Boltzmann Attention: Learnable Ising Couplings for Cooperative Attention

Attention mechanisms are central to modern sequence models, yet standard attention computes relevance primarily through individual query--key similarities. Although softmax normalization introduces competition among positions, a standard attention layer does not explicitly parameterize learnable interactions between attention decisions. This limits its ability to directly model cooperative or antagonistic co-attention structure within the attention mechanism itself. We propose Boltzmann attention, an energy-based generalization in which attention patterns are governed by an interacting Ising model. The method augments the usual data-dependent local fields with learnable pairwise couplings, allowing the model to represent inter-position correlations beyond those captured by softmax or sigmoid attention. Experiments on character-level language modeling and synthetic bracket matching show that Boltzmann attention consistently improves over standard softmax attention within a standard Transformer architecture, with the advantage becoming more pronounced as sequence length increases. A four-way ablation confirms that the improvement arises from the learnable pairwise couplings. These results suggest that explicit inter-position interactions provide a principled enhancement for attention-based sequence modeling. Moreover, the Ising formulation opens a natural path toward quantum-computing-based sampling strategies: we demonstrate that diabatic quantum annealing provides a practical training method while maintaining competitive performance with exact Boltzmann computation.

Information-Geometric Decomposition of Generalization Error in Unsupervised Learning

We decompose the Kullback--Leibler generalization error (GE) -- the expected KL divergence from the data distribution to the trained model -- of unsupervised learning into three non-negative components: model error, data bias, and variance. The decomposition is exact for any e-flat model class and follows from two identities of information geometry: the generalized Pythagorean theorem and a dual e-mixture variance identity. As an analytically tractable demonstration, we apply the framework to $ε$-PCA, a regularized principal component analysis in which the empirical covariance is truncated at rank $N_K$ and discarded directions are pinned at a fixed noise floor $ε$. Although rank-constrained $ε$-PCA is not itself e-flat, it admits a technical reformulation with the same total GE on isotropic Gaussian data, under which each component of the decomposition takes closed form. The optimal rank emerges as the cutoff $λ_{\mathrm{cut}}^{*} = ε$ -- the model retains exactly those empirical eigenvalues exceeding the noise floor -- with the cutoff reflecting a marginal-rate balance between model-error gain and data-bias cost. A boundary comparison further yields a three-regime phase diagram -- retain-all, interior, and collapse -- separated by the lower Marchenko--Pastur edge and an analytically computable collapse threshold $ε_{*}(α)$, where $α$ is the dimension-to-sample-size ratio. All claims are verified numerically.

Multi-Mode Quantum Annealing for Variational Autoencoders with General Boltzmann Priors

Variational autoencoders (VAEs) learn compact latent representations of complex data, but their generative capacity is fundamentally constrained by the choice of prior distribution over the latent space. Energy-based priors offer a principled way to move beyond factorized assumptions and capture structured interactions among latent variables, yet training such priors at scale requires accurate and efficient sampling from intractable distributions. Here we present Boltzmann-machine--prior VAEs (BM-VAEs) trained using quantum annealing--based sampling in three distinct operational modes within a single generative system. During training, diabatic quantum annealing (DQA) provides unbiased Boltzmann samples for gradient estimation of the energy-based prior; for unconditional generation, slower quantum annealing (QA) concentrates samples near low-energy minima; for conditional generation, bias fields are added to direct sampling toward attribute-specific regions of the energy landscape (c-QA). Using up to 2000 qubits on a D-Wave Advantage2 processor, we demonstrate stable and efficient training across multiple datasets, with faster convergence and lower reconstruction loss than a Gaussian-prior VAE. The learned Boltzmann prior enables unconditional generation by sampling directly from the energy-based latent distribution, a capability that plain autoencoders lack, and conditional generation through latent biasing that leverages the learned pairwise interactions.

Tradeoff of generalization error in unsupervised learning

Finding the optimal model complexity that minimizes the generalization error (GE) is a key issue of machine learning. For the conventional supervised learning, this task typically involves the bias-variance tradeoff: lowering the bias by making the model more complex entails an increase in the variance. Meanwhile, little has been studied about whether the same tradeoff exists for unsupervised learning. In this study, we propose that unsupervised learning generally exhibits a two-component tradeoff of the GE, namely the model error and the data error -- using a more complex model reduces the model error at the cost of the data error, with the data error playing a more significant role for a smaller training dataset. This is corroborated by training the restricted Boltzmann machine to generate the configurations of the two-dimensional Ising model at a given temperature and the totally asymmetric simple exclusion process with given entry and exit rates. Our results also indicate that the optimal model tends to be more complex when the data to be learned are more complex.