6,689 research outputs found
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
On Known-Plaintext Attacks to a Compressed Sensing-based Encryption: A Quantitative Analysis
Despite the linearity of its encoding, compressed sensing may be used to
provide a limited form of data protection when random encoding matrices are
used to produce sets of low-dimensional measurements (ciphertexts). In this
paper we quantify by theoretical means the resistance of the least complex form
of this kind of encoding against known-plaintext attacks. For both standard
compressed sensing with antipodal random matrices and recent multiclass
encryption schemes based on it, we show how the number of candidate encoding
matrices that match a typical plaintext-ciphertext pair is so large that the
search for the true encoding matrix inconclusive. Such results on the practical
ineffectiveness of known-plaintext attacks underlie the fact that even
closely-related signal recovery under encoding matrix uncertainty is doomed to
fail.
Practical attacks are then exemplified by applying compressed sensing with
antipodal random matrices as a multiclass encryption scheme to signals such as
images and electrocardiographic tracks, showing that the extracted information
on the true encoding matrix from a plaintext-ciphertext pair leads to no
significant signal recovery quality increase. This theoretical and empirical
evidence clarifies that, although not perfectly secure, both standard
compressed sensing and multiclass encryption schemes feature a noteworthy level
of security against known-plaintext attacks, therefore increasing its appeal as
a negligible-cost encryption method for resource-limited sensing applications.Comment: IEEE Transactions on Information Forensics and Security, accepted for
publication. Article in pres
From algorithms to connectivity and back: finding a giant component in random k-SAT
We take an algorithmic approach to studying the solution space geometry of
relatively sparse random and bounded degree -CNFs for large . In the
course of doing so, we establish that with high probability, a random -CNF
with variables and clause density
has a giant component of solutions that are connected in a graph where
solutions are adjacent if they have Hamming distance and that a
similar result holds for bounded degree -CNFs at similar densities. We are
also able to deduce looseness results for random and bounded degree -CNFs in
a similar regime.
Although our main motivation was understanding the geometry of the solution
space, our methods have algorithmic implications. Towards that end, we
construct an idealized block dynamics that samples solutions from a random
-CNF with density . We show this
Markov chain can with high probability be implemented in polynomial time and by
leveraging spectral independence, we also observe that it mixes relatively
fast, giving a polynomial time algorithm to with high probability sample a
uniformly random solution to a random -CNF. Our work suggests that the
natural route to pinning down when a giant component exists is to develop
sharper algorithms for sampling solutions in random -CNFs.Comment: 41 pages, 1 figur
Sampling Lov\'asz Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time
We give a fast algorithm for sampling uniform solutions of general constraint
satisfaction problems (CSPs) in a local lemma regime. The expected running time
of our algorithm is near-linear in and a fixed polynomial in ,
where is the number of variables and is the max degree of
constraints. Previously, up to similar conditions, sampling algorithms with
running time polynomial in both and , only existed for the almost
atomic case, where each constraint is violated by a small number of forbidden
local configurations.
Our sampling approach departs from all previous fast algorithms for sampling
LLL, which were based on Markov chains. A crucial step of our algorithm is a
recursive marginal sampler that is of independent interests. Within a local
lemma regime, this marginal sampler can draw a random value for a variable
according to its marginal distribution, at a local cost independent of the size
of the CSP
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