While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data--model cross-entropy (CE) can be characterized exactly as an integral of local information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete and continuous diffusion, with the Gaussian case reducing to the well-known mutual information--minimum mean-square error (I-MMSE) relationship. An immediate implication is a locality property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including text8 and CIFAR-10.
Score-based diffusion models have achieved remarkable empirical success in generative modeling, yet their approximation-theoretic foundations remain incomplete. In particular, although classical universal approximation theorems guarantee that neural networks can approximate score…
What does a discrete diffusion model learn: a denoiser, a score ratio, or a bridge plug-in predictor? At the level of jump rates, these are one object in different coordinates, and reading a neural network in the wrong coordinate changes the process being trained and sampled. Sta…
In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width para…
Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first…
We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this pro…
Cooperative multi-task semantic communication (CMT-SemCom) improves task execution performance by leveraging shared representations. However, as we demonstrated in [1], cooperative multi-tasking can be either constructive or destructive, depending on the semantic relationships am…