arXiv subjects

Sridhar Mahadevan

Publications and source records attributed to Sridhar Mahadevan.

At least 19 recordsLinked to original sources

arXiv2026

Learning in Infinitesimal Non-Compositional Sketches

This paper develops a categorical framework -- Learning in Infinitesimal Non-Compositional Sketches (LINCS) -- as the repair of non-compositionality: failures of diagrams to factor through quotient sketches lifted to the tangent category setting. Machine learning problems are specified as sketches: graphs with commutativity conditions $\mathcal D$, limit cones $\mathcal L$, and colimit cocones $\mathcal K$, generalizing the usual scalarization of loss functions or vector space assumptions. Non-compositionality is defined purely as failure of a universal factorization problem, not as arithmetic error between the desired and actual predictions. Given a learning sketch $\mathbb S=(S,\mathcal D,\mathcal L,\mathcal K)$, whose underlying graph is $S$, and a model $D:J \rightarrow C$, the base defect is the obstruction to factorization $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(D))$. The tangent lift applies the tangent functor $T$ to obtain $TD:J \rightarrow C$, and LINCS is defined as the obstruction $\mbox{Obs}(\mbox{Fact}_{\mathbb S}(TD))$ -- asking whether infinitesimal perturbations preserve the compositionality constraints.The paper also introduces Tangent Learning Sketches, which are sketches equipped with Cockett-Cruttwell tangent structure. The paper defines the INC endofunctor, which iterates the tangent lift, producing a tower $D,TD,T^2D, \cdots$ of factorization problems. ML is thereby formulated as the search for a coalgebraic fixed point where successive tangent unfoldings stabilize ($\nu T_{\mbox{INC}}$). Using the Aczel--Mendler theorem, we prove existence of a final INC coalgebra whenever $T_{\mbox{INC}}$ admits a set-based class realization that creates its final carrier. A detailed experimental evaluation of LINCS is underway in a number of concrete ML settings, including deep learning, large language models, and reinforcement learning, and is described in companion papers.

cs.LG↗

arXiv2026

Agentic Skill Optimization over Lie Algebroids

Agentic systems increasingly improve themselves by editing skills: prompts, rubrics, plans, tool contracts, examples, validators, and traces. Skill edits are not independent coordinates in a vector space: they are local repairs to structured artifacts whose effects are observed only after rollout, validation, and critique. Distinct edits can have the same immediate visible effect while differing in routing context, template state, guardrail scope, or future composability. The order of edits can matter as well: repairing a schema before a normalization rule need not be equivalent to applying the same edits in the reverse order. This paper introduces a new framework for skill optimization called LASKO, for Lie Algebroid SKill Optimization. LASKO models typed, anchored Markdown skills as the base category and available edit policies as sections of a controlled Lie algebroid with anchor $\rho$. The anchor maps an edit policy to its visible Markdown effect; the kernel $\ker(\rho)$ represents latent template, routing, or implementation structure; and the algebroid bracket measures noncommuting edit composition. As shown in the paper, LASKO achieves order-of-magnitude speedups in skill optimization in our preliminary benchmark results, primarily because it substitutes inexpensive Lie-bracket screening tests that run in microseconds, before investing in expensive validations that require running large language models. On a causal extraction from natural language task, LASKO achieved a speedup of almost $15 \times$ compared to a brute-force approach that validated all edits by running them through a DeepSeek V3.1 4-bit model with 671B parameters.

