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We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of $\mathbb{R}$. An order-$1$ autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one-step conditional Fr\'echet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalises or extends certain existing models of transportation-based regression and autoregression, and in doing so also provides some additional insights on existing models.

We address the problem of integrating data from multiple, possibly biased, observational and interventional studies, to eventually compute counterfactuals in structural causal models. We start from the case of a single observational dataset affected by a selection bias. We show that the likelihood of the available data has no local maxima. This enables us to use the causal expectation-maximisation scheme to compute approximate bounds for partially identifiable counterfactual queries, which are the focus of this paper. We then show how the same approach can solve the general case of multiple datasets, no matter whether interventional or observational, biased or unbiased, by remapping it into the former one via graphical transformations. Systematic numerical experiments and a case study on palliative care show the effectiveness and accuracy of our approach, while hinting at the benefits of integrating heterogeneous data to get informative bounds in case of partial identifiability.

Symbolic regression (SR) is a challenging task in machine learning that involves finding a mathematical expression for a function based on its values. Recent advancements in SR have demonstrated the efficacy of pretrained transformer-based models for generating equations as sequences, which benefit from large-scale pretraining on synthetic datasets and offer considerable advantages over GP-based methods in terms of inference time. However, these models focus on supervised pretraining goals borrowed from text generation and ignore equation-specific objectives like accuracy and complexity. To address this, we propose TPSR, a Transformer-based Planning strategy for Symbolic Regression that incorporates Monte Carlo Tree Search into the transformer decoding process. TPSR, as opposed to conventional decoding strategies, allows for the integration of non-differentiable feedback, such as fitting accuracy and complexity, as external sources of knowledge into the equation generation process. Extensive experiments on various datasets show that our approach outperforms state-of-the-art methods, enhancing the model's fitting-complexity trade-off, extrapolation abilities, and robustness to noise. We also demonstrate that the utilization of various caching mechanisms can further enhance the efficiency of TPSR.

We provide a unified operational framework for the study of causality, non-locality and contextuality, in a fully device-independent and theory-independent setting. We define causaltopes, our chosen portmanteau of "causal polytopes", for arbitrary spaces of input histories and arbitrary choices of input contexts. We show that causaltopes are obtained by slicing simpler polytopes of conditional probability distributions with a set of causality equations, which we fully characterise. We provide efficient linear programs to compute the maximal component of an empirical model supported by any given sub-causaltope, as well as the associated causal fraction. We introduce a notion of causal separability relative to arbitrary causal constraints. We provide efficient linear programs to compute the maximal causally separable component of an empirical model, and hence its causally separable fraction, as the component jointly supported by certain sub-causaltopes. We study causal fractions and causal separability for several novel examples, including a selection of quantum switches with entangled or contextual control. In the process, we demonstrate the existence of "causal contextuality", a phenomenon where causal inseparability is clearly correlated to, or even directly implied by, non-locality and contextuality.

With the rising complexity of numerous novel applications that serve our modern society comes the strong need to design efficient computing platforms. Designing efficient hardware is, however, a complex multi-objective problem that deals with multiple parameters and their interactions. Given that there are a large number of parameters and objectives involved in hardware design, synthesizing all possible combinations is not a feasible method to find the optimal solution. One promising approach to tackle this problem is statistical modeling of a desired hardware performance. Here, we propose a model-based active learning approach to solve this problem. Our proposed method uses Bayesian models to characterize various aspects of hardware performance. We also use transfer learning and Gaussian regression bootstrapping techniques in conjunction with active learning to create more accurate models. Our proposed statistical modeling method provides hardware models that are sufficiently accurate to perform design space exploration as well as performance prediction simultaneously. We use our proposed method to perform design space exploration and performance prediction for various hardware setups, such as micro-architecture design and OpenCL kernels for FPGA targets. Our experiments show that the number of samples required to create performance models significantly reduces while maintaining the predictive power of our proposed statistical models. For instance, in our performance prediction setting, the proposed method needs 65% fewer samples to create the model, and in the design space exploration setting, our proposed method can find the best parameter settings by exploring less than 50 samples.

