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In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science. 

Alzheimer’s disease (AD) is the most prevalent neurodegenerative disease, yet its current treatments are limited to stopping disease progression. Moreover, the effectiveness of these treatments remains uncertain due to the heterogeneity of the disease. Therefore, it is essential to identify disease subtypes at a very early stage. Current data-driven approaches can be used to classify subtypes during later stages of AD or related disorders, but making predictions in the asymptomatic or prodromal stage is challenging. Furthermore, the classifications of most existing models lack explainability, and these models rely solely on a single modality for assessment, limiting the scope of their analysis. Thus, we propose a multimodal framework that utilizes early-stage indicators, including imaging, genetics, and clinical assessments, to classify AD patients into progression-specific subtypes at an early stage. In our framework, we introduce a tri-modal co-attention mechanism (Tri-COAT) to explicitly capture cross-modal feature associations. Data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (slow progressing = 177, intermediate = 302, and fast = 15) were used to train and evaluate Tri-COAT using a 10-fold stratified cross-testing approach. Our proposed model outperforms baseline models and sheds light on essential associations across multimodal features supported by known biological mechanisms. The multimodal design behind Tri-COAT allows it to achieve the highest classification area under the receiver operating characteristic curve while simultaneously providing interpretability to the model predictions through the co-attention mechanism. 

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.