Transformers Are Universally Consistent

Despite their central role in the success of foundational models and large-scale language modeling, the theoretical foundations governing the operation of Transformers remain only partially understood. Contemporary research has largely focused on their representational capacity for language comprehension and their prowess in in-context learning, frequently under idealized assumptions such as linearized attention mechanisms. Initially conceived to model sequence-to-sequence transformations, a fundamental and unresolved question is whether Transformers can robustly perform functional regression over sequences of input tokens. This question assumes heightened importance given the inherently non-Euclidean geometry underlying real-world data distributions. In this work, we establish that Transformers equipped with softmax-based nonlinear attention are uniformly consistent when tasked with executing Ordinary Least Squares (OLS) regression, provided both the inputs and outputs are embedded in hyperbolic space. We derive deterministic upper bounds on the empirical error which, in the asymptotic regime, decay at a provable rate of $\mathcal{O}(t^{-1/2d})$, where $t$ denotes the number of input tokens and $d$ the embedding dimensionality. Notably, our analysis subsumes the Euclidean setting as a special case, recovering analogous convergence guarantees parameterized by the intrinsic dimensionality of the data manifold. These theoretical insights are corroborated through empirical evaluations on real-world datasets involving both continuous and categorical response variables.

On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks

The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincar\'{e} Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.

Consistent Spectral Clustering in Hyperbolic Spaces

Clustering, as an unsupervised technique, plays a pivotal role in various data analysis applications. Among clustering algorithms, Spectral Clustering on Euclidean Spaces has been extensively studied. However, with the rapid evolution of data complexity, Euclidean Space is proving to be inefficient for representing and learning algorithms. Although Deep Neural Networks on hyperbolic spaces have gained recent traction, clustering algorithms or non-deep machine learning models on non-Euclidean Spaces remain underexplored. In this paper, we propose a spectral clustering algorithm on Hyperbolic Spaces to address this gap. Hyperbolic Spaces offer advantages in representing complex data structures like hierarchical and tree-like structures, which cannot be embedded efficiently in Euclidean Spaces. Our proposed algorithm replaces the Euclidean Similarity Matrix with an appropriate Hyperbolic Similarity Matrix, demonstrating improved efficiency compared to clustering in Euclidean Spaces. Our contributions include the development of the spectral clustering algorithm on Hyperbolic Spaces and the proof of its weak consistency. We show that our algorithm converges at least as fast as Spectral Clustering on Euclidean Spaces. To illustrate the efficacy of our approach, we present experimental results on the Wisconsin Breast Cancer Dataset, highlighting the superior performance of Hyperbolic Spectral Clustering over its Euclidean counterpart. This work opens up avenues for utilizing non-Euclidean Spaces in clustering algorithms, offering new perspectives for handling complex data structures and improving clustering efficiency.