Discovery of new natural laws has for a long time relied on the inspiration of some genius. Recently, however, machine learning technologies, which analyze big data without human prejudice and bias, are expected to find novel natural laws. Here we demonstrate that our proposed machine learning, recursive-LASSO-based symbolic (RLS) regression, enables data-driven formulation of natural laws from noisy data. The RLS regression recurrently repeats feature generation and feature selection, eventually constructing a data-driven model with highly nonlinear features. This data-driven formulation method is quite general and thus can discover new laws in various scientific fields.

We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which provide high dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real datasets.

We address the problem of learning of continuous exponential family distributions with unbounded support. While a lot of progress has been made on learning of Gaussian graphical models, we are still lacking scalable algorithms for reconstructing general continuous exponential families modeling higher-order moments of the data beyond the mean and the covariance. Here, we introduce a computationally efficient method for learning continuous graphical models based on the Interaction Screening approach. Through a series of numerical experiments, we show that our estimator maintains similar requirements in terms of accuracy and sample complexity compared to alternative approaches such as maximization of conditional likelihood, while considerably improving upon the algorithm's run-time.

The importance of aggregated count data, which is calculated from the data of multiple individuals, continues to increase. Collective Graphical Model (CGM) is a probabilistic approach to the analysis of aggregated data. One of the most important operations in CGM is maximum a posteriori (MAP) inference of unobserved variables under given observations. Because the MAP inference problem for general CGMs has been shown to be NP-hard, an approach that solves an approximate problem has been proposed. However, this approach has two major drawbacks. First, the quality of the solution deteriorates when the values in the count tables are small, because the approximation becomes inaccurate. Second, since continuous relaxation is applied, the integrality constraints of the output are violated. To resolve these problems, this paper proposes a new method for MAP inference for CGMs on path graphs. First we show that the MAP inference problem can be formulated as a (non-linear) minimum cost flow problem. Then, we apply Difference of Convex Algorithm (DCA), which is a general methodology to minimize a function represented as the sum of a convex function and a concave function. In our algorithm, important subroutines in DCA can be efficiently calculated by minimum convex cost flow algorithms. Experiments show that the proposed method outputs higher quality solutions than the conventional approach.

When releasing database statistics on a collection of entries from individuals, we would ideally like to make it impossible to reverse-engineer each individual's potentially sensitive information. Privacy-preserving techniques add just enough randomness tailored to the statistical task to guarantee protection. At the same time, it is becoming increasingly common to apply such techniques to databases collected from multiple sources, not all of which can be trusted. Emerging data access frameworks, such as federated analyses across users' devices or data silos [51], make it easier to temper with this collected dataset, leaving private statistical analyses vulnerable to a malicious corruption of a fraction of the data. Differential privacy has emerged as a widely accepted de facto measure of privacy, which is now a standard in releasing the statistics of the U.S. Census data [2] statistics and also deployed in real-world commercial systems [76, 41, 42]. A statistical analysis is said to be differentially private if the likelihood of the (randomized) outcome does not change significantly when a single entry is replaced by another arbitrary entry (formally defined in §1.1). This provides a strong privacy guarantee: even a powerful adversary who knows all the other entries in the database cannot confidently identify whether a particular individual is participating in the database based on the outcome of the analysis, providing plausible deniability, central to protecting an individual's privacy. Despite more than a decade of literature focused in designing private mechanisms for various statistical and learning tasks, only recently have some of the most fundamental questions been resolved. In this paper, we focus on one of the most canonical problems in statistics: estimating the mean of a distribution from i.i.d.

Unsupervised learning seeks to uncover patterns in data. However, different kinds of noise may impede the discovery of useful substructure from real-world time-series data. In this work, we focus on mitigating the interference of left-censorship in the task of clustering. We provide conditions under which clusters and left-censorship may be identified; motivated by this result, we develop a deep generative, continuous-time model of time-series data that clusters while correcting for censorship time. We demonstrate accurate, stable, and interpretable results on synthetic data that outperform several benchmarks. To showcase the utility of our framework on real-world problems, we study how left-censorship can adversely affect the task of disease phenotyping, resulting in the often incorrect assumption that longitudinal patient data are aligned by disease stage. In reality, patients at the time of diagnosis are at different stages of the disease -- both late and early due to differences in when patients seek medical care and such discrepancy can confound unsupervised learning algorithms. On two clinical datasets, our model corrects for this form of censorship and recovers known clinical subtypes.

