74 research outputs found
Advances in variational Bayesian nonlinear blind source separation
Linear data analysis methods such as factor analysis (FA), independent component analysis (ICA) and blind source separation (BSS) as well as state-space models such as the Kalman filter model are used in a wide range of applications. In many of these, linearity is just a convenient approximation while the underlying effect is nonlinear. It would therefore be more appropriate to use nonlinear methods.
In this work, nonlinear generalisations of FA and ICA/BSS are presented. The methods are based on a generative model, with a multilayer perceptron (MLP) network to model the nonlinearity from the latent variables to the observations. The model is estimated using variational Bayesian learning. The variational Bayesian method is well-suited for the nonlinear data analysis problems. The approach is also theoretically interesting, as essentially the same method is used in several different fields and can be derived from several different starting points, including statistical physics, information theory, Bayesian statistics, and information geometry. These complementary views can provide benefits for interpretation of the operation of the learning method and its results.
Much of the work presented in this thesis consists of improvements that make the nonlinear factor analysis and blind source separation methods faster and more stable, while being applicable to other learning problems as well. The improvements include methods to accelerate convergence of alternating optimisation algorithms such as the EM algorithm and an improved approximation of the moments of a nonlinear transform of a multivariate probability distribution. These improvements can be easily applied to other models besides FA and ICA/BSS, such as nonlinear state-space models. A specialised version of the nonlinear factor analysis method for post-nonlinear mixtures is presented as well.reviewe
Statistical models for natural sounds
It is important to understand the rich structure of natural sounds in order to solve important
tasks, like automatic speech recognition, and to understand auditory processing
in the brain. This thesis takes a step in this direction by characterising the statistics of
simple natural sounds. We focus on the statistics because perception often appears to
depend on them, rather than on the raw waveform. For example the perception of auditory
textures, like running water, wind, fire and rain, depends on summary-statistics,
like the rate of falling rain droplets, rather than on the exact details of the physical
source.
In order to analyse the statistics of sounds accurately it is necessary to improve a
number of traditional signal processing methods, including those for amplitude demodulation,
time-frequency analysis, and sub-band demodulation. These estimation tasks
are ill-posed and therefore it is natural to treat them as Bayesian inference problems.
The new probabilistic versions of these methods have several advantages. For example,
they perform more accurately on natural signals and are more robust to noise,
they can also fill-in missing sections of data, and provide error-bars. Furthermore,
free-parameters can be learned from the signal. Using these new algorithms we demonstrate
that the energy, sparsity, modulation depth and modulation time-scale in each
sub-band of a signal are critical statistics, together with the dependencies between the
sub-band modulators. In order to validate this claim, a model containing co-modulated
coloured noise carriers is shown to be capable of generating a range of realistic sounding
auditory textures.
Finally, we explored the connection between the statistics of natural sounds and perception.
We demonstrate that inference in the model for auditory textures qualitatively
replicates the primitive grouping rules that listeners use to understand simple acoustic
scenes. This suggests that the auditory system is optimised for the statistics of natural
sounds
Blind image deconvolution: nonstationary Bayesian approaches to restoring blurred photos
High quality digital images have become pervasive in modern scientific and everyday life —
in areas from photography to astronomy, CCTV, microscopy, and medical imaging. However
there are always limits to the quality of these images due to uncertainty and imprecision in the
measurement systems. Modern signal processing methods offer the promise of overcoming
some of these problems by postprocessing
these blurred and noisy images. In this thesis,
novel methods using nonstationary statistical models are developed for the removal of blurs
from out of focus and other types of degraded photographic images.
The work tackles the fundamental problem blind image deconvolution (BID); its goal is
to restore a sharp image from a blurred observation when the blur itself is completely unknown.
This is a “doubly illposed”
problem — extreme lack of information must be countered
by strong prior constraints about sensible types of solution. In this work, the hierarchical
Bayesian methodology is used as a robust and versatile framework to impart the required prior
knowledge.
The thesis is arranged in two parts. In the first part, the BID problem is reviewed, along
with techniques and models for its solution. Observation models are developed, with an
emphasis on photographic restoration, concluding with a discussion of how these are reduced
to the common linear spatially-invariant
(LSI) convolutional model. Classical methods for the
solution of illposed
problems are summarised to provide a foundation for the main theoretical
ideas that will be used under the Bayesian framework. This is followed by an indepth
review
and discussion of the various prior image and blur models appearing in the literature, and then
their applications to solving the problem with both Bayesian and nonBayesian
techniques.
The second part covers novel restoration methods, making use of the theory presented in Part I.
