Elimination geometry is developed, a typed, native-loss, audit-oriented framework for studying when locally optimal objects can be realized by a shared deployment rule and derives native defects from the original objective.
Abstract
This monograph develops elimination geometry (EG), a typed, native-loss, audit-oriented framework for studying when locally optimal objects can be realized by a shared deployment rule. Elimination and compression may erase distinctions required by prediction, inference, control, or representation. EG asks which distinctions are lost, whether the induced defect is visible to the declared task, and whether changing information, architecture, action space, or deployment domain can repair it. EG separates local solvability, global realizability, and finite-sample certifiability. It derives native defects from the original objective and distinguishes architecture obstruction from model approximation, generalization, and implementation error. The monograph synthesizes tools from geometry, optimization, information theory, statistics, and machine learning into interfaces for integrability, representation admissibility, resource constraints, observational overlap, and common deployment. Formal results address regular, coordination, singular, compositional, and resource-limited mechanisms with explicit antecedents and claim boundaries. Applications include sparse model selection, distribution-free prediction, observational treatment policies, routed expert and retrieval systems, and learned score fields. Obstruction-Aware Learning and Inference links structural diagnosis to finite-data authorization, mechanism-matched intervention, and independent validation. Reproducible synthetic and real-data studies illustrate how certificates can guide architecture repair while recording failed gates and unresolved cases. The framework requires the deployment contract, native endpoint, competing explanations, information and compute budgets, and validation rule to be fixed before a persistent performance floor is attributed to architecture.
Progress in mathematics often requires more than a certificate of truth: it requires proof structures that are transparent, checkable, and reusable. Automated systems can increasingly certify that a result is true; what they typically return, however, is a dense certificate rather than an interpretable, reusable proof structure. Recent work on performance estimation problems has shown that performance bounds and complexity analyses of first-order optimization methods can be discovered by searching over a structured space of Lagrangian dual certificates. We cast the search for simpler proof structures as a second-stage optimization problem over these certificates. Starting from dual certificates, we develop post-processing procedures using tools from sparse optimization and statistical learning. We measure complexity through features such as active hypotheses and residual structure, and introduce methods based on exhaustive sparsification, weighted $\ell_1$-type heuristics, and semidefinite programming (SDP) formulations for discovering simple proofs and intermediate lemmas. Examples on gradient descent, proximal methods, and fast-gradient methods show that these procedures can autonomously prune redundant inequalities, reveal structured proof patterns, and, in the proximal setting, recover Lyapunov functions as intermediate lemmas that lead to simple, streamlined proofs. By distilling dense machine-generated certificates into compact proof structures, this workflow acts as a pre-processing step for the final proof, reducing the complexity that must be managed during human interpretation, reuse, and formalization.
Daniel Berg Thomsen, Manu Upadhyaya, Baptiste Goujaud et al.· 1 citation
Influence functions provide a principled framework for tracing model predictions back to training data, yet existing methods remain impractical for large language models due to prohibitive storage and I/O costs. Prior approaches face two critical barriers: the Fidelity Gap from coarse curvature approximations, and the I/O bottleneck from materializing per-example gradients, which renders million-scale datasets infeasible. We propose StructInf, a structure-aware influence estimation framework that addresses both gaps through system-algorithm co-design. For fidelity, we first replace the indefinite Hessian with an adaptive block-diagonal damped empirical Fisher surrogate, computed via quadratically convergent Newton--Schulz iterations. For efficiency, we then introduce a streaming algorithm that eliminates per-example gradient storage entirely, reducing space complexity from O(N • d) to O(d) and enabling processing of million-scale training instances on consumer hardware, where N denotes the number of training instances and d the parameter size. We further identify and resolve Split-View Bias, a distributed pitfall where naive averaging of local curvatures fails, via a distributed gradient calibration strategy that synchronizes global statistics with minimal communication. Finally, we conduct comprehensive experiments to evaluate StructInf against other baselines. Unlike prior methods requiring hours of pre-computation or TB-scale storage, StructInf enables real-time data selection during training. Across GLUE and instruction-tuning benchmarks, StructInf achieves up to +26% AUC on MRPC and average 7× speedup for 7B-scale models, making high-fidelity influence analysis practical on a single RTX 4090. These results suggest that practical IF for LoRA-tuned LLMs benefits from joint optimization of curvature modeling, memory management, and distributed coordination.
