A First Step Towards Even More Sparse Encodings of Probability Distributions

Despite the widespread use and success of generative AI techniques today, theoretical guarantees on learning a distribution supported in $d$ dimensions from $n$ samples degrade as $O(n^{-1/Θ(d)})$, though shown to be minimax optimal. We hypothesize that present bounds are too pessimistic because smoothness assumptions are not enough to capture the structure of distributions that often appear in real applications. Consequently, we introduce the class of sparse priors and define the "Sparse Dimension" as a measure of sparsity of a prior over the space of all distributions. We show that distribution learning under a $k$-sparse prior achieves a Bayesian risk lower bound of $Ω(\sqrt{k/n})$ under common distance metrics, and show a matching (up to logarithmic terms asymptotically in $n,k$) upper bound for the TV distance under mild additional assumptions. We show the statistical equivalence of distribution learning and learning to sample in the Bayesian setting so that our results apply to learning to sample as well. While $k$ can still depend on the dimension $d$, or a notion of intrinsic dimension, our results show that learning under an appropriate prior overcomes the curse of dimensionality with respect to the dependence on $n$.

The motivation for this paper is the investigation of the trade-offs implicit in probabilistic models used in machine learning. Models are often used to make predictions in the form of conditional probabilities. However, a pair of conditional distributions p(x|y) and p(y|x) may not be compatible with any joint distribution p(x,y). Given two such conditionals, determining if there exists a compatible joint is known as the compatibility problem. For discrete random variables, when the conditionals are encoded as probability tables, the compatibility problem has a known solution, which is computationally tractable. In this paper, we formalise and study a succinct version of the problem, encoding conditional distributions as arithmetic circuits. This is applicable to practical applications of probabilistic modelling in high-dimensional settings, including neural network models. We show that, for succinct circuit representations of conditionals, the compatibility problem is intractable. In the case that all probabilities are non-zero, the problem is co-NP-complete. In the case that probabilities can be zero, we give examples to demonstrate that several notions of compatibility can be distinguished, and we prove that multiple versions of the problem are PSPACE-complete. Furthermore, we show that, assuming the polynomial hierarchy does not collapse, there exist compatible succinct conditionals whose joint cannot be expressed succinctly. Implications of these results for probabilistic modelling and machine learning are discussed.

We introduce probabilistic language tries (PLTs), a unified representation that makes explicit the prefix structure implicitly defined by any generative model over sequences. By assigning to each outgoing edge the conditional probability of the corresponding token or action, a PLT simultaneously serves as: (i) an optimal lossless compressor via frequency-weighted interval encoding, generalizing arithmetic coding to model-conditioned distributions; (ii) a policy representation for sequential decision problems including games, search, and robotic control; and (iii) a memoization index that lets repeated inference queries be answered by structured retrieval rather than full model execution. The central technical result is a prior-guided caching theorem: under a stationary generative distribution, a PLT-guided cache achieves strictly lower expected inference cost than any empirical-frequency cache for all query counts below a threshold that grows with the concentration of the prior. This converts O(n^2) transformer attention cost into an expected cost of p_r * O(log N) + (1 - p_r) * O(n^2), where p_r is the prior-estimated reuse probability and N is the artifact store size. We further introduce a hybrid compression architecture decomposing any dataset into a PLT-covered majority and a sparse residual store, connecting arithmetic coding with Kolmogorov-style program representations and rate-distortion theory. We instantiate the framework across chess, web search, robotics, organizational workflows, and LLM inference, demonstrating that compression, decision making, and computational reuse are all derived from a single probability measure on sequence space.