While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

It is not clear, however, whether these theoretical results apply to actual distributions such as images.

The attention mechanism within the transformer architecture enables the model to weigh and combine tokens based on their relevance to the query.

We can expand the loss (4), i.e., Table 5 shows more results of our method with different configurations and hyperparameter settings.

We can expand the loss (4), i.e., Table 5 shows more results of our method with different configurations and hyperparameter settings.

Blockchain technology enables secure, transparent data management in decentralized systems, supporting applications from cryptocurrencies like Bitcoin to tokenizing real-world assets like property. Its scalability and sustainability hinge on consensus mechanisms balancing security and efficiency. Proof of Work (PoW), used by Bitcoin, ensures security through energy-intensive computations but demands significant resources. Proof of Stake (PoS), as in Ethereum post-Merge, selects validators based on staked cryptocurrency, offering energy efficiency but risking centralization from wealth concentration. With AI models straining computational resources, we propose Proof of Useful Intelligence (PoUI), a hybrid consensus mechanism. In PoUI, workers perform AI tasks like language processing or image analysis to earn coins, which are staked to secure the network, blending security with practical utility. Decentralized nodes--job posters, market coordinators, workers, and validators --collaborate via smart contracts to manage tasks and rewards.

We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is distinguished from most commonly used statistical models. Limited preliminary numerical results are provided to test the efficiency of some specific algorithms derived from our paradigm, which we hope to stimulate further practical interest.

Most modern LLMs use Transformers [30] as their backbones, which demonstrate significant advantages over many existing neural network models. Transformers achieve many state-of-the-art performances in learning tasks including natural language processing [33] and computer vision [18]. However, the underlying mechanism for the success of Transformers remains largely a mystery to theoretical researchers. It has been discussed in a line of recent works [2, 4, 15, 38] that, instead of learning simple prediction rules (such as a linear model) Transformers are capable of learning to perform learning algorithms that can automatically generate new prediction rules. For instance, when a new dataset is organized as the input of a Transformer, the model can automatically perform linear regression on this new dataset to produce a newly fitted linear model and make predictions accordingly. This idea of treating Transformers as "algorithm approximators" has provided insights into the power of large language models. However, these existing works only provide guarantees for the in-context supervised learning capacities of Transformers. It remains unclear whether Transformers are capable of handling unsupervised tasks as well.

In recent years, diffusion models, and more generally score-based deep generative models, have achieved remarkable success in various applications, including image and audio generation. In this paper, we view diffusion models as an implicit approach to nonparametric density estimation and study them within a statistical framework to analyze their surprising performance. A key challenge in high-dimensional statistical inference is leveraging low-dimensional structures inherent in the data to mitigate the curse of dimensionality. We assume that the underlying density exhibits a low-dimensional structure by factorizing into low-dimensional components, a property common in examples such as Bayesian networks and Markov random fields. Under suitable assumptions, we demonstrate that an implicit density estimator constructed from diffusion models adapts to the factorization structure and achieves the minimax optimal rate with respect to the total variation distance. In constructing the estimator, we design a sparse weight-sharing neural network architecture, where sparsity and weight-sharing are key features of practical architectures such as convolutional neural networks and recurrent neural networks.

Symbolic regression is a task aimed at identifying patterns in data and representing them through mathematical expressions, generally involving skeleton prediction and constant optimization. Many methods have achieved some success, however they treat variables and symbols merely as characters of natural language without considering their mathematical essence. This paper introduces the operator feature neural network (OF-Net) which employs operator representation for expressions and proposes an implicit feature encoding method for the intrinsic mathematical operational logic of operators. By substituting operator features for numeric loss, we can predict the combination of operators of target expressions. We evaluate the model on public datasets, and the results demonstrate that the model achieves superior recovery rates and high $R^2$ scores. With the discussion of the results, we analyze the merit and demerit of OF-Net and propose optimizing schemes.