Understanding identifiability of latent content and style variables from unaligned multi-domain data is essential for tasks such as domain translation and data generation. Existing works on content-style identification were often developed under somewhat stringent conditions, e.g., that all latent components are mutually independent and that the dimensions of the content and style variables are known. We introduce a new analytical framework via cross-domain *latent distribution matching* (LDM), which establishes content-style identifiability under substantially more relaxed conditions. Specifically, we show that restrictive assumptions such as component-wise independence of the latent variables can be removed. Most notably, we prove that prior knowledge of the content and style dimensions is not necessary for ensuring identifiability, if sparsity constraints are properly imposed onto the learned latent representations. Bypassing the knowledge of the exact latent dimension has been a longstanding aspiration in unsupervised representation learning---our analysis is the first to underpin its theoretical and practical viability. On the implementation side, we recast the LDM formulation into a regularized multi-domain GAN loss with coupled latent variables. We show that the reformulation is equivalent to LDM under mild conditions---yet requiring considerably less computational resource. Experiments corroborate with our theoretical claims.

Diversified distribution matching (DDM) finds a unified translation function mapping a diverse collection of conditional source distributions to their target counterparts. DDM was proposed to resolve content misalignment issues in unpaired domain translation, achieving translation identifiability. However, DDM has only been implemented using GANs due to its constraints on the translation function. GANs are often unstable to train and do not provide the transport trajectory information---yet such trajectories are useful in applications such as single-cell evolution analysis and robot route planning. This work introduces *diversified flow matching* (DFM), an ODE-based framework for DDM. Adapting flow matching (FM) to enforce a unified translation function as in DDM is challenging, as FM learns the translation function's velocity rather than the translation function itself. A custom bilevel optimization-based training loss, a nonlinear interpolant, and a structural reformulation are proposed to address these challenges, offering a tangible implementation. To our knowledge, DFM is the first ODE-based approach guaranteeing translation identifiability. Experiments on synthetic and real-world datasets validate the proposed method.

Spectrum cartography(SC) aims to estimate the radio power map of multiple emitters over space and frequency using limited sensor data. Recent advances leverage learneddeep generative models(DGMs) as structural priors, achieving state-of-the-art performance by capturing complex spatial-spectral patterns. However, DGMs require large training datasets and may suffer under distribution shifts. To address these limitations, we propose atraining-freeSC approach based onuntrained neural networks(UNNs), which encode structural priors through architectural design. Our custom UNN exploits a spatio-spectral factorization model rooted in the physical structure of radio maps, enabling low sample complexity. Experiments show that our method matches the performance of DGM-based SC without any training data.

Unsupervised domain translation (UDT) aims to find functions that convert samples from one domain (e.g., sketches) to another domain (e.g., photos) without changing the high-level semantic meaning (also referred to as "content"). The translation functions are often sought by probability distribution matching of the transformed source domain and target domain. CycleGAN stands as arguably the most representative approach among this line of work. However, it was noticed in the literature that CycleGAN and variants could fail to identify the desired translation functions and produce content-misaligned translations. This limitation arises due to the presence of multiple translation functions---referred to as ``measure-preserving automorphism" (MPA)---in the solution space of the learning criteria. Despite awareness of such identifiability issues, solutions have remained elusive. This study delves into the core identifiability inquiry and introduces an MPA elimination theory. Our analysis shows that MPA is unlikely to exist, if multiple pairs of diverse cross-domain conditional distributions are matched by the learning function. Our theory leads to a UDT learner using distribution matching over auxiliary variable-induced subsets of the domains---other than over the entire data domains as in the classical approaches. The proposed framework is the first to rigorously establish translation identifiability under reasonable UDT settings, to our best knowledge. Experiments corroborate with our theoretical claims.

A core task in multi-modal learning is to integrate information from multiple feature spaces (e.g., text and audio), offering modality-invariant essential representations of data. Recent research showed that, classical tools such as canonical correlation analysis (CCA) provably identify the shared components up to minor ambiguities, when samples in each modality are generated from a linear mixture of shared and private components. Such identifiability results were obtained under the condition that the cross-modality samples are aligned/paired according to their shared information. This work takes a step further, investigating shared component identifiability from multi-modal linear mixtures where cross-modality samples are unaligned. A distribution divergence minimization-based loss is proposed, under which a suite of sufficient conditions ensuring identifiability of the shared components are derived. Our conditions are based on cross-modality distribution discrepancy characterization and density-preserving transform removal, which are much milder than existing studies relying on independent component analysis. More relaxed conditions are also provided via adding reasonable structural constraints, motivated by available side information in various applications. The identifiability claims are thoroughly validated using synthetic and real-world data.

