We study learning of recurrent neural networks that produce temporal sequences consisting of the concatenation of re-usable "motifs". In the context of neuroscience or robotics, these motifs would be the motor primitives from which complex behavior is generated. Given a known set of motifs, can a new motif be learned without affecting the performance of the known set and then used in new sequences without first explicitly learning every possible transition? Two requirements enable this: (i) parameter updates while learning a new motif do not interfere with the parameters used for the previously acquired ones; and (ii) a new motif can be robustly generated when starting from the network state reached at the end of any of the other motifs, even if that state was not present during training. We meet the first requirement by investigating artificial neural networks (ANNs) with specific architectures, and attempt to meet the second by training them to generate motifs from random initial states. We find that learning of single motifs succeeds but that sequence generation is not robust: transition failures are observed. We then compare these results with a model whose architecture and analytically-tractable dynamics are inspired by the motor thalamocortical circuit, and that includes a specific module used to implement motif transitions. The synaptic weights of this model can be adjusted without requiring stochastic gradient descent (SGD) on the simulated network outputs, and we have asymptotic guarantees that transitions will not fail. Indeed, in simulations, we achieve single-motif accuracy on par with the previously studied ANNs and have improved sequencing robustness with no transition failures. Finally, we show that insights obtained by studying the transition subnetwork of this model can also improve the robustness of transitioning in the traditional ANNs previously studied.

Segmented models are widely used to describe non-stationary sequential data with discrete change points. Their estimation usually requires solving a mixed discrete-continuous optimization problem, where the segmentation is the discrete part and all other model parameters are continuous. A number of estimation algorithms have been developed that are highly specialized for their specific model assumptions. The dependence on non-standard algorithms makes it hard to integrate segmented models in state-of-the-art deep learning architectures that critically depend on gradient-based optimization techniques. In this work, we formulate a relaxed variant of segmented models that enables joint estimation of all model parameters, including the segmentation, with gradient descent. We build on recent advances in learning continuous warping functions and propose a novel family of warping functions based on the two-sided power (TSP) distribution. TSP-based warping functions are differentiable, have simple closed-form expressions, and can represent segmentation functions exactly. Our formulation includes the important class of segmented generalized linear models as a special case, which makes it highly versatile. We use our approach to model the spread of COVID-19 by segmented Poisson regression, perform logistic regression on Fashion-MNIST with artificial concept drift, and demonstrate its capacities for phoneme segmentation.

With the abundance of machine learning methods available and the temptation of using them all in an ensemble method, having a model-agnostic method of feature selection is incredibly alluring. Principal component analysis was developed in 1901 and has been a strong contender in this role since, but in the end is an unsupervised method. It offers no guarantee that the features that are selected have good predictive power because it does not know what is being predicted. To this end, Peng et al. developed the minimum redundancy-maximum relevance (mRMR) method in 2005. It uses the mutual information not only between predictors but also includes the mutual information with the response in its calculation. Estimating mutual information and entropy tend to be expensive and problematic endeavors, which leads to excessive processing times even for dataset that is approximately 750 by 750 in a Leave-One-Subject-Out jackknife situation. To remedy this, we use a method from 2012 called Distance Correlation Sure Independence Screening (DC-SIS) which uses the distance correlation measure of Sz\'ekely et al. to select features that have the greatest dependence with the response. We show that this method produces statistically indistinguishable results to the mRMR selection method on Parkinson's Disease vocal diagnosis data 90 times faster.

BIRCH clustering is a widely known approach for clustering, that has influenced much subsequent research and commercial products. The key contribution of BIRCH is the Clustering Feature tree (CF-Tree), which is a compressed representation of the input data. As new data arrives, the tree is eventually rebuilt to increase the compression. Afterward, the leaves of the tree are used for clustering. Because of the data compression, this method is very scalable. The idea has been adopted for example for k-means, data stream, and density-based clustering. Clustering features used by BIRCH are simple summary statistics that can easily be updated with new data: the number of points, the linear sums, and the sum of squared values. Unfortunately, how the sum of squares is then used in BIRCH is prone to catastrophic cancellation. We introduce a replacement cluster feature that does not have this numeric problem, that is not much more expensive to maintain, and which makes many computations simpler and hence more efficient. These cluster features can also easily be used in other work derived from BIRCH, such as algorithms for streaming data. In the experiments, we demonstrate the numerical problem and compare the performance of the original algorithm compared to the improved cluster features.

