7,467 research outputs found
Approximately Sampling Elements with Fixed Rank in Graded Posets
Graded posets frequently arise throughout combinatorics, where it is natural
to try to count the number of elements of a fixed rank. These counting problems
are often -complete, so we consider approximation algorithms for
counting and uniform sampling. We show that for certain classes of posets,
biased Markov chains that walk along edges of their Hasse diagrams allow us to
approximately generate samples with any fixed rank in expected polynomial time.
Our arguments do not rely on the typical proofs of log-concavity, which are
used to construct a stationary distribution with a specific mode in order to
give a lower bound on the probability of outputting an element of the desired
rank. Instead, we infer this directly from bounds on the mixing time of the
chains through a method we call .
A noteworthy application of our method is sampling restricted classes of
integer partitions of . We give the first provably efficient Markov chain
algorithm to uniformly sample integer partitions of from general restricted
classes. Several observations allow us to improve the efficiency of this chain
to require space, and for unrestricted integer partitions,
expected time. Related applications include sampling permutations
with a fixed number of inversions and lozenge tilings on the triangular lattice
with a fixed average height.Comment: 23 pages, 12 figure
Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models
Mainstream machine-learning techniques such as deep learning and
probabilistic programming rely heavily on sampling from generally intractable
probability distributions. There is increasing interest in the potential
advantages of using quantum computing technologies as sampling engines to speed
up these tasks or to make them more effective. However, some pressing
challenges in state-of-the-art quantum annealers have to be overcome before we
can assess their actual performance. The sparse connectivity, resulting from
the local interaction between quantum bits in physical hardware
implementations, is considered the most severe limitation to the quality of
constructing powerful generative unsupervised machine-learning models. Here we
use embedding techniques to add redundancy to data sets, allowing us to
increase the modeling capacity of quantum annealers. We illustrate our findings
by training hardware-embedded graphical models on a binarized data set of
handwritten digits and two synthetic data sets in experiments with up to 940
quantum bits. Our model can be trained in quantum hardware without full
knowledge of the effective parameters specifying the corresponding quantum
Gibbs-like distribution; therefore, this approach avoids the need to infer the
effective temperature at each iteration, speeding up learning; it also
mitigates the effect of noise in the control parameters, making it robust to
deviations from the reference Gibbs distribution. Our approach demonstrates the
feasibility of using quantum annealers for implementing generative models, and
it provides a suitable framework for benchmarking these quantum technologies on
machine-learning-related tasks.Comment: 17 pages, 8 figures. Minor further revisions. As published in Phys.
Rev.
Improvements in the reconstruction of time-varying gene regulatory networks: dynamic programming and regularization by information sharing among genes
<b>Method:</b> Dynamic Bayesian networks (DBNs) have been applied widely to reconstruct the structure of regulatory processes from time series data, and they have established themselves as a standard modelling tool in computational systems biology. The conventional approach is based on the assumption of a homogeneous Markov chain, and many recent research efforts have focused on relaxing this restriction. An approach that enjoys particular popularity is based on a combination of a DBN with a multiple changepoint process, and the application of a Bayesian inference scheme via reversible jump Markov chain Monte Carlo (RJMCMC). In the present article, we expand this approach in two ways. First, we show that a dynamic programming scheme allows the changepoints to be sampled from the correct conditional distribution, which results in improved convergence over RJMCMC. Second, we introduce a novel Bayesian clustering and information sharing scheme among nodes, which provides a mechanism for automatic model complexity tuning.
<b>Results:</b> We evaluate the dynamic programming scheme on expression time series for Arabidopsis thaliana genes involved in circadian regulation. In a simulation study we demonstrate that the regularization scheme improves the network reconstruction accuracy over that obtained with recently proposed inhomogeneous DBNs. For gene expression profiles from a synthetically designed Saccharomyces cerevisiae strain under switching carbon metabolism we show that the combination of both: dynamic programming and regularization yields an inference procedure that outperforms two alternative established network reconstruction methods from the biology literature
Polynomial tuning of multiparametric combinatorial samplers
Boltzmann samplers and the recursive method are prominent algorithmic
frameworks for the approximate-size and exact-size random generation of large
combinatorial structures, such as maps, tilings, RNA sequences or various
tree-like structures. In their multiparametric variants, these samplers allow
to control the profile of expected values corresponding to multiple
combinatorial parameters. One can control, for instance, the number of leaves,
profile of node degrees in trees or the number of certain subpatterns in
strings. However, such a flexible control requires an additional non-trivial
tuning procedure. In this paper, we propose an efficient polynomial-time, with
respect to the number of tuned parameters, tuning algorithm based on convex
optimisation techniques. Finally, we illustrate the efficiency of our approach
using several applications of rational, algebraic and P\'olya structures
including polyomino tilings with prescribed tile frequencies, planar trees with
a given specific node degree distribution, and weighted partitions.Comment: Extended abstract, accepted to ANALCO2018. 20 pages, 6 figures,
colours. Implementation and examples are available at [1]
https://github.com/maciej-bendkowski/boltzmann-brain [2]
https://github.com/maciej-bendkowski/multiparametric-combinatorial-sampler
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