An l1-norm function is introduced as a computational tool to estimate the Dehn function of a finite presentation and settle certain imbedding questions for finitely presented groups.

Let l = [l0, l1] be the directed line segment from l0 ∈ IR to l1 ∈ IR. Suppose l̄ = [l̄0, l̄1] is a second segment of equal length such that l, l̄ satisfy the “two sticks condition”: ∥∥l1 − l̄0∥∥ ≥ ‖l1 − l0‖ ,∥∥l̄1 − l0∥∥ ≥ ∥∥l̄1 − l̄0∥∥ . Here ‖·‖ is a norm on IR. We explore the manner in which l1− l̄1 is then constrained when assumptions are made about “intermediate points” l∗ ∈ l, l̄∗ ∈ l̄. Roughly spe...

Rank K Binary Matrix Factorization (BMF) approximates a binary matrix by the product of two binary matrices of lower rank, K, using either L1 or L2 norm. In this paper, we first show that the BMFwithL2 norm can be reformulated as an Unconstrained Binary Quadratic Programming (UBQP) problem. We then review several local search strategies that can be used to improve the BMF solutions obtained by ...

We prove the Gagliardo–Nirenberg-type multiplicative interpolation inequality ‖v‖L1(Rn) ≤ C‖v‖ 1/2 Lip′(Rn)‖v‖ 1/2 BV(Rn) ∀v ∈ Lip ′(Rn) ∩ BV(R), where C is a positive constant, independent of v. Here ‖·‖Lip′(Rn) is the norm of the dual to the Lipschitz space Lip 0(R) := C 0,1 0 (Rn) = C0,1(Rn) ∩C0(R) and ‖ · ‖BV(Rn) signifies the norm in the space BV(Rn) consisting of functions of bounded vari...

It is well known that RN has subspaces of dimension proportional to N on which the l1 norm is equivalent to the l2 norm; however, no explicit constructions are known. Extending earlier work by Artstein– Avidan and Milman, we prove that such a subspace can be generated using O(N) random bits.

We obtain estimates in Besov, Triebel–Lizorkin and Lorentz spaces of differential forms on Rn in terms of their L1 norm.

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does l1-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necess...

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does l1-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necess...

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based C1-smooth univariate cubic L1 splines. An L1 spline minimizes the L1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating t...

Regularization of ill-posed linear inverse problems via l1 penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an l1 penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to l1-constraints, using a gradient method, with projection on l1-balls. The correspondin...