Our experimental results demonstrate that this approach significantly improves the training efficiency as compared to maximum likelihood training.

In this section, we provide proof for the disentanglement identifiability of the inferred exogenous variable. Our proof consists of three main components. Then we have ( f, T, λ) ( f, T, λ) . The conditional V AE, in this case, inherits all the properties of maximum likelihood estimation. The following proof is based on the reduction to absurdity.

Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles.

The causal revolution has stimulated interest in understanding complex relationships in various fields.

Clustered Federated Learning (CFL) improves performance under non-IID client heterogeneity by clustering clients and training one model per cluster, thereby balancing between a global model and fully personalized models. However, most CFL methods require the number of clusters K to be fixed a priori, which is impractical when the latent structure is unknown. We propose DPMM-CFL, a CFL algorithm that places a Dirichlet Process (DP) prior over the distribution of cluster parameters. This enables nonparametric Bayesian inference to jointly infer both the number of clusters and client assignments, while optimizing per-cluster federated objectives. This results in a method where, at each round, federated updates and cluster inferences are coupled, as presented in this paper. The algorithm is validated on benchmark datasets under Dirichlet and class-split non-IID partitions.

Abstract--We present a method that models the evolution of an unbounded number of time series clusters by switching among an unknown number of regimes with linear dynamics. We develop a Bayesian non-parametric approach using a hierarchical Dirichlet process as a prior on the parameters of a Switching Linear Dynamical System and a Gaussian process prior to model the statistical variations in amplitude and temporal alignment within each cluster . By modeling the evolution of time series patterns, the method avoids unnecessary proliferation of clusters in a principled manner . We perform inference by formulating a variational lower bound for off-line and on-line scenarios, enabling efficient learning through optimization. We illustrate the versatility and effectiveness of the approach through several case studies of electrocardiogram analysis using publicly available databases. Index T erms--Time series analysis, Bayesian methods, Gaussian processes, linear dynamical systems, Dirichlet processes, unsupervised learning, electrocardiogram, arrhythmia detection. IME series data analysis has come to pervade all scientific and technological domains, driven by the need to understand change over time. With the growing availability of such data, machine learning has assumed an increasingly central role in a wide variety of tasks which fall under the category of pattern recognition. Particularly, there is growing interest in identifying similar behaviors in time series data as a preliminary step towards generating insights into the dynamics of the underlying processes. Some recent methodologies can be found for characterizing sea wave conditions [1], transcriptome-wide gene expression profiling [2], selecting stocks with different share price performance [3], and discovering human motion primitives [4].

Learning from temporally-correlated data is a core facet of modern machine learning. Yet our understanding of sequential learning remains incomplete, particularly in the multi-trajectory setting where data consists of many independent realizations of a time-indexed stochastic process. This important regime both reflects modern training pipelines such as for large foundation models, and offers the potential for learning without the typical mixing assumptions made in the single-trajectory case. However, instance-optimal bounds are known only for least-squares regression with dependent covariates; for more general models or loss functions, the only broadly applicable guarantees result from a reduction to either i.i.d. learning, with effective sample size scaling only in the number of trajectories, or an existing single-trajectory result when each individual trajectory mixes, with effective sample size scaling as the full data budget deflated by the mixing-time. In this work, we significantly broaden the scope of instance-optimal rates in multi-trajectory settings via the Hellinger localization framework, a general approach for maximum likelihood estimation. Our method proceeds by first controlling the squared Hellinger distance at the path-measure level via a reduction to i.i.d. learning, followed by localization as a quadratic form in parameter space weighted by the trajectory Fisher information. This yields instance-optimal bounds that scale with the full data budget under a broad set of conditions. We instantiate our framework across four diverse case studies: a simple mixture of Markov chains, dependent linear regression under non-Gaussian noise, generalized linear models with non-monotonic activations, and linear-attention sequence models. In all cases, our bounds nearly match the instance-optimal rates from asymptotic normality, substantially improving over standard reductions.

Channel state information (CSI) acquisition is a challenging problem in massive multiple-input multiple-output (MIMO) networks. Radio maps provide a promising solution for radio resource management by reducing online CSI acquisition. However, conventional approaches for radio map construction require location-labeled CSI data, which is challenging in practice. This paper investigates unsupervised angular power map construction based on large timescale CSI data collected in a massive MIMO network without location labels. A hidden Markov model (HMM) is built to connect the hidden trajectory of a mobile with the CSI evolution of a massive MIMO channel. As a result, the mobile location can be estimated, enabling the construction of an angular power map. We show that under uniform rectilinear mobility with Poisson-distributed base stations (BSs), the Cramer-Rao Lower Bound (CRLB) for localization error can vanish at any signal-to-noise ratios (SNRs), whereas when BSs are confined to a limited region, the error remains nonzero even with infinite independent measurements. Based on reference signal received power (RSRP) data collected in a real multi-cell massive MIMO network, an average localization error of 18 meters can be achieved although measurements are mainly obtained from a single serving cell.