This work revisits the joint transmit beamforming and antenna selection problem. Existing approaches find approximate solutions to this NP-hard problem via various heuristics, e.g., convex/nonconvex relaxation, greedy method, and (deep) supervised learning. However, optimality (or even feasibility) of these heuristics is not guaranteed. To avoid sub-optimal solutions, an effective branch and bound (B&B) algorithm is proposed. B&B algorithms are ensured to return optimal solutions, but have scalability challenges. In order to enhance efficiency, a graph neural network (GNN)-based classfier is trained with imitation learning to accelerate the B&B algorithm— where the GNN is carefully designed to suit the dynamic nature of wireless communication scenarios. The GNN-based acceleration is shown to provably retain the optimality of B&B with high probability, while substantially reducing the computational burden, under reasonable conditions. Numerical experiments show that our GNN-based method always finds near-optimal and feasible solutions with significantly reduced complexity relative to the plain-vanilla B&B.

Spectrum cartography (SC) techniques craft multi-domain (e.g., space and frequency) radio maps from limited measurements, which is an ill-posed inverse problem. Recent works used low-dimensional priors such as a low tensor rank structure and a deep generative model to assist radio map estimation—with provable guarantees. However, a premise of these approaches is that the sensors are able to send real-valued feedback to a fusion center for SC—yet practical communication systems often use (heavy) quantization for signaling. This work puts forth a limited feedback-based SC framework. Similar to a prior work, a generative adversarial network (GAN)-based deep prior is used in our framework for fending against heavy shadowing. However, instead of using real-valued feedback, a random quantization strategy is adopted and a maximum likelihood estimation (MLE) criterion is proposed. Analysis shows that the MLE provably recovers the radio map, under reasonable conditions. Simulations are conducted to showcase the effectiveness of the proposed approach.

Generalized canonical correlation analysis (GCCA) aims to learn common low-dimensional representations from multiple "views" of the data (e.g., audio and video of the same event). In the era of big data, GCCA computation encounters many new challenges. In particular, distributed optimization for GCCA—which is well-motivated in applications like internet of things and parallel computing—may incur prohibitively high communication costs. To address this challenge, this work proposes a communication-efficient distributed GCCA algorithm under the popular MAX-VAR GCCA paradigm. A quantization strategy for information exchange among the computing agents is employed in the proposed algorithm. It is observed that our design, leveraging the idea of error feedback-based quantization, can reduce communication cost by at least 90% while maintaining essentially the same GCCA performance as the unquantized version. Furthermore, the proposed method is guar-anteed to converge to a neighborhood of the optimal solution in a geometric rate—even under aggressive quantization. The effectiveness of our method is demonstrated using both synthetic and real data experiments.

Spectrum cartography (SC) aims at estimating the multi-aspect (e.g., space, frequency, and time) interference level caused by multiple emitters from limited measurements. Early SC approaches rely on model assumptions about the radio map, e.g., sparsity and smoothness, which may be grossly violated under critical scenarios, e.g., in the presence of severe shadowing. More recent data-driven methods train deep generative networks to distill parsimonious representations of complex scenarios, in order to enhance performance of SC. The challenge is that the state space of this learning problem is extremely large—induced by different combinations of key problem constituents, e.g., the number of emitters, the emitters’ carrier frequencies, and the emitter locations. Learning over such a huge space can be costly in terms of sample complexity and training time; it also frequently leads to generalization problems. Our method integrates the favorable traits of model and data-driven approaches, which substantially ‘shrinks’ the state space. Specifically, the proposed learning paradigm only needs to learn a generative model for the radio map of a single emitter (as opposed to numerous combinations of multiple emitters), leveraging a nonnegative matrix factorization (NMF)-based emitter disaggregation process. Numerical evidence shows that the proposed method outperforms state-of-the-art purely model-driven and purely data-driven approaches.