This paper develops a novel weak multilevel Monte-Carlo (MLMC) approximation scheme for L\'evy-driven Stochastic Differential Equations (SDEs). The scheme is based on the state space discretization (via a continuous-time Markov chain approximation) of the pure-jump component of the driving L\'evy process and is particularly suited if the multidimensional driver is given by a L\'evy copula. The multilevel version of the algorithm requires a new coupling of the approximate L\'evy drivers in the consecutive levels of the scheme, which is defined via a coupling of the corresponding Poisson point processes. The multilevel scheme is weak in the sense that the bound on the level variances is based on the coupling alone without requiring strong convergence. Moreover, the coupling is natural for the proposed discretization of jumps and is easy to simulate. The approximation scheme and its multilevel analogous are applied to examples taken from mathematical finance, including the pricing of credit, equity and interest rate derivatives.
Discrete Determinantal Point Processes (DPPs) have a wide array of potential applications for subsampling datasets. They are however held back in some cases by the high cost of sampling. In the worst-case scenario, the sampling cost scales as O(n^3) where n is the number of elements of the ground set. A popular workaround to this prohibitive cost is to sample DPPs defined by low-rank kernels. In such cases, the cost of standard sampling algorithms scales as O(np^2 + nm^2) where m is the (average) number of samples of the DPP (usually m << n) and p the rank of the kernel used to define the DPP (m \leq p \leq n). The first term, O(np^2), comes from a SVD-like step. We focus here on the second term of this cost, O(nm^2), and show that it can be brought down to O(nm + m^3 log m) without loss on the sampling's exactness. In practice, we observe very substantial speedups compared to the classical algorithm as soon as n > 1000. The algorithm described here is a close variant of the standard algorithm for sampling continuous DPPs, and uses rejection sampling. In the specific case of projection DPPs, we also show that any additional sample can be drawn in time O(m^3 log m). Finally, an interesting by-product of the analysis is that a realisation from a DPP is typically contained in a subset of size O(m log m) formed using leverage score i.i.d. sampling.
Temporal Point Processes (TPP) are probabilistic generative frameworks. They model discrete event sequences localized in continuous time. Generally, real-life events reveal descriptive information, known as marks. Marked TPPs model time and marks of the event together for practical relevance. Conditioned on past events, marked TPPs aim to learn the joint distribution of the time and the mark of the next event. For simplicity, conditionally independent TPP models assume time and marks are independent given event history. They factorize the conditional joint distribution of time and mark into the product of individual conditional distributions. This structural limitation in the design of TPP models hurt the predictive performance on entangled time and mark interactions. In this work, we model the conditional inter-dependence of time and mark to overcome the limitations of conditionally independent models. We construct a multivariate TPP conditioning the time distribution on the current event mark in addition to past events. Besides the conventional intensity-based models for conditional joint distribution, we also draw on flexible intensity-free TPP models from the literature. The proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks. Our experimentation on various datasets with multiple evaluation metrics highlights the merit of the proposed approach.
Temporal point processes (TPP) are a natural tool for modeling event-based data. Among all TPP models, Hawkes processes have proven to be the most widely used, mainly due to their adequate modeling for various applications, particularly when considering exponential or non-parametric kernels. Although non-parametric kernels are an option, such models require large datasets. While exponential kernels are more data efficient and relevant for specific applications where events immediately trigger more events, they are ill-suited for applications where latencies need to be estimated, such as in neuroscience. This work aims to offer an efficient solution to TPP inference using general parametric kernels with finite support. The developed solution consists of a fast $\ell_2$ gradient-based solver leveraging a discretized version of the events. After theoretically supporting the use of discretization, the statistical and computational efficiency of the novel approach is demonstrated through various numerical experiments. Finally, the method's effectiveness is evaluated by modeling the occurrence of stimuli-induced patterns from brain signals recorded with magnetoencephalography (MEG). Given the use of general parametric kernels, results show that the proposed approach leads to an improved estimation of pattern latency than the state-of-the-art.
As a fundamental information fusion approach, the arithmetic average (AA) fusion has recently been investigated for various random finite set (RFS) filter fusion in the context of multi-sensor multi-target tracking. It is not a straightforward extension of the ordinary density-AA fusion to the RFS distribution but has to preserve the form of the fusing multi-target density. In this work, we first propose a statistical concept, probability hypothesis density (PHD) consistency, and explain how it can be achieved by the PHD-AA fusion and lead to more accurate and robust detection and localization of the present targets. This forms a both theoretically sound and technically meaningful reason for performing inter-filter PHD AA-fusion/consensus, while preserving the form of the fusing RFS filter. Then, we derive and analyze the proper AA fusion formulations for most existing unlabeled/labeled RFS filters basing on the (labeled) PHD-AA/consistency. These derivations are theoretically unified, exact, need no approximation and greatly enable heterogenous unlabeled and labeled RFS density fusion which is separately demonstrated in two consequent companion papers.