Firstly, two new nonstationary image models are presented. The first models local variance in
the image, and the second extends this with locally adaptive noncausal
autoregressive (AR)
texture estimation and local mean components. These models allow for recovery of image
details including edges and texture, whilst preserving smooth regions. Most existing methods
do not model the boundary conditions correctly for deblurring of natural photographs, and a
Chapter is devoted to exploring Bayesian solutions to this topic.
Due to the complexity of the models used and the problem itself, there are many challenges
which must be overcome for tractable inference. Using the new models, three different inference
strategies are investigated: firstly using the Bayesian maximum marginalised a posteriori
(MMAP) method with deterministic optimisation; proceeding with the stochastic methods
of variational Bayesian (VB) distribution approximation, and simulation of the posterior distribution
using the Gibbs sampler. Of these, we find the Gibbs sampler to be the most effective
way to deal with a variety of different types of unknown blurs. Along the way, details are given
of the numerical strategies developed to give accurate results and to accelerate performance.
Finally, the thesis demonstrates state of the art
results in blind restoration of synthetic and real
degraded images, such as recovering details in out of focus photographs
Advances in identifiability of nonlinear probabilistic models
Identifiability is a highly prized property of statistical models. This thesis investigates this property in nonlinear models encountered in two fields of statistics: representation learning and causal discovery. In representation learning, identifiability leads to learning interpretable and reproducible representations, while in causal discovery, it is necessary for the estimation of correct causal directions.
We begin by leveraging recent advances in nonlinear ICA to show that the latent space of a VAE is identifiable up to a permutation and pointwise nonlinear transformations of its components. A factorized prior distribution over the latent variables conditioned on an auxiliary observed variable, such as a class label or nearly any other observation, is required for our result. We also extend previous identifiability results in nonlinear ICA to the case of noisy or undercomplete observations, and incorporate them into a maximum likelihood framework.
Our second contribution is to develop the Independently Modulated Component Analysis (IMCA) framework, a generalization of nonlinear ICA to non-independent latent variables. We show that we can drop the independence assumption in ICA while maintaining identifiability, resulting in a very flexible and generic framework for principled disentangled representation learning. This finding is predicated on the existence of an auxiliary variable that modulates the joint distribution of the latent variables in a factorizable manner.
As a third contribution, we extend the identifiability theory to a broad family of conditional energy-based models (EBMs). This novel model generalizes earlier results by removing any distributional assumptions on the representations, which are ubiquitous in the latent variable setting. The conditional EBM can learn identifiable overcomplete representations and has universal approximation capabilities/.
Finally, we investigate a connection between the framework of autoregressive normalizing flow models and causal discovery. Causal models derived from affine autoregressive flows are shown to be identifiable, generalizing the wellknown additive noise model. Using normalizing flows, we can compute the exact likelihood of the causal model, which is subsequently used to derive a likelihood ratio measure for causal discovery. They are also invertible, making them perfectly suitable for performing causal inference tasks like interventions and counterfactuals
GTM: the generative topographic mapping
This thesis describes the Generative Topographic Mapping (GTM) --- a non-linear latent variable model, intended for modelling continuous, intrinsically low-dimensional probability distributions, embedded in high-dimensional spaces. It can be seen as a non-linear form of principal component analysis or factor analysis. It also provides a principled alternative to the self-organizing map --- a widely established neural network model for unsupervised learning --- resolving many of its associated theoretical problems. An important, potential application of the GTM is visualization of high-dimensional data. Since the GTM is non-linear, the relationship between data and its visual representation may be far from trivial, but a better understanding of this relationship can be gained by computing the so-called magnification factor. In essence, the magnification factor relates the distances between data points, as they appear when visualized, to the actual distances between those data points. There are two principal limitations of the basic GTM model. The computational effort required will grow exponentially with the intrinsic dimensionality of the density model. However, if the intended application is visualization, this will typically not be a problem. The other limitation is the inherent structure of the GTM, which makes it most suitable for modelling moderately curved probability distributions of approximately rectangular shape. When the target distribution is very different to that, theaim of maintaining an `interpretable' structure, suitable for visualizing data, may come in conflict with the aim of providing a good density model. The fact that the GTM is a probabilistic model means that results from probability theory and statistics can be used to address problems such as model complexity. Furthermore, this framework provides solid ground for extending the GTM to wider contexts than that of this thesis
Variational Approximate Inference in Latent Linear Models
