The role of $L^2$ regularization, in the specific case of deep neural networks rather than more traditional machine learning models, is still not fully elucidated. We hypothesize that this complex interplay is due to the combination of overparameterization and high dimensional phenomena that take place during training and make it unamenable to standard convex optimization methods. Using insights from statistical physics and random fields theory, we introduce a parameter factoring in both the level of the loss function and its remaining nonconvexity: the \emph{complexity}. We proceed to show that it is desirable to proceed with \emph{complexity gradient descent}. We then show how to use this intuition to derive novel and efficient annealing schemes for the strength of $L^2$ regularization when performing standard stochastic gradient descent in deep neural networks.

This paper studies the supervised learning of the conditional distribution of a high-dimensional output given an input, where the output and input belong to two different modalities, e.g., the output is an image and the input is a sketch. We solve this problem by learning two models that bear similarities to those in reinforcement learning and optimal control. One model is policy-like. It generates the output directly by a non-linear transformation of the input and a noise vector. This amounts to fast thinking because the conditional generation is accomplished by direct sampling. The other model is planner-like. It learns an objective function in the form of a conditional energy function, so that the output can be generated by optimizing the objective function, or more rigorously by sampling from the conditional energy-based model. This amounts to slow thinking because the sampling process is accomplished by an iterative algorithm such as Langevin dynamics. We propose to learn the two models jointly, where the fast thinking policy-like model serves to initialize the sampling of the slow thinking planner-like model, and the planner-like model refines the initial output by an iterative algorithm. The planner-like model learns from the difference between the refined output and the observed output, while the policy-like model learns from how the planner-like model refines its initial output. We demonstrate the effectiveness of the proposed method on various image generation tasks.

We propose a neural hybrid model consisting of a linear model defined on a set of features computed by a deep, invertible transformation (i.e. a normalizing flow). An attractive property of our model is that both p(features), the features' density, and p(targets | features), the predictive distribution, can be computed exactly in a single feed-forward pass. We show that our hybrid model, despite the invertibility constraints, achieves similar accuracy to purely predictive models. Yet the generative component remains a good model of the input features despite the hybrid optimization objective. This offers additional capabilities such as detection of out-of-distribution inputs and enabling semi-supervised learning. The availability of the exact joint density p(targets, features) also allows us to compute many quantities readily, making our hybrid model a useful building block for downstream applications of probabilistic deep learning.

We study the performance of sparse regression methods and propose new techniques to distill the governing equations of dynamical systems from data. We first look at the generic methodology of learning interpretable equation forms from data, proposed by Brunton et al., followed by performance of LASSO for this purpose. We then propose a new algorithm that uses the dual of LASSO optimization for higher accuracy and stability. In the second part, we propose a novel algorithm that learns the candidate function library in a completely data-driven manner to distill the governing equations of the dynamical system. This is achieved via sequentially thresholded ridge regression (STRidge) over a orthogonal polynomial space. The performance of the three discussed methods is illustrated by looking the Lorenz 63 system and the quadratic Lorenz system.

With the dawn of the Big Data era, data sets are growing rapidly. Data is streaming from everywhere - from cameras, mobile phones, cars, and other electronic devices. Clustering streaming data is a very challenging problem. Unlike the traditional clustering algorithms where the dataset can be stored and scanned multiple times, clustering streaming data has to satisfy constraints such as limit memory size, real-time response, unknown data statistics and an unknown number of clusters. In this paper, we present a novel online clustering algorithm which can be used to cluster streaming data without knowing the number of clusters a priori. Results on both synthetic and real datasets show that the proposed algorithm produces partitions which are close to what you could get if you clustered the whole data at one time.

Motivating study and statistical approaches Crohn's disease (CD) is a chronic debilitating condition characterized by periods of inflammatory activity in the bowel that causes symptoms such as abdominal pain, diarrhea, andweight loss. Pharmacologic treatment for CD includes medications such as steroids, immunomodulating drugs, and biological therapy. Despite these available medications, many people with CD are escalated to surgical interventions from small to extensive resections of the bowel or colon (Gomollón et al., 2016). Previous studies have estimated that up to 50% of patients with CD undergo surgery within 10 years after diagnosis; however, surgical rates have decreased over time, possibly due to the introduction of modern treatments such as thiopurines and anti-TNF (Lakatos et al., 2012; Ramadas et al., 2010). The aim of this study is to determine whether clinical and demographic characteristics observed at the time of diagnosis can be used to predict the occurrence of major abdominal surgery within 5 years, with the goal of personalized disease management.

The ability to generalize quickly from few observations is crucial for intelligent systems. In this paper we introduce APL, an algorithm that approximates probability distributions by remembering the most surprising observations it has encountered. These past observations are recalled from an external memory module and processed by a decoder network that can combine information from different memory slots to generalize beyond direct recall. We show this algorithm can perform as well as state of the art baselines on few-shot classification benchmarks with a smaller memory footprint. In addition, its memory compression allows it to scale to thousands of unknown labels. Finally, we introduce a meta-learning reasoning task which is more challenging than direct classification. In this setting, APL is able to generalize with fewer than one example per class via deductive reasoning.

We propose a novel approach to model selection for simulator-based statistical models. The proposed approach defines a mixture of candidate models, and then iteratively updates the weight coefficients for those models as well as the parameters in each model simultaneously; this is done by recursively applying Bayes' rule, using the recently proposed kernel recursive ABC algorithm. The practical advantage of the method is that it can be used even when a modeler lacks appropriate prior knowledge about the parameters in each model. We demonstrate the effectiveness of the proposed approach with a number of experiments, including model selection for dynamical systems in ecology and epidemiology.

Sparsity promoting norms are frequently used in high dimensional regression. A limitation of Lasso-type estimators is that the regulariza-tion parameter depends on the noise level which varies between datasets and experiments. Esti-mators such as the concomitant Lasso address this dependence by jointly estimating the noise level and the regression coefficients. As sample sizes are often limited in high dimensional regimes, simplified heteroscedastic models are customary. However, in many experimental applications , data is obtained by averaging multiple measurements. This helps reducing the noise variance, yet it dramatically reduces sample sizes, preventing refined noise modeling. In this work, we propose an estimator that can cope with complex heteroscedastic noise structures by using non-averaged measurements and a con-comitant formulation. The resulting optimization problem is convex, so thanks to smoothing theory, it is amenable to state-of-the-art proximal coordinate descent techniques that can leverage the expected sparsity of the solutions. Practical benefits are demonstrated on simulations and on neuroimaging applications.

Humans perceive the seemingly chaotic world in a structured and compositional way with the prerequisite of being able to segregate conceptual entities from the complex visual scenes. The mechanism of grouping basic visual elements of scenes into conceptual entities is termed as perceptual grouping. In this work, we propose a new type of spatial mixture models with learnable priors for perceptual grouping. Different from existing methods, the proposed method disentangles the representation of an object into `shape' and `appearance' which are modeled separately by the mixture weights and the conditional probability distributions. More specifically, each object in the visual scene is modeled by one mixture component, whose mixture weights and the parameter of the conditional probability distribution are generated by two neural networks, respectively. The mixture weights focus on modeling spatial dependencies (i.e., shape) and the conditional probability distributions deal with intra-object variations (i.e., appearance). In addition, the background is separately modeled as a special component complementary to the foreground objects. Our extensive empirical tests on two perceptual grouping datasets demonstrate that the proposed method outperforms the state-of-the-art methods under most experimental configurations. The learned conceptual entities are generalizable to novel visual scenes and insensitive to the diversity of objects.