The comparison of local characteristics of two random processes can shed light on periods of time or space at which the processes differ the most. This paper proposes a method that learns about regions with a certain volume, where the marginal attributes of two processes are less similar. The proposed methods are devised in full generality for the setting where the data of interest are themselves stochastic processes, and thus the proposed method can be used for pointing out the regions of maximum dissimilarity with a certain volume, in the contexts of functional data, time series, and point processes. The parameter functions underlying both stochastic processes of interest are modeled via a basis representation, and Bayesian inference is conducted via an integrated nested Laplace approximation. The numerical studies validate the proposed methods, and we showcase their application with case studies on criminology, finance, and medicine.
A treatment policy defines when and what treatments are applied to affect some outcome of interest. Data-driven decision-making requires the ability to predict what happens if a policy is changed. Existing methods that predict how the outcome evolves under different scenarios assume that the tentative sequences of future treatments are fixed in advance, while in practice the treatments are determined stochastically by a policy and may depend, for example, on the efficiency of previous treatments. Therefore, the current methods are not applicable if the treatment policy is unknown or a counterfactual analysis is needed. To handle these limitations, we model the treatments and outcomes jointly in continuous time, by combining Gaussian processes and point processes. Our model enables the estimation of a treatment policy from observational sequences of treatments and outcomes, and it can predict the interventional and counterfactual progression of the outcome after an intervention on the treatment policy (in contrast with the causal effect of a single treatment). We show with real-world and semi-synthetic data on blood glucose progression that our method can answer causal queries more accurately than existing alternatives.
We propose a new sparse Granger-causal learning framework for temporal event data. We focus on a specific class of point processes called the Hawkes process. We begin by pointing out that most of the existing sparse causal learning algorithms for the Hawkes process suffer from a singularity in maximum likelihood estimation. As a result, their sparse solutions can appear only as numerical artifacts. In this paper, we propose a mathematically well-defined sparse causal learning framework based on a cardinality-regularized Hawkes process, which remedies the pathological issues of existing approaches. We leverage the proposed algorithm for the task of instance-wise causal event analysis, where sparsity plays a critical role. We validate the proposed framework with two real use-cases, one from the power grid and the other from the cloud data center management domain.
The event camera is a novel bio-inspired vision sensor. When the brightness change exceeds the preset threshold, the sensor generates events asynchronously. The number of valid events directly affects the performance of event-based tasks, such as reconstruction, detection, and recognition. However, when in low-brightness or slow-moving scenes, events are often sparse and accompanied by noise, which poses challenges for event-based tasks. To solve these challenges, we propose an event temporal up-sampling algorithm1 to generate more effective and reliable events. The main idea of our algorithm is to generate up-sampling events on the event motion trajectory. First, we estimate the event motion trajectory by contrast maximization algorithm and then up-sampling the events by temporal point processes. Experimental results show that up-sampling events can provide more effective information and improve the performance of downstream tasks, such as improving the quality of reconstructed images and increasing the accuracy of object detection.
In this work, we identify open research opportunities in applying Neural Temporal Point Process (NTPP) models to industry scale customer behavior data by carefully reproducing NTPP models published up to date on known literature benchmarks as well as applying NTPP models to a novel, real world consumer behavior dataset that is twice as large as the largest publicly available NTPP benchmark. We identify the following challenges. First, NTPP models, albeit their generative nature, remain vulnerable to dataset imbalances and cannot forecast rare events. Second, NTPP models based on stochastic differential equations, despite their theoretical appeal and leading performance on literature benchmarks, do not scale easily to large industry-scale data. The former is in light of previously made observations on deep generative models. Additionally, to combat a cold-start problem, we explore a novel addition to NTPP models - a parametrization based on static user features.
We present a novel Recurrent Graph Network (RGN) approach for predicting discrete marked event sequences by learning the underlying complex stochastic process. Using the framework of Point Processes, we interpret a marked discrete event sequence as the superposition of different sequences each of a unique type. The nodes of the Graph Network use LSTM to incorporate past information whereas a Graph Attention Network (GAT Network) introduces strong inductive biases to capture the interaction between these different types of events. By changing the self-attention mechanism from attending over past events to attending over event types, we obtain a reduction in time and space complexity from $\mathcal{O}(N^2)$ (total number of events) to $\mathcal{O}(|\mathcal{Y}|^2)$ (number of event types). Experiments show that the proposed approach improves performance in log-likelihood, prediction and goodness-of-fit tasks with lower time and space complexity compared to state-of-the art Transformer based architectures.
