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127
Sparse nonnegative solutions of underdetermined linear equations by linear programming
 Proceedings of the National Academy of Sciences
, 2005
"... Consider an underdetermined system of linear equations y = Ax with known d×n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NPhard. However, for many matrices A there is a threshold phenomeno ..."
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Cited by 110 (6 self)
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Consider an underdetermined system of linear equations y = Ax with known d×n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NPhard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. In classical convex polytope theory, a polytope P is called kneighborly if every set of k vertices of P span a face of P. Let aj denote the jth column of A, 1 ≤ j ≤ n, let a0 = 0 and let P denote the convex hull of the aj. We say P is outwardly kneighborly if every subset of k vertices not including 0 spans a face of P. We show that outward kneighborliness is completely equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. Using this and classical results on polytope neighborliness we obtain two types of corollaries. First, because many ⌊d/2⌋neighborly polytopes are known, there are many systems where the sparsest solution is available by convex optimization rather than combinatorial
Neighborly Polytopes and Sparse Solutions of Underdetermined Linear Equations
, 2005
"... Consider a d × n matrix A, with d < n. The problem of solving for x in y = Ax is underdetermined, and has many possible solutions (if there are any). In several fields it is of interest to find the sparsest solution – the one with fewest nonzeros – but in general this involves combinatorial optim ..."
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Cited by 87 (12 self)
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Consider a d × n matrix A, with d < n. The problem of solving for x in y = Ax is underdetermined, and has many possible solutions (if there are any). In several fields it is of interest to find the sparsest solution – the one with fewest nonzeros – but in general this involves combinatorial optimization. Let ai denote the ith column of A, 1 ≤ i ≤ n. Associate to A the quotient polytope P formed by taking the convex hull of the 2n points (±ai) in R d. P is centrosymmetric and is called (centrally) kneighborly if every subset of k + 1 elements (±ilail)k+1 l=1 are the vertices of a face of P. We show that if P is kneighborly, then if a system y = Ax has a solution with at most k nonzeros, that solution is also the unique solution of the convex optimization problem min �x�1 subject to y = Ax; the converse holds as well. This complete equivalence between the study of sparse solution by ℓ 1 minimization and neighborliness of convex polytopes immediately gives new results in each field. On the one
Reconstruction and subgaussian operators in Asymptotic Geometric Analysis
 FUNCT. ANAL
"... We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probabilit ..."
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Cited by 36 (5 self)
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We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T, vectors (Xi) k i=1 of R n and k scalar products (〈Xi, v〉) k i=1, where (Xi) k i=1 are i.i.d. isotropic subgaussian random vectors in R n, and k ≪ n. We show that with high probability, any y ∈ T for which (〈Xi, y〉) k i=1 is close to the data vector (〈Xi, v〉) k i=1 will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}valued vectors with i.i.d, symmetric entries, yields new information on the geometry of faces of random {−1, 1}polytope; we show that a kdimensional random {−1, 1}polytope with n vertices is mneighborly for very large m ≤ ck / log(c ′ n/k). The proofs are � based on new estimates on the behavior of the empirical process supf∈F �k−1 �k i=1 f 2 (Xi) − Ef 2 � when F is a subset of the L2 sphere. The estimates are given in terms of the γ2 functional with respect to the ψ2 metric on F, and hold both in exponential probability and in expectation.
New Lower Bounds for Convex Hull Problems in Odd Dimensions
 SIAM J. Comput
, 1996
"... We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result fo ..."
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Cited by 27 (7 self)
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We show that in the worst case, &Omega;(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasisimplicial nvertex polytope with &Omega;(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that ddimensional convex hulls can have &Omega;(n bd=2c ) facets, the previously best lower bound for these problems is only &Omega;(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is &lceil;d/2&rceil;hard, in the in the sense of Gajentaan and Overmars.
Tangential structures on toric manifolds, and connected sums of polytopes
 MATH. RES. NOTICES
, 2001
"... We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an elem ..."
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Cited by 22 (9 self)
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We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension> 2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch’s famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple ndimensional polytopes; when P n is a product of simplices, we describe P n #Q n by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Q n.
Control of PiecewiseLinear Hybrid Systems on Simplices and Rectangles
, 2001
"... A necessary and sucient condition for the reachability of a piecewiselinear hybrid system is formulated in terms of reachability of a nitestate discreteevent system and of a nite family of ane systems on a polyhedral set. As a subproblem, the reachability of an ane system on a polytope is c ..."
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Cited by 20 (2 self)
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A necessary and sucient condition for the reachability of a piecewiselinear hybrid system is formulated in terms of reachability of a nitestate discreteevent system and of a nite family of ane systems on a polyhedral set. As a subproblem, the reachability of an ane system on a polytope is considered, with the control objective of reaching a particular facet of the polytope. If the polytope is a simplex, necessary and sucient conditions for the solvability of this problem by ane state feedback are described. If the polytope is a multidimensional rectangle, then a solution is obtained using continuous piecewiseane state feedback.
gELEMENTS, FINITE BUILDINGS AND HIGHER COHENMACAULAY CONNECTIVITY
, 2005
"... Chari proved that if ∆ is a (d − 1)dimensional simplicial complex with a convex ear decomposition, then h0 ≤ · · · ≤ h ⌊d/2 ⌋ [7]. Nyman and Swartz raised the problem of whether or not the corresponding gvector is an Mvector [18]. This is proved to be true by showing that the set of pairs (ω, ..."
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Cited by 17 (1 self)
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Chari proved that if ∆ is a (d − 1)dimensional simplicial complex with a convex ear decomposition, then h0 ≤ · · · ≤ h ⌊d/2 ⌋ [7]. Nyman and Swartz raised the problem of whether or not the corresponding gvector is an Mvector [18]. This is proved to be true by showing that the set of pairs (ω, Θ), where Θ is a l.s.o.p. for k[∆], the face ring of ∆, and ω is a gelement for k[∆]/Θ, is nonempty whenever the characteristic of k is zero. Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag hvector of such spaces similar in spirit to those examined in [18] for order complexes of geometric lattices. This also leads to connections between higher CohenMacaulay connectivity and conditions which insure that h0 < · · · < hi for a predetermined i.
Convexity Recognition of the Union of Polyhedra
, 2000
"... In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R d , P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (Hpolyh ..."
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Cited by 17 (6 self)
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In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R d , P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (Hpolyhedra) (2) when they are given by vertices and extreme rays (Vpolyhedra) (3) when both H and Vpolyhedral representations are available. Both the bounded (polytopes) and the unbounded case are considered. We show that the first two problems are polynomially solvable, and that the third problem is stronglypolynomially solvable.