Motivated by parallel decoding in masked diffusion models, we study adaptive parallel sampling of discrete vectors: in each round, a deterministic policy selects unrevealed coordinates on the basis of the values observed so far, and the selected coordinates are sampled independently from their exact conditional marginals. Approximation error is measured by forward Kullback-Leibler divergence, and serial depth is the minimum target-averaged number of rounds meeting a prescribed error budget. Our central result is an exact identity: the divergence of every policy equals the expected conditional total correlation accumulated over its reveal rounds, so conditional total correlation is the exact information cost of within-round parallelism. The identity yields zero-error schedules for finite-order Markov chains with round complexity proportional to the Markov order and logarithmic in sequence length, a matching logarithmic characterization of the Bernoulli walk at every fixed error budget, and a linear-versus-logarithmic separation between left-to-right and hierarchical reveal orders. Uniform random permutations require linearly many expected rounds at every fixed budget; their hard-cap round-error tradeoff is an exact integer-composition problem whose fixed-round asymptotics and joint-scaling frontier we determine. Uniform balanced binary strings have depth of order squared logarithm, and binary one-hot blocks have square-root depth, with rectangular versions realizing every polynomial exponent up to one half. These results separate serial depth from entropy and negative log-likelihood, and establish conditional-dependence structure as a fundamental determinant of parallelizability. Experiments with a masked diffusion language model show that the pseudo-cost distinguishes deployed decoding rules and that its policy rankings agree closely with the quality of self-sampled outputs.
This article presents the abridged core of \emph{A Mathematical Theory of Interpretation} (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes.
Many algorithms spend an internal resource before returning a decision and are evaluated only by the quality of that terminal output. We formalize such procedures as terminal computation-allocation problems: costly computations produce observations, update beliefs about a latent environment, and matter only through terminal decision loss. Bellman equations characterize optimal allocation under fixed budgets, priced computation, and exact certification. We then relate value of computation (VOC) to information. Mutual information equals myopic VOC under log loss, whereas under simple regret VOC is a knowledge-gradient quantity; moreover, information gain can rank computations arbitrarily poorly, although it gives a one-sided upper bound on VOC. Bandit pulls, tree simulations, and node expansions illustrate the same model under different computation topologies. Finally, under an explicit frontier-resolution and heuristic-error model, maximizing approximate VOC recovers weighted A*, with A* and greedy best-first search as limiting cases. The theory identifies a shared decision problem without asserting that one acquisition rule is universally optimal.
Photios A. Stavrou, Giuseppe Serra, Marios Kountouriscs.IT cs.LG eess.SY
Classical rate-distortion (RD) theory has long established the fundamental limits of lossy compression by quantifying the minimum number of bits required to represent a source under a prescribed distortion constraint. However, widely used distortion measures such as mean-squared error often fail to capture perceptual quality or semantic validity, which are increasingly central in modern learning-driven applications. Rate-distortion-perception (RDP) theory extends the RD framework by introducing perception as a third fundamental axis, quantified via distributional similarity between the source and reconstructed signals, leading to the rate-distortion-perception function (RDPF). This tutorial provides a structured overview of the coding principles underlying perception-aware lossy compression and surveys recent achievability results under different randomness assumptions. It then presents a unifying optimization viewpoint for computing the RDPF as defined by Blau and Michaeli, for both discrete and continuous sources under broad families of perceptual constraints, including f-divergences, alpha-divergences, and Wasserstein-based metrics. Special attention is given to computational tools such as alternating minimization schemes, Newton-based methods, and convex optimization formulations, as well as to analytically tractable cases such as Gaussian sources and the perfect-realism regime. Unlike recent broad surveys that emphasize generative architectures and AI-empowered communication systems, this tutorial focuses on the coding-theoretic and computational machinery needed to characterize, compute, and interpret the RDP limits. Finally, the tutorial outlines promising research directions at the intersection of information theory, neural compression, robust source coding, and perception-aware networked control systems.
Wenhui Chen, Jianlin Chen, Ziyao Lin +1cs.AI cs.IT
The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.