This work introduces DADAO: the first decentralized, accelerated, asynchronous, primal, first-order algorithm to minimize a sum of $L$-smooth and $\mu$-strongly convex functions distributed over a given network of size $n$. Our key insight is based on modeling the local gradient updates and gossip communication procedures with separate independent Poisson Point Processes. This allows us to decouple the computation and communication steps, which can be run in parallel, while making the whole approach completely asynchronous. This leads to communication acceleration compared to synchronous approaches. Our new method employs primal gradients and does not use a multi-consensus inner loop nor other ad-hoc mechanisms such as Error Feedback, Gradient Tracking, or a Proximal operator. By relating the inverse of the smallest positive eigenvalue of the Laplacian matrix $\chi_1$ and the maximal resistance $\chi_2\leq \chi_1$ of the graph to a sufficient minimal communication rate between the nodes of the network, we show that our algorithm requires $\mathcal{O}(n\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ local gradients and only $\mathcal{O}(n\sqrt{\chi_1\chi_2}\sqrt{\frac{L}{\mu}}\log(\frac{1}{\epsilon}))$ communications to reach a precision $\epsilon$, up to logarithmic terms. Thus, we simultaneously obtain an accelerated rate for both computations and communications, leading to an improvement over state-of-the-art works, our simulations further validating the strength of our relatively unconstrained method.
Machine learning algorithms minimizing the average training loss usually suffer from poor generalization performance due to the greedy exploitation of correlations among the training data, which are not stable under distributional shifts. It inspires various works for domain generalization (DG), where a series of methods, such as Causal Matching and FISH, work by pairwise domain operations. They would need $O(n^2)$ pairwise domain operations with $n$ domains, where each one is often highly expensive. Moreover, while a common objective in the DG literature is to learn invariant representations against domain-induced spurious correlations, we highlight the importance of mitigating spurious correlations caused by objects. Based on the observation that diversity helps mitigate spurious correlations, we propose a Diversity boosted twO-level saMplIng framework (DOMI) utilizing Determinantal Point Processes (DPPs) to efficiently sample the most informative ones among large number of domains. We show that DOMI helps train robust models against spurious correlations from both domain-side and object-side, substantially enhancing the performance of the backbone DG algorithms on rotated MNIST, rotated Fashion MNIST, and iwildcam datasets.
A determinantal point process (DPP) is an elegant model that assigns a probability to every subset of a collection of $n$ items. While conventionally a DPP is parameterized by a symmetric kernel matrix, removing this symmetry constraint, resulting in nonsymmetric DPPs (NDPPs), leads to significant improvements in modeling power and predictive performance. Recent work has studied an approximate Markov chain Monte Carlo (MCMC) sampling algorithm for NDPPs restricted to size-$k$ subsets (called $k$-NDPPs). However, the runtime of this approach is quadratic in $n$, making it infeasible for large-scale settings. In this work, we develop a scalable MCMC sampling algorithm for $k$-NDPPs with low-rank kernels, thus enabling runtime that is sublinear in $n$. Our method is based on a state-of-the-art NDPP rejection sampling algorithm, which we enhance with a novel approach for efficiently constructing the proposal distribution. Furthermore, we extend our scalable $k$-NDPP sampling algorithm to NDPPs without size constraints. Our resulting sampling method has polynomial time complexity in the rank of the kernel, while the existing approach has runtime that is exponential in the rank. With both a theoretical analysis and experiments on real-world datasets, we verify that our scalable approximate sampling algorithms are orders of magnitude faster than existing sampling approaches for $k$-NDPPs and NDPPs.
A large fraction of data generated via human activities such as online purchases, health records, spatial mobility etc. can be represented as a sequence of events over a continuous-time. Learning deep learning models over these continuous-time event sequences is a non-trivial task as it involves modeling the ever-increasing event timestamps, inter-event time gaps, event types, and the influences between different events within and across different sequences. In recent years neural enhancements to marked temporal point processes (MTPP) have emerged as a powerful framework to model the underlying generative mechanism of asynchronous events localized in continuous time. However, most existing models and inference methods in the MTPP framework consider only the complete observation scenario i.e. the event sequence being modeled is completely observed with no missing events -- an ideal setting that is rarely applicable in real-world applications. A recent line of work which considers missing events while training MTPP utilizes supervised learning techniques that require additional knowledge of missing or observed label for each event in a sequence, which further restricts its practicability as in several scenarios the details of missing events is not known apriori. In this work, we provide a novel unsupervised model and inference method for learning MTPP in presence of event sequences with missing events. Specifically, we first model the generative processes of observed events and missing events using two MTPP, where the missing events are represented as latent random variables. Then, we devise an unsupervised training method that jointly learns both the MTPP by means of variational inference. Such a formulation can effectively impute the missing data among the observed events and can identify the optimal position of missing events in a sequence.