cs.LG↗

arXiv2026

Odyssey: Constructing Verifiable Local Truth-Preserving Foundation Models

We introduce a categorical framework called ODYSSEY for constructing verifiable, local truth-preserving foundation models as compositions of foundries: building-block architectural components that specify a cover of local contexts, local representation families, restriction maps, gluing rules, obstruction policies, update obligations, and human-facing views. A foundry is an organized sheaf of knowledge that carries within it an argumentation component. Concrete foundries are built from generic foundries such as evidence/argument, operational decision, institutional/financial, market meaning, scientific challenge, research-program, assistant-build, and evaluation-harness foundries. Universal Foundry Learning (UFL) formalizes foundry construction as a composition of left and right Kan extensions, with left Kan extension rolling local artifacts into candidate foundries and right Kan extension enforcing the restriction, gluing, obstruction, and argumentation conditions required for promotion. Foundry SQL (FSQL) is a small typed query surface for slicing maintained foundry artifacts that uses TICKET (Topos Integration using Causal Kan Extension Transformers) certification for admitting external or pre-built models into durable ODYSSEY state. ODYSSEY is fully implemented and tested across a wide spectrum of concrete foundries, showing that the same categorical machinery supports domain construction, artifact replay, sheaf diagnostics, grounded Toulmin/local-LLM scrutiny, residual-obstruction ledgers, and optimized TICKET-compatible causal-claim extraction across heterogeneous sources. This paper is to be presented as a 2.5 hour tutorial at ICML 2026. The tutorial home page is at https://bit.ly/4ajS0nA.

cs.AI↗

arXiv2026

Infinitesimal Causality

Interventions can be varied continuously in many causal models. Differentiating a specified smooth intervention protocol produces vector fields on a statistical model, and their Lie brackets describe the noncommutativity of the corresponding local perturbations. We formulate this differential geometry of interventions on smooth statistical models and call the resulting framework infinitesimal causality (IC). Given a constant-rank distribution spanned by visible intervention fields, we define the normal Lie-bracket residual and show that its vanishing is exactly the involutivity condition in the classical Frobenius theorem. We establish the coordinate invariance of the zero-residual property and characterize its dependence on the intervention protocol, visible span, and metric. Fully observed and latent-variable examples delineate the additional structural assumptions needed to interpret bracket residuals causally. We also distinguish tangent vectors on a statistical parameter manifold from derivatives of stochastic kernels. In the finite-state linearization of a Markov category, normalization and copy compatibility yield well-typed first-order defects. Normalization is automatic for differentiable paths of stochastic kernels, whereas copy compatibility characterizes a more restrictive deterministic or comonoid-preserving perturbation. Together, the geometric and kernel-level constructions make IC a precise foundation for Lie-bracket-based causal diagnostics and identify the assumptions required to pass from local intervention geometry to causal conclusions.

math.CT↗

arXiv2026

Latent Confounded Causal Discovery via Lie Bracket Geometry

We study causal discovery from observational and interventional regimes when latent variables may affect the measured system. Our first algorithm, BRIDGE (Bracket Residuals for Interventional Discovery and Geometric Estimation), combines a density-ratio or transport engine with a high-recall geometric screen and passes the retained arrows to a score-based or differentiable discovery method. The main formulation and experiments use known single-node intervention targets; in that regime the screen is designed to retain candidate directed effects, while a downstream learner determines the final graph or equivalence-class representation. Our second algorithm, Spectral Kernel Flow Matching (SKFM), amortizes the response fields, summarizes residual nonclosure by a spectral visible-footprint subspace, and applies an order-dependent graph extractor. Direct extraction succeeds on calibrated chains and selected motifs, but is unstable on harder random DAGs when the order must be learned. On ten-node nonlinear random DAGs, the more reliable hybrid role of the geometry is as a candidate generator: calibrated SKFM/Bridge fields followed by local BIC scoring achieve mean directed $F_1\simeq0.86$. Sachs protein signaling provides a real-data stress test and supports a diagnostic, not fully identified, interpretation. The contribution is therefore a practical interventional screening pipeline, explicit guarantees for screen retention and residual-footprint rank under stated assumptions, and a falsifiable account of the boundary between geometric diagnostics and causal identification.

