We consider the problem of regularized Poisson Non-negative Matrix Factorization (NMF) problem, encompassing various regularization terms such as Lipschitz and relatively smooth functions, alongside linear constraints. This problem holds significant relevance in numerous Machine Learning applications, particularly within the domain of physical linear unmixing problems. A notable challenge arises from the main loss term in the Poisson NMF problem being a KL divergence, which is non-Lipschitz, rendering traditional gradient descent-based approaches inefficient. In this contribution, we explore the utilization of Block Successive Upper Minimization (BSUM) to overcome this challenge. We build approriate majorizing function for Lipschitz and relatively smooth functions, and show how to introduce linear constraints into the problem. This results in the development of two novel algorithms for regularized Poisson NMF. We conduct numerical simulations to showcase the effectiveness of our approach.

In this technical report we compare different deep learning models for prediction of water depth rasters at high spatial resolution. Efficient, accurate, and fast methods for water depth prediction are nowadays important as urban floods are increasing due to higher rainfall intensity caused by climate change, expansion of cities and changes in land use. While hydrodynamic models models can provide reliable forecasts by simulating water depth at every location of a catchment, they also have a high computational burden which jeopardizes their application to real-time prediction in large urban areas at high spatial resolution. Here, we propose to address this issue by using data-driven techniques. Specifically, we evaluate deep learning models which are trained to reproduce the data simulated by the CADDIES cellular-automata flood model, providing flood forecasts that can occur at different future time horizons. The advantage of using such models is that they can learn the underlying physical phenomena a priori, preventing manual parameter setting and computational burden. We perform experiments on a dataset consisting of two catchments areas within Switzerland with 18 simpler, short rainfall patterns and 4 long, more complex ones. Our results show that the deep learning models present in general lower errors compared to the other methods, especially for water depths $>0.5m$. However, when testing on more complex rainfall events or unseen catchment areas, the deep models do not show benefits over the simpler ones.

Classification algorithms using RNA-Sequencing (RNA-Seq) data as input are used in a variety of biological applications. By nature, RNA-Seq data is subject to uncontrolled fluctuations both within and especially across datasets, which presents a major difficulty for a trained classifier to generalize to an external dataset. Replacing raw gene counts with the rank of gene counts inside an observation has proven effective to mitigate this problem. However, the rank of a feature is by definition relative to all other features, including highly variable features that introduce noise in the ranking. To address this problem and obtain more robust ranks, we propose a logistic regression model, optirank, which learns simultaneously the parameters of the model and the genes to use as a reference set in the ranking. We show the effectiveness of this method on simulated data. We also consider real classification tasks, which present different kinds of distribution shifts between train and test data. Those tasks concern a variety of applications, such as cancer of unknown primary classification, identification of specific gene signatures, and determination of cell type in single-cell RNA-Seq datasets. On those real tasks, optirank performs at least as well as the vanilla logistic regression on classical ranks, while producing sparser solutions. In addition, to increase the robustness against dataset shifts, we propose a multi-source learning scheme and demonstrate its effectiveness when used in combination with rank-based classifiers.

Fingerprints are key tools in climate change detection and attribution (D&A) that are used to determine whether changes in observations are different from internal climate variability (detection), and whether observed changes can be assigned to specific external drivers (attribution). We propose a direct D&A approach based on supervised learning to extract fingerprints that lead to robust predictions under relevant interventions on exogenous variables, i.e., climate drivers other than the target. We employ anchor regression, a distributionally-robust statistical learning method inspired by causal inference that extrapolates well to perturbed data under the interventions considered. The residuals from the prediction achieve either uncorrelatedness or mean independence with the exogenous variables, thus guaranteeing robustness. We define D&A as a unified hypothesis testing framework that relies on the same statistical model but uses different targets and test statistics. In the experiments, we first show that the CO2 forcing can be robustly predicted from temperature spatial patterns under strong interventions on the solar forcing. Second, we illustrate attribution to the greenhouse gases and aerosols while protecting against interventions on the aerosols and CO2 forcing, respectively. Our study shows that incorporating robustness constraints against relevant interventions may significantly benefit detection and attribution of climate change.

The edge structure of the graph defining an undirected graphical model describes precisely the structure of dependence between the variables in the graph. In many applications, the dependence structure is unknown and it is desirable to learn it from data, often because it is a preliminary step to be able to ascertain causal effects. This problem, known as structure learning, is hard in general, but for Gaussian graphical models it is slightly easier because the structure of the graph is given by the sparsity pattern of the precision matrix of the joint distribution, and because independence coincides with decorrelation. A major difficulty too often ignored in structure learning is the fact that if some variables are not observed, the marginal dependence graph over the observed variables will possibly be significantly more complex and no longer reflect the direct dependencies that are potentially associated with causal effects. In this work, we consider a family of latent variable Gaussian graphical models in which the graph of the joint distribution between observed and unobserved variables is sparse, and the unobserved variables are conditionally independent given the others. Prior work was able to recover the connectivity between observed variables, but could only identify the subspace spanned by unobserved variables, whereas we propose a convex optimization formulation based on structured matrix sparsity to estimate the complete connectivity of the complete graph including unobserved variables, given the knowledge of the number of missing variables, and a priori knowledge of their level of connectivity. Our formulation is supported by a theoretical result of identifiability of the latent dependence structure for sparse graphs in the infinite data limit. We propose an algorithm leveraging recent active set methods, which performs well in the experiments on synthetic data.

The problem of Knowledge Base Completion can be framed as a 3rd-order binary tensor completion problem. In this light, the Canonical Tensor Decomposition (CP) (Hitchcock, 1927) seems like a natural solution; however, current implementations of CP on standard Knowledge Base Completion benchmarks are lagging behind their competitors. In this work, we attempt to understand the limits of CP for knowledge base completion. First, we motivate and test a novel regularizer, based on tensor nuclear $p$-norms. Then, we present a reformulation of the problem that makes it invariant to arbitrary choices in the inclusion of predicates or their reciprocals in the dataset. These two methods combined allow us to beat the current state of the art on several datasets with a CP decomposition, and obtain even better results using the more advanced ComplEx model.

Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications. We compute slow rates and an upper bound on the statistical dimension of the suggested norm for rank 1 matrices, showing that its statistical dimension is an order of magnitude smaller than the usual $\ell\_1$-norm, trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we propose an active set algorithm leveraging the structure of the convex problem to solve it and show promising numerical results.

We show that the herding procedure of Welling (2009) takes exactly the form of a standard convex optimization algorithm--namely a conditional gradient algorithm minimizing a quadratic moment discrepancy. This link enables us to invoke convergence results from convex optimization and to consider faster alternatives for the task of approximating integrals in a reproducing kernel Hilbert space. We study the behavior of the different variants through numerical simulations. The experiments indicate that while we can improve over herding on the task of approximating integrals, the original herding algorithm tends to approach more often the maximum entropy distribution, shedding more light on the learning bias behind herding.

In this paper, we propose an unifying view of several recently proposed structured sparsity-inducing norms. We consider the situation of a model simultaneously (a) penalized by a set- function de ned on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in Lp-norm. We show that the natural combinatorial optimization problems obtained may be relaxed into convex optimization problems and introduce a notion, the lower combinatorial envelope of a set-function, that characterizes the tightness of our relaxations. We moreover establish links with norms based on latent representations including the latent group Lasso and block-coding, and with norms obtained from submodular functions.

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the $\ell_1$-norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the $\ell_1$-norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of non-linear variable selection.