共查询到20条相似文献,搜索用时 15 毫秒
1.
For a bounded system of linear equalities and inequalities, we show that the NP-hard ℓ
0-norm minimization problem is completely equivalent to the concave ℓ
p
-norm minimization problem, for a sufficiently small p. A local solution to the latter problem can be easily obtained by solving a provably finite number of linear programs. Computational
results frequently leading to a global solution of the ℓ
0-minimization problem and often producing sparser solutions than the corresponding ℓ
1-solution are given. A similar approach applies to finding minimal ℓ
0-solutions of linear programs. 相似文献
2.
We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b
ℓ
(n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds
to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients
of ∏
m=1∞(1−q
m
)
ℓ−1.
相似文献
3.
A well known argument of James yields that if a Banach spaceX contains ℓ
1
n
’s uniformly, thenX contains ℓ
1
n
’s almost isometrically. In the first half of the paper we extend this idea to the ordinal ℓ1-indices of Bourgain. In the second half we use our results to calculate the ℓ1-index of certain Banach spaces. Furthermore we show that the ℓ1-index of a separable Banach space not containing ℓ1 must be of the form ωα for some countable ordinal α.
Research supported by the NSF and TARP. 相似文献
4.
We discuss necessary and sufficient conditions for a sensing matrix to be “s-good”—to allow for exact ℓ
1-recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect ℓ
1-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the ℓ
1-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal
result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions
for exact sparse ℓ
1-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like
Restricted Isometry or Restricted Eigenvalue properties. 相似文献
5.
We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by
reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution
turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in
the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s
algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented
efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.
相似文献
6.
Given a frame F = {f
j
} for a separable Hilbert space H, we introduce the linear subspace HpFH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ
p
-space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation
to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as HpFH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in HFpH_{F}^{p} converges in both the Hilbert space norm and the ||·||
F, p
-norm which is induced by the ℓ
p
-norm. 相似文献
7.
Uri Srebro 《Israel Journal of Mathematics》1995,89(1-3):61-70
It is shown that the only balls in the Carathéodory distance onH
n
={z ε ℂ
n
:‖z‖1<1},n≥2, which are balls with respect to the ℓ1 norm in ℂ
n
are those centered at the origin.
In memory of Albert Pfluger
The research was partially supported by the Fund for the Promotion of Research at the Technion. 相似文献
8.
In applications such as signal processing and statistics, many problems involve finding sparse solutions to under-determined
linear systems of equations. These problems can be formulated as a structured nonsmooth optimization problems, i.e., the problem
of minimizing ℓ
1-regularized linear least squares problems. In this paper, we propose a block coordinate gradient descent method (abbreviated
as CGD) to solve the more general ℓ
1-regularized convex minimization problems, i.e., the problem of minimizing an ℓ
1-regularized convex smooth function. We establish a Q-linear convergence rate for our method when the coordinate block is
chosen by a Gauss-Southwell-type rule to ensure sufficient descent. We propose efficient implementations of the CGD method
and report numerical results for solving large-scale ℓ
1-regularized linear least squares problems arising in compressed sensing and image deconvolution as well as large-scale ℓ
1-regularized logistic regression problems for feature selection in data classification. Comparison with several state-of-the-art
algorithms specifically designed for solving large-scale ℓ
1-regularized linear least squares or logistic regression problems suggests that an efficiently implemented CGD method may
outperform these algorithms despite the fact that the CGD method is not specifically designed just to solve these special
classes of problems. 相似文献
9.
We study hypercyclicity and supercyclicity of weighted shifts on ℓ∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be
weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c0 or ℓp (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ∞ is norm nowhere dense. Finally, we show that ℓ∞ supports an isometry that is weak * sequentially supercyclic. 相似文献
10.
Saharon Shelah 《Israel Journal of Mathematics》2008,166(1):61-96
We show that, consistently, there is an ultrafilter on ω such that if N
nℓ = (P
nℓ ∪ Q
nℓ, P
nℓ, Q
nℓ, R
nℓ) (for ℓ = 1, 2, n < ω), P
nℓ ∪ Q
nℓ ⊆ ω, and are models of the canonical theory t
ind of the strong independence property, then every isomorphism from onto is a product isomorphism.
The first version of this work done in 93; First typed: May 1993.
This research was partially supported by the United States-Israel Binational Science Foundation. Publication 509 相似文献
11.
