In the wild, we often encounter collections of sequential data such as electrocardiograms, motion capture, genomes, and natural language, and sequences may be multichannel or symbolic with nonlinear dynamics. We introduce a new method to learn low-dimensional representations of nonlinear time series without supervision and can have provable recovery guarantees. The learned representation can be used for downstream machine-learning tasks such as clustering and classification. The method is based on the assumption that the observed sequences arise from a common domain, but each sequence obeys its own autoregressive models that are related to each other through low-rank regularization. We cast the problem as a computationally efficient convex matrix parameter recovery problem using monotone Variational Inequality and encode the common domain assumption via low-rank constraint across the learned representations, which can learn the geometry for the entire domain as well as faithful representations for the dynamics of each individual sequence using the domain information in totality. We show the competitive performance of our method on real-world time-series data with the baselines and demonstrate its effectiveness for symbolic text modeling and RNA sequence clustering.

The consistency of a learning method is usually established under the assumption that the observations are a realization of an independent and identically distributed (i.i.d.) or mixing process. Yet, kernel methods such as support vector machines (SVMs), Gaussian processes, or conditional kernel mean embeddings (CKMEs) all give excellent performance under sampling schemes that are obviously non-i.i.d., such as when data comes from a dynamical system. We propose the new notion of empirical weak convergence (EWC) as a general assumption explaining such phenomena for kernel methods. It assumes the existence of a random asymptotic data distribution and is a strict weakening of previous assumptions in the field. Our main results then establish consistency of SVMs, kernel mean embeddings, and general Hilbert-space valued empirical expectations with EWC data. Our analysis holds for both finite- and infinite-dimensional outputs, as we extend classical results of statistical learning to the latter case. In particular, it is also applicable to CKMEs. Overall, our results open new classes of processes to statistical learning and can serve as a foundation for a theory of learning beyond i.i.d. and mixing.

Markov jump processes are continuous-time stochastic processes which describe dynamical systems evolving in discrete state spaces. These processes find wide application in the natural sciences and machine learning, but their inference is known to be far from trivial. In this work we introduce a methodology for zeroshot inference of Markov jump processes (MJPs), on bounded state spaces, from noisy and sparse observations, which consists of two components. First, a broad probability distribution over families of MJPs, as well as over possible observation times and noise mechanisms, with which we simulate a synthetic dataset of hidden MJPs and their noisy observation process. Second, a neural network model that processes subsets of the simulated observations, and that is trained to output the initial condition and rate matrix of the target MJP in a supervised way. We empirically demonstrate that one and the same (pretrained) model can infer, in a zero-shot fashion, hidden MJPs evolving in state spaces of different dimensionalities. Specifically, we infer MJPs which describe (i) discrete flashing ratchet systems, which are a type of Brownian motors, and the conformational dynamics in (ii) molecular simulations, (iii) experimental ion channel data and (iv) simple protein folding models. What is more, we show that our model performs on par with state-of-the-art models which are finetuned to the target datasets. Our pretrained model is available online.

This paper studies federated learning for nonparametric regression in the context of distributed samples across different servers, each adhering to distinct differential privacy constraints. The setting we consider is heterogeneous, encompassing both varying sample sizes and differential privacy constraints across servers. Within this framework, both global and pointwise estimation are considered, and optimal rates of convergence over the Besov spaces are established. Distributed privacy-preserving estimators are proposed and their risk properties are investigated. Matching minimax lower bounds, up to a logarithmic factor, are established for both global and pointwise estimation. Together, these findings shed light on the tradeoff between statistical accuracy and privacy preservation. In particular, we characterize the compromise not only in terms of the privacy budget but also concerning the loss incurred by distributing data within the privacy framework as a whole. This insight captures the folklore wisdom that it is easier to retain privacy in larger samples, and explores the differences between pointwise and global estimation under distributed privacy constraints.

Federated learning has attracted significant recent attention due to its applicability across a wide range of settings where data is collected and analyzed across disparate locations. In this paper, we study federated nonparametric goodness-of-fit testing in the white-noise-with-drift model under distributed differential privacy (DP) constraints. We first establish matching lower and upper bounds, up to a logarithmic factor, on the minimax separation rate. This optimal rate serves as a benchmark for the difficulty of the testing problem, factoring in model characteristics such as the number of observations, noise level, and regularity of the signal class, along with the strictness of the $(\epsilon,\delta)$-DP requirement. The results demonstrate interesting and novel phase transition phenomena. Furthermore, the results reveal an interesting phenomenon that distributed one-shot protocols with access to shared randomness outperform those without access to shared randomness. We also construct a data-driven testing procedure that possesses the ability to adapt to an unknown regularity parameter over a large collection of function classes with minimal additional cost, all while maintaining adherence to the same set of DP constraints.