cs.LG↗

arXiv2026

Causal Density Functions

We study the full density ratio between a specified intervention regime $P_a$ and an observational regime $P_0$, $\rho_a=dP_a/dP_0$, under the prerequisite $P_a\ll P_0$. We call the regime-indexed ratio a causal density function when $P_a$ is an identified or directly observed interventional law. The underlying Radon-Nikodym derivative and the identity $ \mathbb{E}_{a}[f(Z)] = \mathbb{E}_{0}\!\left[f(Z)\rho_a(Z)\right] $ are classical importance weighting, not new identification results. Our narrower question is whether retaining the entire pointwise ratio is useful as a reusable diagnostic across several downstream functionals. We evaluate a two-density plug-in baseline through held-out moment transport and overlap stress tests on synthetic and perturbation data. We also report a pairwise graph-scoring heuristic as a negative result: its F1 is $0.10$ on a synthetic DAG, $0.12$ on Sachs, and $0.33$ on a multi-regime chain. These experiments do not establish an estimation advantage over direct density-ratio, inverse-probability, Riesz, or doubly robust methods; they instead delimit what the pointwise ratio target and the present plug-in estimator do and do not provide.

stat.ME↗

arXiv2026

Universal Decision Learners

Many theories of decision making -- planning, reinforcement learning, causal intervention, online learning, and game-theoretic equilibrium -- turn local information into globally coherent behavior. This paper proposes a common categorical formulation: a Universal Decision Learner (UDL) extends a partially specified decision functor from observed contexts to new contexts by a pair of universal constructions. Left Kan extensions express rollout, aggregation, and candidate generation; right Kan extensions express consistency, constraint satisfaction, and fixed-point semantics. The central claim is not that every decision problem has the same algorithm, but that many decision formalisms instantiate the same universal problem: extend local behavioral data canonically, then characterize the globally coherent extensions. We give the abstract UDL construction, prove its universal comparison property, define Kan-invariant behavioral equivalence and minimal abstractions, and show how Bellman equations, planning recursions, causal interventions, online regret, and equilibria arise as special cases. The supplementary material develops the reinforcement-learning specialization in more detail.

cs.LG↗

arXiv2026

Kan Extension Transformers: A Categorical Unification of Attention, Diffusion, and Predict-Detach Self-Conditioning

We propose Kan Extension Transformers (KETs) as a categorical design language for a diverse group of Transformer implementations. A layer can be viewed generally as a weighted structured extension operator: attention uses token neighborhoods, geometric mixing uses sparse incidences, and KET uses simplicial sources. This operator is an actual enriched left Kan extension only when the source values are functorial, the weights are representable hom-objects (or the specified profunctor action), and aggregation realizes the corresponding coend; otherwise ``Kan-style'' denotes an interpretation rather than an identity theorem. Predict-detach blocks gradients through a predictive carrier and avoids transporting teacher-forced hidden states, but detach alone does not make a noncausal update strictly autoregressive: every carrier consumed at target $t$ must also be measurable from the prefix available at $t$. We evaluate 12 implementations on Penn Treebank, WikiText-2, and WikiText-103 across strict-causal and self-conditioned regimes, using widths $d=64,256$ and depths $L=2,8,16$ across the reported studies. Quadratic KET is strongest among the compared strict-causal architectures on WikiText-2 and WikiText-103; the largest cross-regime gains arise from additional self-conditioning information, not neighborhood design alone.