Mengyi Yan, Yaoshu Wang, Guangyi Zhang et al.· Proceedings of the 32nd ACM...· 0 citations
Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.
Brendan Kennedy, T. Emerson, G. Roek et al.· 0 citations
Agent harnesses combine retrieval, routing, state, provenance, and verification, but locally successful components may disagree on shared state. We model this failure with a finite \emph{capability sheaf}: stalks encode typed behavior signatures, restriction maps retain shared fields, and accepted runs are useful global sections. An exact finite constraint-satisfaction problem (CSP) defines acceptance, while a linearized relative cohomology class provides a diagnostic and search feature. A controlled experiment over 20 task clusters introduces hidden interior mediators whose raw states are nuisance variables. Quotienting their coboundaries reduces the candidate budget from 2,000 to 1,000 per cluster; aligning the hidden state removes the gap. Exact CSP matches the quotient, so the result demonstrates invariance to stale representatives, not superiority over exact reasoning. We then test the method on a discovery split from the SWE-bench Multilingual pool of PatchFuseBench: 160 issues from 20 repositories, 875 real candidate patches, 2,579 source-aware edit atoms, and 153 newly executed patches. A first pool-level construction is constant because $[b-Dx]=[b]$ in $\operatorname{coker}D$ and therefore cannot rank configurations. A candidate-indexed repair is nontrivial on 848/875 candidates and varies within 120/160 issues. It resolves 118 issues versus 116 for a matched noncohomological selector, but the difference is not supported across repositories (exact sign-flip $p=0.75$). A leave-one-repository-out abstention gate reaches 127/160, tying the strong anchor and exceeding its matched gate by one issue ($p=1.0$). The discovery gate therefore fails and the confirmatory split remains sealed. The study supports the controlled invariance mechanism and an identifiability correction, but not a real-world cohomological advantage.
Reasoning shortcuts are rule solutions that reach correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings, asking when rules pin concepts down. Its key definition, one value permutation shared across all positions, does not apply as stated to any of its four heterogeneous benchmarks, and the most direct embedding, padding, produces confident false pathology: 90.91% of solution pairs unexplained on CLE4EVR, versus 0% under every well-defined rung of the componentwise hierarchy we introduce; the padded verdict rotates under configuration-file ordering. Across eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure. For circuit-given rules, symmetry-inertness of a coordinate is coNP-complete; automorphism existence is coNP-hard under randomized reductions, lies in $\Sigma_2^p$, is not $\Sigma_2^p$-complete in the Boolean case unless PH collapses, and is coNP-complete on monotone circuits. Boolean transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the theory flags, none at the 48 it certifies transitive, and none at twelve typed-ambiguous levels. Relocating the absorbing element moves every shortcut with it; a confusion null attributes the location to geometry while the observed rate exceeds it by half again. Trained end to end on CLE4EVR's rule and heterogeneous domains through a synthetic prototype front end, models produce 20,223 label-preserving errors with zero different-orbit exceptions, as transitivity predicts, where the padded instrument would misreport 78-88% of them.
When data sources are integrated through a shared interface, a downstream query may or may not be determined by what the interface exposes: two globally consistent worlds can agree on every shared attribute yet disagree on the query answer. This ambiguity is structural -- a property of the interface design, not the data volume -- and cannot be resolved by collecting more records or training a larger model. We formalize query identifiability for data integration under interface laws (functional dependencies that hold uniformly across all legal worlds rather than within a single instance) and prove three results. (i) A polynomial-time certificate (CheckCert) decides identifiability via attribute closure, and is exact on instances that expose any residual ambiguity (closure-separable). (ii) Non-identifiable queries face an irreducible 1/2 minimax error floor for any estimator using only interface evidence, bounding multi-view pretraining systems from below. (iii) A minimum-augmentation algorithm (Greedy-MinAug) finds the smallest set of interface additions to certify a query, reducing to Set Cover (logarithmic approximation). Experiments on synthetic benchmarks, real integration datasets spanning three domains (scholarly, product, restaurant), and schemas up to 10^3 attributes confirm CheckCert is exact, both algorithms run in single-digit milliseconds, and ML classifiers exhibit the predicted error floor and abrupt capability gains.
A new machine-learning framework aims to improve the success rate of computational protein design while moving away from results that reproduce sequences found in nature.
MIT News · Artificial Intelligence· news.mit.eduAug 24, 2026
