101 research outputs found
Explicit measurements with almost optimal thresholds for compressed sensing
We consider the deterministic construction of a measurement
matrix and a recovery method for signals that are block
sparse. A signal that has dimension N = nd, which consists
of n blocks of size d, is called (s, d)-block sparse if
only s blocks out of n are nonzero. We construct an explicit
linear mapping Φ that maps the (s, d)-block sparse signal
to a measurement vector of dimension M, where s•d <N(1-(1-M/N)^(d/(d+1))-o(1).
We show that if the (s, d)-
block sparse signal is chosen uniformly at random then the
signal can almost surely be reconstructed from the measurement
vector in O(N^3) computations
Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem
Graph G is the square of graph H if two vertices x,y have an edge in G if and
only if x,y are of distance at most two in H. Given H it is easy to compute its
square H^2. Determining if a given graph G is the square of some graph is not
easy in general. Motwani and Sudan proved that it is NP-complete to determine
if a given graph G is the square of some graph. The graph introduced in their
reduction is a graph that contains many triangles and is relatively dense.
Farzad et al. proved the NP-completeness for finding a square root for girth 4
while they gave a polynomial time algorithm for computing a square root of
girth at least six. Adamaszek and Adamaszek proved that if a graph has a square
root of girth six then this square root is unique up to isomorphism. In this
paper we consider the characterization and recognition problem of graphs that
are square of graphs of girth at least five. We introduce a family of graphs
with exponentially many non-isomorphic square roots, and as the main result of
this paper we prove that the square root finding problem is NP-complete for
square roots of girth five. This proof is providing the complete dichotomy
theorem for square root problem in terms of the girth of the square roots
Recovering Sparse Signals Using Sparse Measurement Matrices in Compressed DNA Microarrays
Microarrays (DNA, protein, etc.) are massively parallel affinity-based biosensors capable of detecting and quantifying a large number of different genomic particles simultaneously. Among them, DNA microarrays comprising tens of thousands of probe spots are currently being employed to test multitude of targets in a single experiment. In conventional microarrays, each spot contains a large number of copies of a single probe designed to capture a single target, and, hence, collects only a single data point. This is a wasteful use of the sensing resources in comparative DNA microarray experiments, where a test sample is measured relative to a reference sample. Typically, only a fraction of the total number of genes represented by the two samples is differentially expressed, and, thus, a vast number of probe spots may not provide any useful information. To this end, we propose an alternative design, the so-called compressed microarrays, wherein each spot contains copies of several different probes and the total number of spots is potentially much smaller than the number of targets being tested. Fewer spots directly translates to significantly lower costs due to cheaper array manufacturing, simpler image acquisition and processing, and smaller amount of genomic material needed for experiments. To recover signals from compressed microarray measurements, we leverage ideas from compressive sampling. For sparse measurement matrices, we propose an algorithm that has significantly lower computational complexity than the widely used linear-programming-based methods, and can also recover signals with less sparsity
Computing Graph Roots Without Short Cycles
Graph G is the square of graph H if two vertices x, y have an edge in G if
and only if x, y are of distance at most two in H. Given H it is easy to
compute its square H2, however Motwani and Sudan proved that it is NP-complete
to determine if a given graph G is the square of some graph H (of girth 3). In
this paper we consider the characterization and recognition problems of graphs
that are squares of graphs of small girth, i.e. to determine if G = H2 for some
graph H of small girth. The main results are the following. - There is a graph
theoretical characterization for graphs that are squares of some graph of girth
at least 7. A corollary is that if a graph G has a square root H of girth at
least 7 then H is unique up to isomorphism. - There is a polynomial time
algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is
NP-complete to recognize if G = H2 for some graph H of girth 4. These results
almost provide a dichotomy theorem for the complexity of the recognition
problem in terms of girth of the square roots. The algorithmic and graph
theoretical results generalize previous results on tree square roots, and
provide polynomial time algorithms to compute a graph square root of small
girth if it exists. Some open questions and conjectures will also be discussed
On the reconstruction of block-sparse signals with an optimal number of measurements
Let A be an M by N matrix (M < N) which is an instance of a real random
Gaussian ensemble. In compressed sensing we are interested in finding the
sparsest solution to the system of equations A x = y for a given y. In general,
whenever the sparsity of x is smaller than half the dimension of y then with
overwhelming probability over A the sparsest solution is unique and can be
found by an exhaustive search over x with an exponential time complexity for
any y. The recent work of Cand\'es, Donoho, and Tao shows that minimization of
the L_1 norm of x subject to A x = y results in the sparsest solution provided
the sparsity of x, say K, is smaller than a certain threshold for a given
number of measurements. Specifically, if the dimension of y approaches the
dimension of x, the sparsity of x should be K < 0.239 N. Here, we consider the
case where x is d-block sparse, i.e., x consists of n = N / d blocks where each
block is either a zero vector or a nonzero vector. Instead of L_1-norm
relaxation, we consider the following relaxation min x \| X_1 \|_2 + \| X_2
\|_2 + ... + \| X_n \|_2, subject to A x = y where X_i = (x_{(i-1)d+1},
x_{(i-1)d+2}, ..., x_{i d}) for i = 1,2, ..., N. Our main result is that as n
-> \infty, the minimization finds the sparsest solution to Ax = y, with
overwhelming probability in A, for any x whose block sparsity is k/n < 1/2 -
O(\epsilon), provided M/N > 1 - 1/d, and d = \Omega(\log(1/\epsilon)/\epsilon).
The relaxation can be solved in polynomial time using semi-definite
programming
A hierarchical network formation model
We present a network formation model based on a particularly interesting class of networks in social settings, where individuals' positions are determined according to a topic-based or hierarchical taxonomy. In this game-theoretic model, players are located in the leaves of a complete b-ary tree as the seed network with the objective of minimizing their collective distances to others in the network. In the grid-based model of Even-Dar and Kearns [3], they demonstrate the existence of small diameter networks with the threshold of a = 2 where the cost of a new link depends on the distance between the two endpoints to the power of a. We show the appearance of small diameter equilibrium networks with the threshold of a = 1/4 in the hierarchical or tree-based networks. Moreover, the general set of equilibrium networks in our model are guaranteed to exist and they are pairwise Nash stable with transfers [2]
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