cs.LG↗

arXiv2026

PROMETHEUS: Automating Deep Causal Research Integrating Text, Data and Models

Large language models can extract local causal claims from text, but those claims become more useful when organized as persistent, navigable world models rather than as flat summaries. We introduce PROMETHEUS, a framework that turns retrieved literature, filings, reviews, reports, agent traces, source data, code, simulations, and scientific models into causal atlases: sheaf-like families of local causal predictive-state models over an explicit cover of a research substrate. Each local region contains causal episodes, structured claim tables, predictive tests, support statistics, and provenance; restriction maps compare overlapping regions; gluing diagnostics expose agreement, drift, contradiction, and underdetermination. The resulting Topos World Model is not a single universal graph. It is a research instrument for navigating what a corpus says, where it says it, how strongly it is supported, and where local claims fail to assemble into a coherent global view. Three literature-atlas case studies -- ocean-temperature impacts on marine populations, GLP-1 weight-loss evidence, and resveratrol/red-wine health-benefit claims -- illustrate deep causal research from text with explicit locality, evidence, persistent state, and gluing tension. Four grounded-counterfactual case studies -- a Nature Climate Change microplastics forcing paper, an Indus Valley hydrology paper with VIC-derived figure data and model code, the canonical Sachs protein-signaling study with single-cell perturbation data, and a Nature singing-mouse study with MAPseq projection matrices -- show a stronger mode: when a paper ships source data, simulation outputs, or code, PROMETHEUS can evaluate a counterfactual against that scientific substrate and then rebuild the sheaf world model around the

cs.AI↗

arXiv2026

CSQL: Mapping Documents into Causal Databases

We describe a novel system, CSQL, which automatically converts a collection of unstructured text documents into an SQL-queryable causal database (CDB). A CDB differs from a traditional DB: it is designed to answer "why'' questions via causal interventions and structured causal queries. CSQL builds on our earlier system, DEMOCRITUS, which converts documents into thousands of local causal models derived from causal discourse. Unlike RAG-based systems or knowledge-graph based approaches, CSQL supports causal analysis over document collections rather than purely associative retrieval. For example, given an article on the origins of human bipedal walking, CSQL enables queries such as: "What are the strongest causal influences on bipedalism?'' or "Which variables act as causal hubs with the largest downstream influence?'' Beyond single-document case studies, we show that CSQL can also ingest RAG/IE-compiled causal corpora at scale by compiling the Testing Causal Claims (TCC) dataset of economics papers into a causal database containing 265,656 claim instances spanning 45,319 papers, 44 years, and 1,575 reported method strings, thereby enabling corpus-level causal queries and longitudinal analyses in CSQL. Viewed abstractly, CSQL functions as a compiler from unstructured documents into a causal database equipped with a principled algebra of queries, and can be applied broadly across many domains ranging from business, humanities, and science.

cs.DB↗

arXiv2026

Categorical Belief Propagation: Sheaf-Theoretic Inference via Descent and Holonomy

We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category $\Syn_\Sigma$ on a typed signature and prove its universal property, yielding compositional semantics via a unique functor to the matrix category $\cat{Mat}_R$. Message-passing is formulated using a Grothendieck fibration $\int\Msg \to \cat{FG}_\Sigma$ over polarized factor graphs, with schedule-indexed endomorphisms defining BP updates. We characterize exact inference as effective descent: local beliefs form a descent datum when compatibility conditions hold on overlaps. This framework unifies tree exactness, junction tree algorithms, and loopy BP failures under sheaf-theoretic obstructions. We introduce HATCC (Holonomy-Aware Tree Compilation), an algorithm that detects descent obstructions via holonomy computation on the factor nerve, compiles non-trivial holonomy into mode variables, and reduces to tree BP on an augmented graph. Complexity is $O(n^2 d_{\max} + c \cdot k_{\max} \cdot \delta_{\max}^3 + n \cdot \delta_{\max}^2)$ for $n$ factors and $c$ fundamental cycles. Experimental results demonstrate exact inference with significant speedup over junction trees on grid MRFs and random graphs, along with UNSAT detection on satisfiability instances.

