Isotropic Multiple Scattering Processes on Hyperspheres

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres ${\mathbb S}^{p-1}$. It allows one to derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions (pdf) on ${\mathbb S}^{p-1}$ is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic approximation allows us to compute estimation bounds for the parameters of Compound Cox Processes (CCP) on ${\mathbb S}^{p-1}$.Comment: 16 pages, 4 figure

Kernel Methods in Computer-Aided Constructive Drug Design

A drug is typically a small molecule that interacts with the binding site of some target protein. Drug design involves the optimization of this interaction so that the drug effectively binds with the target protein while not binding with other proteins (an event that could produce dangerous side effects). Computational drug design involves the geometric modeling of drug molecules, with the goal of generating similar molecules that will be more effective drug candidates. It is necessary that algorithms incorporate strategies to measure molecular similarity by comparing molecular descriptors that may involve dozens to hundreds of attributes. We use kernel-based methods to define these measures of similarity. Kernels are general functions that can be used to formulate similarity comparisons. The overall goal of this thesis is to develop effective and efficient computational methods that are reliant on transparent mathematical descriptors of molecules with applications to affinity prediction, detection of multiple binding modes, and generation of new drug leads. While in this thesis we derive computational strategies for the discovery of new drug leads, our approach differs from the traditional ligandbased approach. We have developed novel procedures to calculate inverse mappings and subsequently recover the structure of a potential drug lead. The contributions of this thesis are the following: 1. We propose a vector space model molecular descriptor (VSMMD) based on a vector space model that is suitable for kernel studies in QSAR modeling. Our experiments have provided convincing comparative empirical evidence that our descriptor formulation in conjunction with kernel based regression algorithms can provide sufficient discrimination to predict various biological activities of a molecule with reasonable accuracy. 2. We present a new component selection algorithm KACS (Kernel Alignment Component Selection) based on kernel alignment for a QSAR study. Kernel alignment has been developed as a measure of similarity between two kernel functions. In our algorithm, we refine kernel alignment as an evaluation tool, using recursive component elimination to eventually select the most important components for classification. We have demonstrated empirically and proven theoretically that our algorithm works well for finding the most important components in different QSAR data sets. 3. We extend the VSMMD in conjunction with a kernel based clustering algorithm to the prediction of multiple binding modes, a challenging area of research that has been previously studied by means of time consuming docking simulations. The results reported in this study provide strong empirical evidence that our strategy has enough resolving power to distinguish multiple binding modes through the use of a standard k-means algorithm. 4. We develop a set of reverse engineering strategies for QSAR modeling based on our VSMMD. These strategies include: (a) The use of a kernel feature space algorithm to design or modify descriptor image points in a feature space. (b) The deployment of a pre-image algorithm to map the newly defined descriptor image points in the feature space back to the input space of the descriptors. (c) The design of a probabilistic strategy to convert new descriptors to meaningful chemical graph templates. The most important aspect of these contributions is the presentation of strategies that actually generate the structure of a new drug candidate. While the training set is still used to generate a new image point in the feature space, the reverse engineering strategies just described allows us to develop a new drug candidate that is independent of issues related to probability distribution constraints placed on test set molecules

Constructive Approximation and Learning by Greedy Algorithms

This thesis develops several kernel-based greedy algorithms for different machine learning problems and analyzes their theoretical and empirical properties. Greedy approaches have been extensively used in the past for tackling problems in combinatorial optimization where finding even a feasible solution can be a computationally hard problem (i.e., not solvable in polynomial time). A key feature of greedy algorithms is that a solution is constructed recursively from the smallest constituent parts. In each step of the constructive process a component is added to the partial solution from the previous step and, thus, the size of the optimization problem is reduced. The selected components are given by optimization problems that are simpler and easier to solve than the original problem. As such schemes are typically fast at constructing a solution they can be very effective on complex optimization problems where finding an optimal/good solution has a high computational cost. Moreover, greedy solutions are rather intuitive and the schemes themselves are simple to design and easy to implement. There is a large class of problems for which greedy schemes generate an optimal solution or a good approximation of the optimum. In the first part of the thesis, we develop two deterministic greedy algorithms for optimization problems in which a solution is given by a set of functions mapping an instance space to the space of reals. The first of the two approaches facilitates data understanding through interactive visualization by providing means for experts to incorporate their domain knowledge into otherwise static kernel principal component analysis. This is achieved by greedily constructing embedding directions that maximize the variance at data points (unexplained by the previously constructed embedding directions) while adhering to specified domain knowledge constraints. The second deterministic greedy approach is a supervised feature construction method capable of addressing the problem of kernel choice. The goal of the approach is to construct a feature representation for which a set of linear hypotheses is of sufficient capacity — large enough to contain a satisfactory solution to the considered problem and small enough to allow good generalization from a small number of training examples. The approach mimics functional gradient descent and constructs features by fitting squared error residuals. We show that the constructive process is consistent and provide conditions under which it converges to the optimal solution. In the second part of the thesis, we investigate two problems for which deterministic greedy schemes can fail to find an optimal solution or a good approximation of the optimum. This happens as a result of making a sequence of choices which take into account only the immediate reward without considering the consequences onto future decisions. To address this shortcoming of deterministic greedy schemes, we propose two efficient randomized greedy algorithms which are guaranteed to find effective solutions to the corresponding problems. In the first of the two approaches, we provide a mean to scale kernel methods to problems with millions of instances. An approach, frequently used in practice, for this type of problems is the Nyström method for low-rank approximation of kernel matrices. A crucial step in this method is the choice of landmarks which determine the quality of the approximation. We tackle this problem with a randomized greedy algorithm based on the K-means++ cluster seeding scheme and provide a theoretical and empirical study of its effectiveness. In the second problem for which a deterministic strategy can fail to find a good solution, the goal is to find a set of objects from a structured space that are likely to exhibit an unknown target property. This discrete optimization problem is of significant interest to cyclic discovery processes such as de novo drug design. We propose to address it with an adaptive Metropolis–Hastings approach that samples candidates from the posterior distribution of structures conditioned on them having the target property. The proposed constructive scheme defines a consistent random process and our empirical evaluation demonstrates its effectiveness across several different application domains

