Two impossibility theorems show formal alignment certification is impossible for AI systems.
problem Formal certification of AI alignment over open-ended domains is impossible.
method Two independent impossibility theorems: Semantic and Statistical barriers.
result No procedure can simultaneously satisfy soundness, completeness, and tractability.
Machine fairness is impossible to achieve fully due to historical biases.
problem Machine learning models inherit biases from historical data, making it impossible to satisfy fairness metrics simultaneously.
method Presented a causal perspective to the impossibility theorem of fairness.
result It is impossible to satisfy fairness metrics like demographic parity, equal opportunity, and equalized odds simultaneously.
Paper proves impossibility of three desirable properties in node embedding.
problem Understanding limitations of node embedding methods.
method Axiomatic approach to node embedding, proving impossibility of three properties.
result No node embedding method can satisfy all three desirable properties simultaneously.
No feature ranking can be faithful, stable, and complete when features are collinear.
problem The impossibility of creating a feature ranking that is simultaneously faithful, stable, and complete when features are collinear.
method Proving the impossibility, quantifying it for four model classes, resolving it via ensemble averaging (DASH), and machine-verifying it with Lean 4 theorems.
result No method lies outside the dichotomy of faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods (stable, reporting ties for symmetric features).
Paper investigates learnability of OOD detection under various conditions.
problem Learnability of OOD detection under different scenarios.
method Investigates PAC learning theory, proves impossibility theorems, and provides necessary and sufficient conditions.
result Some conditions for learnability of OOD detection may not hold in practical scenarios.
Paper investigates OOD detection learnability under various conditions.
problem Learnability of OOD detection under diverse and unknown test data.
method PAC learning theory applied to OOD detection, proving impossibility theorems and necessary conditions.
result Some conditions for learnability hold in practical scenarios.
Theoretical limits on verifying self-improving systems without risking unbounded utility.
problem Formalizing and proving the limits of safety verification for self-improving systems.
method Developed dual conditions and used Holder's inequality, NP counting method, and Lipschitz bounds to establish impossibility and ceiling results.
result A classifier-based safety gate cannot simultaneously permit unbounded beneficial self-modification and bounded cumulative risk.
An impossibility result shows limitations in learning symmetries and equivariant functions.
problem Learning symmetries and equivariant functions simultaneously is impossible under certain conditions.
method Careful study of approximation for groups and semigroups, analysis of neural networks.
result Linearly equivariant networks can be used to learn equivariant functions, but group-convolutional networks have limitations.
No universal trading strategy exists due to mathematical impossibilities.
problem The impossibility of universally winning trading strategies in competitive markets.
method Three mathematical paradigms: measure-theoretic, No-Free-Lunch theorem, and adversarial Cantor diagonalization.
result No-arbitrage and free-lunch principles are mathematically precluded in competitive markets.
Paper proves impossible for large language models to control hallucinations without sacrificing other properties.
problem Achieving truthful knowledge representation, semantic information conservation, and knowledge-constrained optimality simultaneously in large language models.
method Modeling inference as an auction of ideas, using mechanism design, proper scoring rules, and transformer architecture analysis.
result No LLM can simultaneously achieve all four essential properties without violating at least one.
New protocol identifies impossible edge orientations in causal graphs.
problem Causal-discovery algorithms cannot distinguish edge directions without assumptions.
method Discrete impossibility certificates and oracle queries.
result Upper bound of 1+K expert interactions for DAG recovery. This paper develops a cohomological hierarchy for bistable visual paradoxes.
problem Understanding the hierarchy of visual paradoxes built from bistable elements.
method Develops a cohomological hierarchy using Z2 coefficients and a discrete Stokes theorem. result Reveals a hierarchy of paradox classes from H0 through H2, refined at each degree by the relative/absolute distinction. The study defines backdoor detection in ML and proves its infeasibility.
problem Backdoor detection in machine learning systems.
method Formal statistical definition and analysis of feasibility.
result Backdoor detection is impossible except for very small alphabet sizes.
