1,751 research outputs found
Approximate Sparse Recovery: Optimizing Time and Measurements
An approximate sparse recovery system consists of parameters , an
-by- measurement matrix, , and a decoding algorithm, .
Given a vector, , the system approximates by , which must satisfy , where denotes the optimal -term approximation to . For
each vector , the system must succeed with probability at least 3/4. Among
the goals in designing such systems are minimizing the number of
measurements and the runtime of the decoding algorithm, .
In this paper, we give a system with
measurements--matching a lower bound, up to a constant factor--and decoding
time , matching a lower bound up to factors.
We also consider the encode time (i.e., the time to multiply by ),
the time to update measurements (i.e., the time to multiply by a
1-sparse ), and the robustness and stability of the algorithm (adding noise
before and after the measurements). Our encode and update times are optimal up
to factors
Recovering Structured Probability Matrices
We consider the problem of accurately recovering a matrix B of size M by M ,
which represents a probability distribution over M2 outcomes, given access to
an observed matrix of "counts" generated by taking independent samples from the
distribution B. How can structural properties of the underlying matrix B be
leveraged to yield computationally efficient and information theoretically
optimal reconstruction algorithms? When can accurate reconstruction be
accomplished in the sparse data regime? This basic problem lies at the core of
a number of questions that are currently being considered by different
communities, including building recommendation systems and collaborative
filtering in the sparse data regime, community detection in sparse random
graphs, learning structured models such as topic models or hidden Markov
models, and the efforts from the natural language processing community to
compute "word embeddings".
Our results apply to the setting where B has a low rank structure. For this
setting, we propose an efficient algorithm that accurately recovers the
underlying M by M matrix using Theta(M) samples. This result easily translates
to Theta(M) sample algorithms for learning topic models and learning hidden
Markov Models. These linear sample complexities are optimal, up to constant
factors, in an extremely strong sense: even testing basic properties of the
underlying matrix (such as whether it has rank 1 or 2) requires Omega(M)
samples. We provide an even stronger lower bound where distinguishing whether a
sequence of observations were drawn from the uniform distribution over M
observations versus being generated by an HMM with two hidden states requires
Omega(M) observations. This precludes sublinear-sample hypothesis tests for
basic properties, such as identity or uniformity, as well as sublinear sample
estimators for quantities such as the entropy rate of HMMs
Dynamic Graph Stream Algorithms in Space
In this paper we study graph problems in dynamic streaming model, where the
input is defined by a sequence of edge insertions and deletions. As many
natural problems require space, where is the number of
vertices, existing works mainly focused on designing space
algorithms. Although sublinear in the number of edges for dense graphs, it
could still be too large for many applications (e.g. is huge or the graph
is sparse). In this work, we give single-pass algorithms beating this space
barrier for two classes of problems.
We present space algorithms for estimating the number of connected
components with additive error and
-approximating the weight of minimum spanning tree, for any
small constant . The latter improves previous
space algorithm given by Ahn et al. (SODA 2012) for connected graphs with
bounded edge weights.
We initiate the study of approximate graph property testing in the dynamic
streaming model, where we want to distinguish graphs satisfying the property
from graphs that are -far from having the property. We consider
the problem of testing -edge connectivity, -vertex connectivity,
cycle-freeness and bipartiteness (of planar graphs), for which, we provide
algorithms using roughly space, which is
for any constant .
To complement our algorithms, we present space
lower bounds for these problems, which show that such a dependence on
is necessary.Comment: ICALP 201
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