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One primary focus in multimodal feature extraction is to find the representations of individual modalities that are maximally correlated. As a well-known measure of dependence, the Hirschfeld-Gebelein-Rényi (HGR) maximal correlation be-´ comes an appealing objective because of its operational meaning and desirable properties. However, the strict whitening constraints formalized in the HGR maximal correlation limit its application. To address this problem, this paper proposes Soft-HGR, a novel framework to extract informative features from multiple data modalities. Specifically, our framework prevents the “hard” whitening constraints, while simultaneously preserving the same feature geometry as in the HGR maximal correlation. The objective of Soft-HGR is straightforward, only involving two inner products, which guarantees the efficiency and stability in optimization. We further generalize the framework to handle more than two modalities and missing modalities. When labels are partially available, we enhance the discriminative power of the feature representations by making a semi-supervised adaptation. Empirical evaluation implies that our approach learns more informative feature mappings and is more efficient to optimize. 

Multi-label annotation is challenging since a large amount of well-labeled training data are required to achieve promising performance. However, providing such data is expensive while unlabeled data are widely available. To this end, we propose a novel Adaptive Graph Guided Embedding (AG2E) approach for multi-label annotation in a semi-supervised fashion, which utilizes limited labeled data associating with large-scale unlabeled data to facilitate learning performance. Specifically, a multi-label propagation scheme and an effective embedding are jointly learned to seek a latent space where unlabeled instances tend to be well assigned multiple labels. Furthermore, a locality structure regularizer is designed to preserve the intrinsic structure and enhance the multi-label annotation. We evaluate our model in both conventional multi-label learning and zero-shot learning scenario. Experimental results demonstrate that our approach outperforms other compared state-of-the-art methods. 

Abstract Charge density waves (CDWs) in the cuprate high-temperature superconductors have evoked much interest, yet their typical short-range nature has raised questions regarding the role of disorder. Here we report a resonant X-ray diffraction study of ZrTe$${}_{3}$$ ${}_3$, a model CDW system, with focus on the influence of disorder. Near the CDW transition temperature, we observe two independent signals that arise concomitantly, only to become clearly separated in momentum while developing very different correlation lengths in the well-ordered state that is reached at a distinctly lower temperature. Anomalously slow dynamics of mesoscopic charge domains are further found near the transition temperature, in spite of the expected strong thermal fluctuations. Our observations signify the presence of distinct experimental fingerprints of pristine and disorder-perturbed CDWs. We discuss the latter also in the context of Friedel oscillations, which we argue might promote CDW formation via a self-amplifying process.