112 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
Detecting SQL Injection Attacks by Binary Gray Wolf Optimizer and Machine Learning Algorithms
SQL injection is one of the important security issues in web applications because it allows an attacker to interact with the
application’s database. SQL injection attacks can be detected using machine learning algorithms. The effective features
should be employed in the training stage to develop an optimal classifier with optimal accuracy. Identifying the most
effective features is an NP-complete combinatorial optimization problem. Feature selection is the process of selecting the
training dataset’s smallest and most effective features. The main objective of this study is to enhance the accuracy,
precision, and sensitivity of the SQLi detection method. In this study, an effective method to detect SQL injection attacks
has been proposed. In the first stage, a specific training dataset consisting of 13 features was prepared. In the second stage,
two different binary versions of the Gray-Wolf algorithm were developed to select the most effective features of the
dataset. The created optimal datasets were used by different machine learning algorithms. Creating a new SQLi training
dataset with 13 numeric features, developing two different binary versions of the gray wolf optimizer to optimally select
the features of the dataset, and creating an effective and efficient classifier to detect SQLi attacks are the main contributions
of this study. The results of the conducted tests indicate that the proposed SQL injection detector obtain 99.68% accuracy,
99.40% precision, and 98.72% sensitivity. The proposed method increases the efficiency of attack detection methods by
selecting 20% of the most effective features
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