<p>In recent years there has been a growing interest in data mining and graph machine learning for techniques that can obtain frequencies of <em>k</em>-node Connected Induced Subgraphs (<em>k</em>-CIS) contained in large real-world graphs. While recent work has shown that 5-CISs can be counted exactly, no exact polynomial-time algorithms are known that solve this task for <em>k </em>> 5. In the past, sampling-based algorithms that work well in moderately-sized graphs for <em>k</em> ≤ 8 have been proposed. In this thesis I push this boundary up to <em>k</em> ≤ 16 for graphs containing up to 120M edges, and to <em>k</em> ≤ 25 for smaller graphs containing between a million to 20M edges. I do so by re-imagining two older, but elegant and memory-efficient algorithms -- FANMOD and PSRW -- which have large estimation errors by modern standards. This is because FANMOD produces highly correlated k-CIS samples and the cost of sampling the PSRW Markov chain becomes prohibitively expensive for k-CIS’s larger than <em>k </em>> 8.</p> <p>In this thesis, I introduce:</p> <p>(a)  <strong>RTS:</strong> a novel regenerative Markov chain Monte Carlo (MCMC) sampling procedure on the tree, generated on-the-fly by the FANMOD algorithm. RTS is able to run on multiple cores and multiple machines (embarrassingly parallel) and compute confidence intervals of estimates, all this while preserving the memory-efficient nature of FANMOD. RTS is thus able to estimate subgraph statistics for <em>k</em> ≤ 16 for larger graphs containing up to 120M edges, and for <em>k</em> ≤ 25 for smaller graphs containing between a million to 20M edges.</p> <p>(b) <strong>R-PSRW:</strong> which scales the PSRW algorithm to larger CIS-sizes using a rejection sampling procedure to efficiently sample transitions from the PSRW Markov chain. R-PSRW matches RTS in terms of scaling to larger CIS sizes.</p> <p>(c) <strong>Ripple:</strong> which achieves unprecedented scalability by stratifying the R-PSRW Markov chain state-space into ordered strata via a new technique that I call <em>sequential stratified regeneration</em>. I show that the Ripple estimator is consistent, highly parallelizable, and scales well. Ripple is able to <em>count</em> CISs of size up to <em>k </em>≤ 12 in real world graphs containing up to 120M edges.</p> <p>My empirical results show that the proposed methods offer a considerable improvement over the state-of-the-art. Moreover my methods are able to run at a scale that has been considered unreachable until now, not only by prior MCMC-based methods but also by other sampling approaches. </p> <p><strong>Optimization of Restricted Boltzmann Machines. </strong>In addition, I also propose a regenerative transformation of MCMC samplers of Restricted Boltzmann Machines RBMs. My approach, Markov Chain Las Vegas (MCLV) gives statistical guarantees in exchange for random running times. MCLV uses a stopping set built from the training data and has a maximum number of Markov chain step-count <em>K</em> (referred as MCLV-<em>K</em>). I present a MCLV-<em>K</em> gradient estimator (LVS-<em>K</em>) for RBMs and explore the correspondence and differences between LVS-<em>K</em> and Contrastive Divergence (CD-<em>K</em>). LVS-<em>K</em> significantly outperforms CD-<em>K</em> in the task of training RBMs over the MNIST dataset, indicating MCLV to be a promising direction in learning generative models.</p>

We consider a mechanistic non-linear machine learning approach to learning signals in financial time series data. A modularised and decoupled algorithm framework is established and is proven on daily sampled closing time-series data for JSE equity markets. The input patterns are based on input data vectors of data windows preprocessed into a sequence of daily, weekly and monthly or quarterly sampled feature measurement changes (log feature fluctuations). The data processing is split into a batch processed step where features are learnt using a Stacked AutoEncoder (SAE) via unsupervised learning, and then both batch and online supervised learning are carried out on Feedforward Neural Networks (FNNs) using these features. The FNN output is a point prediction of measured time-series feature fluctuations (log differenced data) in the future (ex-post). Weight initializations for these networks are implemented with restricted Boltzmann machine pretraining, and variance based initializations. The validity of the FNN backtest results are shown under a rigorous assessment of backtest overfitting using both Combinatorially Symmetrical Cross Validation and Probabilistic and Deflated Sharpe Ratios. Results are further used to develop a view on the phenomenology of financial markets and the value of complex historical data under unstable dynamics.

Advances in unsupervised learning and deep neural networks have led to increased performance in a number of domains, and to the ability to draw strong comparisons between the biological method of self-organization conducted by the brain and computational mechanisms. This thesis aims to use real-world data to tackle two areas in the domain of computer vision which have biological equivalents: feature detection and motion tracking. The aforementioned advances have allowed efficient learning of feature representations directly from large sets of unlabeled data instead of using traditional handcrafted features. The first part of this thesis evaluates such representations by comparing regularization and preprocessing methods which incorporate local neighbouring information during training on a single-layer neural network. The networks are trained and tested on the Hollywood2 video dataset, as well as the static CIFAR-10, STL-10, COIL-100, and MNIST image datasets. The induction of topography or simple image blurring via Gaussian filters during training produces better discriminative features as evidenced by the consistent and notable increase in classification results that they produce. In the visual domain, invariant features are desirable such that objects can be classified despite transformations. It is found that most of the compared methods produce more invariant features, however, classification accuracy does not correlate to invariance. The second, and paramount, contribution of this thesis is a biologically-inspired model to explain the emergence of motion tracking behaviour in early development using unsupervised learning. The model’s self-organization is biased by an original concept called retinal constancy, which measures how similar visual contents are between successive frames. In the proposed two-layer deep network, when exposed to real-world video, the first layer learns to encode visual motion, and the second layer learns to relate that motion to gaze movements, which it perceives and creates through bi-directional nodes. This is unique because it uses general machine learning algorithms, and their inherent generative properties, to learn from real-world data. It also implements a biological theory and learns in a fully unsupervised manner. An analysis of its parameters and limitations is conducted, and its tracking performance is evaluated. Results show that this model is able to successfully follow targets in real-world video, despite being trained without supervision on real-world video.

This work is focused on objects and scenes recognition using machine learning and computer vision tools. Before the solution of this problem has been studied basic phases of the machine learning concept and statistical models with accent on their division into discriminative and generative method. Further, the Bag-of-words method and its modification have been investigated and described. In the practical part of this work, the implementation of the Bag-of-words method with the SVM classifier was created in the Matlab environment and the model was tested on various sets of publicly available images.