Do Large Language Models Capture the Diversity in their Training Data?

Large language models are trained to model conditional distributions over text, yet it remains inadequately understood whether they capture the full diversity of plausible outputs present in their training data. We study this question through an information-theoretic lens by comparing the conditional entropy of model-generated outputs with that of the corresponding training data. Given paired input-output samples, we use conditional entropy and its matrix-based analogue based on von Neumann entropy to measure output variability beyond what is explained by the conditioning input, without requiring multiple reference outputs for the same prompt. Across LLM families with publicly available training data, including OLMo, Pythia, and GPT-Neo, we consistently find that model-generated outputs exhibit lower conditional entropy than their training data, across different model scales, sequence lengths, and decoding strategies. We observe a similar conditional diversity gap beyond language modeling, including class-conditioned ImageNet generators and text-conditioned models trained on MS-COCO. To address this gap, we propose a post-hoc correction mechanism that generates multiple outputs for each input and reweights them through a matrix-entropy projection, increasing conditional diversity while remaining close to the original model distribution. We prove the concavity of the matrix-based conditional entropy functional, which makes the resulting entropy-constrained projection a convex optimization problem, and develop a scalable mirror-descent algorithm for its implementation. Our results reveal a systematic conditional diversity gap between modern generative models and their training data, and provide an information-theoretic framework for measuring and mitigating this gap.

Tracing distinguishability through transformer processing with stochastic LayerNorm

Representational similarity is foundational to analyses of deep networks, yet distances between point-valued representations are not intrinsically tied to downstream function: nearby states may produce different behaviors, while distant states may behave similarly. We instead give representations volume, turning similarity into statistical distinguishability. Overlapping stochastic representations necessarily induce overlapping downstream distributions, grounding latent comparison in model function and bringing it under information-theoretic tools such as the data-processing inequality. We realize this idea in pretrained transformers through a light-touch modification to LayerNorm: at each residual-stream read, we normalize the state, add isotropic Gaussian noise, and renormalize. During distillation fine-tuning, one learned allocation parameter per residual-stream read distributes a fixed global rate budget across the processing stack. The resulting model can be viewed as transformer blocks reading the residual stream with learned finite precision under a shared global rate budget. Using the Bhattacharyya coefficient, we trace which counterfactual distinctions are preserved through MLP blocks or selectively exposed to the query, key, and value computations of individual attention heads. Experiments on ViT-S and GPT-2 small reveal the depthwise propagation of continuous visual perturbations and head-specific sensitivity to token distinctions aligned with known attention motifs. These results establish distinguishability as a functionally grounded lens on transformer computation that complements existing interpretability approaches.

A Formal Limitation on Learning Human Language From Textual Corpora

Can a listener recover what a speaker means from the form of an utterance alone? We answer this question information-theoretically, and for a listener given by any featurizer of text, including the hidden states of contemporary large language models. Modeling language use as a joint distribution over meanings, contexts, and utterances, we derive upper bounds on the probability that a decoder recovers a speaker's intended meaning from a representation of the utterance. The bounds are governed by the uncertainty that form leaves about meaning, which splits into an irreducible part and a part that only (extralinguistic) context, but never the utterance alone, can resolve. Because these quantities are intrinsic to language, no representation, however much text or supervision produced it, can surpass them; the bounds hold whether the space of meanings is discrete or continuous. Experiments on artificial languages, Mandarin zero-pronoun resolution, and color reference provide empirical evidence in support of the theory.

Beyond Parallel Blindness: Information Floors and Model Gaps in Block Drafting

Block drafters propose several tokens in one forward pass, before earlier target tokens are realised. Their rejection mixes two losses: missing within-block path information and imperfect modelling of observable information. Accepted length cannot distinguish them. We separate the two with an information floor, the minimum expected rejection at a specified conditioning order; rejection above this floor is the model gap. Estimating both from target rollouts across four domains, four open-weight targets, and a frontier API target yields three findings. First, the all-parallel floor reaches $0.286$ at the final slot on Qwen3-4B, limiting even the best proposal to $71\%$ per-slot acceptance. Second, one realised token removes $86$--$100\%$ of this floor, a locality also recovered by an independent mutual-information analysis. Third, current drafters remain far above their floors: the final-slot model gap accounts for $43$--$64\%$ of DFlash rejection and $85$--$92\%$ of DSpark's oracle-conditioned rejection. These findings separate the value of short-range conditioning from proposal quality.