Hawkes processes are a special class of temporal point processes which exhibit a natural notion of causality, as occurrence of events in the past may increase the probability of events in the future. Discovery of the underlying influence network among the dimensions of multi-dimensional temporal processes is of high importance in disciplines where a high-frequency data is to model, e.g. in financial data or in seismological data. This paper approaches the problem of learning Granger-causal network in multi-dimensional Hawkes processes. We formulate this problem as a model selection task in which we follow the minimum description length (MDL) principle. Moreover, we propose a general algorithm for MDL-based inference using a Monte-Carlo method and we use it for our causal discovery problem. We compare our algorithm with the state-of-the-art baseline methods on synthetic and real-world financial data. The synthetic experiments demonstrate superiority of our method incausal graph discovery compared to the baseline methods with respect to the size of the data. The results of experiments with the G-7 bonds price data are consistent with the experts knowledge.
The maximum a posteriori (MAP) inference for determinantal point processes (DPPs) is crucial for selecting diverse items in many machine learning applications. Although DPP MAP inference is NP-hard, the greedy algorithm often finds high-quality solutions, and many researchers have studied its efficient implementation. One classical and practical method is the lazy greedy algorithm, which is applicable to general submodular function maximization, while a recent fast greedy algorithm based on the Cholesky factorization is more efficient for DPP MAP inference. This paper presents how to combine the ideas of "lazy" and "fast", which have been considered incompatible in the literature. Our lazy and fast greedy algorithm achieves almost the same time complexity as the current best one and runs faster in practice. The idea of "lazy + fast" is extendable to other greedy-type algorithms. We also give a fast version of the double greedy algorithm for unconstrained DPP MAP inference. Experiments validate the effectiveness of our acceleration ideas.
Identifying causality behind complex systems plays a significant role in different domains, such as decision making, policy implementations, and management recommendations. However, existing causality studies on temporal event sequences data mainly focus on individual causal discovery, which is incapable of exploiting combined causality. To fill the absence of combined causes discovery on temporal event sequence data,eliminating and recruiting principles are defined to balance the effectiveness and controllability on cause combinations. We also leverage the Granger causality algorithm based on the reactive point processes to describe impelling or inhibiting behavior patterns among entities. In addition, we design an informative and aesthetic visual metaphor of "electrocircuit" to encode aggregated causality for ensuring our causality visualization is non-overlapping and non-intersecting. Diverse sorting strategies and aggregation layout are also embedded into our parallel-based, directed and weighted hypergraph for illustrating combined causality. Our developed combined causality visual analysis system can help users effectively explore combined causes as well as an individual cause. This interactive system supports multi-level causality exploration with diverse ordering strategies and a focus and context technique to help users obtain different levels of information abstraction. The usefulness and effectiveness of the system are further evaluated by conducting a pilot user study and two case studies on event sequence data.
Temporal Point Processes (TPPs) have recently become increasingly interesting for learning dynamics in graph data. A reason for this is that learning on dynamic graph data is becoming more relevant, since data from many scientific fields, ranging from mathematics, biology, social sciences, and physics to computer science, is naturally related and inherently dynamic. In addition, TPPs provide a meaningful characterization of event streams and a prediction mechanism for future events. Therefore, (semi-)parameterized Neural TPPs have been introduced whose characterization can be (partially) learned and, thus, enable the representation of more complex phenomena. However, the research on modeling dynamic graphs with TPPs is relatively young, and only a few models for node attribute changes or evolving edges have been proposed yet. To allow for learning on fully dynamic graph streams, i.e., graphs that can change in their structure (addition/deletion of nodes/edge) and in their node/edge attributes, we propose a Marked Neural Spatio-Temporal Point Process (MNSTPP). It leverages a Dynamic Graph Neural Network to learn a Marked TPP that handles attributes and spatial data to model and predict any event in a graph stream.
Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2011) that the problem of finding a maximum likelihood DPP model for a given data set is NP-complete. In this work we prove Kulesza's conjecture. In fact, we prove the following stronger hardness of approximation result: even computing a $\left(1-O(\frac{1}{\log^9{N}})\right)$-approximation to the maximum log-likelihood of a DPP on a ground set of $N$ elements is NP-complete. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation to the optimal log-likelihood: the approximation factor is $\frac{1}{(1+o(1))\log{m}}$ unconditionally (for data sets that consist of $m$ subsets), and can be improved to $1-\frac{1+o(1)}{\log N}$ if all $N$ elements appear in a $O(1/N)$-fraction of the subsets. In terms of techniques, we reduce approximating the maximum log-likelihood of DPPs on a data set to solving a gap instance of a "vector coloring" problem on a hypergraph. Such a hypergraph is built on a bounded-degree graph construction of Bogdanov, Obata and Trevisan (FOCS 2002), and is further enhanced by the strong expanders of Alon and Capalbo (FOCS 2007) to serve our purposes.