of conv(U) and [Formula omitted] to those of conv(S) and [Formula omitted], we derive analogous complexity bounds for probabilistic computation of the vertex set of conv(U) or those of [Formula omitted], or an approximation to them. Finally, we apply AVTA to design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models, our new algorithm leads to significantly better reconstruction of the topic-word matrix than state of the art approaches of Arora et al. (International conference on machine learning, pp 280-288, 2013) and Bansal et al. (Advances in neural information processing systems, pp 1997-2005, 2014). Additionally, we provide a robust analysis of AVTA and empirically demonstrate that it can handle larger amounts of noise than existing methods. For non-negative matrix factorization we show that AVTA is competitive with existing methods that are specialized for this task in Arora et al. (Proceedings of the forty-fourth annual ACM symposium on theory of computing, ACM, pp 145-162, 2012a). We also contrast AVTA with Blum et al. (Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, Society for Industrial and Applied Mathematics, pp 548-557, 2016) Greedy Clustering coreset algorithm for computing approximation to the set of vertices and argue that not only there are regimes where AVTA outperforms that algorithm but it can also be used as a pre-processing step to their algorithm. Thus the two algorithms in fact complement each other. The problem of computing the vertices of the convex hull of a given input set [Formula omitted] is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Triangle Algorithm (AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset [Formula omitted] of all K vertices of the convex hull of S so that the convex hull of the approximate subset of vertices is as close to conv(S) as desired. On the other hand, assuming a known lower bound [Formula omitted] on the ratio [Formula omitted], where [Formula omitted] the minimum of the distances from each vertex to the convex hull of the remaining vertices and R the diameter of S, AVTA can recover all of [Formula omitted]. Furthermore, assuming that instead of S the input is an [Formula omitted]-perturbation of S, [Formula omitted], where [Formula omitted], AVTA can compute approximation to [Formula omitted], to any prescribed accuracy. Also, given a lower bound to the ratio [Formula omitted], where [Formula omitted] is the minimum of the distances from each vertex to the convex hull of the remaining point of S, AVTA can recover all of [Formula omitted]. We show [Formula omitted], where [Formula omitted] is the minimum distance between distinct pair of points in S and prove the following main results: Given any [Formula omitted], AVTA computes a subset [Formula omitted] of [Formula omitted] of cardinality [Formula omitted] in [Formula omitted] operations so that for any [Formula omitted] its Euclidean distance to [Formula omitted] is at most tR. The problem of computing the vertices of the convex hull of a given input set S = { v i ∈ R m : i = 1 , ⋯ , n } is a classic and fundamental problem, studied in the context of computational geometry, linear and convex programming, machine learning and more. In this article we present All Vertex Triangle Algorithm (AVTA), a robust and efficient algorithm for this problem. On the one hand, without any assumptions, AVTA computes approximation to the subset S ¯ of all K vertices of the convex hull of S so that the convex hull of the approximate subset of vertices is as close to conv ( S ) as desired. On the other hand, assuming a known lower bound γ on the ratio Γ ∗ / R , where Γ ∗ the minimum of the distances from each vertex to the convex hull of the remaining vertices and R the diameter of S , AVTA can recover all of S ¯ . Furthermore, assuming that instead of S the input is an ε -perturbation of S , S ¯ ε , where ‖ v i - v i ε ‖ ≤ ε R , AVTA can compute approximation to c o n v ( S ¯ ε ) , to any prescribed accuracy. Also, given a lower bound to the ratio Σ ∗ / R , where Σ ∗ is the minimum of the distances from each vertex to the convex hull of the remaining point of S , AVTA can recover all of