Gaussian processes (GPs) are a well-known nonparametric Bayesian inference technique, but they suffer from scalability problems for large sample sizes, and their performance can degrade for non-stationary or spatially heterogeneous data. In this work, we seek to overcome these issues through (i) employing variational free energy approximations of GPs operating in tandem with online expectation propagation steps; and (ii) introducing a local splitting step which instantiates a new GP whenever the posterior distribution changes significantly as quantified by the Wasserstein metric over posterior distributions. Over time, then, this yields an ensemble of sparse GPs which may be updated incrementally, and adapts to locality, heterogeneity, and non-stationarity in training data.

This paper studies how to construct confidence regions for principal component analysis (PCA) in high dimension, a problem that has been vastly under-explored. While computing measures of uncertainty for nonlinear/nonconvex estimators is in general difficult in high dimension, the challenge is further compounded by the prevalent presence of missing data and heteroskedastic noise. We propose a suite of solutions to perform valid inference on the principal subspace based on two estimators: a vanilla SVD-based approach, and a more refined iterative scheme called $\textsf{HeteroPCA}$ (Zhang et al., 2018). We develop non-asymptotic distributional guarantees for both estimators, and demonstrate how these can be invoked to compute both confidence regions for the principal subspace and entrywise confidence intervals for the spiked covariance matrix. Particularly worth highlighting is the inference procedure built on top of $\textsf{HeteroPCA}$, which is not only valid but also statistically efficient for broader scenarios (e.g., it covers a wider range of missing rates and signal-to-noise ratios). Our solutions are fully data-driven and adaptive to heteroskedastic random noise, without requiring prior knowledge about the noise levels and noise distributions.

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on $\mathbb{R}^d$. When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive a central limit theorem for a density plugin estimator of the squared Wasserstein distance, which is centered at its population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for Wasserstein distances.

In a recent methodological paper, we have shown how to learn chaotic dynamics along with the state trajectory from sequentially acquired observations, using local ensemble Kalman filters. Here, we more systematically investigate the possibilty to use a local ensemble Kalman filter with either covariance localization or local domains, in order to retrieve the state and a mix of key global and local parameters. Global parameters are meant to represent the surrogate dynamics, for instance through a neural network, which is reminiscent of data-driven machine learning of dynamics, while the local parameters typically stand for the forcings of the model. A family of algorithms for covariance and local domain localization is proposed in this joint state and parameter filter context. In particular, we show how to rigorously update global parameters using a local domain EnKF such as the LETKF, an inherently local method. The approach is tested with success on the 40-variable Lorenz model using several of the local EnKF flavors. A two-dimensional illustration based on a multi-layer Lorenz model is finally provided. It uses radiance-like non-local observations, and both local domains and covariance localization in order to learn the chaotic dynamics, the local forcings, and the couplings between layers. This paper more generally addresses the key question of online estimation of both global and local model parameters.

We use graphical methods to probe neural nets that classify images. Plots of t-SNE outputs at successive layers in a network reveal increasingly organized arrangement of the data points. They can also reveal how a network can diminish or even forget about within-class structure as the data proceeds through layers. We use class-specific analogues of principal components to visualize how succeeding layers separate the classes. These allow us to sort images from a given class from most typical to least typical (in the data) and they also serve as very useful projection coordinates for data visualization. We find them especially useful when defining versions guided tours for animated data visualization.

In this work we propose a new biophysical computational model of brain regions relevant to Parkinson's Disease based on local field potential data collected from the brain of marmoset monkeys. Parkinson's disease is a neurodegenerative disorder, linked to the death of dopaminergic neurons at the substantia nigra pars compacta, which affects the normal dynamics of the basal ganglia-thalamus-cortex neuronal circuit of the brain. Although there are multiple mechanisms underlying the disease, a complete description of those mechanisms and molecular pathogenesis are still missing, and there is still no cure. To address this gap, computational models that resemble neurobiological aspects found in animal models have been proposed. In our model, we performed a data-driven approach in which a set of biologically constrained parameters is optimised using differential evolution. Evolved models successfully resembled single-neuron mean firing rates and spectral signatures of local field potentials from healthy and parkinsonian marmoset brain data. As far as we are concerned, this is the first computational model of Parkinson's Disease based on simultaneous electrophysiological recordings from seven brain regions of Marmoset monkeys. Results show that the proposed model could facilitate the investigation of the mechanisms of PD and support the development of techniques that can indicate new therapies. It could also be applied to other computational neuroscience problems in which biological data could be used to fit multi-scale models of brain circuits.