Learning probability measures based on an i.i.d. sample is a fundamental inference task, but is challenging when the sample space is high-dimensional. Inspired by the success of tree boosting in high-dimensional classification and regression, we propose a tree boosting method for learning high-dimensional probability distributions. We formulate concepts of "addition'' and "residuals'' on probability distributions in terms of compositions of a new, more general notion of multivariate cumulative distribution functions (CDFs) than classical CDFs. This then gives rise to a simple boosting algorithm based on forward-stagewise (FS) fitting of an additive ensemble of measures. The output of the FS algorithm allows analytic computation of the probability density function for the fitted distribution. It also provides an exact simulator for drawing independent Monte Carlo samples from the fitted measure. Typical considerations in applying boosting -- namely choosing the number of trees, setting the appropriate level of shrinkage/regularization in the weak learner, and the evaluation of variable importance -- can be accomplished in an analogous fashion to traditional boosting in supervised learning. Numerical experiments confirm that boosting can substantially improve the fit to multivariate distributions compared to the state-of-the-art single-tree learner and is computationally efficient. We illustrate through an application to a data set from mass cytometry how the simulator can be used to investigate various aspects of the underlying distribution.

We propose dynamic curriculum learning via data parameters for noise robust keyword spotting. Data parameter learning has recently been introduced for image processing, where weight parameters, so-called data parameters, for target classes and instances are introduced and optimized along with model parameters. The data parameters scale logits and control importance over classes and instances during training, which enables automatic curriculum learning without additional annotations for training data. Similarly, in this paper, we propose using this curriculum learning approach for acoustic modeling, and train an acoustic model on clean and noisy utterances with the data parameters. The proposed approach automatically learns the difficulty of the classes and instances, e.g. due to low speech to noise ratio (SNR), in the gradient descent optimization and performs curriculum learning. This curriculum learning leads to overall improvement of the accuracy of the acoustic model. We evaluate the effectiveness of the proposed approach on a keyword spotting task. Experimental results show 7.7% relative reduction in false reject ratio with the data parameters compared to a baseline model which is simply trained on the multiconditioned dataset.

Epigenetic alterations have an important role in the development of several types of cancer. Epigenetic studies generate a large amount of data, which makes it essential to develop novel models capable of dealing with large-scale data. In this work, we propose a deep embedded refined clustering method for breast cancer differentiation based on DNA methylation. In concrete, the deep learning system presented here uses the levels of CpG island methylation between 0 and 1. The proposed approach is composed of two main stages. The first stage consists in the dimensionality reduction of the methylation data based on an autoencoder. The second stage is a clustering algorithm based on the soft-assignment of the latent space provided by the autoencoder. The whole method is optimized through a weighted loss function composed of two terms: reconstruction and classification terms. To the best of the authors' knowledge, no previous studies have focused on the dimensionality reduction algorithms linked to classification trained end-to-end for DNA methylation analysis. The proposed method achieves an unsupervised clustering accuracy of 0.9927 and an error rate (%) of 0.73 on 137 breast tissue samples. After a second test of the deep-learning-based method using a different methylation database, an accuracy of 0.9343 and an error rate (%) of 6.57 on 45 breast tissue samples is obtained. Based on these results, the proposed algorithm outperforms other state-of-the-art methods evaluated under the same conditions for breast cancer classification based on DNA methylation data.

The clustering methods have been used in a variety of fields such as image processing, data mining, pattern recognition, and statistical analysis. Generally, the clustering algorithms consider all variables equally relevant or not correlated for the clustering task. Nevertheless, in real situations, some variables can be correlated or may be more or less relevant or even irrelevant for this task. This paper proposes partitioning fuzzy clustering algorithms based on Euclidean, City-block and Mahalanobis distances and entropy regularization. These methods are an iterative three steps algorithms which provide a fuzzy partition, a representative for each fuzzy cluster, and the relevance weight of the variables or their correlation by minimizing a suitable objective function. Several experiments on synthetic and real datasets, including its application to noisy image texture segmentation, demonstrate the usefulness of these adaptive clustering methods.