Permutation invariant neural networks are a promising tool for predictive modeling of set data. We show, however, that existing architectures struggle to perform well when they are deep. In this work, we mathematically and empirically analyze normalization layers and residual connections in the context of deep permutation invariant neural networks. We develop set norm, a normalization tailored for sets, and introduce the “clean path principle” for equivariant residual connections alongside a novel benefit of such connections, the reduction of information loss. Based on our analysis, we propose Deep Sets++ and Set Transformer++, deep models that reach comparable or better performance than their original counterparts on a diverse suite of tasks. We additionally introduce Flow-RBC, a new single-cell dataset and real-world application of permutation invariant prediction. We open-source our data and code here: https://github.com/rajesh-lab/deep_permutation_invariant.

The increased availability of multivariate time-series asks for the development of suitable methods able to holistically analyse them. To this aim, we propose a novel flexible method for data-mining, forecasting and causal patterns detection that leverages the coupling of Hidden Markov Models and Gaussian Graphical Models. Given a multivariate non-stationary time-series, the proposed method simultaneously clusters time points while understanding probabilistic relationships among variables. The clustering divides the time points into stationary sub-groups whose underlying distribution can be inferred through a graphical model. Such coupling can be further exploited to build a time-varying regression model which allows to both make predictions and obtain insights on the presence of causal patterns. We extensively validate the proposed approach on synthetic data showing that it has better performance than the state of the art on clustering, graphical models inference and prediction. Finally, to demonstrate the applicability of our approach in real-world scenarios, we exploit its characteristics to build a profitable investment portfolio. Results show that we are able to improve the state of the art, by going from a -%20 profit to a noticeable 80%.

Graphical models allow to describe the interplay among variables of a system through a compact representation, suitable when relations evolve over time. For example, in a biological setting, genes interact differently depending on external environmental or metabolic factors. To incorporate this dynamics a viable strategy is to estimate a sequence of temporally related graphs assuming similarity among samples in different time points. While adjacent time points may direct the analysis towards a robust estimate of the underlying graph, the resulting model will not incorporate long-term or recurrent temporal relationships. In this work we propose a dynamical network inference model that leverages on kernels to consider general temporal patterns (such as circadian rhythms or seasonality). We show how our approach may also be exploited when the recurrent patterns are unknown, by coupling the network inference with a clustering procedure that detects possibly non-consecutive similar networks. Such clusters are then used to build similarity kernels. The convexity of the functional is determined by whether we impose or infer the kernel. In the first case, the optimisation algorithm exploits efficiently proximity operators with closed-form solutions. In the other case, we resort to an alternating minimisation procedure which jointly learns the temporal kernel and the underlying network. Extensive analysis on synthetic data shows the efficacy of our models compared to state-of-the-art methods. Finally, we applied our approach on two realworld applications to show how considering long-term patterns is fundamental to have insights on the behaviour of a complex system.

In many applications of finance, biology and sociology, complex systems involve entities interacting with each other. These processes have the peculiarity of evolving over time and of comprising latent factors, which influence the system without being explicitly measured. In this work we present latent variable time-varying graphical lasso (LTGL), a method for multivariate time-series graphical modelling that considers the influence of hidden or unmeasurable factors. The estimation of the contribution of the latent factors is embedded in the model which produces both sparse and low-rank components for each time point. In particular, the first component represents the connectivity structure of observable variables of the system, while the second represents the influence of hidden factors, assumed to be few with respect to the observed variables. Our model includes temporal consistency on both components, providing an accurate evolutionary pattern of the system. We derive a tractable optimisation algorithm based on alternating direction method of multipliers, and develop a scalable and efficient implementation which exploits proximity operators in closed form. LTGL is extensively validated on synthetic data, achieving optimal performance in terms of accuracy, structure learning and scalability with respect to ground truth and state-of-the-art methods for graphical inference. We conclude with the application of LTGL to real case studies, from biology and finance, to illustrate how our method can be successfully employed to gain insights on multivariate time-series data.