Recent learning approaches that implicitly represent surface geometry using coordinate-based neural representations have shown impressive results in the problem of multi-view 3D reconstruction. The effectiveness of these techniques is, however, subject to the availability of a large number (several tens) of input views of the scene, and computationally demanding optimizations. In this paper, we tackle these limitations for the specific problem of few-shot full 3D head reconstruction, by endowing coordinate-based representations with a probabilistic shape prior that enables faster convergence and better generalization when using few input images (down to three). First, we learn a shape model of 3D heads from thousands of incomplete raw scans using implicit representations. At test time, we jointly overfit two coordinate-based neural networks to the scene, one modeling the geometry and another estimating the surface radiance, using implicit differentiable rendering. We devise a two-stage optimization strategy in which the learned prior is used to initialize and constrain the geometry during an initial optimization phase. Then, the prior is unfrozen and fine-tuned to the scene. By doing this, we achieve high-fidelity head reconstructions, including hair and shoulders, and with a high level of detail that consistently outperforms both state-of-the-art 3D Morphable Models methods in the few-shot scenario, and non-parametric methods when large sets of views are available.

Advanced tensor decomposition, such as Tensor train (TT) and Tensor ring (TR), has been widely studied for deep neural network (DNN) model compression, especially for recurrent neural networks (RNNs). However, compressing convolutional neural networks (CNNs) using TT/TR always suffers significant accuracy loss. In this paper, we propose a systematic framework for tensor decomposition-based model compression using Alternating Direction Method of Multipliers (ADMM). By formulating TT decomposition-based model compression to an optimization problem with constraints on tensor ranks, we leverage ADMM technique to systemically solve this optimization problem in an iterative way. During this procedure, the entire DNN model is trained in the original structure instead of TT format, but gradually enjoys the desired low tensor rank characteristics. We then decompose this uncompressed model to TT format and fine-tune it to finally obtain a high-accuracy TT-format DNN model. Our framework is very general, and it works for both CNNs and RNNs, and can be easily modified to fit other tensor decomposition approaches. We evaluate our proposed framework on different DNN models for image classification and video recognition tasks. Experimental results show that our ADMM-based TT-format models demonstrate very high compression performance with high accuracy. Notably, on CIFAR-100, with 2.3X and 2.4X compression ratios, our models have 1.96% and 2.21% higher top-1 accuracy than the original ResNet-20 and ResNet-32, respectively. For compressing ResNet-18 on ImageNet, our model achieves 2.47X FLOPs reduction without accuracy loss.

Narrative generation and analysis are still on the fringe of modern natural language processing yet are crucial in a variety of applications. This paper proposes a feature extraction method for plot dynamics. We present a dataset that consists of the plot descriptions for thirteen thousand TV shows alongside meta-information on their genres and dynamic plots extracted from them. We validate the proposed tool for plot dynamics extraction and discuss possible applications of this method to the tasks of narrative analysis and generation.

When humans solve complex problems, they rarely come up with a decision right-away. Instead, they start with an intuitive decision, reflect upon it, spot mistakes, resolve contradictions and jump between different hypotheses. Thus, they create a sequence of ideas and follow a train of thought that ultimately reaches a conclusive decision. Contrary to this, today's neural classification models are mostly trained to map an input to one single and fixed output. In this paper, we investigate how we can give models the opportunity of a second, third and $k$-th thought. We take inspiration from Hegel's dialectics and propose a method that turns an existing classifier's class prediction (such as the image class forest) into a sequence of predictions (such as forest $\rightarrow$ tree $\rightarrow$ mushroom). Concretely, we propose a correction module that is trained to estimate the model's correctness as well as an iterative prediction update based on the prediction's gradient. Our approach results in a dynamic system over class probability distributions $\unicode{x2014}$ the thought flow. We evaluate our method on diverse datasets and tasks from computer vision and natural language processing. We observe surprisingly complex but intuitive behavior and demonstrate that our method (i) can correct misclassifications, (ii) strengthens model performance, (iii) is robust to high levels of adversarial attacks, (iv) can increase accuracy up to 4% in a label-distribution-shift setting and (iv) provides a tool for model interpretability that uncovers model knowledge which otherwise remains invisible in a single distribution prediction.