Sándor Baran 《Lithuanian Mathematical Journal》2011,51(2):122-140
Asymptotic properties of the variances of the spatial autoregressive model X
k,ℓ
= αX
k−1,ℓ
+ βX
k,ℓ−1 + γX
k−1,ℓ−1 + ε
k,ℓ
are investigated in the unit root case, that is, where the parameters are on the boundary of the domain of stability that
forms a tetrahedron in [−1, 1]3. The limit of the variance of n
−ϱ
X
[ns],[nt] is determined, where ϱ = 1/4 on the interior of the faces of the domain of stability, ϱ = 1/2 on the edges, and ϱ = 1 on the vertices. 相似文献
12.
LetY be a Banach space, 1<p<∞. We give a simple criterion for embedding ℓ
p
⊂Y, namely it suffices that the positive cone ℓ
p
+
⊂Y. This result is applied to the study of highly smooth operators from ℓ
p
intoY (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless ℓ
p/K
⊂Y for someK ∈ ℕ.
Supported by grants GAUK 277/2001, GAČR 201-01-1198, A1019205. 相似文献
13.
Let X be an infinite-dimensional Banach space with weight τ. By Cld
AW
(X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin
AW
(X), Comp
AW
(X) and Bdd
AW
(X), we denote the subspaces of Cld
AW
(X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that
Fin
AW
(X)≈Comp
AW
(X)≈ℓ2(τ)×ℓ2
f
ℓandℓBdd
AW
(X)≈ℓ2(2τ)×ℓ2
f
where ≈ means ‘is homeomorphic to’, ℓ2(τ) is the Hilbert space with weight τ (ℓ2(ℵ0)=ℓ2 the separable Hilbert space) and
ℓ2
f
={(x
i
)
iεN
εℓ2∣x
i
=0 except for finitely many iεN}. 相似文献
14.
Let B
w
(ℓ
p
) denote the space of infinite matrices A for which A(x) ∈ ℓ
p
for all x = {x
k
}
k=1∞ ∈ ℓ
p
with |x
k
| ↘ 0. We characterize the upper triangular positive matrices from B
w
(ℓ
p
), 1 < p < ∞, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are
stated and discussed. 相似文献
15.
Krzysztof C. Kiwiel 《Mathematical Programming》2001,90(1):1-25
We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint
in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not
be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and
to the generalized solution set for classical stepsizes t
k
→0, ∑t
k
=∞, and weak or strong convergence of the iterates to a solution for {t
k
}∈ℓ2∖ℓ1 under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds ε-solutions with an
efficiency estimate of O(ε-2), thus being optimal in the sense of Nemirovskii.
Received: October 4, 1998 / Accepted: July 24, 2000?Published online January 17, 2001 相似文献
16.
A finitely presented group G is hyperbolic iff H
(1)
1(G,ℝ)=0=(1)
2(G, ℝ), where H
(1)
* (resp. (1)
*) denotes the ℓ1-homology (resp. reduced ℓ1-homology). If Γ is a graph, then every ℓ1 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H
(1)
1(G,ℝ)=0.
Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999 相似文献
17.
Paul Ramsden 《Semigroup Forum》2009,79(3):515-530
We shall study the biflatness of the convolution algebra ℓ
1(S) for a semigroup S. We show that for any semigroup S such that ℓ
1(S) is biflat the canonical partial ordering on the idempotents must be uniformly locally finite. We use this to characterize
the biflatness of ℓ
1(S) for an inverse semigroup S. 相似文献
18.
19.
We establish a strong regularity property for the distributions of the random sums Σ±λ
n
, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w
n+1...,w
n+ℓ) given the sum Σ
n=0
∞
w
n
λ
n
, tends to the uniform distribution on {±1}ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system
of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.
The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.). 相似文献
20.
Nicholas Sparks 《Israel Journal of Mathematics》1998,104(1):125-128
The Banach space ℓ
∞
c
(ω
1) is the space of boundedω
1-sequences of countable support. A pointwise-closed subspaceV≤ℓ
∞
c
(ω
1) will be calledunbounded if lcub;min(supp(υ)):υ ∈Vrcub; is unbounded inω
1. It is shown that there are Lipshitz functionsf: Sph(ℓ
∞
c
(ω
1)) → ℝ which have large variation on the unit sphere of any unbounded subspace. This answers a question implicit in Partington
[P 80]. 相似文献