19,078 research outputs found
Entropy Message Passing
The paper proposes a new message passing algorithm for cycle-free factor
graphs. The proposed "entropy message passing" (EMP) algorithm may be viewed as
sum-product message passing over the entropy semiring, which has previously
appeared in automata theory. The primary use of EMP is to compute the entropy
of a model. However, EMP can also be used to compute expressions that appear in
expectation maximization and in gradient descent algorithms.Comment: 5 pages, 1 figure, to appear in IEEE Transactions on Information
Theor
Tree-AMP: Compositional Inference with Tree Approximate Message Passing
We introduce Tree-AMP, standing for Tree Approximate Message Passing, a
python package for compositional inference in high-dimensional tree-structured
models. The package provides a unifying framework to study several approximate
message passing algorithms previously derived for a variety of machine learning
tasks such as generalized linear models, inference in multi-layer networks,
matrix factorization, and reconstruction using non-separable penalties. For
some models, the asymptotic performance of the algorithm can be theoretically
predicted by the state evolution, and the measurements entropy estimated by the
free entropy formalism. The implementation is modular by design: each module,
which implements a factor, can be composed at will with other modules to solve
complex inference tasks. The user only needs to declare the factor graph of the
model: the inference algorithm, state evolution and entropy estimation are
fully automated.Comment: Source code available at https://github.com/sphinxteam/tramp and
documentation at https://sphinxteam.github.io/tramp.doc
Multi Terminal Probabilistic Compressed Sensing
In this paper, the `Approximate Message Passing' (AMP) algorithm, initially
developed for compressed sensing of signals under i.i.d. Gaussian measurement
matrices, has been extended to a multi-terminal setting (MAMP algorithm). It
has been shown that similar to its single terminal counterpart, the behavior of
MAMP algorithm is fully characterized by a `State Evolution' (SE) equation for
large block-lengths. This equation has been used to obtain the rate-distortion
curve of a multi-terminal memoryless source. It is observed that by spatially
coupling the measurement matrices, the rate-distortion curve of MAMP algorithm
undergoes a phase transition, where the measurement rate region corresponding
to a low distortion (approximately zero distortion) regime is fully
characterized by the joint and conditional Renyi information dimension (RID) of
the multi-terminal source. This measurement rate region is very similar to the
rate region of the Slepian-Wolf distributed source coding problem where the RID
plays a role similar to the discrete entropy.
Simulations have been done to investigate the empirical behavior of MAMP
algorithm. It is observed that simulation results match very well with
predictions of SE equation for reasonably large block-lengths.Comment: 11 pages, 13 figures. arXiv admin note: text overlap with
arXiv:1112.0708 by other author
Towards Optimal Moment Estimation in Streaming and Distributed Models
One of the oldest problems in the data stream model is to approximate the p-th moment ||X||_p^p = sum_{i=1}^n X_i^p of an underlying non-negative vector X in R^n, which is presented as a sequence of poly(n) updates to its coordinates. Of particular interest is when p in (0,2]. Although a tight space bound of Theta(epsilon^-2 log n) bits is known for this problem when both positive and negative updates are allowed, surprisingly there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is O(epsilon^-2 log n) bits, while the lower bound is only Omega(epsilon^-2 + log n) bits. Recently, an upper bound of O~(epsilon^-2 + log n) bits was obtained under the assumption that the updates arrive in a random order.
We show that for p in (0, 1], the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of O~(epsilon^-2 + log n) bits for estimating |X |_p^p. Our techniques also give new upper bounds for estimating the empirical entropy in a stream. On the other hand, we show that for p in (1,2], in the natural coordinator and blackboard distributed communication topologies, there is an O~(epsilon^-2) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies G, obtaining an O~(epsilon^2 log d) max-communication upper bound, where d is the diameter of G. Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an Omega(epsilon^-2 log n) bit lower bound for p in (1,2] for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter
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