The intrinsic dimensionality refers to the ``true'' dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower than the number of variables. Local intrinsic dimensionality refers to the observation that this property can vary for different parts of the data set; and intrinsic dimensionality can serve as a proxy for the local difficulty of the data set. Most popular methods for estimating the local intrinsic dimensionality are based on distances, and the rate at which the distances to the nearest neighbors increase, a concept known as ``expansion dimension''. In this paper we introduce an orthogonal concept, which does not use any distances: we use the distribution of angles between neighbor points. We derive the theoretical distribution of angles and use this to construct an estimator for intrinsic dimensionality. Experimentally, we verify that this measure behaves similarly, but complementarily, to existing measures of intrinsic dimensionality. By introducing a new idea of intrinsic dimensionality to the research community, we hope to contribute to a better understanding of intrinsic dimensionality and to spur new research in this direction.

When a missing process depends on the missing values themselves, it needs to be explicitly modelled and taken into account while doing likelihood-based inference. We present an approach for building and fitting deep latent variable models (DLVMs) in cases where the missing process is dependent on the missing data. Specifically, a deep neural network enables us to flexibly model the conditional distribution of the missingness pattern given the data. This allows for incorporating prior information about the type of missingness (e.g. self-censoring) into the model. Our inference technique, based on importance-weighted variational inference, involves maximising a lower bound of the joint likelihood. Stochastic gradients of the bound are obtained by using the reparameterisation trick both in latent space and data space. We show on various kinds of data sets and missingness patterns that explicitly modelling the missing process can be invaluable.

Hawkes process provides an effective statistical framework for analyzing the time-dependent interaction of neuronal spiking activities. Although utilized in many real applications, the classical Hawkes process is incapable of modelling inhibitory interactions among neurons. Instead, the nonlinear Hawkes process allows for a more flexible influence pattern with excitatory or inhibitory interactions. In this paper, three sets of auxiliary latent variables (P\'{o}lya-Gamma variables, latent marked Poisson processes and sparsity variables) are augmented to make synapses connection weights in a Gaussian form, which allows for a simple iterative algorithm with analytical updates. As a result, an efficient expectation-maximization (EM) algorithm is derived to obtain the maximum a posteriori (MAP) estimate. We demonstrate the accuracy and efficiency performance of our algorithm on synthetic and real data. For real neural recordings, we show our algorithm can estimate the temporal dynamics of interaction and reveal the interpretable synaptic structure underlying neural spike trains.

This paper introduces the \textit{weighted partial copula} function for testing conditional independence. The proposed test procedure results from these two ingredients: (i) the test statistic is an explicit Cramer-von Mises transformation of the \textit{weighted partial copula}, (ii) the regions of rejection are computed using a bootstrap procedure which mimics conditional independence by generating samples from the product measure of the estimated conditional marginals. Under conditional independence, the weak convergence of the \textit{weighted partial copula proces}s is established when the marginals are estimated using a smoothed local linear estimator. Finally, an experimental section demonstrates that the proposed test has competitive power compared to recent state-of-the-art methods such as kernel-based test.

Predictive power has always been the main research focus of learning algorithms. While the general approach for these algorithms is to consider all possible attributes in a dataset to best predict the response of interest, an important branch of research is focused on sparse learning. Indeed, in many practical settings we believe that only an extremely small combination of different attributes affect the response. However even sparse-learning methods can still preserve a high number of attributes in high-dimensional settings and possibly deliver inconsistent prediction performance. The latter methods can also be hard to interpret for researchers and practitioners, a problem which is even more relevant for the ``black-box''-type mechanisms of many learning approaches. Finally, there is often a problem of replicability since not all data-collection procedures measure (or observe) the same attributes and therefore cannot make use of proposed learners for testing purposes. To address all the previous issues, we propose to study a procedure that combines screening and wrapper methods and aims to find a library of extremely low-dimensional attribute combinations (with consequent low data collection and storage costs) in order to (i) match or improve the predictive performance of any particular learning method which uses all attributes as an input (including sparse learners); (ii) provide a low-dimensional network of attributes easily interpretable by researchers and practitioners; and (iii) increase the potential replicability of results due to a diversity of attribute combinations defining strong learners with equivalent predictive power. We call this algorithm ``Sparse Wrapper AlGorithm'' (SWAG).

The notion of concept drift refers to the phenomenon that the distribution, which is underlying the observed data, changes over time; as a consequence machine learning models may become inaccurate and need adjustment. While there do exist methods to detect concept drift or to adjust models in the presence of observed drift, the question of explaining drift has hardly been considered so far. This problem is of importance, since it enables an inspection of the most prominent features where drift manifests itself; hence it enables human understanding of the necessity of change and it increases acceptance of life-long learning models. In this paper we present a novel technology, which characterizes concept drift in terms of the characteristic change of spatial features represented by typical examples based on counterfactual explanations. We establish a formal definition of this problem, derive an efficient algorithmic solution based on counterfactual explanations, and demonstrate its usefulness in several examples.