633 research outputs found
Quantum Computation by Adiabatic Evolution
We give a quantum algorithm for solving instances of the satisfiability
problem, based on adiabatic evolution. The evolution of the quantum state is
governed by a time-dependent Hamiltonian that interpolates between an initial
Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian,
whose ground state encodes the satisfying assignment. To ensure that the system
evolves to the desired final ground state, the evolution time must be big
enough. The time required depends on the minimum energy difference between the
two lowest states of the interpolating Hamiltonian. We are unable to estimate
this gap in general. We give some special symmetric cases of the satisfiability
problem where the symmetry allows us to estimate the gap and we show that, in
these cases, our algorithm runs in polynomial time.Comment: 24 pages, 12 figures, LaTeX, amssymb,amsmath, BoxedEPS packages;
email to [email protected]
How many functions can be distinguished with k quantum queries?
Suppose an oracle is known to hold one of a given set of D two-valued
functions. To successfully identify which function the oracle holds with k
classical queries, it must be the case that D is at most 2^k. In this paper we
derive a bound for how many functions can be distinguished with k quantum
queries.Comment: 5 pages. Lower bound on sorting n items improved to (1-epsilon)n
quantum queries. Minor changes to text and corrections to reference
Conditional Noise-Contrastive Estimation of Unnormalised Models
Many parametric statistical models are not properly normalised and only
specified up to an intractable partition function, which renders parameter
estimation difficult. Examples of unnormalised models are Gibbs distributions,
Markov random fields, and neural network models in unsupervised deep learning.
In previous work, the estimation principle called noise-contrastive estimation
(NCE) was introduced where unnormalised models are estimated by learning to
distinguish between data and auxiliary noise. An open question is how to best
choose the auxiliary noise distribution. We here propose a new method that
addresses this issue. The proposed method shares with NCE the idea of
formulating density estimation as a supervised learning problem but in contrast
to NCE, the proposed method leverages the observed data when generating noise
samples. The noise can thus be generated in a semi-automated manner. We first
present the underlying theory of the new method, show that score matching
emerges as a limiting case, validate the method on continuous and discrete
valued synthetic data, and show that we can expect an improved performance
compared to NCE when the data lie in a lower-dimensional manifold. Then we
demonstrate its applicability in unsupervised deep learning by estimating a
four-layer neural image model.Comment: Accepted to ICML 201
Robust Optimisation Monte Carlo
This paper is on Bayesian inference for parametric statistical models that
are defined by a stochastic simulator which specifies how data is generated.
Exact sampling is then possible but evaluating the likelihood function is
typically prohibitively expensive. Approximate Bayesian Computation (ABC) is a
framework to perform approximate inference in such situations. While basic ABC
algorithms are widely applicable, they are notoriously slow and much research
has focused on increasing their efficiency. Optimisation Monte Carlo (OMC) has
recently been proposed as an efficient and embarrassingly parallel method that
leverages optimisation to accelerate the inference. In this paper, we
demonstrate an important previously unrecognised failure mode of OMC: It
generates strongly overconfident approximations by collapsing regions of
similar or near-constant likelihood into a single point. We propose an
efficient, robust generalisation of OMC that corrects this. It makes fewer
assumptions, retains the main benefits of OMC, and can be performed either as
post-processing to OMC or as a stand-alone computation. We demonstrate the
effectiveness of the proposed Robust OMC on toy examples and tasks in
inverse-graphics where we perform Bayesian inference with a complex image
renderer.Comment: 8 pages + 6 page appendix; v2: made clarifications, added a second
possible algorithm implementation and its results; v3: small clarifications,
to be published in AISTATS 202
Bayesian Experimental Design for Implicit Models by Mutual Information Neural Estimation
Implicit stochastic models, where the data-generation distribution is
intractable but sampling is possible, are ubiquitous in the natural sciences.
The models typically have free parameters that need to be inferred from data
collected in scientific experiments. A fundamental question is how to design
the experiments so that the collected data are most useful. The field of
Bayesian experimental design advocates that, ideally, we should choose designs
that maximise the mutual information (MI) between the data and the parameters.
For implicit models, however, this approach is severely hampered by the high
computational cost of computing posteriors and maximising MI, in particular
when we have more than a handful of design variables to optimise. In this
paper, we propose a new approach to Bayesian experimental design for implicit
models that leverages recent advances in neural MI estimation to deal with
these issues. We show that training a neural network to maximise a lower bound
on MI allows us to jointly determine the optimal design and the posterior.
Simulation studies illustrate that this gracefully extends Bayesian
experimental design for implicit models to higher design dimensions.Comment: Accepted at the thirty-seventh International Conference on Machine
Learning (ICML) 2020. Camera-ready versio
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