S ¯ ε . We show Σ ∗ ≥ ρ ∗ Γ ∗ / R , where ρ ∗ is the minimum distance between distinct pair of points in S and prove the following main results: Given any t ∈ ( 0 , 1 ) , AVTA computes a subset S ¯ t of S ¯ of cardinality K ( t ) in O ( n K ( t ) ( m + t - 2 ) ) operations so that for any p ∈ c o n v ( S ) its Euclidean distance to c o n v ( S ¯ t ) is at most tR . Given γ ≤ γ ∗ = Γ ∗ / R , AVTA computes S ¯ in O ( n K ( m + γ - 2 ) ) operations. If K is known, the complexity of AVTA is O ( n K ( m + γ ∗ - 2 ) log ( γ ∗ - 1 ) ) . Assuming instead of S , its ε -perturbation, S ε is given, we prove Given any t ∈ ( 0 , 1 ) , AVTA computes a subset S ¯ ε t ⊂ S ¯ ε of cardinality K ε ( t ) in O ( n K ε ( t ) ( m + t - 2 ) ) operations so that for any p ∈ c o n v ( S ) its distance to c o n v ( S ¯ ε t ) is at most ( t + ε ) R . Given σ ∈ [ 4 ε , σ ∗ = Γ ∗ / R ] , AVTA computes S ¯ ε in O ( n K ε ( m + σ - 2 ) ) operations, where K ≤ K ε ≤ n . If γ ≤ γ ∗ = Γ ∗ / R is known satisfying 4 ε ≤ γ ρ ∗ / R , AVTA computes S ¯ ε in O ( n K ε ( m + ( γ ρ ∗ ) - 2 ) ) operations. Given σ ∈ [ 4 ε , σ ∗ ] , if K is known, AVTA computes S ¯ ε in O ( n K ( m + σ ∗ - 2 ) log ( σ ∗ - 1 ) ) operations. We also consider the application of AVTA in the recovery of vertices through the projection of S or S ε under a Johnson–Lindenstrauss randomized linear projection L : R m → R m ′ . Denoting U = L ( S ) and U ε = L ( S ε ) , by relating the robustness parameters of conv ( U ) and c o n v ( U ε ) to those of conv ( S ) and c o n v ( S ε ) , we derive analogous complexity bounds for probabilistic computation of the vertex set of conv ( U ) or those of c o n v ( U ε ) , or an approximation to them. Finally, we apply AVTA to design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models, our new algorithm leads to significantly better reconstruction of the topic-word matrix than state of the art approaches of Arora et al. (International conference on machine learning, pp 280–288, 2013) and Bansal et al. (Advances in neural information processing systems, pp 1997–2005, 2014). Additionally, we provide a robust analysis of AVTA and empirically demonstrate that it can handle larger amounts of noise than existing methods. For non-negative matrix factorization we show that AVTA is competitive with existing methods that are specialized for this task in Arora et al. (Proceedings of the forty-fourth annual ACM symposium on theory of computing, ACM, pp 145–162, 2012a). We also contrast AVTA with Blum et al. (Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, Society for Industrial and Applied Mathematics, pp 548–557, 2016) Greedy Clustering coreset algorithm for computing approximation to the set of vertices and argue that not only there are regimes where AVTA outperforms that algorithm but it can also be used as a pre-processing step to their algorithm. Thus the two algorithms in fact complement each other. |
| Audience | Academic |
| Author | Zhang, Yikai Kalantari, Bahman Awasthi, Pranjal |
| Author_xml | – sequence: 1 givenname: Pranjal surname: Awasthi fullname: Awasthi, Pranjal organization: Rutgers University – sequence: 2 givenname: Bahman surname: Kalantari fullname: Kalantari, Bahman email: kalantar@cs.rutgers.edu organization: Rutgers University – sequence: 3 givenname: Yikai surname: Zhang fullname: Zhang, Yikai organization: Rutgers University |
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| CitedBy_id | crossref_primary_10_1145_3516520 crossref_primary_10_1016_j_ejor_2022_08_040 crossref_primary_10_1109_TKDE_2025_3552644 |
| Cites_doi | 10.1007/BF02712874 10.1007/BF02573985 10.1201/9781420035315 10.1145/2213977.2213994 10.1137/1.9781611974331.ch40 10.1007/BF02712873 10.1145/235815.235821 10.1137/0304007 10.1145/800057.808695 10.1002/nav.3800030109 10.1109/FOCS.2012.49 10.1145/2133806.2133826 10.1109/SFCS.1994.365723 10.1007/s10479-014-1707-2 10.1145/1542362.1542370 10.1016/j.laa.2005.12.022 10.1145/1557019.1557121 10.1023/A:1009715923555 10.1109/TIT.2002.808136 10.1016/0041-5553(80)90061-0 |
| ContentType | Journal Article |
| Copyright | Springer Science+Business Media, LLC, part of Springer Nature 2020 COPYRIGHT 2020 Springer Springer Science+Business Media, LLC, part of Springer Nature 2020. |