In modern deep learning, transformer networks have established themselves as a fundamental building block, showcasing their versatility across diverse tasks such as language modeling [40], computer vision [16], and reinforcement learning [27]. At the core of transformers is the selfattention layer [57], a critical component assigning varying attention weights to different segments of the input sequence by discerning relevance between tokens. The success of transformers is closely tied to their representational capabilities in extracting structural information encoded in token embeddings. Empirical observations reveal that transformers trained with GD-based algorithms exhibit biases towards certain algorithmic solutions in some arithmetic tasks [18, 33, 61, 35]. However, few works have presented rigorous mathematical evidence that substantiates their superiority over alternative architectures. In recent work, Sanford et al. [48] introduced a simple task known as the q-sparse averaging for an input context sequence, where the target output is an average of a q-subset of the input tokens, specified in the input.

In this paper, we propose a local squared Wasserstein-2 (W_2) method to solve the inverse problem of reconstructing models with uncertain latent variables or parameters. A key advantage of our approach is that it does not require prior information on the distribution of the latent variables or parameters in the underlying models. Instead, our method can efficiently reconstruct the distributions of the output associated with different inputs based on empirical distributions of observation data. We demonstrate the effectiveness of our proposed method across several uncertainty quantification (UQ) tasks, including linear regression with coefficient uncertainty, training neural networks with weight uncertainty, and reconstructing ordinary differential equations (ODEs) with a latent random variable.

A simple and intuitive method for feature selection consists of choosing the feature subset that maximizes a nonparametric measure of dependence between the response and the features. A popular proposal from the literature uses the Hilbert-Schmidt Independence Criterion (HSIC) as the nonparametric dependence measure. The rationale behind this approach to feature selection is that important features will exhibit a high dependence with the response and their inclusion in the set of selected features will increase the HSIC. Through counterexamples, we demonstrate that this rationale is flawed and that feature selection via HSIC maximization can miss critical features.

Latent variable models serve as powerful tools to infer underlying dynamics from observed neural activity. However, due to the absence of ground truth data, prediction benchmarks are often employed as proxies. In this study, we reveal the limitations of the widely-used 'co-smoothing' prediction framework and propose an improved few-shot prediction approach that encourages more accurate latent dynamics. Utilizing a student-teacher setup with Hidden Markov Models, we demonstrate that the high co-smoothing model space can encompass models with arbitrary extraneous dynamics within their latent representations. To address this, we introduce a secondary metric -- a few-shot version of co-smoothing. This involves performing regression from the latent variables to held-out channels in the data using fewer trials. Our results indicate that among models with near-optimal co-smoothing, those with extraneous dynamics underperform in the few-shot co-smoothing compared to 'minimal' models devoid of such dynamics. We also provide analytical insights into the origin of this phenomenon. We further validate our findings on real neural data using two state-of-the-art methods: LFADS and STNDT. In the absence of ground truth, we suggest a proxy measure to quantify extraneous dynamics. By cross-decoding the latent variables of all model pairs with high co-smoothing, we identify models with minimal extraneous dynamics. We find a correlation between few-shot co-smoothing performance and this new measure. In summary, we present a novel prediction metric designed to yield latent variables that more accurately reflect the ground truth, offering a significant improvement for latent dynamics inference.

The causal dependence in data is often characterized by Directed Acyclic Graphical (DAG) models, widely used in many areas. Causal discovery aims to recover the DAG structure using observational data. This paper focuses on causal discovery with multi-variate count data. We are motivated by real-world web visit data, recording individual user visits to multiple websites. Building a causal diagram can help understand user behavior in transitioning between websites, inspiring operational strategy. A challenge in modeling is user heterogeneity, as users with different backgrounds exhibit varied behaviors. Additionally, social network connections can result in similar behaviors among friends. We introduce personalized Binomial DAG models to address heterogeneity and network dependency between observations, which are common in real-world applications. To learn the proposed DAG model, we develop an algorithm that embeds the network structure into a dimension-reduced covariate, learns each node's neighborhood to reduce the DAG search space, and explores the variance-mean relation to determine the ordering. Simulations show our algorithm outperforms state-of-the-art competitors in heterogeneous data. We demonstrate its practical usefulness on a real-world web visit dataset.