2,601 research outputs found
Bayesian Inference of Log Determinants
The log-determinant of a kernel matrix appears in a variety of machine
learning problems, ranging from determinantal point processes and generalized
Markov random fields, through to the training of Gaussian processes. Exact
calculation of this term is often intractable when the size of the kernel
matrix exceeds a few thousand. In the spirit of probabilistic numerics, we
reinterpret the problem of computing the log-determinant as a Bayesian
inference problem. In particular, we combine prior knowledge in the form of
bounds from matrix theory and evidence derived from stochastic trace estimation
to obtain probabilistic estimates for the log-determinant and its associated
uncertainty within a given computational budget. Beyond its novelty and
theoretic appeal, the performance of our proposal is competitive with
state-of-the-art approaches to approximating the log-determinant, while also
quantifying the uncertainty due to budget-constrained evidence.Comment: 12 pages, 3 figure
Testing quantum mechanics: a statistical approach
As experiments continue to push the quantum-classical boundary using
increasingly complex dynamical systems, the interpretation of experimental data
becomes more and more challenging: when the observations are noisy, indirect,
and limited, how can we be sure that we are observing quantum behavior? This
tutorial highlights some of the difficulties in such experimental tests of
quantum mechanics, using optomechanics as the central example, and discusses
how the issues can be resolved using techniques from statistics and insights
from quantum information theory.Comment: v1: 2 pages; v2: invited tutorial for Quantum Measurements and
Quantum Metrology, substantial expansion of v1, 19 pages; v3: accepted; v4:
corrected some errors, publishe
Variance Reduction for Matrix Computations with Applications to Gaussian Processes
In addition to recent developments in computing speed and memory,
methodological advances have contributed to significant gains in the
performance of stochastic simulation. In this paper, we focus on variance
reduction for matrix computations via matrix factorization. We provide insights
into existing variance reduction methods for estimating the entries of large
matrices. Popular methods do not exploit the reduction in variance that is
possible when the matrix is factorized. We show how computing the square root
factorization of the matrix can achieve in some important cases arbitrarily
better stochastic performance. In addition, we propose a factorized estimator
for the trace of a product of matrices and numerically demonstrate that the
estimator can be up to 1,000 times more efficient on certain problems of
estimating the log-likelihood of a Gaussian process. Additionally, we provide a
new estimator of the log-determinant of a positive semi-definite matrix where
the log-determinant is treated as a normalizing constant of a probability
density.Comment: 20 pages, 3 figure
An adaptive scheme for quantum state tomography
The process of inferring and reconstructing the state of a quantum system from the results of measurements, better known as quantum state tomography, constitutes a crucial task in the emerging field of quantum technologies. Today it is possible to experimentally control quantum systems containing tens of entangled qubits and perform measurements of arbitrary observables with great accuracy.
However, in order to complete characterize an unknown -qubit state, quantum state tomography requires a number of measurements which grows exponentially with . A possible way to avoid this problem consists in performing an incomplete tomographic procedure able to provide a good estimate of the true state with few measurements. This thesis proposes a scheme for -qubit state tomography which aims to improve the fidelity between the reconstructed state and the target state. In particular, the scheme identifies the next measurement to perform based on the knowledge already acquired from the previous measurements on the experimental prepared state. The performance of this scheme was finally analyzed by means of simulations of quantum state tomography with product measurements as well as with entangled measurements. In both cases one observes that the here proposed adaptive scheme significantly outperforms a standard scheme in terms of the fidelity of the reconstructed state
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