The Darboux theorem in symplectic geometry implies that any two points in a connected symplectic manifold have neighbourhoods symplectomorphic to each other. The impossibility of such a theorem in the more general multisymplectic framework appears to be, at least, folkloristic, but no explicit counterexample seems to e…
New theorem on embedding Moebius bands in 3D space.
problem Proving the impossibility of placing uncountably many disjoint Moebius bands in 3D space.
method Generalization of Grushin and Palamodov's result to tame subsets in R^N and arbitrary topological embeddings in R^3.
result The impossibility of embedding uncountably many pairwise disjoint Moebius bands in 3D space, even for arbitrary topological embeddings.
There are many theorems in the differential geometry literature of the following sort. Let M be a complete Riemannian manifold with some conditions on various curvatures, diameters, volumes, etc. Then M is homotopy equivalent to a finite CW complex, or M is the interior of a compact, topological manifold with boundary.…
Determining the quality of the results obtained by clustering techniques is a key issue in unsupervised machine learning. Many authors have discussed the desirable features of good clustering algorithms. However, Jon Kleinberg established an impossibility theorem for clustering. As a consequence, a wealth of studies ha…
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
New theorem shows embedding impossibility for certain 3-manifolds.
problem Embedding restrictions in 3-manifolds.
method Construction of hyperbolic 3-manifolds with complicated ideals.
result Existence of non-embeddable manifolds in certain 3-manifolds.
We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We us…
In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.
TWM doesn't reduce delta in PDLPs, proving impossibility.
problem TWM in PDLPs doesn't uniformly reduce portfolio delta.
method Proved TWM's condition is self-contradictory and showed impossibility.
result No TWM can uniformly reduce portfolio delta.
Large language models can't efficiently reason conditionally in a distribution-free setting.
problem Impossibility of conditional PAC-efficient reasoning in large language models.
method Proof of impossibility in a distribution-free setting for non-atomic input spaces.
result Any algorithm achieving conditional PAC efficiency must defer to the expert model with high probability.
Theoretical limits show experimental data can falsify but not validate causal estimates from observational studies.
problem Fundamental limits on validating causal estimates using experimental data in observational studies.
method Impossible inference framework, Gaussian Process based approach.
result Experimental data can falsify but not validate causal estimates from observational studies.
Paper shows data poisoning and Byzantine attacks are equivalent, impacting federated learning security.
problem Resilience of federated learning systems to adversarial attacks.
method Proved equivalence between data poisoning and Byzantine gradient attacks.
result Equivalence between data poisoning and Byzantine attacks in federated learning.
Study on curvature functions for compact manifolds with boundary.
problem Understanding curvature functions on compact manifolds with boundary.
method Proves necessary and sufficient conditions for geodesic and Gaussian curvature, solves problems in the pointwise conformal case.
result New existence and nonexistence results for metrics with prescribed curvature, depending on Euler characteristic.
Bell's theorem shows quantum correlations can't be explained by classical causal models, even with some measurement dependence.
problem Quantum correlations violate classical causal models.
method Using causal networks, the study bounds the level of measurement dependence and derives nonlinear Bell inequalities.
result Quantum correlations can't be explained by classical causal models even with some measurement dependence.
A new method for variable importance measures without impossible data.
problem Using impossible data for variable importance measures in black box models.
method Cohort Shapley, a method grounded in economic game theory using only observed data.
result Cohort Shapley provides a more trustworthy explanation of black box models' decisions.
Previous studies have used a specific success metric within an algorithmic search framework to prove machine learning impossibility results. However, this specific success metric prevents us from applying these results on other forms of machine learning, e.g. transfer learning. We define decomposable metrics as a categ…
The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
problem Proving the non-existence of polyomino tilings on specific square-tiled surfaces.
method Study of homology groups for topological tilings, using coloring proofs.
result Several results about the non-existence of polyomino tilings on certain square-tiled surfaces.
Researchers prove it's impossible to partially recover graph alignments in certain conditions.
problem Recovering vertex correspondence between two random graphs with correlated edges.
method Used the probabilistic method to build automorphisms between tree components of a subcritical Erdös-Rényi graph.
result Proved an impossibility result for partial recovery in the sparse regime with constant average degree and correlation.
