We study the estimation of the latent variable Gaussian graphical model (LVGGM), where the precision matrix is the superposition of a sparse matrix and a low-rank matrix. In order to speed up the estimation of the sparse plus low-rank components, we propose a sparsity constrained maximum likelihood estimator based on matrix factorization, and an efficient alternating gradient descent algorithm with hard thresholding to solve it. Our algorithm is orders of magnitude faster than the convex relaxation based methods for LVGGM. In addition, we prove that our algorithm is guaranteed to linearly converge to the unknown sparse and low-rank components up to the optimal statistical precision. Experiments on both synthetic and genomic data demonstrate the superiority of our algorithm over the state-of-the-art algorithms and corroborate our theory.

Techniques known as Nonlinear Set Membership prediction, Kinky Inference or Lipschitz Interpolation are fast and numerically robust approaches to nonparametric machine learning that have been proposed to be utilised in the context of system identification and learning-based control. They utilise presupposed Lipschitz properties in order to compute inferences over unobserved function values. Unfortunately, most of these approaches rely on exact knowledge about the input space metric as well as about the Lipschitz constant. Furthermore, existing techniques to estimate the Lipschitz constants from the data are not robust to noise or seem to be ad-hoc and typically are decoupled from the ultimate learning and prediction task. To overcome these limitations, we propose an approach for optimising parameters of the presupposed metrics by minimising validation set prediction errors. To avoid poor performance due to local minima, we propose to utilise Lipschitz properties of the optimisation objective to ensure global optimisation success. The resulting approach is a new flexible method for nonparametric black-box learning. We provide experimental evidence of the competitiveness of our approach on artificial as well as on real data.

Tree ensembles, such as random forests and boosted trees, are renowned for their high prediction performance. However, their interpretability is critically limited due to the enormous complexity. In this study, we present a method to make a complex tree ensemble interpretable by simplifying the model. Specifically, we formalize the simplification of tree ensembles as a model selection problem. Given a complex tree ensemble, we aim at obtaining the simplest representation that is essentially equivalent to the original one. To this end, we derive a Bayesian model selection algorithm that optimizes the simplified model while maintaining the prediction performance. Our numerical experiments on several datasets showed that complicated tree ensembles were reasonably approximated as interpretable.

Recently, technologies such as face detection, facial landmark localisation and face recognition and verification have matured enough to provide effective and efficient solutions for imagery captured under arbitrary conditions (referred to as "in-the-wild"). This is partially attributed to the fact that comprehensive "in-the-wild" benchmarks have been developed for face detection, landmark localisation and recognition/verification. A very important technology that has not been thoroughly evaluated yet is deformable face tracking "in-the-wild". Until now, the performance has mainly been assessed qualitatively by visually assessing the result of a deformable face tracking technology on short videos. In this paper, we perform the first, to the best of our knowledge, thorough evaluation of state-of-the-art deformable face tracking pipelines using the recently introduced 300VW benchmark. We evaluate many different architectures focusing mainly on the task of on-line deformable face tracking. In particular, we compare the following general strategies: (a) generic face detection plus generic facial landmark localisation, (b) generic model free tracking plus generic facial landmark localisation, as well as (c) hybrid approaches using state-of-the-art face detection, model free tracking and facial landmark localisation technologies. Our evaluation reveals future avenues for further research on the topic.

I have a multi-class machine learning problem for which I will try different methods on such as logistic regression, decision trees, multilayer perceptron etc. The observations in the data set have an attribute which is an index from 1-5 which defines how important it is that a certain observation gets correctly classified (index 1 very important, 5 not important at all). Question 1: How should I emphasize to the models that the lower index observations have greater importance? I am thinking of duplicating these observations so the models fit the lower index observations more well, what other approaches are possible? Question 2: What performance evaluation criterias can I use to find the models that predict these low index observations well?

