5 research outputs found
On Model-Based RIP-1 Matrices
The Restricted Isometry Property (RIP) is a fundamental property of a matrix
enabling sparse recovery. Informally, an m x n matrix satisfies RIP of order k
in the l_p norm if ||Ax||_p \approx ||x||_p for any vector x that is k-sparse,
i.e., that has at most k non-zeros. The minimal number of rows m necessary for
the property to hold has been extensively investigated, and tight bounds are
known. Motivated by signal processing models, a recent work of Baraniuk et al
has generalized this notion to the case where the support of x must belong to a
given model, i.e., a given family of supports. This more general notion is much
less understood, especially for norms other than l_2. In this paper we present
tight bounds for the model-based RIP property in the l_1 norm. Our bounds hold
for the two most frequently investigated models: tree-sparsity and
block-sparsity. We also show implications of our results to sparse recovery
problems.Comment: Version 3 corrects a few errors present in the earlier version. In
particular, it states and proves correct upper and lower bounds for the
number of rows in RIP-1 matrices for the block-sparse model. The bounds are
of the form k log_b n, not k log_k n as stated in the earlier versio
On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation
We study classic streaming and sparse recovery problems using deterministic
linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the
latter also being known as l1-heavy hitters), norm estimation, and approximate
inner product. We focus on devising a fixed matrix A in R^{m x n} and a
deterministic recovery/estimation procedure which work for all possible input
vectors simultaneously. Our results improve upon existing work, the following
being our main contributions:
* A proof that linf/l1 sparse recovery and inner product estimation are
equivalent, and that incoherent matrices can be used to solve both problems.
Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log
n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms
by making use of the Fast Johnson-Lindenstrauss transform. Both our running
times and number of measurements improve upon previous work. We can also obtain
better error guarantees than previous work in terms of a smaller tail of the
input vector.
* A new lower bound for the number of linear measurements required to solve
l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are
required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where
x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude.
* A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of
measurements required to solve deterministic norm estimation, i.e., to recover
|x|_2 +/- eps|x|_1.
For all the problems we study, tight bounds are already known for the
randomized complexity from previous work, except in the case of l1/l1 sparse
recovery, where a nearly tight bound is known. Our work thus aims to study the
deterministic complexities of these problems
On a game in manufacturing
We analyse a non-zero sum two-person game introduced by Teraoka and Yamada to model the strategic aspects of production development in manufacturing. In particular we investigate how sensitive their solution concept (Nash equilibrium) is to small variations in their assumptions. It is proved that a Nash equilibrium is unique if it exists and that a Nash equilibrium exists when the capital costs of the players are zero or when the players are equal in every respect. However, when the capital costs differ, in general a Nash equilibrium exists only when the players' capital costs are high compared to their profit rates