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| DOI | 10.1007/s10479-020-03557-0 |
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| References | Arora, S., Ge, R., Halpern, Y., Mimno, D., Moitra, A., Sontag, D., et al. (2013). A practical algorithm for topic modeling with provable guarantees. In International conference on machine learning (pp. 280–288). BurgesCJCA tutorial on support vector machines for pattern recognitionData Mining and Knowledge Discovery19982212116710.1023/A:1009715923555 Vu, K., Poirion, P.-L., & Liberti, L. (2017). Random projections for linear programming. arXiv preprint arXiv:1706.02768. Yao, L., Mimno, D., & McCallum, A. (2009). Efficient methods for topic model inference on streaming document collections. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 937–946). ACM. KalantariBA characterization theorem and an algorithm for a convex hull problemAnnals of Operations Research2015226130134910.1007/s10479-014-1707-2 ChanTMOptimal output-sensitive convex hull algorithms in two and three dimensionsDiscrete & Computational Geometry199616436136810.1007/BF02712873 DingWRohbanMHIshwarPSaligramaVTopic discovery through data dependent and random projectionsICML2013312021210 Arora, S., Ge, R., Kannan, R., & Moitra, A. (2012a). Computing a nonnegative matrix factorization—provably. In Proceedings of the forty-fourth annual ACM symposium on theory of computing (pp. 145–162). ACM. BarberCBDobkinDPHuhdanpaaHThe quickhull algorithm for convex hullsACM Transactions on Mathematical Software (TOMS)199622446948310.1145/235815.235821 BleiDMProbabilistic topic modelsCommunications of the ACM2012554778410.1145/2133806.2133826 ChvatalVLinear programming1983New YorkMacmillan Bansal, T., Bhattacharyya, C., & Kannan, R. (2014) A provable SVD-based algorithm for learning topics in dominant admixture corpus. In Advances in neural information processing systems (pp. 1997–2005). Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. In Proceedings of the sixteenth annual ACM symposium on theory of computing (pp. 302–311). ACM. ZhangTSequential greedy approximation for certain convex optimization problemsIEEE Transactions on Information Theory200349368269110.1109/TIT.2002.808136 JinYKalantariBA procedure of chvátal for testing feasibility in linear programming and matrix scalingLinear Algebra and its Applications20064162–379579810.1016/j.laa.2005.12.022 Stevens, K., Kegelmeyer, P., Andrzejewski, D., & Buttler, D. (2012). Exploring topic coherence over many models and many topics. In Proceedings of the 2012 joint conference on empirical methods in natural language processing and computational natural language learning (pp. 952–961). Association for Computational Linguistics. Donoho, D., & Stodden, V. (2003). When does non-negative matrix factorization give a correct decomposition into parts? In Advances in neural information processing systems. ChanTMOutput-sensitive results on convex hulls, extreme points, and related problemsDiscrete & Computational Geometry199616436938710.1007/BF02712874 ClarksonKLCoresets, sparse greedy approximation, and the Frank-Wolfe algorithmACM Transactions on Algorithms (TALG)20106463 FrankMWolfePAn algorithm for quadratic programmingNaval Research Logistics (NRL)195631–29511010.1002/nav.3800030109 Gärtner, B., & Jaggi, M. (2009). Coresets for polytope distance. In Proceedings of the twenty-fifth annual symposium on computational geometry (pp. 33–42). ACM. JohnsonWBLindenstraussJExtensions of lipschitz mappings into a hilbert spaceContemporary Mathematics198426189–2061 KhachiyanLGPolynomial algorithms in linear programmingUSSR Computational Mathematics and Mathematical Physics1980201537210.1016/0041-5553(80)90061-0 Anandkumar, A., Foster, D. P., Hsu, D. J., Kakade, S. M. & Liu, Y.-K. (2012). A spectral algorithm for latent dirichlet allocation. In Advances in neural information processing systems (pp. 917–925). Blum, A., Har-Peled, S., & Raichel, B. (2016). Sparse approximation via generating point sets. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms (pp. 548–557). Society for Industrial and Applied Mathematics. GilbertEGAn iterative procedure for computing the minimum of a quadratic form on a convex setSIAM Journal on Control196641618010.1137/0304007 Clarkson, K. L. (1994). More output-sensitive geometric algorithms. In 1994 Proceedings of the 35th annual symposium on foundations of computer science (pp. 695–702). IEEE. Arora, S., Ge, R., & Moitra, A. (2012b). Learning topic models—going beyond SVD. In 2012 IEEE 53rd annual symposium on foundations of computer science (FOCS), (pp. 1–10). IEEE. Lee, D. D. & Seung, H. S. (2001). Algorithms for non-negative matrix factorization. In Advances in neural information processing systems (pp. 556–562). TothCDO’RourkeJGoodmanJEHandbook of discrete and computational geometry2004Boca RatonCRC Press10.1201/9781420035315 ChazelleBAn optimal convex hull algorithm in any fixed dimensionDiscrete & Computational Geometry199310137740910.1007/BF02573985 Jaggi, M. (2013). Revisiting Frank–Wolfe: projection-free sparse convex optimization. BleiDMNgAYJordanMILatent dirichlet allocationJournal of Machine Learning Research20033Jan9931022 3557_CR4 M Frank (3557_CR19) 1956; 3 3557_CR5 EG Gilbert (3557_CR21) 1966; 4 3557_CR2 3557_CR3 3557_CR9 LG Khachiyan (3557_CR27) 1980; 20 TM Chan (3557_CR12) 1996; 16 3557_CR1 3557_CR29 3557_CR28 Y Jin (3557_CR23) 2006; 416 3557_CR26 DM Blei (3557_CR7) 2012; 55 3557_CR22 WB Johnson (3557_CR24) 1984; 26 CJC Burges (3557_CR10) 1998; 2 3557_CR20 V Chvatal (3557_CR14) 1983 B Chazelle (3557_CR13) 1993; 10 DM Blei (3557_CR8) 2003; 3 CD Toth (3557_CR30) 2004 B Kalantari (3557_CR25) 2015; 226 3557_CR18 W Ding (3557_CR17) 2013; 3 3557_CR15 CB Barber (3557_CR6) 1996; 22 T Zhang (3557_CR33) 2003; 49 3557_CR32 3557_CR31 TM Chan (3557_CR11) 1996; 16 KL Clarkson (3557_CR16) 2010; 6 |
| References_xml | – reference: Vu, K., Poirion, P.-L., & Liberti, L. (2017). Random projections for linear programming. arXiv preprint arXiv:1706.02768. – reference: Jaggi, M. (2013). Revisiting Frank–Wolfe: projection-free sparse convex optimization. – reference: ZhangTSequential greedy approximation for certain convex optimization problemsIEEE Transactions on Information Theory200349368269110.1109/TIT.2002.808136 – reference: JinYKalantariBA procedure of chvátal for testing feasibility in linear programming and matrix scalingLinear Algebra and its Applications20064162–379579810.1016/j.laa.2005.12.022 – reference: Arora, S., Ge, R., & Moitra, A. (2012b). Learning topic models—going beyond SVD. In 2012 IEEE 53rd annual symposium on foundations of computer science (FOCS), (pp. 1–10). IEEE. – reference: KalantariBA characterization theorem and an algorithm for a convex hull problemAnnals of Operations Research2015226130134910.1007/s10479-014-1707-2 – reference: Karmarkar, N. (1984). A new polynomial-time algorithm for linear programming. In Proceedings of the sixteenth annual ACM symposium on theory of computing (pp. 302–311). ACM. – reference: BleiDMProbabilistic topic modelsCommunications of the ACM2012554778410.1145/2133806.2133826 – reference: FrankMWolfePAn algorithm for quadratic programmingNaval Research Logistics (NRL)195631–29511010.1002/nav.3800030109 – reference: BleiDMNgAYJordanMILatent dirichlet allocationJournal of Machine Learning Research20033Jan9931022 – reference: JohnsonWBLindenstraussJExtensions of lipschitz mappings into a hilbert spaceContemporary Mathematics198426189–2061 – reference: TothCDO’RourkeJGoodmanJEHandbook of discrete and computational geometry2004Boca RatonCRC Press10.1201/9781420035315 – reference: Anandkumar, A., Foster, D. P., Hsu, D. J., Kakade, S. M. & Liu, Y.-K. (2012). A spectral algorithm for latent dirichlet allocation. In Advances in neural information processing systems (pp. 917–925). – reference: ClarksonKLCoresets, sparse greedy approximation, and the Frank-Wolfe algorithmACM Transactions on Algorithms (TALG)20106463 – reference: Blum, A., Har-Peled, S., & Raichel, B. (2016). Sparse approximation via generating point sets. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms (pp. 548–557). Society for Industrial and Applied Mathematics. – reference: ChvatalVLinear programming1983New YorkMacmillan – reference: Bansal, T., Bhattacharyya, C., & Kannan, R. (2014) A provable SVD-based algorithm for learning topics in dominant admixture corpus. In Advances in neural information processing systems (pp. 1997–2005). – reference: Donoho, D., & Stodden, V. (2003). When does non-negative matrix factorization give a correct decomposition into parts? In Advances in neural information processing systems. – reference: BarberCBDobkinDPHuhdanpaaHThe quickhull algorithm for convex hullsACM Transactions on Mathematical Software (TOMS)199622446948310.1145/235815.235821 – reference: Arora, S., Ge, R., Kannan, R., & Moitra, A. (2012a). Computing a nonnegative matrix factorization—provably. In Proceedings of the forty-fourth annual ACM symposium on theory of computing (pp. 145–162). ACM. – reference: DingWRohbanMHIshwarPSaligramaVTopic discovery through data dependent and random projectionsICML2013312021210 – reference: Clarkson, K. L. (1994). More output-sensitive geometric algorithms. In 1994 Proceedings of the 35th annual symposium on foundations of computer science (pp. 695–702). IEEE. – reference: GilbertEGAn iterative procedure for computing the minimum of a quadratic form on a convex setSIAM Journal on Control196641618010.1137/0304007 – reference: ChanTMOptimal output-sensitive convex hull algorithms in two and three dimensionsDiscrete & Computational Geometry199616436136810.1007/BF02712873 – reference: Yao, L., Mimno, D., & McCallum, A. (2009). Efficient methods for topic model inference on streaming document collections. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 937–946). ACM. – reference: Gärtner, B., & Jaggi, M. (2009). Coresets for polytope distance. In Proceedings of the twenty-fifth annual symposium on computational geometry (pp. 33–42). ACM. – reference: Arora, S., Ge, R., Halpern, Y., Mimno, D., Moitra, A., Sontag, D., et al. (2013). A practical algorithm for topic modeling with provable guarantees. In International conference on machine learning (pp. 280–288). – reference: ChanTMOutput-sensitive results on convex hulls, extreme points, and related problemsDiscrete & Computational Geometry199616436938710.1007/BF02712874 – reference: Lee, D. D. & Seung, H. S. (2001). Algorithms for non-negative matrix factorization. In Advances in neural information processing systems (pp. 556–562). – reference: ChazelleBAn optimal convex hull algorithm in any fixed dimensionDiscrete & Computational Geometry199310137740910.1007/BF02573985 – reference: KhachiyanLGPolynomial algorithms in linear programmingUSSR Computational Mathematics and Mathematical Physics1980201537210.1016/0041-5553(80)90061-0 – reference: BurgesCJCA tutorial on support vector machines for pattern recognitionData Mining and Knowledge Discovery19982212116710.1023/A:1009715923555 – reference: Stevens, K., Kegelmeyer, P., Andrzejewski, D., & Buttler, D. (2012). Exploring topic coherence over many models and many topics. In Proceedings of the 2012 joint conference on empirical methods in natural language processing and computational natural language learning (pp. 952–961). 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is a classic and fundamental problem, studied... The problem of computing the vertices of the convex hull of a given input set [Formula omitted] is a classic and fundamental problem, studied in the context of... The problem of computing the vertices of the convex hull of a given input set S={vi∈Rm:i=1,⋯,n} is a classic and fundamental problem, studied in the context of... |
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| SubjectTerms | Algorithms Angle Apexes Applications of mathematics Approximation Authorship Business and Management Clustering Combinatorics Complexity Computational geometry Convex surfaces Convexity Data processing Enumeration Euclidean geometry Factorization Greedy algorithms Hulls Lower bounds Machine learning Mathematical programming Matrix methods Measurement Operations research Operations Research/Decision Theory Original Research Parameter robustness Perturbation Robustness (mathematics) Technology application Theory of Computation Triangles Vertex sets |
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