cs.AI↗

arXiv2025

Large Causal Models from Large Language Models

We introduce a new paradigm for building large causal models (LCMs) that exploits the enormous potential latent in today's large language models (LLMs). We describe our ongoing experiments with an implemented system called DEMOCRITUS (Decentralized Extraction of Manifold Ontologies of Causal Relations Integrating Topos Universal Slices) aimed at building, organizing, and visualizing LCMs that span disparate domains extracted from carefully targeted textual queries to LLMs. DEMOCRITUS is methodologically distinct from traditional narrow domain and hypothesis centered causal inference that builds causal models from experiments that produce numerical data. A high-quality LLM is used to propose topics, generate causal questions, and extract plausible causal statements from a diverse range of domains. The technical challenge is then to take these isolated, fragmented, potentially ambiguous and possibly conflicting causal claims, and weave them into a coherent whole, converting them into relational causal triples and embedding them into a LCM. Addressing this technical challenge required inventing new categorical machine learning methods, which we can only briefly summarize in this paper, as it is focused more on the systems side of building DEMOCRITUS. We describe the implementation pipeline for DEMOCRITUS comprising of six modules, examine its computational cost profile to determine where the current bottlenecks in scaling the system to larger models. We describe the results of using DEMOCRITUS over a wide range of domains, spanning archaeology, biology, climate change, economics, medicine and technology. We discuss the limitations of the current DEMOCRITUS system, and outline directions for extending its capabilities.

cs.AI↗

arXiv2025

Decentralized Causal Discovery using Judo Calculus

We describe a theory and implementation of an intuitionistic decentralized framework for causal discovery using judo calculus, which is formally defined as j-stable causal inference using j-do-calculus in a topos of sheaves. In real-world applications -- from biology to medicine and social science -- causal effects depend on regime (age, country, dose, genotype, or lab protocol). Our proposed judo calculus formalizes this context dependence formally as local truth: a causal claim is proven true on a cover of regimes, not everywhere at once. The Lawvere-Tierney modal operator j chooses which regimes are relevant; j-stability means the claim holds constructively and consistently across that family. We describe an algorithmic and implementation framework for judo calculus, combining it with standard score-based, constraint-based, and gradient-based causal discovery methods. We describe experimental results on a range of domains, from synthetic to real-world datasets from biology and economics. Our experimental results show the computational efficiency gained by the decentralized nature of sheaf-theoretic causal discovery, as well as improved performance over classical causal discovery methods.

cs.AI↗

arXiv2025

Intuitionistic $j$-Do-Calculus in Topos Causal Models

In this paper, we generalize Pearl's do-calculus to an Intuitionistic setting called $j$-stable causal inference inside a topos of sheaves. Our framework is an elaboration of the recently proposed framework of Topos Causal Models (TCMs), where causal interventions are defined as subobjects. We generalize the original setting of TCM using the Lawvere-Tierney topology on a topos, defined by a modal operator $j$ on the subobject classifier $\Omega$. We introduce $j$-do-calculus, where we replace global truth with local truth defined by Kripke-Joyal semantics, and formalize causal reasoning as structure-preserving morphisms that are stable along $j$-covers. $j$-do-calculus is a sound rule system whose premises and conclusions are formulas of the internal Intuitionistic logic of the causal topos. We define $j$-stability for conditional independences and interventional claims as local truth in the internal logic of the causal topos. We give three inference rules that mirror Pearl's insertion/deletion and action/observation exchange, and we prove soundness in the Kripke-Joyal semantics. A companion paper in preparation will describe how to estimate the required entities from data and instantiate $j$-do with standard discovery procedures (e.g., score-based and constraint-based methods), and will include experimental results on how to (i) form data-driven $j$-covers (via regime/section constructions), (ii) compute chartwise conditional independences after graph surgeries, and (iii) glue them to certify the premises of the $j$-do rules in practice

cs.LO↗

arXiv2025

Consciousness as a Functor

We propose a novel theory of consciousness as a functor (CF) that receives and transmits contents from unconscious memory into conscious memory. Our CF framework can be seen as a categorial formulation of the Global Workspace Theory proposed by Baars. CF models the ensemble of unconscious processes as a topos category of coalgebras. The internal language of thought in CF is defined as a Multi-modal Universal Mitchell-Benabou Language Embedding (MUMBLE). We model the transmission of information from conscious short-term working memory to long-term unconscious memory using our recently proposed Universal Reinforcement Learning (URL) framework. To model the transmission of information from unconscious long-term memory into resource-constrained short-term memory, we propose a network economic model.