Mathematical techniques for shape modelling in computer graphics: A distance-based approach.

This research is concerned with shape modelling in computer graphics. The dissertation provides a review of the main research topics and developments in shape modelling and discusses current visualisation techniques required for the display of the models produced. In computer graphics surfaces are normally defined using analytic functions. Geometry however, supplies many shapes without providing their analytic descriptions. These are defined implicitly through fundamental relationships between primitive geometrical objects. Transferring this approach in computer graphics, opens new directions in shape modelling by enabling the definition of new objects or supplying a rigorous alternative to analytical definitions of objects with complex analytical descriptions. We review, in this dissertation, relevant works in the area of implicit modelling. Based on our observations on the shortcomings of these works, we develop an implicit modelling approach which draws on a seminal technique in this area: the distance based object definition. We investigate the principles, potential and applications of this technique both in conceptual terms (modelling aspects) and on technical merit (visualisation issues). This is the context of this PhD research. The conceptual and technological frameworks developed are presented in terms of a comprehensive investigation of an object's constituent primitives and modelling constraints on the one hand, and software visualisation platforms on the other. Finally, we adopt a critical perspective of our work to discuss possible directions for further improvements and exploitation for the modelling approach we have developed

The application of statistical methods to problems relating to the chemical industry

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Probabilistic Machine Learning in the Age of Deep Learning: New Perspectives for Gaussian Processes, Bayesian Optimisation and Beyond

Advances in artificial intelligence (AI) are rapidly transforming our world, with systems now surpassing human capabilities in numerous domains. Much of this progress traces back to machine learning (ML), particularly deep learning, and its ability to uncover meaningful patterns in data. However, true intelligence in AI demands more than raw predictive power; it requires a principled approach to making decisions under uncertainty. Probabilistic ML offers a framework for reasoning about the unknown in ML models through probability theory and Bayesian inference. Gaussian processes (GPs) are a quintessential probabilistic model known for their flexibility, data efficiency, and well-calibrated uncertainty estimates. GPs are integral to sequential decision-making algorithms like Bayesian optimisation (BO), which optimises expensive black-box objective functions. Despite efforts to improve GP scalability, performance gaps persist compared to neural networks (NNs) due to their lack of representation learning capabilities. This thesis aims to integrate deep learning with probabilistic methods and lend probabilistic perspectives to deep learning. Key contributions include: (1) Extending orthogonally-decoupled sparse GP approximations to incorporate nonlinear NN activations as inter-domain features, bringing predictive performance closer to NNs. (2) Framing cycle-consistent adversarial networks (CYCLEGANs) for unpaired image-to-image translation as variational inference (VI) in an implicit latent variable model, providing a Bayesian perspective on these deep generative models. (3) Introducing a model-agnostic reformulation of BO based on binary classification, enabling the integration of powerful modelling approaches like deep learning for complex optimisation tasks. By enriching the interplay between deep learning and probabilistic ML, this thesis advances the foundations of AI, facilitating the development of more capable and dependable automated decision-making systems

Risk assessment tool for diabetic neuropathy.

