312 research outputs found
Inflation with a graceful exit in a random landscape
We develop a stochastic description of small-field inflationary histories
with a graceful exit in a random potential whose Hessian is a Gaussian random
matrix as a model of the unstructured part of the string landscape. The
dynamical evolution in such a random potential from a small-field inflation
region towards a viable late-time de Sitter (dS) minimum maps to the dynamics
of Dyson Brownian motion describing the relaxation of non-equilibrium
eigenvalue spectra in random matrix theory. We analytically compute the
relaxation probability in a saddle point approximation of the partition
function of the eigenvalue distribution of the Wigner ensemble describing the
mass matrices of the critical points. When applied to small-field inflation in
the landscape, this leads to an exponentially strong bias against small-field
ranges and an upper bound on the number of light fields
participating during inflation from the non-observation of negative spatial
curvature.Comment: Published versio
Generalized Direct Sampling for Hierarchical Bayesian Models
We develop a new method to sample from posterior distributions in
hierarchical models without using Markov chain Monte Carlo. This method, which
is a variant of importance sampling ideas, is generally applicable to
high-dimensional models involving large data sets. Samples are independent, so
they can be collected in parallel, and we do not need to be concerned with
issues like chain convergence and autocorrelation. Additionally, the method can
be used to compute marginal likelihoods
Inapproximability for Antiferromagnetic Spin Systems in the Tree Non-Uniqueness Region
A remarkable connection has been established for antiferromagnetic 2-spin
systems, including the Ising and hard-core models, showing that the
computational complexity of approximating the partition function for graphs
with maximum degree D undergoes a phase transition that coincides with the
statistical physics uniqueness/non-uniqueness phase transition on the infinite
D-regular tree. Despite this clear picture for 2-spin systems, there is little
known for multi-spin systems. We present the first analog of the above
inapproximability results for multi-spin systems.
The main difficulty in previous inapproximability results was analyzing the
behavior of the model on random D-regular bipartite graphs, which served as the
gadget in the reduction. To this end one needs to understand the moments of the
partition function. Our key contribution is connecting: (i) induced matrix
norms, (ii) maxima of the expectation of the partition function, and (iii)
attractive fixed points of the associated tree recursions (belief propagation).
The view through matrix norms allows a simple and generic analysis of the
second moment for any spin system on random D-regular bipartite graphs. This
yields concentration results for any spin system in which one can analyze the
maxima of the first moment. The connection to fixed points of the tree
recursions enables an analysis of the maxima of the first moment for specific
models of interest.
For k-colorings we prove that for even k, in the tree non-uniqueness region
(which corresponds to k<D) it is NP-hard, unless NP=RP, to approximate the
number of colorings for triangle-free D-regular graphs. Our proof extends to
the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic
model under a mild condition
RISE: An Incremental Trust-Region Method for Robust Online Sparse Least-Squares Estimation
Many point estimation problems in robotics, computer vision, and machine learning can be formulated as instances of the general problem of minimizing a sparse nonlinear sum-of-squares objective function. For inference problems of this type, each input datum gives rise to a summand in the objective function, and therefore performing online inference corresponds to solving a sequence of sparse nonlinear least-squares minimization problems in which additional summands are added to the objective function over time. In this paper, we present Robust Incremental least-Squares Estimation (RISE), an incrementalized version of the Powell's Dog-Leg numerical optimization method suitable for use in online sequential sparse least-squares minimization. As a trust-region method, RISE is naturally robust to objective function nonlinearity and numerical ill-conditioning and is provably globally convergent for a broad class of inferential cost functions (twice-continuously differentiable functions with bounded sublevel sets). Consequently, RISE maintains the speed of current state-of-the-art online sparse least-squares methods while providing superior reliability.United States. Office of Naval Research (Grant N00014-12-1-0093)United States. Office of Naval Research (Grant N00014-11-1-0688)United States. Office of Naval Research (Grant N00014-06-1-0043)United States. Office of Naval Research (Grant N00014-10-1-0936)United States. Air Force Research Laboratory (Contract FA8650-11-C-7137
An Efficient Approach for Computing Optimal Low-Rank Regularized Inverse Matrices
Standard regularization methods that are used to compute solutions to
ill-posed inverse problems require knowledge of the forward model. In many
real-life applications, the forward model is not known, but training data is
readily available. In this paper, we develop a new framework that uses training
data, as a substitute for knowledge of the forward model, to compute an optimal
low-rank regularized inverse matrix directly, allowing for very fast
computation of a regularized solution. We consider a statistical framework
based on Bayes and empirical Bayes risk minimization to analyze theoretical
properties of the problem. We propose an efficient rank update approach for
computing an optimal low-rank regularized inverse matrix for various error
measures. Numerical experiments demonstrate the benefits and potential
applications of our approach to problems in signal and image processing.Comment: 24 pages, 11 figure
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