Business knowldege (domain exeprtise) could defintely help in pruning the set of variables from the starting set of 300 to a smaller set. But even if you cut it down to a 100 variables, taking those and lags of different orders on these variables, you could have an overwhelming number of "explanatory" variables to forecast the dependent variable (daily sales). Sometimes a model in which the lag of the dependent variable is used as an explanatory variable along with the other selected variables among the 300 (perhaps with lags of a few of a them, based on intuition) will not only reduce the number of explantory variables and thereby increase the degrees of freedom for the prediction model but also provide more stable predictions. Also one can make use of the first so many principal components among the chosen predictor variables to deal with multicollinearity issues which typically arise in such probelms. This also cuts down the number of parameters and thereby increases the df of the model predictions.

Bike Sharing Systems (BSSs) are widely adopted in major cities of the world due to concerns associated with extensive private vehicle usage, namely, increased carbon emissions, traffic congestion and usage of nonrenewable resources. In a BSS, base stations are strategically placed throughout a city and each station is stocked with a pre-determined number of bikes at the beginning of the day. Customers hire the bikes from one station and return them at another station. Due to unpredictable movements of customers hiring bikes, there is either congestion (more than required) or starvation (fewer than required) of bikes at base stations. Existing data has shown that congestion/starvation is a common phenomenon that leads to a large number of unsatisfied customers resulting in a significant loss in customer demand. In order to tackle this problem, we propose an optimisation formulation to reposition bikes using vehicles while also considering the routes for vehicles and future expected demand. Furthermore, we contribute two approaches that rely on decomposability in the problem (bike repositioning and vehicle routing) and aggregation of base stations to reduce the computation time significantly. Finally, we demonstrate the utility of our approach by comparing against two benchmark approaches on two real-world data sets of bike sharing systems. These approaches are evaluated using a simulation where the movements of customers are generated from real-world data sets.

Multimodal clustering is an unsupervised technique for mining interesting patterns in $n$-adic binary relations or $n$-mode networks. Among different types of such generalized patterns one can find biclusters and formal concepts (maximal bicliques) for 2-mode case, triclusters and triconcepts for 3-mode case, closed $n$-sets for $n$-mode case, etc. Object-attribute biclustering (OA-biclustering) for mining large binary datatables (formal contexts or 2-mode networks) arose by the end of the last decade due to intractability of computation problems related to formal concepts; this type of patterns was proposed as a meaningful and scalable approximation of formal concepts. In this paper, our aim is to present recent advance in OA-biclustering and its extensions to mining multi-mode communities in SNA setting. We also discuss connection between clustering coefficients known in SNA community for 1-mode and 2-mode networks and OA-bicluster density, the main quality measure of an OA-bicluster. Our experiments with 2-, 3-, and 4-mode large real-world networks show that this type of patterns is suitable for community detection in multi-mode cases within reasonable time even though the number of corresponding $n$-cliques is still unknown due to computation difficulties. An interpretation of OA-biclusters for 1-mode networks is provided as well.

Uniform deviation bounds limit the difference between a model's expected loss and its loss on an empirical sample uniformly for all models in a learning problem. As such, they are a critical component to empirical risk minimization. In this paper, we provide a novel framework to obtain uniform deviation bounds for loss functions which are *unbounded*. In our main application, this allows us to obtain bounds for $k$-Means clustering under weak assumptions on the underlying distribution. If the fourth moment is bounded, we prove a rate of $\mathcal{O}\left(m^{-\frac12}\right)$ compared to the previously known $\mathcal{O}\left(m^{-\frac14}\right)$ rate. Furthermore, we show that the rate also depends on the kurtosis - the normalized fourth moment which measures the "tailedness" of a distribution. We further provide improved rates under progressively stronger assumptions, namely, bounded higher moments, subgaussianity and bounded support.

We propose a network structure discovery model for continuous observations that generalizes linear causal models by incorporating a Gaussian process (GP) prior on a network-independent component, and random sparsity and weight matrices as the network-dependent parameters. This approach provides flexible modeling of network-independent trends in the observations as well as uncertainty quantification around the discovered network structure. We establish a connection between our model and multi-task GPs and develop an efficient stochastic variational inference algorithm for it. Furthermore, we formally show that our approach is numerically stable and in fact numerically easy to carry out almost everywhere on the support of the random variables involved. Finally, we evaluate our model on three applications, showing that it outperforms previous approaches. We provide a qualitative and quantitative analysis of the structures discovered for domains such as the study of the full genome regulation of the yeast Saccharomyces cerevisiae.