21,564 research outputs found
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
Conditional Density Estimation with Dimensionality Reduction via Squared-Loss Conditional Entropy Minimization
Regression aims at estimating the conditional mean of output given input.
However, regression is not informative enough if the conditional density is
multimodal, heteroscedastic, and asymmetric. In such a case, estimating the
conditional density itself is preferable, but conditional density estimation
(CDE) is challenging in high-dimensional space. A naive approach to coping with
high-dimensionality is to first perform dimensionality reduction (DR) and then
execute CDE. However, such a two-step process does not perform well in practice
because the error incurred in the first DR step can be magnified in the second
CDE step. In this paper, we propose a novel single-shot procedure that performs
CDE and DR simultaneously in an integrated way. Our key idea is to formulate DR
as the problem of minimizing a squared-loss variant of conditional entropy, and
this is solved via CDE. Thus, an additional CDE step is not needed after DR. We
demonstrate the usefulness of the proposed method through extensive experiments
on various datasets including humanoid robot transition and computer art
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
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
Verification Under Increasing Dimensionality
Verification decisions are often based on second order statistics estimated from a set of samples. Ongoing growth of computational resources allows for considering more and more features, increasing the dimensionality of the samples. If the dimensionality is of the same order as the number of samples used in the estimation or even higher, then the accuracy of the estimate decreases significantly. In particular, the eigenvalues of the covariance matrix are estimated with a bias and the estimate of the eigenvectors differ considerably from the real eigenvectors. We show how a classical approach of verification in high dimensions is severely affected by these problems, and we show how bias correction methods can reduce these problems
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