Peripheral neuropathy is one of the serious complications of diabetes. Symptoms such as tingling and loss of touch sensation are commonly associated with the early stages of neuropathy causing numbness in the feet. Early detection of this condition is necessary in order to prevent the progression of the disease. Out of many detection techniques vibration perception is becoming the gold standard for neuropathy assessment. Devices like tuning fork, Biothesiometer and Neurothesiometer use this technology but require an operator to record and manually interpret the results. The results are user-dependent and are not consistent. To overcome these limitations, a platform-based device “VibraScan” was developed that can be self-operated and results displayed on a user interface. The development of the device is based on studying the effect of the vibration on the human subject by identifying the receptors responsible for sensation. The requirement of generating vibration was achieved by selecting a specific actuator that creates vibration perpendicular to the contact surface. The battery operated VibraScan is wirelessly controlled by software to generate vibration for determining the vibration perception threshold (VPT). Care has been taken while developing the user interface for human safety with the vibration intensity. The device can be operated without any assistance and results are automatically interpreted in terms of severity level indicated similar to the traffic-light classification. In order to provide consistent results with the existing devices a study was undertaken between Neurothesiometer and VibraScan with 20 healthy subjects. The results were compared using Bland-Altman plot and a close agreement was found between the two measurements. VibraScan accurately measures VPT based on the perceived vibration threshold, however, it does not predict any risk associated with neuropathy. In order to supplement this device with the progression of neuropathy a risk assessment tool was developed for automated prediction of neuropathy based on the clinical history of patients. The smart tool is based on the research related to the risk factors of diabetic neuropathy which was studied and analysed using summarised patient data. Box-Cox regression was used with the response variable (VPT) and a set of clinical variables as potential predictors. Significant predictors were: age, height, weight, urine albumin to creatinine ratio (ACR), HbA1c, cholesterol and duration of diabetes. Ordinary Least Squares Regression was then used with logarithmic (VPT) and the significant predictor set (Box-Cox transformed) to obtain additional fit estimates. With the aim to improving the precision of VPT prediction, a simulated patient data set (n = 4158) was also generated using the mean and the covariance of the original patient variables, but with reduced standard errors. For clinical or patient use, providing direct knowledge of VPT was considered less helpful than providing a simple risk category corresponding to a range of VPT values. To achieve this, the continuous scale VPT was recoded into three categories based on the following clinical thresholds in volts (V): low risk (0 to 20.99 V), medium risk (21 to 30.99 V) and high risk (≥ 31 V). Ordinal Logistic Regression was then used with this categorical outcome variable to confirm the original predictor set. Having established the effectiveness of this “classical” baseline, attention turned to Neural Network modelling. This showed that a carefully tuned Neural Network based Proportional Odds Model (NNPOM) could achieve a classification success >70%, somewhat higher than that obtained with the classical modelling. A version of this model was implemented in the VibraScan risk assessment tool. Integrating VibraScan and the risk assessment software has created a comprehensive diagnostic tool for diabetic neuropathy

Harnessing Evolution in-Materio as an Unconventional Computing Resource

This thesis illustrates the use and development of physical conductive analogue systems for unconventional computing using the Evolution in-Materio (EiM) paradigm. EiM uses an Evolutionary Algorithm to configure and exploit a physical material (or medium) for computation. While EiM processors show promise, fundamental questions and scaling issues remain. Additionally, their development is hindered by slow manufacturing and physical experimentation. This work addressed these issues by implementing simulated models to speed up research efforts, followed by investigations of physically implemented novel in-materio devices. Initial work leveraged simulated conductive networks as single substrate ‘monolithic’ EiM processors, performing classification by formulating the system as an optimisation problem, solved using Differential Evolution. Different material properties and algorithm parameters were isolated and investigated; which explained the capabilities of configurable parameters and showed ideal nanomaterial choice depended upon problem complexity. Subsequently, drawing from concepts in the wider Machine Learning field, several enhancements to monolithic EiM processors were proposed and investigated. These ensured more efficient use of training data, better classification decision boundary placement, an independently optimised readout layer, and a smoother search space. Finally, scalability and performance issues were addressed by constructing in-Materio Neural Networks (iM-NNs), where several EiM processors were stacked in parallel and operated as physical realisations of Hidden Layer neurons. Greater flexibility in system implementation was achieved by re-using a single physical substrate recursively as several virtual neurons, but this sacrificed faster parallelised execution. These novel iM-NNs were first implemented using Simulated in-Materio neurons, and trained for classification as Extreme Learning Machines, which were found to outperform artificial networks of a similar size. Physical iM-NN were then implemented using a Raspberry Pi, custom Hardware Interface and Lambda Diode based Physical in-Materio neurons, which were trained successfully with neuroevolution. A more complex AutoEncoder structure was then proposed and implemented physically to perform dimensionality reduction on a handwritten digits dataset, outperforming both Principal Component Analysis and artificial AutoEncoders. This work presents an approach to exploit systems with interesting physical dynamics, and leverage them as a computational resource. Such systems could become low power, high speed, unconventional computing assets in the future