Henry Hunt, Mason Kamb, Surya Gangulics.LG cond-mat.dis-nn
How diffusion models circumvent the curse of dimensionality to learn complex distributions over high dimensional spaces from a finite training set, instead of memorizing it, remains a fundamental mystery. To address this, we introduce analytically tractable Bayesian information restricted diffusion (BIRD) models, in which each pixel observes restricted information about noisy data. A BIRD model time-reverses diffusion by inferring which past training sample produced its current restricted observation using the Bayesian posterior. This model class generalizes existing analytical diffusion models that use spatially local information restriction. We show that spatially local BIRD models closely approximate trained diffusion models \textit{early in training}, across different architectures such as UNets and DiTs. Under minimal assumptions on the data distribution, we identify an information-theoretic phase boundary between memorization and generalization in the joint space of amount of training data, time in the reverse generative process, and amount of information restriction: a BIRD model memorizes when the mutual information between its restricted noisy observations and the training data exceeds the log number of training points, and it generalizes otherwise. Experiments across a range of datasets confirm our theoretically predicted location for the transition. We find that generation proceeds near the edge of memorization: both spatially local BIRD models and early-training diffusion models track the memorization-generalization phase boundary by increasingly restricting information over time. Overall, our results reveal a fundamental role for information restriction in generative AI to circumvent the curse of dimensionality.
Srinivasa Rao P., Vangmayi P Reddycs.LG cs.AI cs.CG
Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary, balanced by the von Neumann entropy of the network's weight space. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.
We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $π$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm{Stab}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G - 1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2, \mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.
Thomas Dittrich, Oliver Potocki, Philipp Grohscs.LG cs.IT math.FA
Modern deep learning architectures are increasingly multi-task and multi-modal, using a pretrained foundation model combined with task-specific, fine-tuned models. Empirically, exploiting similarity across different problems, instead of solving them individually, can significantly improve overall performance. While the generalization and sample complexity properties of multitask learning have been widely studied, the parametric complexity of joint approximation in comparison to separate approximation remains less well understood. The question is particularly relevant in modern deep learning, where models are increasingly required to satisfy structural constraints such as equivariance, conservation laws, or orthogonality. We prove lower and upper bounds on the description-length for separate and joint approximation classes, respectively, in uniform norm. We build a class of orthogonal functions by composing a shared hard feature, realized by a Rademacher-Haar wavelet series, with Sawtooth-Walsh readouts to enforce orthogonality of output coordinates. The dyadic tree structure of the Rademacher-Haar wavelet concentrates the approximation hardness in the common feature component, while the readouts act as task-specific heads. Using an information-theoretic framework, we obtain a sharp gap between the optimal approximation rates achievable by joint and separate coding. Finally, we realize this separation in a neural network model using Heaviside activations via reduction to triangle-wave approximation. Our results show that even under an orthogonality constraint joint approximation requires strictly fewer bits in compositional architectures, provided the tasks share a latent hard feature. This provides theoretical insight into the description-length-efficiency of compositional multi-output architectures and clarifies how neural networks can retain expressivity under geometric constraints.
In the current era of deep learning and especially generative models, there is significant investment in training very large deep neural networks. Thus far, such models have been "black boxes" that are difficult to understand in the sense that they have opaque internal mechanisms, leading to difficulties in interpretability, reliability, and control. Naturally, this lack of understanding has led to both hype and fear. This book is an attempt to "open the black box" and understand the mechanisms of large deep networks, through the perspective of representation learning, which is a major factor - arguably the single most important one - in the empirical power of deep learning models. A brief outline of this book is as follows. Chapter 1 will summarize the threads that underlie the whole text. Chapters 2, 3, 4, 5, and 6 will explain the design principles of modern neural network architectures through optimization and information theory, reducing the process of architecture development (long having been described as a sort of "alchemy") to undergraduate-level linear algebra and calculus exercises once the underlying principles are introduced. Chapters 7 and 8 will discuss applications of these principles to solve problems in more paradigmatic ways, obtaining new methods and models which are efficient, interpretable, and controllable by design, and yet no less - sometimes even more - powerful than the black-box models they resemble. Chapter 9 will discuss potential future directions for deep learning, the role of representation learning, as well as some open problems.
We identify and prove a fundamental trade-off governing long-sequence models: no model can simultaneously achieve (i) per-step computation independent of sequence length (Efficiency), (ii) state size independent of sequence length (Compactness), and (iii) the ability to recall a number of historical facts proportional to sequence length (Recall). We formalize this trade-off within an Online Sequence Processor abstraction that unifies Transformers, state space models, linear recurrent networks, and their hybrids. Using the Data Processing Inequality and Fano's Inequality, we prove that any model satisfying Efficiency and Compactness can recall at most O(poly(d)/log V) key-value pairs from a sequence of arbitrary length, where d is the model dimension and V is the vocabulary size. We classify 52 architectures published before March 2026 into the triangle, showing that each achieves at most two of the three properties and that hybrid architectures trace continuous trajectories in the interior. Experiments on synthetic associative recall tasks with five representative architectures validate the theoretical bound: empirical recall capacity lies strictly below the information-theoretic limit, and no architecture escapes the triangle.