cs.AI↗

arXiv2025

Universal Reinforcement Learning in Coalgebras: Asynchronous Stochastic Computation via Conduction

In this paper, we introduce a categorial generalization of RL, termed universal reinforcement learning (URL), building on powerful mathematical abstractions from the study of coinduction on non-well-founded sets and universal coalgebras, topos theory, and categorial models of asynchronous parallel distributed computation. In the first half of the paper, we review the basic RL framework, illustrate the use of categories and functors in RL, showing how they lead to interesting insights. In particular, we also introduce a standard model of asynchronous distributed minimization proposed by Bertsekas and Tsitsiklis, and describe the relationship between metric coinduction and their proof of the Asynchronous Convergence Theorem. The space of algorithms for MDPs or PSRs can be modeled as a functor category, where the co-domain category forms a topos, which admits all (co)limits, possesses a subobject classifier, and has exponential objects. In the second half of the paper, we move on to universal coalgebras. Dynamical system models, such as Markov decision processes (MDPs), partially observed MDPs (POMDPs), a predictive state representation (PSRs), and linear dynamical systems (LDSs) are all special types of coalgebras. We describe a broad family of universal coalgebras, extending the dynamic system models studied previously in RL. The core problem in finding fixed points in RL to determine the exact or approximate (action) value function is generalized in URL to determining the final coalgebra asynchronously in a parallel distributed manner.

cs.LG↗

arXiv2025

A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models

Natural language is replete with superficially different statements, such as ``Charles Darwin wrote" and ``Charles Darwin is the author of", which carry the same meaning. Large language models (LLMs) should generate the same next-token probabilities in such cases, but usually do not. Empirical workarounds have been explored, such as using k-NN estimates of sentence similarity to produce smoothed estimates. In this paper, we tackle this problem more abstractly, introducing a categorical homotopy framework for LLMs. We introduce an LLM Markov category to represent probability distributions in language generated by an LLM, where the probability of a sentence, such as ``Charles Darwin wrote" is defined by an arrow in a Markov category. However, this approach runs into difficulties as language is full of equivalent rephrases, and each generates a non-isomorphic arrow in the LLM Markov category. To address this fundamental problem, we use categorical homotopy techniques to capture ``weak equivalences" in an LLM Markov category. We present a detailed overview of application of categorical homotopy to LLMs, from higher algebraic K-theory to model categories, building on powerful theoretical results developed over the past half a century.

cs.CL↗

arXiv2025

Topos Theory for Generative AI and LLMs

We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical results on the Transformer model have shown that it is a universal sequence-to-sequence function approximator, and dense in the space of all continuous functions with compact support on the Euclidean space of embeddings of tokens. Building on this theoretical result, we explore novel architectures for LLMs that exploit the property that the category of LLMs, viewed as functions, forms a topos. Previous studies of large language models (LLMs) have focused on daisy-chained linear architectures or mixture-of-experts. In this paper, we use universal constructions in category theory to construct novel LLM architectures based on new types of compositional structures. In particular, these new compositional structures are derived from universal properties of LLM categories, and include pullback, pushout, (co) equalizers, exponential objects, and subobject classifiers. We theoretically validate these new compositional structures by showing that the category of LLMs is (co)complete, meaning that all diagrams have solutions in the form of (co)limits. Building on this completeness result, we then show that the category of LLMs forms a topos, a ``set-like" category, which requires showing the existence of exponential objects as well as subobject classifiers. We use a functorial characterization of backpropagation to define a potential implementation of an LLM topos architecture.

cs.AI↗