Projection Pursuit through Φ\Phi-Divergence Minimisation

Consider a defined density on a set of very large dimension. It is quite difficult to find an estimate of this density from a data set. However, it is possible through a projection pursuit methodology to solve this problem. Touboul's article "Projection Pursuit Through Relative Entropy Minimization", 2009, demonstrates the interest of the author's method in a very simple given case. He considers the factorization of a density through an Elliptical component and some residual density. The above Touboul's work is based on minimizing relative entropy. In the present article, our proposal will aim at extending this very methodology to the $\Phi-$divergence. Furthermore, we will also consider the case when the density to be factorized is estimated from an i.i.d. sample. We will then propose a test for the factorization of the estimated density. Applications include a new test of fit pertaining to the Elliptical copulas.Comment: 32 pages, 4 figures, 5 tableaux, elsarticle clas

Improving the Practicality of Model-Based Reinforcement Learning: An Investigation into Scaling up Model-Based Methods in Online Settings

This thesis is a response to the current scarcity of practical model-based control algorithms in the reinforcement learning (RL) framework. As of yet there is no consensus on how best to integrate imperfect transition models into RL whilst mitigating policy improvement instabilities in online settings. Current state-of-the-art policy learning algorithms that surpass human performance often rely on model-free approaches that enjoy unmitigated sampling of transition data. Model-based RL (MBRL) instead attempts to distil experience into transition models that allow agents to plan new policies without needing to return to the environment and sample more data. The initial focus of this investigation is on kernel conditional mean embeddings (CMEs) (Song et al., 2009) deployed in an approximate policy iteration (API) algorithm (Grünewälder et al., 2012a). This existing MBRL algorithm boasts theoretically stable policy updates in continuous state and discrete action spaces. The Bellman operator’s value function and (transition) conditional expectation are modelled and embedded respectively as functions in a reproducing kernel Hilbert space (RKHS). The resulting finite-induced approximate pseudo-MDP (Yao et al., 2014a) can be solved exactly in a dynamic programming algorithm with policy improvement suboptimality guarantees. However model construction and policy planning scale cubically and quadratically respectively with the training set size, rendering the CME impractical for sampleabundant tasks in online settings. Three variants of CME API are investigated to strike a balance between stable policy updates and reduced computational complexity. The first variant models the value function and state-action representation explicitly in a parametric CME (PCME) algorithm with favourable computational complexity. However a soft conservative policy update technique is developed to mitigate policy learning oscillations in the planning process. The second variant returns to the non-parametric embedding and contributes (along with external work) to the compressed CME (CCME); a sparse and computationally more favourable CME. The final variant is a fully end-to-end differentiable embedding trained with stochastic gradient updates. The value function remains modelled in an RKHS such that backprop is driven by a non-parametric RKHS loss function. Actively compressed CME (ACCME) satisfies the pseudo-MDP contraction constraint using a sparse softmax activation function. The size of the pseudo-MDP (i.e. the size of the embedding’s last layer) is controlled by sparsifying the last layer weight matrix by extending the truncated gradient method (Langford et al., 2009) with group lasso updates in a novel ‘use it or lose it’ neuron pruning mechanism. Surprisingly this technique does not require extensive fine-tuning between control tasks

Sparse machine learning methods with applications in multivariate signal processing

This thesis details theoretical and empirical work that draws from two main subject areas: Machine Learning (ML) and Digital Signal Processing (DSP). A unified general framework is given for the application of sparse machine learning methods to multivariate signal processing. In particular, methods that enforce sparsity will be employed for reasons of computational efficiency, regularisation, and compressibility. The methods presented can be seen as modular building blocks that can be applied to a variety of applications. Application specific prior knowledge can be used in various ways, resulting in a flexible and powerful set of tools. The motivation for the methods is to be able to learn and generalise from a set of multivariate signals. In addition to testing on benchmark datasets, a series of empirical evaluations on real world datasets were carried out. These included: the classification of musical genre from polyphonic audio files; a study of how the sampling rate in a digital radar can be reduced through the use of Compressed Sensing (CS); analysis of human perception of different modulations of musical key from Electroencephalography (EEG) recordings; classification of genre of musical pieces to which a listener is attending from Magnetoencephalography (MEG) brain recordings. These applications demonstrate the efficacy of the framework and highlight interesting directions of future research

Multiarray Signal Processing: Tensor decomposition meets compressed sensing

We discuss how recently discovered techniques and tools from compressed sensing can be used in tensor decompositions, with a view towards modeling signals from multiple arrays of multiple sensors. We show that with appropriate bounds on a measure of separation between radiating sources called coherence, one could always guarantee the existence and uniqueness of a best rank-r approximation of the tensor representing the signal. We also deduce a computationally feasible variant of Kruskal's uniqueness condition, where the coherence appears as a proxy for k-rank. Problems of sparsest recovery with an infinite continuous dictionary, lowest-rank tensor representation, and blind source separation are treated in a uniform fashion. The decomposition of the measurement tensor leads to simultaneous localization and extraction of radiating sources, in an entirely deterministic manner.Comment: 10 pages, 1 figur

Automatic Music Transcription using Structure and Sparsity

PhdAutomatic Music Transcription seeks a machine understanding of a musical signal in terms of pitch-time activations. One popular approach to this problem is the use of spectrogram decompositions, whereby a signal matrix is decomposed over a dictionary of spectral templates, each representing a note. Typically the decomposition is performed using gradient descent based methods, performed using multiplicative updates based on Non-negative Matrix Factorisation (NMF). The final representation may be expected to be sparse, as the musical signal itself is considered to consist of few active notes. In this thesis some concepts that are familiar in the sparse representations literature are introduced to the AMT problem. Structured sparsity assumes that certain atoms tend to be active together. In the context of AMT this affords the use of subspace modelling of notes, and non-negative group sparse algorithms are proposed in order to exploit the greater modelling capability introduced. Stepwise methods are often used for decomposing sparse signals and their use for AMT has previously been limited. Some new approaches to AMT are proposed by incorporation of stepwise optimal approaches with promising results seen. Dictionary coherence is used to provide recovery conditions for sparse algorithms. While such guarantees are not possible in the context of AMT, it is found that coherence is a useful parameter to consider, affording improved performance in spectrogram decompositions

Epigraphical Projection and Proximal Tools for Solving Constrained Convex Optimization Problems: Part I

We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set. In this paper, we focus on a family of constraints involving linear transforms of l_1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm and the sup norm. The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems

Vectorial variational problems in L∞ and applications to data assimilation

This thesis is a collection of published, submitted and developing papers. Each paper is presented as a chapter of this thesis, in each paper we advance the field of vectorial Calculus of Variations in L ∞. This new progress includes constrained problems, such as the constraint of the Navier-Stokes equations studied in Chapter 2. Additionally the combination of constraints, including a nonlinear operator and a supremal functional, deliberated in Chapter 3. Finally, Chapter 4 presents an alternative supremal constraint, in the contemplation of the second order generalised ∞-eigenvalue problem. In Chapter 2 we introduce the joint paper with Nikos Katzourakis and Boris Muha. We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler-Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergence form counterpart of the corresponding nondivergence Aronsson-Euler system. In Chapter 3 we present the joint paper with Nikos Katzourakis. We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞, we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. Chapter 4 provides part of the corresponding developing preprint, joint work with Nikos Katzourakis. We consider the problem of minimising the L ∞ norm of a function of the Hessian over a class of maps, subject to a mass constraint involving the L ∞ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of L p approximations, we establish the existence of a special L ∞ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem