45,066 research outputs found
High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation
The ratio between two probability density functions is an important component
of various tasks, including selection bias correction, novelty detection and
classification. Recently, several estimators of this ratio have been proposed.
Most of these methods fail if the sample space is high-dimensional, and hence
require a dimension reduction step, the result of which can be a significant
loss of information. Here we propose a simple-to-implement, fully nonparametric
density ratio estimator that expands the ratio in terms of the eigenfunctions
of a kernel-based operator; these functions reflect the underlying geometry of
the data (e.g., submanifold structure), often leading to better estimates
without an explicit dimension reduction step. We show how our general framework
can be extended to address another important problem, the estimation of a
likelihood function in situations where that function cannot be
well-approximated by an analytical form. One is often faced with this situation
when performing statistical inference with data from the sciences, due the
complexity of the data and of the processes that generated those data. We
emphasize applications where using existing likelihood-free methods of
inference would be challenging due to the high dimensionality of the sample
space, but where our spectral series method yields a reasonable estimate of the
likelihood function. We provide theoretical guarantees and illustrate the
effectiveness of our proposed method with numerical experiments.Comment: With supplementary materia
The equivalence of information-theoretic and likelihood-based methods for neural dimensionality reduction
Stimulus dimensionality-reduction methods in neuroscience seek to identify a
low-dimensional space of stimulus features that affect a neuron's probability
of spiking. One popular method, known as maximally informative dimensions
(MID), uses an information-theoretic quantity known as "single-spike
information" to identify this space. Here we examine MID from a model-based
perspective. We show that MID is a maximum-likelihood estimator for the
parameters of a linear-nonlinear-Poisson (LNP) model, and that the empirical
single-spike information corresponds to the normalized log-likelihood under a
Poisson model. This equivalence implies that MID does not necessarily find
maximally informative stimulus dimensions when spiking is not well described as
Poisson. We provide several examples to illustrate this shortcoming, and derive
a lower bound on the information lost when spiking is Bernoulli in discrete
time bins. To overcome this limitation, we introduce model-based dimensionality
reduction methods for neurons with non-Poisson firing statistics, and show that
they can be framed equivalently in likelihood-based or information-theoretic
terms. Finally, we show how to overcome practical limitations on the number of
stimulus dimensions that MID can estimate by constraining the form of the
non-parametric nonlinearity in an LNP model. We illustrate these methods with
simulations and data from primate visual cortex
Approximating Likelihood Ratios with Calibrated Discriminative Classifiers
In many fields of science, generalized likelihood ratio tests are established
tools for statistical inference. At the same time, it has become increasingly
common that a simulator (or generative model) is used to describe complex
processes that tie parameters of an underlying theory and measurement
apparatus to high-dimensional observations .
However, simulator often do not provide a way to evaluate the likelihood
function for a given observation , which motivates a new class of
likelihood-free inference algorithms. In this paper, we show that likelihood
ratios are invariant under a specific class of dimensionality reduction maps
. As a direct consequence, we show that
discriminative classifiers can be used to approximate the generalized
likelihood ratio statistic when only a generative model for the data is
available. This leads to a new machine learning-based approach to
likelihood-free inference that is complementary to Approximate Bayesian
Computation, and which does not require a prior on the model parameters.
Experimental results on artificial problems with known exact likelihoods
illustrate the potential of the proposed method.Comment: 35 pages, 5 figure
Approximated and User Steerable tSNE for Progressive Visual Analytics
Progressive Visual Analytics aims at improving the interactivity in existing
analytics techniques by means of visualization as well as interaction with
intermediate results. One key method for data analysis is dimensionality
reduction, for example, to produce 2D embeddings that can be visualized and
analyzed efficiently. t-Distributed Stochastic Neighbor Embedding (tSNE) is a
well-suited technique for the visualization of several high-dimensional data.
tSNE can create meaningful intermediate results but suffers from a slow
initialization that constrains its application in Progressive Visual Analytics.
We introduce a controllable tSNE approximation (A-tSNE), which trades off speed
and accuracy, to enable interactive data exploration. We offer real-time
visualization techniques, including a density-based solution and a Magic Lens
to inspect the degree of approximation. With this feedback, the user can decide
on local refinements and steer the approximation level during the analysis. We
demonstrate our technique with several datasets, in a real-world research
scenario and for the real-time analysis of high-dimensional streams to
illustrate its effectiveness for interactive data analysis
Compressive sensing adaptation for polynomial chaos expansions
Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of
the underlying Gaussian germ. Several rotations have been proposed in the
literature resulting in adaptations with different convergence properties. In
this paper we present a new adaptation mechanism that builds on compressive
sensing algorithms, resulting in a reduced polynomial chaos approximation with
optimal sparsity. The developed adaptation algorithm consists of a two-step
optimization procedure that computes the optimal coefficients and the input
projection matrix of a low dimensional chaos expansion with respect to an
optimally rotated basis. We demonstrate the attractive features of our
algorithm through several numerical examples including the application on
Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE
scramjet engine.Comment: Submitted to Journal of Computational Physic
Non-linear regression models for Approximate Bayesian Computation
Approximate Bayesian inference on the basis of summary statistics is
well-suited to complex problems for which the likelihood is either
mathematically or computationally intractable. However the methods that use
rejection suffer from the curse of dimensionality when the number of summary
statistics is increased. Here we propose a machine-learning approach to the
estimation of the posterior density by introducing two innovations. The new
method fits a nonlinear conditional heteroscedastic regression of the parameter
on the summary statistics, and then adaptively improves estimation using
importance sampling. The new algorithm is compared to the state-of-the-art
approximate Bayesian methods, and achieves considerable reduction of the
computational burden in two examples of inference in statistical genetics and
in a queueing model.Comment: 4 figures; version 3 minor changes; to appear in Statistics and
Computin
Error Metrics for Learning Reliable Manifolds from Streaming Data
Spectral dimensionality reduction is frequently used to identify
low-dimensional structure in high-dimensional data. However, learning
manifolds, especially from the streaming data, is computationally and memory
expensive. In this paper, we argue that a stable manifold can be learned using
only a fraction of the stream, and the remaining stream can be mapped to the
manifold in a significantly less costly manner. Identifying the transition
point at which the manifold is stable is the key step. We present error metrics
that allow us to identify the transition point for a given stream by
quantitatively assessing the quality of a manifold learned using Isomap. We
further propose an efficient mapping algorithm, called S-Isomap, that can be
used to map new samples onto the stable manifold. We describe experiments on a
variety of data sets that show that the proposed approach is computationally
efficient without sacrificing accuracy
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