39 research outputs found
Summary Based Structures with Improved Sublinear Recovery for Compressed Sensing
We introduce a new class of measurement matrices for compressed sensing,
using low order summaries over binary sequences of a given length. We prove
recovery guarantees for three reconstruction algorithms using the proposed
measurements, including minimization and two combinatorial methods. In
particular, one of the algorithms recovers -sparse vectors of length in
sublinear time , and requires at most
measurements. The empirical oversampling constant
of the algorithm is significantly better than existing sublinear recovery
algorithms such as Chaining Pursuit and Sudocodes. In particular, for and , the oversampling factor is between 3 to 8. We provide
preliminary insight into how the proposed constructions, and the fast recovery
scheme can be used in a number of practical applications such as market basket
analysis, and real time compressed sensing implementation
CrossWalk: Fairness-enhanced Node Representation Learning
The potential for machine learning systems to amplify social inequities and
unfairness is receiving increasing popular and academic attention. Much recent
work has focused on developing algorithmic tools to assess and mitigate such
unfairness. However, there is little work on enhancing fairness in graph
algorithms. Here, we develop a simple, effective and general method, CrossWalk,
that enhances fairness of various graph algorithms, including influence
maximization, link prediction and node classification, applied to node
embeddings. CrossWalk is applicable to any random walk based node
representation learning algorithm, such as DeepWalk and Node2Vec. The key idea
is to bias random walks to cross group boundaries, by upweighting edges which
(1) are closer to the groups' peripheries or (2) connect different groups in
the network. CrossWalk pulls nodes that are near groups' peripheries towards
their neighbors from other groups in the embedding space, while preserving the
necessary structural properties of the graph. Extensive experiments show the
effectiveness of our algorithm to enhance fairness in various graph algorithms,
including influence maximization, link prediction and node classification in
synthetic and real networks, with only a very small decrease in performance.Comment: Association for the Advancement of Artificial Intelligence (AAAI)
202
Sparse Recovery of Positive Signals with Minimal Expansion
We investigate the sparse recovery problem of reconstructing a
high-dimensional non-negative sparse vector from lower dimensional linear
measurements. While much work has focused on dense measurement matrices, sparse
measurement schemes are crucial in applications, such as DNA microarrays and
sensor networks, where dense measurements are not practically feasible. One
possible construction uses the adjacency matrices of expander graphs, which
often leads to recovery algorithms much more efficient than
minimization. However, to date, constructions based on expanders have required
very high expansion coefficients which can potentially make the construction of
such graphs difficult and the size of the recoverable sets small.
In this paper, we construct sparse measurement matrices for the recovery of
non-negative vectors, using perturbations of the adjacency matrix of an
expander graph with much smaller expansion coefficient. We present a necessary
and sufficient condition for optimization to successfully recover the
unknown vector and obtain expressions for the recovery threshold. For certain
classes of measurement matrices, this necessary and sufficient condition is
further equivalent to the existence of a "unique" vector in the constraint set,
which opens the door to alternative algorithms to minimization. We
further show that the minimal expansion we use is necessary for any graph for
which sparse recovery is possible and that therefore our construction is tight.
We finally present a novel recovery algorithm that exploits expansion and is
much faster than optimization. Finally, we demonstrate through
theoretical bounds, as well as simulation, that our method is robust to noise
and approximate sparsity.Comment: 25 pages, submitted for publicatio
On the recovery of nonnegative sparse vectors from sparse measurements inspired by expanders
This paper studies compressed sensing for the recovery of non-negative sparse vectors from a smaller number of measurements than the ambient dimension of the unknown vector. We focus on measurement matrices that are sparse, i.e., have only a constant number of nonzero (and non-negative) entries in each column. For such measurement matrices we give a simple necessary and sufficient condition for l1 optimization to successfully recover the unknown vector. Using a simple ldquoperturbationrdquo to the adjacency matrix of an unbalanced expander, we obtain simple closed form expressions for the threshold relating the ambient dimension n, number of measurements m and sparsity level k, for which l1 optimization is successful with overwhelming probability. Simulation results suggest that the theoretical thresholds are fairly tight and demonstrate that the ldquoperturbationsrdquo significantly improve the performance over a direct use of the adjacency matrix of an expander graph
Divide-and-conquer: Approaching the capacity of the two-pair bidirectional Gaussian relay network
The capacity region of multi-pair bidirectional relay networks, in which a
relay node facilitates the communication between multiple pairs of users, is
studied. This problem is first examined in the context of the linear shift
deterministic channel model. The capacity region of this network when the relay
is operating at either full-duplex mode or half-duplex mode for arbitrary
number of pairs is characterized. It is shown that the cut-set upper-bound is
tight and the capacity region is achieved by a so called divide-and-conquer
relaying strategy. The insights gained from the deterministic network are then
used for the Gaussian bidirectional relay network. The strategy in the
deterministic channel translates to a specific superposition of lattice codes
and random Gaussian codes at the source nodes and successive interference
cancelation at the receiving nodes for the Gaussian network. The achievable
rate of this scheme with two pairs is analyzed and it is shown that for all
channel gains it achieves to within 3 bits/sec/Hz per user of the cut-set
upper-bound. Hence, the capacity region of the two-pair bidirectional Gaussian
relay network to within 3 bits/sec/Hz per user is characterized.Comment: IEEE Trans. on Information Theory, accepte