Distinguishing cause from effect using observations of a pair of random variables is a core problem in causal discovery. Most approaches proposed for this task, namely additive noise models (ANM), are only adequate for quantitative data. We propose a criterion to address the cause-effect problem with categorical variables (living in sets with no meaningful order), inspired by seeing a conditional probability mass function (pmf) as a discrete memoryless channel. We select as the most likely causal direction the one in which the conditional pmf is closer to a uniform channel (UC). The rationale is that, in a UC, as in an ANM, the conditional entropy (of the effect given the cause) is independent of the cause distribution, in agreement with the principle of independence of cause and mechanism. Our approach, which we call the uniform channel model (UCM), thus extends the ANM rationale to categorical variables. To assess how close a conditional pmf (estimated from data) is to a UC, we use statistical testing, supported by a closed-form estimate of a UC channel. On the theoretical front, we prove identifiability of the UCM and show its equivalence with a structural causal model with a low-cardinality exogenous variable. Finally, the proposed method compares favorably with recent state-of-the-art alternatives in experiments on synthetic, benchmark, and real data.

Contrastive loss has been increasingly used in learning representations from multiple modalities. In the limit, the nature of the contrastive loss encourages modalities to exactly match each other in the latent space. Yet it remains an open question how the modality alignment affects the downstream task performance. In this paper, based on an information-theoretic argument, we first prove that exact modality alignment is sub-optimal in general for downstream prediction tasks. Hence we advocate that the key of better performance lies in meaningful latent modality structures instead of perfect modality alignment. To this end, we propose three general approaches to construct latent modality structures. Specifically, we design 1) a deep feature separation loss for intra-modality regularization; 2) a Brownian-bridge loss for inter-modality regularization; and 3) a geometric consistency loss for both intra- and inter-modality regularization. Extensive experiments are conducted on two popular multi-modal representation learning frameworks: the CLIP-based two-tower model and the ALBEF-based fusion model. We test our model on a variety of tasks including zero/few-shot image classification, image-text retrieval, visual question answering, visual reasoning, and visual entailment. Our method achieves consistent improvements over existing methods, demonstrating the effectiveness and generalizability of our proposed approach on latent modality structure regularization.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

Learning disentanglement aims at finding a low dimensional representation which consists of multiple explanatory and generative factors of the observational data. The framework of variational autoencoder (VAE) is commonly used to disentangle independent factors from observations. However, in real scenarios, factors with semantics are not necessarily independent. Instead, there might be an underlying causal structure which renders these factors dependent. We thus propose a new VAE based framework named CausalVAE, which includes a Causal Layer to transform independent exogenous factors into causal endogenous ones that correspond to causally related concepts in data. We further analyze the model identifiabitily, showing that the proposed model learned from observations recovers the true one up to a certain degree. Experiments are conducted on various datasets, including synthetic and real word benchmark CelebA. Results show that the causal representations learned by CausalVAE are semantically interpretable, and their causal relationship as a Directed Acyclic Graph (DAG) is identified with good accuracy. Furthermore, we demonstrate that the proposed CausalVAE model is able to generate counterfactual data through "do-operation" to the causal factors.

Large knowledge graphs often grow to store temporal facts that model the dynamic relations or interactions of entities along the timeline. Since such temporal knowledge graphs often suffer from incompleteness, it is important to develop time-aware representation learning models that help to infer the missing temporal facts. While the temporal facts are typically evolving, it is observed that many facts often show a repeated pattern along the timeline, such as economic crises and diplomatic activities. This observation indicates that a model could potentially learn much from the known facts appeared in history. To this end, we propose a new representation learning model for temporal knowledge graphs, namely CyGNet, based on a novel timeaware copy-generation mechanism. CyGNet is not only able to predict future facts from the whole entity vocabulary, but also capable of identifying facts with repetition and accordingly predicting such future facts with reference to the known facts in the past. We evaluate the proposed method on the knowledge graph completion task using five benchmark datasets. Extensive experiments demonstrate the effectiveness of CyGNet for predicting future facts with repetition as well as de novo fact prediction.