Finding Clustering Algorithms in the Transformer Architecture

The invention of the transformer architecture has revolutionized Artificial Intelligence (AI), yielding unprecedented success in areas such as natural language processing, computer vision, and multimodal reasoning. Despite these advances, it is unclear whether transformers are able to learn and implement precise algorithms. Here, we demonstrate that transformers can exactly implement a fundamental and widely used algorithm for $k$-means clustering: Lloyd's algorithm. First, we theoretically prove the existence of such a transformer architecture, which we term the $k$-means transformer, that exactly implements Lloyd's algorithm for $k$-means clustering using the standard ingredients of modern transformers: attention and residual connections. Next, we numerically implement this transformer and demonstrate in experiments the exact correspondence between our architecture and Lloyd's algorithm, providing a fully neural implementation of $k$-means clustering. Finally, we demonstrate that interpretable alterations (e.g., incorporating layer normalizations or multilayer perceptrons) to this architecture yields diverse and novel variants of clustering algorithms, such as soft $k$-means, spherical $k$-means, trimmed $k$-means, and more. Collectively, our findings demonstrate how transformer mechanisms can precisely map onto algorithmic procedures, offering a clear and interpretable perspective on implementing precise algorithms in transformers.

Exploiting Observation Bias to Improve Matrix Completion

We consider a variant of matrix completion where entries are revealed in a biased manner, adopting a model akin to that introduced by Ma and Chen. Instead of treating this observation bias as a disadvantage, as is typically the case, our goal is to exploit the shared information between the bias and the outcome of interest to improve predictions. Towards this, we propose a simple two-stage algorithm: (i) interpreting the observation pattern as a fully observed noisy matrix, we apply traditional matrix completion methods to the observation pattern to estimate the distances between the latent factors; (ii) we apply supervised learning on the recovered features to impute missing observations. We establish finite-sample error rates that are competitive with the corresponding supervised learning parametric rates, suggesting that our learning performance is comparable to having access to the unobserved covariates. Empirical evaluation using a real-world dataset reflects similar performance gains, with our algorithm's estimates having 30x smaller mean squared error compared to traditional matrix completion methods.