Latent linear models are core to much of machine learning and statistics. Specific examples of this model class include Bayesian generalised linear models, Gaussian process regression models and unsupervised latent linear models such as factor analysis and principal components analysis. In general, exact inference in this model class is computationally and analytically intractable. Approximations are thus required. In this thesis we consider deterministic approximate inference methods based on minimising the Kullback-Leibler (KL) divergence between a given target density and an approximating `variational' density. First we consider Gaussian KL (G-KL) approximate inference methods where the approximating variational density is a multivariate Gaussian. Regarding this procedure we make a number of novel contributions: sufficient conditions for which the G-KL objective is differentiable and convex are described, constrained parameterisations of Gaussian covariance that make G-KL methods fast and scalable are presented, the G-KL lower-bound to the target density's normalisation constant is proven to dominate those provided by local variational bounding methods. We also discuss complexity and model applicability issues of G-KL and other Gaussian approximate inference methods. To numerically validate our approach we present results comparing the performance of G-KL and other deterministic Gaussian approximate inference methods across a range of latent linear model inference problems. Second we present a new method to perform KL variational inference for a broad class of approximating variational densities. Specifically, we construct the variational density as an affine transformation of independently distributed latent random variables. The method we develop extends the known class of tractable variational approximations for which the KL divergence can be computed and optimised and enables more accurate approximations of non-Gaussian target densities to be obtained
Scalable software and models for large-scale extracellular recordings
The brain represents information about the world through the electrical activity of
populations of neurons. By placing an electrode near a neuron that is firing (spiking), it
is possible to detect the resulting extracellular action potential (EAP) that is transmitted
down an axon to other neurons. In this way, it is possible to monitor the communication
of a group of neurons to uncover how they encode and transmit information. As the
number of recorded neurons continues to increase, however, so do the data processing
and analysis challenges. It is crucial that scalable software and analysis tools are developed
and made available to the neuroscience community to keep up with the large
amounts of data that are already being gathered.
This thesis is composed of three pieces of work which I develop in order to better
process and analyze large-scale extracellular recordings. My work spans all stages of extracellular
analysis from the processing of raw electrical recordings to the development
of statistical models to reveal underlying structure in neural population activity.
In the first work, I focus on developing software to improve the comparison and adoption
of different computational approaches for spike sorting. When analyzing neural
recordings, most researchers are interested in the spiking activity of individual neurons,
which must be extracted from the raw electrical traces through a process called
spike sorting. Much development has been directed towards improving the performance
and automation of spike sorting. This continuous development, while essential,
has contributed to an over-saturation of new, incompatible tools that hinders rigorous
benchmarking and complicates reproducible analysis. To address these limitations, I
develop SpikeInterface, an open-source, Python framework designed to unify preexisting
spike sorting technologies into a single toolkit and to facilitate straightforward
benchmarking of different approaches. With this framework, I demonstrate that modern,
automated spike sorters have low agreement when analyzing the same dataset, i.e.
they find different numbers of neurons with different activity profiles; This result holds
true for a variety of simulated and real datasets. Also, I demonstrate that utilizing a
consensus-based approach to spike sorting, where the outputs of multiple spike sorters
are combined, can dramatically reduce the number of falsely detected neurons.
In the second work, I focus on developing an unsupervised machine learning approach
for determining the source location of individually detected spikes that are
recorded by high-density, microelectrode arrays. By localizing the source of individual
spikes, my method is able to determine the approximate position of the recorded neuriii
ons in relation to the microelectrode array. To allow my model to work with large-scale
datasets, I utilize deep neural networks, a family of machine learning algorithms that
can be trained to approximate complicated functions in a scalable fashion. I evaluate
my method on both simulated and real extracellular datasets, demonstrating that it is
more accurate than other commonly used methods. Also, I show that location estimates
for individual spikes can be utilized to improve the efficiency and accuracy of spike
sorting. After training, my method allows for localization of one million spikes in approximately
37 seconds on a TITAN X GPU, enabling real-time analysis of massive
extracellular datasets.
In my third and final presented work, I focus on developing an unsupervised machine
learning model that can uncover patterns of activity from neural populations
associated with a behaviour being performed. Specifically, I introduce Targeted Neural
Dynamical Modelling (TNDM), a statistical model that jointly models the neural activity
and any external behavioural variables. TNDM decomposes neural dynamics (i.e.
temporal activity patterns) into behaviourally relevant and behaviourally irrelevant dynamics;
the behaviourally relevant dynamics constitute all activity patterns required
to generate the behaviour of interest while behaviourally irrelevant dynamics may be
completely unrelated (e.g. other behavioural or brain states), or even related to behaviour
execution (e.g. dynamics that are associated with behaviour generally but are not
task specific). Again, I implement TNDM using a deep neural network to improve its
scalability and expressivity. On synthetic data and on real recordings from the premotor
(PMd) and primary motor cortex (M1) of a monkey performing a center-out reaching
task, I show that TNDM is able to extract low-dimensional neural dynamics that are
highly predictive of behaviour without sacrificing its fit to the neural data
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