Handover of In-Context Learning State Across Session Boundaries

This study investigates the methodological and theoretical properties of session handover in applications that use large language models. A task may continue in a new session when the context reaches the model's input limit, when the application restarts, or when another agent is asked to finish the task. The application must then decide which information from the earlier session to pass on. We formulate handover as the transfer of a task-relative in-context learning (ICL) state and distinguish exact recovery of earlier material from preservation of the target distribution. Under an exogeneity condition, predictive equivalence characterizes the coarsest deterministic sufficient handover and gives a fixed-length bit requirement. The analysis isolates the effects of the memory constraint, the writer, and the continuation procedure, and quantifies the cost of writing before the realized downstream query is known. We propose a three-part record that stores decisions and constraints exactly, uses task-justified statistics for repeated evidence, and retains original observations whose effect is not preserved by those statistics. Gaussian linear regression gives an exact finite-dimensional handover and finite-bit perturbation bounds, while nonparametric regression gives upper and lower bounds that relate memory to squared prediction error. These results provide a theory and method for deciding what a handover must retain and how its memory requirement depends on the continuation task.

Reflection or Re-Generation? Why LLM Revision Fails Where Human Revision Succeeds

Reflection, the ability to revisit and revise prior reasoning, is central to how humans improve their answers. Large language models (LLMs) are increasingly prompted to "reflect," yet whether this resembles human revision remains unclear. We introduce the Human-LLM Reflection Framework (HRF), a controlled two-pass protocol comparing human and LLM revision under identical conditions across self-, peer-, and cross-agent settings. Using an information-theoretic analysis based on per-iteration cross-entropy reduction, we find two failure modes of LLM reflection. On objective tasks with finite answer spaces, reflection yields near-zero information gain (Delta I approx 0), behaving as neutral re-generation indistinguishable from re-sampling. On subjective tasks, it yields significant negative gain (Delta I < 0), moving predictions away from the target. Human revision, by contrast, yields positive gain in both settings. Cross-agent experiments localize the failure to the revision step, not input quality: LLMs degrade even high-quality human responses. Diagnostic analyses (revision conditioned on first-pass correctness, and oracle-guided revision against a random-reshuffle baseline) show that which sub-step dominates varies by task and by model rather than reducing to a single mechanism: self-error detection is present on objective multiple-choice tasks but weak on subjective ones, and recovery under an oracle error signal exceeds the baseline for some models and falls below it for others. The unifying account is structural: without external information, self-conditioned revision cannot reduce uncertainty about the target, so LLM reflection is better understood as conditioned re-generation than as genuine error-driven revision.

Neural Collapse Is Forbidden: Information Floors in Language Models

Within-class variance in language-model representations is commonly read as incomplete neural collapse. We argue it is allocated information storage, and that the allocation obeys a law. A one-line centering identity voids a family of simplex equiangular-tight-frame claims, including our own earlier ones; in dimensionless variance shares across 14 models, macro-category structure carries only 4-12% of representational variance and within-token context carries 79-91%, stable across a 100x parameter range. On the theory side, token-level weight decay penalizes a category in proportion to its type count, not its occurrence mass, reducing next-token prediction to an imbalanced K-class problem whose optimum orders category norms by type count. A converse floor, proved for binary categories, forces within-category dispersion to be at least proportional to the conditional mutual information I(token; context | category). The law holds: identity dispersion, not total variance, tracks this information across every tested model and partition, under a model-free estimate and even across models, where one model's information predicts another's dispersion; and over pretraining the category share overshoots, decays, and partially recovers, because the information it must carry never left.

Information Dynamics of Language Communication

Quantifying how meaning propagates through communicative exchanges remains underdeveloped in computational linguistics. Here we introduce an information-theoretic framework that quantifies the directed flow of semantic content between interlocutors and decomposes multi-source contributions into redundant, unique, and synergistic components. Our approach leverages large language models as probabilistic estimators of natural language to compute two measures: semantic transfer entropy (STE), which captures directed predictive influence between speakers, and semantic partial information decomposition (SPID), which resolves how multiple sources jointly shape a target's language. Across four experiments we show that the framework detects reduced information flow in cognitively rigid dialogue, captures the dominant role of persuaders in shaping discourse, distinguishes high- from low-quality psychotherapy by the directionality of therapist-client information exchange, and reveals synergistic premise contributions in argumentative essays. This framework opens new avenues for studying information dynamics in digital discourse, pedagogical interactions, clinical dialogues, and any domain in which the structure of linguistic exchange is of research relevance.

FlowEdit: Information-Theoretic Control of LLM Reasoning Flows for Ill-posed Problems Involving Conflicts

Large Language Models (LLMs) perform strongly on well-specified reasoning tasks with a feasible answer. However, problems encountered in the open world can become ill-posed due to inconsistent conditions, conflicting statements, or mutually incompatible requirements, admitting no valid responses. We argue that reasoning of such ill-posed problems involving conflicts require novel LLM capabilities to make hidden conflicts explicit, maintain competing hypotheses via multiple reasoning branches, and generate alternative responses in a single pass, all of which are challenging due to the limitation of the next-token prediction mechanism in LLMs. To this end, we propose FlowEdit, a novel framework that leverages information-theoretic principles to quantify and regulate internal reasoning flows of LLMs, for generating a full set of alternative responses under valid hypotheses. FlowEdit can be viewed as enforcing a branch-aware reasoning process using two dual information-theoretic objectives on the model's internal reasoning representations: maximizing the information flow from each selected hypothesis to the branch outcome, while minimizing the overlap and conditional dependence across sibling branches, to provide a diverse, informative set of responses with broad coverage. We show that this is achieved through tractable variational bounds under boundary embeddings being ε-sufficient, optimizing the underlying conditional mutual information in LLM reasoning process. Extensive experiments demonstrate that FlowEdit outperforms leading proprietary models, improving exact-set-match accuracy by 68%, while boosting overall response informativeness by 24%. We further show that flow regulation surfaces in the token stream as a redistribution of next-token entropy that concentrates inside each branch, amplifies at flow boundaries, and scales with the number of flows the problem requires.

What Makes Effective Supervision in Latent Chain-of-Thought: An Information-Theoretic Analysis

Latent Chain-of-Thought (CoT) internalizes reasoning within continuous hidden states, offering a promising alternative to verbose discrete reasoning traces. However, robust latent reasoning remains difficult because outcome supervision provides weak learning signals and leaves latent trajectories prone to semantic drift. In this work, we analyze Latent CoT from an information-theoretic perspective and identify this failure as a dual collapse: gradient attenuation along the optimization path and representational drift in the latent space. We further decompose process supervision into two complementary dimensions: Trajectory Supervision, which injects dense stepwise reasoning signals, and Space Supervision, which preserves the semantic structure of the latent manifold. Our analysis shows that rigid geometric compression can collapse the reasoning space, whereas generative reconstruction provides a more flexible semantic anchor that better preserves information capacity. To measure these effects, we introduce the Unified Latent Probe (ULP), which quantifies the mutual information between latent trajectories and explicit reasoning steps. Experiments reveal a clear Information-Performance Binding: reasoning accuracy depends on the information fidelity preserved in the latent chain. These findings provide a principled framework for latent reasoning supervision and suggest shifting from geometric imitation toward mutual information maximization. Our code is available at \href{https://github.com/EIT-NLP/Supervision-in-Latent-CoT}{this repository}.

Measuring Semantic Progress in Multi-turn Dialogue via Information Gain

Evaluating multi-turn dialogue is challenging because quality emerges across turns rather than within individual responses. We focus on a key dimension of information-seeking dialogue: semantic progress, defined as the accumulation of new, question-relevant, and non-redundant information over the course of a conversation. We formalize semantic progress as question-conditioned uncertainty reduction and introduce an information-theoretic metric that approximates it in embedding space. Our main estimator uses a tractable Gaussian formulation with closed-form updates, while a complementary maximum-entropy argument shows why log-determinant structure arises more broadly when only second-order embedding information is retained. This formulation yields desirable theoretical properties, including monotonicity, additive decomposition of total information gain across turns, and diminishing returns for redundant evidence. Unlike LLM-as-a-judge approaches, our metric requires no autoregressive inference at evaluation time and is fully reproducible for a fixed embedding model. Experiments on MT-Bench, Chatbot Arena, and UltraFeedback show that the proposed metric achieves competitive agreement with human judgments despite targeting only semantic progress, with improved alignment on MT-Bench and UltraFeedback compared to several LLM-based judges. Notably, the method remains effective with lightweight embedding models under CPU-only execution, indicating that semantic progress can be captured without reliance on large model capacity.

Calibrated Surprise: An Information-Theoretic Account of Creative Quality

The essence of good creative writing is calibrated surprise: when constraints from all relevant dimensions act together, the feasible solution space collapses into a narrow region, and the surviving choices look least predictable from an unconstrained view. "Calibrated" has a precise meaning: the author's intent, the reader's reasonable expectation, and the logic of reality converge. When these three independent judgements agree on every dimension, the set of admissible writing choices is forced into a very small region. A mathematical corollary follows: full-dimensional accuracy and mediocrity are mutually exclusive -- two sides of one constraint structure, not separate goals. We use Shannon's mutual information $I(X;Y) = H(X) - H(X|Y)$ as our analysis tool. "Calibrated" corresponds to conditional entropy going to zero; "surprise" to entropy going up; mutual information is the precise measure of the joint quantity. The argument rests on two pillars. Static: when constraints from ethos, mythos, lexis, and dianoia are imposed together, the admissible set collapses sharply, and surviving solutions show up as low-probability choices from an unconstrained view. Dynamic: the chain rule shows each writing choice is constrained by what came before and constrains what comes after; macro-level decisions naturally contribute a larger share of information, removing the need for hand-tuned weighting. Through case studies and lightweight LLM-logprob computations, we show the framework is both analytically useful and operational, laying the theoretical groundwork for Creative Quality Alignment (CQA) and a professional evaluation benchmark.