The paper proves ADL mechanisms face a trilemma and optimizes them for fairness, revenue, and exchange solvency.
problem The impossibility of a perpetual futures exchange achieving solvency, revenue, and fairness.
method Formal model of ADL, proving trilemma, and analyzing three ADL mechanisms.
result Optimized ADL mechanisms can reduce trader losses while maintaining exchange solvency.
Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus n with k punctures for any n and k.
This paper introduces time-uniform CLT-based confidence intervals for statistical inference.
problem Developing valid statistical inference methods for sequential data.
method Time-uniform central limit theory and strong invariance principles.
result Asymptotic confidence sequences (CSs) that are uniformly valid over time.
We show that many Lorentzian manifolds of dimension >2 do not admit a spacelike codimension-one foliation, and that almost every manifold of dimension >2 which admits a Lorentzian metric at all admits one which satisfies the dominant energy condition and the timelike convergence condition. These two seemingly unrelated…
This manuscript presents some new impossibility results on adversarial robustness in machine learning, a very important yet largely open problem. We show that if conditioned on a class label the data distribution satisfies the W2 Talagrand transportation-cost inequality (for example, this condition is satisfied if t…
New method for zeroth-order stochastic gradient algorithms provides confidence intervals.
problem Lack of inferential capabilities for zeroth-order stochastic gradient algorithms.
method Established central limit theorem and provided online estimators for asymptotic covariance matrix.
result Asymptotically valid confidence sets for parameter estimation and prediction.
Researchers prove inner product recovery is impossible in latent space models.
problem Recovering inner products in latent space models with random geometric graphs.
method Rate-distortion theory applied to Gaussian or spherical latent locations.
result Impossible to recover inner products if dimensionality exceeds nh(p), matching positive results' conditions. We solve the multi-criteria benchmarking problem by formalizing it as a social choice problem and identifying conditions for meaningful rankings.
problem Aggregating multiple metrics into a single ranking for models in benchmarking problems.
method Formalizing multi-criteria benchmarking as a social choice problem and identifying sufficient conditions for meaningful rankings.
result We prove that meaningful multi-criteria benchmarking becomes possible under certain preference conditions (single-peaked, group-separable, distance-restricted).
The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in wh…
We show that if a split link is obtained from a split link L in S3 by 1/n-Dehn surgery along a trivial knot C, then the link L∪C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…
Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.
problem Understanding the behavior of curves with curvature and torsion under curve shortening flow.
method Defined curvature-torsion entropy to analyze the flow of twisted curves.
result Curved curves under curve shortening flow either develop inflection points or exhibit highly irregular singularities.
The AAA credit rating may have been overly precise given available data.
problem The feasibility of achieving high reliability targets for structured credit products.
method Bayes' theorem and historical data analysis.
result High reliability targets for structured products require substantial statistical discrimination, which was not achievable with available data.
Study semi-coarse spaces' homotopy and homology, extending coarse geometry.
problem Extend homotopy and homology concepts to semi-coarse spaces.
method Analyze homotopy and construct homology groups invariant under semi-coarse homotopy equivalence.
result Show semi-coarse homology is isomorphic to Vietoris-Rips homology for graphs.
New welfare-based fairness notions align with existing error rate balance and predictive parity.
problem Aligning fairness notions with welfare-based criteria.
method Discussing and establishing conditions for envy freeness and prejudice freeness.
result Envy freeness and prejudice freeness are equivalent to error rate balance and predictive parity.
New findings show local attributions can't be both robust and provide recourse.
problem Ensuring machine learning systems are accountable and provide actionable recourse options.
method Formal definition of recourse sensitivity and counterexamples for popular attribution methods.
result It is impossible for any single attribution method to be both robust and provide recourse.
LLMs will inevitably hallucinate due to their mathematical structure.
problem The inherent limitations of Large Language Models (LLMs).
method Analysis of LLMs using computational theory and Godel's Incompleteness Theorem.
result Hallucinations in LLMs are an inevitable feature, not just errors.